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+arXiv:2301.00273v1  [math.PR]  31 Dec 2022
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+PETER B¨URGISSER
+Abstract. Consider a system f1(x) = 0, . . . , fn(x) = 0 of n random real polynomials in n
+variables, where each fi has a prescribed set of exponent vectors described by a set Ai ⊆ Zn
+of cardinality ti, whose convex hull is denoted Pi. Assuming that the coefficients of the fi are
+independent standard Gaussian, we prove that the expected number of zeros of the random
+system in the positive orthant is at most (2π)− n
+2 f0 (t1 − 1) . . . (tn − 1). Here f0 denotes the
+number of vertices of the Minkowski sum P1 + . . . + Pn. We also derive a better bound in
+the unmixed case where all supports Ai are equal, improving upon [8]. All arguments equally
+work for real exponent vectors.
+1. Introduction
+In many applications, we want to understand or find the (positive) real solutions of a system
+of multivariate polynomial equations, e.g., see [11, 16, 29]. Bezout’s theorem, which bounds
+the number complex zeros in terms of degrees, usually highly overestimates the number of real
+zeros. This can be already seen from Descartes’ rule of signs [10, p. 42], which implies that
+a real univariate polynomial with t terms has at most t − 1 positive zeros. In 1980, Khovan-
+skii [19] obtained a far reaching generalization of Descartes’ rule. He showed that the number
+of nondegenerate1 positive solutions of a system f1(x) = 0, . . . , fn(x) = 0 of n real polynomial
+equations in n variables is bounded only in terms of n and the number t of distinct exponent
+vectors occurring in the system. This result in fact allows for any real exponents. Following
+Kushnirenko, one speaks of fewnomial systems, with the idea that the number t of terms is small,
+see [20].
+Understanding the complex zeros of fewnomial systems is much simpler: the famous BKK-
+Theorem [2, 24] states that for given finite supports A1, . . . , An ⊆ Zn and Laurent polynomials
+fi(x) = �
+a∈A ci(a)xa1
+1 · · · xan
+n
+with generic complex coefficients ci(a), the number of complex
+solutions in (C×)n of a corresponding system f1(x) = 0, . . . , fn(x) = 0 is given by n! times the
+mixed volume of the Newton polytopes P1, . . . , Pn, where Pi is defined as the convex hull of Ai.
+Note that the number of real zeros has little to do with the metric properties of Pi: indeed,
+replacing Ai by a multiple mAi amounts to substituting xi by xm
+i (m > 0). Clearly, this does not
+change the number of positive real zeros of a fewnomial system, however Pi has been replaced
+by mPi.
+The bound on the number of real zeros obtained by Khovanskii is exponential in the number t.
+It is widely conjectured that this bound is far from optimal: in fact it is conjectured [27] that for
+fixed n, the number of nondegenerate positive solutions of a fewnomial system with t exponent
+vectors is bounded by a polynomial in t. Quite surprisingly, this question is open even for n = 2!
+Date: January 3, 2023.
+2010 Mathematics Subject Classification. 60D05, 14P99.
+Key words and phrases. fewnomials, random polynomials, real algebraic geometry, sparsity.
+Supported by the ERC under the European Union’s Horizon 2020 research and innovation programme (grant
+agreement no. 787840).
+1i.e., the Jacobian of the system does not vanish at the zero.
+1
+
+2
+PETER B¨URGISSER
+For results in special cases, we refer to [4, 1, 29, 22, 23, 3]. Moreover, there is a very interesting
+connection to complexity theory [21, 5].
+Given this state of affairs of real fewnomial theory, a possible way to advance is to ask
+what happens in generic situations. This can be made formal by considering random fewnomial
+systems. Fix supports A1, . . . , An ⊆ Zn of cardinality t1, . . . , tn, respectively, and consider a
+system of n random polynomials fi(x) as above, but now the coefficients ci(a) are assumed
+to be independent standard Gaussian. Let us denote by N(A1, . . . , An) the expectation of the
+number of nondegenerate positive real zeros of such system. Actually, we work in more generality,
+allowing any subsets Ai of Rn; see Section 4. There is a large literature on the real zeros of
+random polynomials: we refer to [8] for references.
+In [8] it was proven that N(A, . . . , A) ≤ 21−n�t
+n
+�
+. The main result of the present paper is an
+extension of this to the mixed case, where the fewnomials may have have different supports Ai.
+Our bound depends on the combinatorial structure of the Minkowski sum P1 + . . .+ Pn through
+the number of its vertices. We remark that our proof is quite different from the one in [8], which
+was quite indirect.
+Theorem 1.1. If Pi denotes the convex hull of the finite nonempty set Ai ⊆ Rn of cardinality ti,
+for i = 1, . . . , n, then
+N(A1, . . . , An) ≤ (2π)− n
+2 f0 (t1 − 1) . . . (tn − 1).
+Here f0 denotes the number of vertices of the Minkowski sum P := P1 + . . . + Pn.
+The bound in this theorem looks similarly to the one in a conjecture attributed to Kush-
+nirenko, which states that the number of positive nondegenerate zeros is always bounded by
+(t1 − 1) · · · (tn − 1). However, this was disproved in [15].2
+In the unmixed situation, where all supports are equal to A, it is well known [12] that the
+expected number of positive zeros can be expressed by the volume of the image of the Veronese
+like map Rn
+>0 → P(RA) sending x to [xa]a∈A. This is a consequence of the kinematic formula
+for real projective spaces. In the mixed situation, there is no such simple characterization: one
+has to work with the more complicated kinematic formula for products of projective spaces
+(Theorem 3.2) that we derive from [17, 9]. After passing to exponential coordinates w = log x,
+we bound the resulting integral over Rn with a strategy inspired by the theory of toric varieties.
+The normal fan of the polytope P affords a decomposition of Rn into the normal cones C at
+the vertices of P. The resulting integral over C can be bounded in terms of the characteristic
+function of the dual cone of C. Finally, an explicit a priori bound on the characteristic function
+(Proposition 2.4) completes the argument.
+1.1. The univariate case and a conjecture. The univariate case (n = 1) was settled by
+Jindal et al. [18]. They showed that for any subset S ⊆ R of cardinality t, we have
+(1.1)
+N(S) ≤ 2
+π
+√t − 1.
+Moreover, they constructed a sequence St ⊆ Z of supports of cardinality t with N(St) ≥ c
+√
+t for
+some constant c > 0. Consider for t1, . . . , tn ≥ 1 the supports A1 := St1 × 0 . . . × 0, . . . , An :=
+0 × . . . × 0 × Stn. These supports describe a system of n equations, where the ith equation
+depends on xi only. Therefore, N(A1, . . . , An) = N(St1) · · · N(Stn), which with the above leads
+to the lower bound
+(1.2)
+N(A1, . . . , An) ≥ cn√t1 · · · tn.
+2This conjecture was never published by Kushnirenko and apparently, he did not believe in it.
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+3
+We complement this by showing that for any A = S1 ×. . .×Sn in product form, the expectation
+N(A, . . . , A) can be expressed in terms of the N(Si) as follows.
+Proposition 1.2. If A = S1 × . . . × Sn for finite Si ⊆ R, then
+N(A, . . . , A) =
+πn
+vol(Pn) N(S1) · · · N(Sn).
+We conjecture that the lower bound (1.2) is optimal in the following sense.
+Conjecture 1. Let Ai ⊆ Rn be finite nonempty sets of cardinality ti with convex hull Pi, for
+i = 1, . . . , n. We denote by f0 the number of vertices of P1 + . . . + Pn. Then
+N(A1, . . . , An) ≤ κ(f0)√t1 · · · tn
+for some function κ : N → N. In particular, N(A, . . . , A) ≤ κ(f0) t
+n
+2 for A ⊆ Rn of cardinality t.
+In the special case A = S1 × . . . × Sn, by combining (1.1) with Proposition 1.2, we obtain
+N(A, . . . , A)vol(Pn) ≤ 2n√
+t, which is much smaller than what Conjecture 1 predicts.
+1.2. Improvement in unmixed case. We can improve the dependence on n in the bound of
+Theorem 1.1 in the case where all supports are equal.
+Theorem 1.3. For A ⊆ Rn of cardinality t ≥ 1 with convex hull P and f0 vertices, we have
+N(A, . . . , A) ≤
+1
+vol(Pn) f0
+�t−1
+n
+�
+.
+To compare this with the bound N(A, . . . , A) ≤ 21−n�t
+n
+�
+from [8], note that vol(Pn)−1 =
+Γ( n+1
+2 )π− n+1
+2 . Hence, for n → ∞, the new bound goes asymptotically faster to zero than the
+old one. However, the bound in [8] also holds for nonstandard centered Gaussian coefficients
+ci(a) ∼ N(0, σi(a)2).
+1.3. Location of zeros. We finish with a result on the typical location of the zeros. It is well
+known that for random real polynomials, the positive reals zeros x tend to accumulate around 1:
+see [12] for the dense and [18] for the sparse case. This means that w = log x accumulates
+around 0. We generalize this to multivariate systems as follows.
+Theorem 1.4. Fix a finite supports A1 . . . , An ⊆ Rn and consider a random system (4.3) with
+independent standard Gaussian coefficients ci(a) for the stretched supports mAi, where m ∈ Z>0.
+Fix ε > 0. Then the probability that the system has a zero w ∈ Rn with ∥w∥ > ε goes to zero, as
+m → ∞.
+2. Preliminaries
+2.1. A metric property of charts of real projective space. Consider the real projective
+space Pm. We shall identify the tangent space T[y]Pm at a point [y] := [y0 : . . . : ym] with Ry⊥.
+The standard Riemannian metric on Pm is defined by ⟨v, w⟩[y] := ∥y∥−2⟨v, w⟩ for v, w ∈ Ry⊥.
+We denote by Py the orthogonal projection onto Ry⊥.
+Consider the affine chart (Pm)y0̸=0 → Rm, which maps [y0 : . . . : ym] to y−1
+0 (y1, . . . , ym). Its
+inverse is given by
+π: Rm → (Pm)y0̸=0, (y1, . . . , ym) �→ [1 : y1 : . . . : ym].
+By [6, Lemma 14.8], the derivative of π at y′ := (y1, . . . , yn) satisfies Dy′π = ∥π(y′)∥−1Py. From
+this we conclude for the spectral norm
+(2.1)
+∥Dy′π∥ ≤ ∥π(y′)∥−1 ≤ 1.
+
+4
+PETER B¨URGISSER
+2.2. On the quantity σ. The relative position of two subspaces of a Euclidean vector space E
+can be quantified by a volume like quantity, which is crucial in the study of integral geometry
+in homogeneous spaces; see [17] and [9, §3.3]. To define this quantity, note first that there is an
+induced inner product on the exterior algebra Λ(E) given by (see [9, (2.1)])
+⟨v1 ∧ · · · ∧ vk, w1 ∧ · · · ∧ wk⟩ = det(⟨vi, wj⟩)1≤i,j≤k.
+More concretely, ∥v1 ∧ . . . ∧ vn∥ = | det[v1, . . . , vn]|, where [v1, . . . , vn] denotes the matrix with
+columns vi ∈ E = Rn
+Let V, W be linear subspaces of E of complementary dimensions. We define (see [9, (3.3)])
+(2.2)
+σ(V, W) := ∥v1 ∧ . . . ∧ vk ∧ w1 ∧ . . . ∧ wm∥ ∈ [0, 1],
+where v1, . . . , vk and w1, . . . , wm are orthonormal bases of V and W, respectively.
+Clearly,
+σ(V, W) = σ(W, V ). The extreme cases are: σ(V, W) = 0 iff V ∩ W ̸= 0 and σ(V, W) = 1 iff v
+and W are orthogonal.
+Proposition 2.1. We have σ(V ⊥, W ⊥) = | det p |, if p: V ⊥ → W denotes the restriction of the
+orthogonal projection E → W to V ⊥. Moreover, σ(V, W) = σ(V ⊥, W ⊥).
+Proof. Let ν1, . . . , νm be an orthonormal basis of V ⊥. We decompose νi = ν′
+i + ν′′
+i according
+to E = W ⊕ W ⊥. Then p(νi) = ν′
+i and | det p | = ∥ν′
+1 ∧ . . . ∧ ν′
+m∥. If ω1, . . . , ωk denotes an
+orthonormal basis of W ⊥, we have
+σ(V ⊥, W ⊥) = ∥ν1 ∧ . . . ∧ νm ∧ ω1 ∧ . . . ∧ ωk∥ = ∥ν′
+1 ∧ . . . ∧ ν′
+m ∧ ω1 ∧ . . . ∧ ωk∥ = ∥ν′
+1 ∧ . . . ∧ ν′
+m∥,
+the last equality holding since the span of the ν′
+i equals W, which is orthogonal to the span of
+the wj, which is W ⊥. This proves σ(V ⊥, W ⊥) = | det p |.
+For the second assertion, we use that | det p | = | det q |, where q: W ⊥ → V denotes the
+restriction of the orthogonal projection E → V to W ⊥, see [7, Lemma 5.4].
+□
+Clearly, the definition (2.2) can be extended to more than two subspaces; see [9, (3.5)]. But if
+W = W1 ⊕. . .⊕Wn is an orthogonal decomposition, we can reduce to the case of two subspaces:
+(2.3)
+σ(V, W1, . . . , Wn) = σ(V, W1 + . . . + Wn).
+This is a consequence of [9, Lemma A.6].
+2.3. Characteristic function of convex cones. We prove here an priori upper bound on the
+characteristic function of a convex cone, which is a key ingredient in the proof of Theorem 1.1.
+A convex cone C ⊆ Rn is called proper if it is n-dimensional and pointed, i.e., contained in a
+halfspace. It is well known that a convex C ⊆ Rn is proper iff its dual cone
+C∗ := {x ∈ Rn | ∀y ∈ C ⟨x, y⟩ ≥ 0}
+is proper. Let g ∈ GL(n, R). Then K := g(C) is a proper cone and gT (K∗) = C∗. We denote
+by int(C) the interior of C.
+We assign to a proper cone C ⊆ Rn the function
+(2.4)
+vC : int(C∗) → R>0, vC(x) :=
+�
+C
+e−⟨x,y⟩ dy.
+One calls vC the characteristic function (or Koszul-Vinberg characteristic) of C∗. It is a useful
+analytic tool for investigating convex cones, e.g., see [13, I.3] and [14]. E.g., Rn
+>0 is self dual and
+vRn
+>0(x) = (x1 · . . . · xn)−1 for x ∈ Rn
+>0.
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+5
+The homogeneity property vC(tx) = t−nvC(x) for t > 0, x ∈ int(C∗) is immediate to check.
+Moreover, the transformation formula implies the following invariance property: if g ∈ GL(n, R)
+and K := g(C), then gT (K∗) = C∗ and
+(2.5)
+vK(z) = | det g| vC(gT z)
+for z ∈ int(K∗).
+Remark 2.2. The function log vC is strictly convex and essentially equals Nesterov and Ne-
+mirowski’s universal self-concordant barrier function [26, §2.5], see [14] for the proof.
+The following result is well known, e.g., see [14, Thm. 4.1]. We provide the simple proof for
+the sake of completeness.
+Lemma 2.3. For x ∈ int(C∗) we have
+vC(x) = n! vol
+�
+y ∈ C | ⟨x, y⟩ ≤ 1
+�
+.
+Proof. We fix x ∈ int(C∗). For t ≥ 0 we define the n − 1-dimensional slice
+Ct :=
+�
+y ∈ C | ⟨x, y⟩ = t∥x∥
+�
+.
+By Fubini, we get
+vC(x) =
+�
+C
+e−⟨x,y⟩ dy =
+� ∞
+0
+voln−1(Ct)e−t∥x∥dt = voln−1(C1)
+� ∞
+0
+tn−1e−t∥x∥dt.
+Note that
+� ∞
+0
+tn−1e−t∥x∥dt =
+1
+∥x∥n
+� ∞
+0
+sn−1e−sds =
+1
+∥x∥n Γ(n) = (n − 1)!
+∥x∥n .
+Moreover, we have
+voln−1(C1) = n voln
+�
+y ∈ C | ⟨x, y⟩ ≤ ∥x|
+�
+= n ∥x∥n voln
+�
+y ∈ C | ⟨x, y⟩ ≤ 1
+�
+.
+It follows that
+vC(x) = n ∥x∥n voln
+�
+y ∈ C | ⟨x, y⟩ ≤ 1
+� (n − 1)!
+∥x∥n
+= n! voln
+�
+y ∈ C | ⟨x, y⟩ ≤ 1
+�
+,
+completing the proof.
+□
+The following technical result is essential for the proof of Theorem 1.1.
+Proposition 2.4. Let C ⊆ Rn be a proper cone and b1, . . . , bn ∈ C∗. Then we have
+∥b1 ∧ . . . ∧ bn∥ · vC(b1 + . . . + bn) ≤ 1.
+This bound is optimal.
+Proof. We denote by cone(b1, . . . , bn) ⊆ C∗ the convex cone generated by b1, . . . , bn. Without
+loss of generality, we may assume that b1, . . . , bn ∈ C∗ is a basis of Rn. Let b∗
+1, . . . , b∗
+n denote its
+dual basis, that is ⟨b∗
+i , bj⟩ = δij. In matrix terminology, [b∗
+1, . . . , b∗
+n]T [b1, . . . , bn] = In, hence
+(2.6)
+det[b∗
+1, . . . , b∗
+n] det[b1, . . . , bn] = ±1.
+Moreover, the definition of the dual basis implies that cone(b∗
+1, . . . , b∗
+n) is the dual cone of
+cone(b1, . . . , bn). Therefore, by duality, we get
+C ⊆ cone(b1, . . . , bn)∗ = cone(b∗
+1, . . . , b∗
+n).
+Put d := b1 + . . . + bn and let y ∈ C such that ⟨d, y⟩ ≤ 1. Since C ⊆ cone(b∗
+1, . . . , b∗
+n), we can
+write y = �
+i tib∗
+i with ti ≥ 0. Moreover �
+i ti = ⟨d, y⟩ ≤ 1. Thus we have shown the inclusion
+{y ∈ C | ⟨d, y⟩ ≤ 1} ⊆ conv{0, b∗
+1, . . . , b∗
+n}.
+
+6
+PETER B¨URGISSER
+This implies the inequality of volumes
+voln{y ∈ C | ⟨d, y⟩ ≤ 1} ≤ volnconv{0, b∗
+1, . . . , b∗
+n} = 1
+n!| det[b∗
+1, . . . , b∗
+n]|.
+Multiplying with n! | det[b1, . . . , bn]|, using (2.6) and taking into account Lemma 2.3, the assertion
+follows.
+The optimality is attained for C = Rn
++ and bi = diei with di > 0. Indeed, we have
+| det[b1, . . . , bn] |vC(d) = d1 · . . . · dn (d1 · . . . · dn)−1 = 1.
+□
+2.4. Vertices and normal fan of sums of polytopes. We recall here some basic facts about
+polytopes and their normal fans; see [30, §7.1] for more details.
+Let P ⊆ Rn be a full-dimensional polytope and v be a vertex of P. The cone Pv of P at v is
+defined as the convex cone generated by P − v. It is a proper cone. The dual cone of Pv, also
+called the inner normal cone of P at v, is defined as
+P ∗
+v := {y ∈ Rn | ∀x ∈ P ⟨x, y⟩ ≥ 0}.
+The cone P ∗
+v is also proper. The union over all P ∗
+v equals Rn. Moreover, for v1 ̸= v2, we have
+dim(P ∗
+v1 ∩ P ∗
+v2) < n. In fact, the P ∗
+v are the n-dimensional cones of the normal fan of P.
+We will need the following result.
+Lemma 2.5. Let P1, . . . , Pn be polytopes in Rn. There is an injective map
+Vert(P1 + . . . + Pn) → Vert(P1) × . . . Vert(Pn), v �→ (v1, . . . , vn)
+satisfying v = v1 + . . . + vn. Moreover, if we denote by Πi the cone of Pi at the vertex vi, then
+Π := Π1 + . . . + Πn is the cone of P1 + . . . + Pn at the vertex v1 + . . . + vn. In particular,
+Π∗ = Π∗
+1 ∩ . . . ∩ Π∗
+n.
+Proof. For the following, see [28, §1.7]. To a nonzero weight ω ∈ Rn we assign the face of Pi,
+given by
+F(Pi, ω) :=
+�
+w ∈ Rn | ⟨w, ω⟩ = min
+w′∈Pi⟨w′, ω⟩
+�
+.
+We have (see [28, Thm. 1.7.5])
+F(P1 + . . . + Pn, ω) = F(P1, ω) + . . . + F(Pn, ω).
+Suppose that F(P1 + . . . + Pn, ω) = {v} is a vertex. Then all F(Pi, ω) = {vi} are vertices
+and v = v1 + . . . vn.
+It is easy to see that the vi are uniquely determined by v.
+The map
+v �→ (v1, . . . , vn) is as required. The remaining assertions are clear.
+□
+3. Random intersections in products of projective spaces
+3.1. The kinematic formula. We specialize here the general kinematic formula for homo-
+geneous spaces from [9, Thm. A.2] to the case of products of real projective spaces (Theo-
+rem 3.2). For this purpose, we define the average scaling factor and we explain how to bound it
+in Lemma 3.5.
+Consider the product Ω := Pm1 × · · · × Pmn of real projective spaces. The product G :=
+O(m1 +1)×· · ·×O(mn +1) of orthogonal groups acts transitively on Ω. So Ω is a homogeneous
+space and we have an induced transitive action of G on the tangent bundle of Ω. We focus on
+the special hypersurfaces H1, . . . , Hn of Ω of the following shape
+(3.1)
+H1 := Pm1−1 × Pm2 × · · · × Pmn, . . . , Hn := Pm1 × Pm2 · · · × Pmn−1.
+They are determined upon selecting hyperplanes Pmi−1 in Pmi. Our goal is to investigate the
+average cardinality of the intersection Z ∩H1 ∩. . .∩Hn of an n-dimensional smooth submanifold
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+7
+Z ⊆ Ω with random Hi corresponding to independently chosen uniform random hyperplanes
+in Pmi.
+Fix a distinguished point ω ∈ Ω and denote by K the stabilizer group of ω.
+E.g., take
+ωi = [1 : 0 . . . : 0] for all i. Notice that we have an induced action of K on the tangent space
+T := TωΩ, which we can identify with the standard action of K = O(m1) × · · · × O(mn) on
+T = Rm1 × · · · × Rmn. This induces an action of K on the Grassmann manifold Gr(d, T ) of
+linear subspaces of T with codimension d. Note that this action is transitive if n = 1, but not
+for n ≥ 2.
+We assign to an n-dimensional smooth submanifold Z ⊆ Ω a map
+(3.2)
+Z → Gr(n, T )/K, z �→ KgNpZ
+as follows. For given p ∈ Z choose any g ∈ G such that gp = ω. The induced action of g maps
+the tangent space TpΩ to TωΩ = T . This transports the normal subspace NpZ ⊆ TpΩ of Z at p
+to gNpZ ⊆ T . Note that the K-orbit of the subspace gNpZ does not depend on the choice of g,
+which shows that the map (3.2) is well defined.
+We call the submanifold Z cohomogeneous if the map (3.2) is constant; see [9, A.5.1] and [25].
+For instance, a product Z = L1 × . . . × Ln of lines Li in Pmi is cohomogeneous: indeed, the
+map (3.2) sends any point p ∈ Z to the K-orbit of R × . . . × R.
+Definition 3.1. The average scaling factor function of the n-dimensional submanifold Z of
+Pm1 × · · · × Pmn is the function ¯σZ : Z → [0, 1] defined at p ∈ Z by
+¯σZ(p) := E Liσ(gNpZ, L1 × . . . × Ln),
+where g ∈ G satisfies gp = ω0, and the expectation is taken over uniformly random lines Li in
+T = Rm1 × · · · × Rmn (see (2.2) for the definition of σ).
+Note that due to the averaging over the K-orbit, the choice of g is irrelevant. The above
+definition is consistent with the one in [9, Def. A.1], since
+(3.3)
+σ(gNpZ, L1 × . . . × Ln) = σ(gNpZ, L1 × 0 × · · · × 0, . . . , 0 × · · · × 0 × Ln)
+by (2.3); indeed note that the n lines L1 × 0 × · · · × 0, ... are pairwise orthogonal.
+We introduce the notation
+ρn := E ∥x∥ =
+√
+2 Γ( n+1
+2 )
+Γ( n
+2 )
+≤ √n
+for standard Gaussian x ∈ Rn. We note that by [6, Lemma 2.25],
+(3.4)
+vol(Pmi−1)
+vol(Pmi)
+=
+1
+√π
+Γ( mi+1
+2
+)
+Γ( mi
+2 )
+=
+1
+√
+2π ρmi,
+We can now explicitly state the kinematic formula for products of real projective spaces.
+Theorem 3.2. For any n-dimensional submanifold Z of Pm1 × · · · × Pmn, we have
+E g∈G#(Z ∩ g1H1 ∩ . . . ∩ gnHn) = (2π)− n
+2 ρm1 · · · ρmn
+�
+Z
+¯σZ dZ,
+where the hypersurfaces Hi are defined in (3.1).
+Proof. If σK : Z × H1 × . . . × Hn → [0, 1] denotes the average scaling function from [9, Def. A.1],
+then [9, Thm. A.2] states that
+E g∈G#(Z ∩ g1H1 ∩ . . . ∩ gnHn) =
+1
+vol(Ω)n
+�
+Z×H1×...×Hn
+σK d(Z × H1 × . . . × Hn).
+
+8
+PETER B¨URGISSER
+By K-invariance and (3.3), we have σK(z, y1, . . . , yn) = ¯σZ(z) for all z ∈ Z and yi ∈ Hi.
+Therefore,
+E g∈G#(Z ∩ g1H1 ∩ . . . ∩ gnHn) = vol(H1) · · · vol(Hn)
+vol(Ω)n
+�
+Z
+¯σZ dZ.
+Finally, (3.4) gives
+vol(H1) · · · vol(Hn)
+vol(Ω)n
+=
+n
+�
+i=1
+vol(Pmi−1)
+vol(Pmi)
+= (2π)− n
+2 ρm1 · · · ρmn,
+which completes the proof.
+□
+Example 3.3. A product Z = L1 × . . . × Ln of lines Li is cohomogeneous. Theorem 3.2 implies
+that ¯σZ = (2/π)n/2(ρm1 · · · ρmn)−1.
+We shall focus on submanifolds Z arising as the image of an injective smooth map
+(3.5)
+ψ: U → Pm1 × · · · × Pmn, ψ(x) := (ψ1(x), . . . , ψn(x)),
+where the ψi : U → Pmi are smooth maps defined on an open subset U ⊆ Rn. Let us denote by
+Jψ(x) :=
+�
+det((Dxψ)T Dxψ)
+the absolute Jacobian of ψ at x. The transformation formula implies that
+(3.6)
+�
+Z
+¯σZ dZ =
+�
+U
+¯σZ(ψ(x))Jψ(x) dx.
+We next analyze the integrand on the right-hand side more closely.
+Lemma 3.4. Let x ∈ U and put Ti := Tψi(x)Pmi.
+Let λ1, . . . , λn be independent standard
+Gaussian linear forms on Ti. This defines the random linear forms λi ◦ Dxψi on Rn. Then
+ρm1 · · · ρmn ¯σZ(ψ(x))Jψ(x) = E λ1,...,λn ∥(λ1 ◦ Dxψ1) ∧ . . . ∧ (λn ◦ Dxψn)∥ .
+Proof. To simplify notation, we assume w.l.o.g. that p = ψ(x) is the distinguished point ω.
+We also identify Ti with Rmi. For ui ∈ Ti with ∥ui∥ = 1 consider the line Li = Rui and the
+orthogonal projection pi : Ti → Li, which is is given by pi(w) = µi(w)ui with the linear form on Ti
+defined by µi(w) := ⟨w, ui⟩. Thus the orthogonal projection pL : T1 × · · · × Tn → L1 × · · · × Ln
+is described by µ1, . . . , µn. This implies that
+(3.7)
+| det(pL ◦ Dxψ)| = ∥(µ1 ◦ Dxψ1) ∧ . . . ∧ (µn ◦ Dxψn)∥ .
+On the other hand, according to Proposition 2.1, we have
+σ(L1 × · · · × Ln, NpZ) = | det p′
+L|,
+where p′
+L : TpZ → L1 × · · · × Ln denotes the restriction of pL to TpZ. Applying the determinant
+to the composition of Dxψ with p′
+L, we get
+Jψ(x) | det p′
+L| = | det(pL ◦ Dxψ)|.
+By averaging over random lines Li, we deduce from the definition of ¯σZ and the above that
+Jψ(x)¯σZ(p) = Jψ(x) E Liσ(NpZ, L1 × · · · × Ln) = Jψ(x) E Li| det p′
+L| = E Li| det(pL ◦ Dψ)|.
+Finally, a standard Gaussian linear form on Ti is obtained as λi = riµi with independent random
+variables ri and ui, where ui is uniformly random in the unit sphere of Ti and r2
+i is χ2-distributed
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+9
+with mi degrees of freedom. Thus E ri = ρmi. Altogether, we obtain, using (3.7),
+ρm1 · · · ρmnJψ(x)¯σZ(p) = ρm1 · · · ρmnE | det(pL ◦ Dψ)|
+= ρm1 · · · ρmnE ∥(µ1 ◦ Dxψ1) ∧ . . . ∧ (µn ◦ Dxψn)∥
+= E ∥(λ1 ◦ Dxψ1) ∧ . . . ∧ (λn ◦ Dxψn)∥ ,
+which completes the proof.
+□
+3.2. Bounding the average scaling factor. In order to bound the quantity in Lemma 3.4,
+we use affine charts for the product of projective spaces. Denote by yi0, . . . , yimi coordinates
+for Pmi. Fix 0 ≤ ri ≤ mi for i = 1, . . . , n, and consider the inverse of the affine chart πiri : Rmi →
+(Pmi)yiri̸=0, see Subsection 2.1. We describe the maps ψi from (3.5) in these charts by smooth
+functions defined on open subsets of Rn,
+(3.8)
+ϕiri : Rn ⊇ Uiri → Rmi,
+satisfying ψi := πiri ◦ ϕiri. In order to simplify notation, we assume w.l.og. ri = 0 and write
+πi := πi0, ϕi := ϕi0. In these charts, the combined map ψ of (3.5) is represented by a map
+ϕ: U → Rm1 × · · · × Rmn, ϕ(x) = (ϕ1(x), . . . , ϕn(x))
+defined on some open subset U ⊆ Rn. We view the derivative M(x) := Dxϕ as a matrix of
+format (m1 + . . . + mn) × n with blocks Mi(x) := Dxϕi ∈ Rmi×n. For 1 ≤ ji ≤ mi, i = 1, . . . , n,
+we denote by M(x)j1,...,jn the n × n submatrix of M(x) obtained by selecting in the ith block
+the jith row.
+Lemma 3.5. Let x ∈ U be such that [yi] := ψi(x) ∈ (Pmi)y0̸=0 for all i. Then
+ρm1 · · · ρmn ¯σZ(ψ(x))Jψ(x) ≤
+�
+j1,...,jn
+| det M(x)j1,...,jn|,
+where the sum is over the n-tuples (j1, . . . , jn) ∈ [m1] × . . . × [mn].
+Proof. From ψi = πi ◦ ϕi we get Dψi = Dπi ◦ Dϕi, where we drop arguments for notational
+simplicity. Let λi : Ti → R be a linear form on Ti = Tψi(x)Pmi. Then, defining wi := λi ◦ Dπi,
+λi ◦ Dψi = λi ◦ Dπi ◦ Dϕi = wi ◦ Dϕi.
+If we identify λi ◦ Dxψi with a vector in Rn and wi with a vector in Rmi, then we have the
+matrix product of formats n × �
+i mi and �
+i mi × n,
+(3.9)
+R(x) :=
+
+
+(λ1 ◦ Dxψ1)T
+...
+(λn ◦ Dxψ1)T
+
+ =
+
+
+wT
+1
+0
+. . . 0
+0
+wT
+2
+. . . 0
+...
+...
+...
+0
+0
+wT
+n
+
+ ·
+
+
+M1(x)
+...
+Mn(x)
+
+ .
+Lemma 3.4 tells us that
+ρm1 · · · ρmn ¯σZ(ψ(x))Jψ(x) = E λi| det R(x))|,
+where the expectation is over independent standard Gaussian λi. Note that the resuling random
+vector wi := λi ◦ Dπi is not standard Gaussian anymore. However ∥Dπi∥ ≤ 1 by (2.1) and
+Lemma 3.6 below implies that E w2
+ij ≤ 1 for the jth component wij of wi.
+From Binet-Cauchy, we obtain from (3.9)
+(det R(x))2 =
+�
+j1,...,jn
+w2
+1j1 · · · w2
+njn(det M(x)j1,...,jn)2,
+
+10
+PETER B¨URGISSER
+where the sum is over all (j1, . . . , jn) ∈ [m1] × . . . × [mn]. Taking expectations yields
+E w(det R(x))2 ≤
+�
+j1,...,jn
+(det M(x)j1,...,jn)2.
+We conclude that
+E w| det R(x))| ≤
+�
+E w(det R(x))2� 1
+2 ≤
+�
+j1,...,jn
+| det M(x)j1,...,jn|,
+which completes the proof.
+□
+Lemma 3.6. Let A ∈ Rp×m with ∥A∥ ≤ 1. If y ∈ Rp is standard Gaussian, then the random
+variable z := yA satisfies E |zj|2 ≤ 1 for all j.
+Proof. From zj = �
+i yiaij we get z2
+j = �
+i,k yiykaijakj. Hence E z2
+j = �
+i a2
+ij. Finally, �
+i a2
+ij =
+∥A(ej)∥2 ≤ ∥A∥2 ≤ 1.
+□
+4. Mixed random fewnomial systems
+The goal here is to provide the proofs of the assertions in the introduction.
+Let us first
+introduce some notation.
+We assign to a real valued function c: A → R on a finite nonempty subset A ⊆ Rn the real
+analytic function FA,c : Rn → R
+(4.1)
+FA,c(w) :=
+�
+a∈A
+c(a)e⟨a,w⟩.
+In the special case where A consists of integer vectors, FA,c arises from the Laurent polynomial
+fA,c(x) = �
+a∈A c(a)xa by a substitution: FA,c(w) = fA,c(ew). Generally, we have the following
+equivariance property: for g ∈ GL(n, R) and b ∈ Rn,
+(4.2)
+FA+b,b.c(w) = e⟨b,w⟩FA,c(w),
+Fg(A),g.c(w) = FA,c(gT w),
+where (g.c)(a) := c(g−1a) and b.c(a) := c(a − b).
+Suppose now we have n such analytic functions encoded by ci : Ai → R, for i = 1, . . . , n.
+Throughout, we denote by ti the cardinality of Ai and by Pi its convex hull. We are interested
+in the number N of nondegenerate zeros w ∈ Rn of the system
+(4.3)
+FA1,c1(w) = 0, . . . , FAn,cn(w) = 0.
+Our goal is to study the expected number of nondegenerate zeros for random coefficient func-
+tions. More specifically, we denote by N(A1, . . . , An) the expectation of N, when all the coeffi-
+cients ci(a), for i ∈ [n] and ai ∈ Ai, are independent standard Gaussians. Clearly, N(A1, . . . , An)
+is invariant under permutations of the Ai. Also, N(A1, . . . , An) = 0 if ti = 1 for some i. More-
+over, we have N(A1, . . . , An) = 0 if dim(P1 + . . . + Pn) < n, see Lemma 4.2.
+Equation (4.2) implies the following invariance properties
+(4.4)
+N(A1 + b1, . . . , An + bn) = N(A1, . . . , An),
+N(g(A1), . . . , g(An)) = N(A1, . . . , An),
+where b1, . . . , bn ∈ Rn and g ∈ GL(n, R).
+Our main result is Theorem 1.1 stated in the introduction. Note that it gives the correct
+answer N(A1, . . . , An) = 0 if ti = 1 for some i.
+Example 4.1. In the case t1 = . . . = tn = 2, the Pi are segments. If they are linearly independent,
+P1 + . . . + Pn is a parallelepiped with 2n vertices. Thus, Theorem 1.1 gives N(A1, . . . , An) ≤
+(2/π)
+n
+2 . This can be easily verified directly as follows. Suppose Ai = {ai, bi}. We claim that
+N(A1, . . . , An) = 2−n if b1 − a1, . . . , bn − an are linearly independent. For showing this, by the
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+11
+invariance properties (4.4), it suffices to consider the case where Ai = {0, ei}. Then (4.3) amounts
+to the system ci(0) + ci(ei)ewi = 0, for i = 1, . . . , n, which has a solution iff ci(0)ci(ei) < 0, for
+all i. This happens with probability 2−n, hence indeed N(A1, . . . , An) = 2−n.
+4.1. Proof of Theorem 1.1. Let us look at a special instance of (3.5). To the given finite
+nonempty subsets A1, . . . , An ⊆ Rn, we assign the maps
+ψi : Rn
+>0 → P(RAi) ≃ Pmi, ψi(x) := [xai]ai∈Ai,
+where mi := #Ai − 1. Recall that Pi denotes the convex hull of Ai and put P := P1 + . . . + Pn.
+We consider the combined map
+(4.5)
+ψ: Rn
+>0 → Pm1 × · · · × Pmn, ψ(x) := (ψ1(x), . . . , ψn(x)).
+Lemma 4.2. The map ψ is injective iff P is n-dimensional. Moreover, if P is not n-dimensional,
+then rankDxψ < n for all x ∈ Rn
+>0.
+Proof. Assume ψ(exp(w)) = ψ(exp(w′)) for w ̸= w′ ∈ Rn Then there are ci ∈ R such that for
+all ai ∈ Ai we have ⟨ai, w − w′⟩ = ci. Hence, ⟨x, w − w′⟩ = ci for all xi ∈ Pi. It follows that
+⟨x, w − w′⟩ = c1 + . . . + cn for all x ∈ P. Hence dim P < n.
+Conversely, assume there is a nonzero w ∈ Rn and c ∈ R such that ⟨x, w⟩ = c for all x ∈ P.
+Then there are ci ∈ R such that ⟨xi, w⟩ = ci for all xi ∈ Pi. It follows that for any x ∈ Rn
+>0 and
+any s ∈ R we have
+ψi(eswx) = [(esw)aixai]ai∈Ai = [es⟨ai,w⟩xai]a∈Ai = [escixai]a∈Ai = ψi(x).
+Hence ψ is not injective. Moreover, w is in the kernel of the derivative of ψi at x.
+□
+We denote by Z the image of ψ. Then we can write
+N(A1, . . . , An) = E g∈G#(Z ∩ g1H1 ∩ . . . ∩ gnHn),
+where the hypersurfaces Hi are defined in (3.1). By Theorem 3.2 and (3.6), this can be expressed
+as
+(4.6)
+N(A1, . . . , An) = (2π)− n
+2 ρm1 · · · ρmn
+�
+Rn
+>
+(¯σZ ◦ ψ)Jψ dx.
+We make the coordinate change Rn → Rn
+>0, (w1, . . . , wn) �→ x = (e−w1, . . . , e−wn), which has
+the absolute Jacobian x1 · · · xn, and obtain (slightly abusing notation)
+(4.7)
+�
+Rn
+>
+(¯σZ ◦ ψ)Jψ dx =
+�
+Rn x1 · · · xn(¯σZ ◦ ψ)Jψ dw.
+Recall from Subsection 2.4 that each vertex v of P defines the inner normal cone Cv := P ∗
+v .
+We can write
+(4.8)
+Rn =
+�
+v
+Cv
+as the union over the vertices v of P. Moreover, we know that dim(Cv ∩ Cv′) < n for different
+vertices v, v′. Therefore, we can rewrite (4.7) as the sum
+�
+v
+�
+Cv
+x1 · · · xn(¯σZ ◦ ψ)Jψ dw.
+over the f0 many vertices v of P.
+Fix now a vertex v of P. According to Lemma 2.5, there are vertices vi of Pi, for i = 1, . . . , n,
+satisfying v = v1 + . . . + vn. Note that ai ∈ Ai.
+
+12
+PETER B¨URGISSER
+We define the map ϕi : Rn
+>0 → RAi\{vi} by
+ϕi(x) = (xai−vi)ai∈Ai\{vi} ∈ RAi\{vi} ≃ Rmi.
+Note that ϕi expresses ψi in the affine chart given by P(RAi)yvi ̸=0 → Rai∈Ai\{vi}, which maps
+[yai]ai∈Ai to y−1
+vi (yai)ai∈Ai\{vi}. So we are in the setting of Subsection 3.2 and ϕi is an instance
+of (3.8). The rows of the matrix M(x) := Dxϕ are labeled by the disjoint union A1⊔. . .⊔An and
+M(x) has n columns. For any n-tuple (a1, . . . , an) with ai ∈ Ai \{vi}, we denote by M(x)a1,...,an
+the n×n submatrix of M(x), obtained by selecting from M(x) the rows numbered by a1, . . . , an.
+We apply Lemma 3.5 to bound
+ρm1 · · · ρmn
+�
+Cv
+x1 · · · xn(¯σZ ◦ ψ)Jψ dw ≤
+�
+a1,...,an
+�
+Cv
+x1 · · · xn| det M(x)a1,...,an| dw,
+where the sum runs over all tuples (a1, . . . , an) with ai ∈ Ai \ {vi}. So there are m1 · · · mn many
+summands. To prove Theorem 1.1, it is sufficient to show that
+(4.9)
+�
+Cv
+x1 · · · xn| det M(x)a1,...,an| dw ≤ 1
+for each vertex v and each selection (a1, . . . , an).
+The component (row) of the derivative Dxϕi corresponding to ai ∈ Ai \ {vi} is given by
+(Dxϕi)ai = xai−vi(ai − vi)diag(x−1
+1 , . . . , x−1
+n ).
+Hence the n × n-submatrix M(x)a1,...,an of M(x) is given by
+M(x)a1,...,an = diag(xa1−v1, . . . , xan−vn)
+
+
+a1 − v1
+...
+an − vn
+
+ diag(x−1
+1 , . . . , x−1
+n ).
+Therefore, setting bi := ai − vi, we get
+x1 · · · xn det(M(x)a1,...,an) = xb1+...+bn det[b1, . . . , bn].
+Let us write Πi for the cone of Pi at the vertex vi. By definition, bi ∈ Π∗
+i . By Lemma 2.5,
+Π := Π1+. . .+Πn equals the cone of the polytope P = P1+. . .+Pn at the vertex v = v1+. . .+vn.
+Hence bi ∈ Π∗
+i ⊆ Π∗
+1 ∩ . . . ∩ Π∗
+n = Π∗ = Cv.
+We can therefore rewrite the left-hand side of (4.9) as
+(4.10)
+�
+Cv
+x1 · · · xn| det M(x)a1,...,an| dw =
+�
+Cv
+e−⟨b1+...+bn,w⟩| det[b1, . . . , bn]| dw.
+By Proposition 2.4, this is at most 1. This shows claim (4.9) and finishes the proof of Theo-
+rem 1.1.
+□
+4.2. Proof of Proposition 1.2. For finite Si ⊆ R, put A := S1 × . . . × Sn, and consider the
+maps
+ψi : R>0 → P(RSi), xi �→ [xai
+i ]ai∈Si,
+ψ: Rn
+>0 → P(RA), x �→ [xa]a∈A
+with images Zi and Z, respectively. The kinematic formula for real projective space gives (see
+[9, Cor. A.3])
+N(Si) = vol(Zi)
+vol(P1),
+N(A, . . . , A) = vol(Z)
+vol(Pn).
+The key insight is that Z is obtained as the image of Z1 × . . . × Zn under the Segre embedding
+P(RS1) × . . . × P(RSn) → P(RS1 ⊗ . . . ⊗ RSn) ≃ P(RA),
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+13
+which is well known to be isometric. Therefore, vol(Z) = vol(Z1) · · · vol(Zn), and this completes
+the proof of Proposition 1.2.
+□
+4.3. Proof of Theorem 1.3. Given is a finite subset A ⊆ Rn with convex hull P. By Lemma 4.2
+we can can w.l.o.g. assume that dim P = n. Consider the injective map
+(4.11)
+ψ: Rn
+>0 → P(RA), ψ(x) := [xa]a∈A
+with image Z ⊆ P(RA). The kinematic formula for real projective space [9, Cor. A.3] is con-
+siderably simpler than the one in Theorem 3.2, since O(m) acts transitively on the Grassmann
+manifolds Gr(k, Rm): we have
+(4.12)
+N(A, . . . , A) = vol(Z)
+vol(Pn) =
+1
+vol(Pn)
+�
+Rn
+>0
+Jψ(x) dx.
+We now proceed as in the proof of Theorem 1.3. We make the coordinate change x = e−w
+and decompose the resulting integral according to the decomposition (4.8) of Rn into the full
+dimensional cones Cv corresponding to vertices v. Thus
+�
+Rn
+>0
+Jψ(x) dx =
+�
+Rn x1 · · · xnJψ(x) dw =
+�
+Cv
+�
+Cv
+x1 · · · xnJψ(x) dw
+For a fixed vertex v of P, we consider the map ϕ: Rn
+>0 → RA\{v} defined by
+(4.13)
+ϕ(x) = (xa−v)a∈A\{v}.
+Then we have ψ(x) = π(ϕ(x)), where π is the inverse of the chart P(RA)yv̸=0 → RA\{v}. It is
+easy to verify that Jψ(x) ≤ Jϕ(x) using ∥Dϕ(x)π∥ ≤ 1, see (2.1).
+Le us view M(x) := Dxϕ as a matrix whose rows are labelled by elements of A \ {v}. and
+denote by M(x)a1,...,an the submatrix of M(x) obtained by selecting the rows labelled by the ai.
+Binet-Cauchy implies that
+Jϕ(x)2 = det(M(x)T M(x)) =
+�
+a1,...,an
+(det M(x)a1,...,an)2,
+with the sum running over all n-element subsets {a1, . . . , an} of A\{v}, of which there are
+�t−1
+n
+�
+many. This implies Jϕ(x) ≤ �
+a1,...,an | det M(x)a1,...,an|. We have arrived at
+�
+Cv
+x1 · · · xnJψ(x) dw ≤
+�
+a1,...,an
+�
+Cv
+x1 · · · xn| det M(x)a1,...,an| dw ≤
+�t − 1
+n
+�
+,
+where the right-hand inequality follows from Proposition 2.4 as in (4.10).
+□
+4.4. Proof of Theorem 1.4. The key observation is the following. Define for ε > 0
+Dε := {x ∈ Rn | ∥x∥ ≥ ε}.
+Lemma 4.3. Let C ⊆ Rn be a proper cone, d ∈ int(C∗), and ε > 0. Then
+lim
+m→∞ mn
+�
+C∩Bε
+e−m⟨d,w⟩ dw = 0
+Proof. Since ∩m≥1Dmε = ∅, basic integration theory implies
+lim
+m→∞
+�
+C∩Dmε
+e−⟨d,u⟩ du = 0.
+Making the change of variables u = mw shows the assertion.
+□
+
+14
+PETER B¨URGISSER
+We now observe the following. Let U ⊆ Rn
+>0 be open. Analogously as for (4.6), one shows
+that
+(2π)− n
+2 ρm1 · · · ρmn
+�
+U
+x1 · · · xn(¯σZ ◦ ψ)Jψ dw.
+equals the expected number of nondegenerate zeros in U of the random system (4.3).
+We follow the proof of Theorem 1.1. Note that stretching the support does not change the
+Newton polytopes Pi and P = P1 + . . . + Pn. Fix a vertex v of P. According to Lemma 2.5,
+there are vertices vi of Pi, for i = 1, . . . , n, satisfying v = v1 + . . . + vn. Tracing the proof
+of Theorem 1.1, one sees that it is sufficient to show that (compare (4.10)) for any selection
+a1 ∈ A1 \ {v1}, . . . , an ∈ An \ {vn}, the vectors bi = ai − vi satisfy
+lim
+m→∞
+�
+Cv
+e−m⟨b1+...+bn,w⟩| det[mb1, . . . , mbn]| dw = 0.
+However, this is a consequence of Lemma 4.3.
+□
+4.5. Additional comment. It is instructive to see how (4.12) directly follows from the more
+general kinematic formula in Theorem 3.2. Consider the injective map ψ from (4.11) with image
+Z ⊆ P(RA). We use ψ to define the map
+(4.14)
+ψd : Rn
+>0 → (P(RA))n, x �→ (ψ(x), . . . , ψ(x)).
+The image Zd = {(y, . . . , y) | y ∈ Z} ⊆ (Pm)n of ψd is the diagonal embedding of Z in the
+product of projective spaces. By Theorem 3.2 and (3.6) we have
+N(A, . . . , A) = (2π)− n
+2 ρn
+m
+�
+Rn
+>0
+(¯σZd ◦ ψd)Jψd dx.
+Via Lemma 4.4 below, we indeed conclude that
+N(A, . . . , A) =
+1
+vol(Pn)
+�
+Rn
+>0
+Jψ dx = vol(Z)
+vol(Pn),
+which is (4.12).
+Lemma 4.4. For x ∈ Rn
+>0 we have
+ρn
+m ¯σZd(ψd(x))Jψd(x) =
+(2π)
+n
+2
+vol(Pn)Jψ(x).
+Proof. Lemma 3.4 applied to the map ψd from (4.14) gives
+(4.15)
+ρn
+m ¯σZd(ψd(x))Jψd(x) = E λ1,...,λn ∥(λ1 ◦ Dxψ) ∧ . . . ∧ (λn ◦ Dxψ)∥
+where the λi are standard Gaussian linear forms on Tψ(x)Pm. Take an isometry Tψ(x)Pm ≃ Rm,
+view λi ∈ Rm as a vector, and view ∆ := Dxψ as a matrix in Rm×n. We note that Jψ(x) =
+�
+det(∆T ∆). The right-hand side of (4.15) can be written as the expectation E λi| det R(x)|,
+with the matrix
+(4.16)
+R(x) :=
+
+
+λT
+1 ◦ Dxψ
+...
+λT
+n ◦ Dxψ
+
+ =
+
+
+λT
+1
+...
+λT
+n
+
+ · ∆.
+We thus need to prove that
+(4.17)
+E λi| det R(x)| =
+(2π)
+n
+2
+vol(Pn)
+�
+det(∆T ∆).
+
+REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS
+15
+In order to show this, by the singular value decomposition, we may assume that ∆ =
+�
+D
+0
+�
+,
+where D = diag(σ1, . . . , σn). Note that
+�
+det(∆T ∆) = σ1 · · · σn. Then (4.16) can be written as
+R(x) = ΛD, where Λ ∈ Rn×n is a standard Gaussian square matrix and we get E Λ| det(R(x))| =
+σ1 · · · σn E w| det Λ|. It is well known that E Λ| det Λ| = ρnρn−1 · · · ρ1, e.g., see [6, Cor. 4.11]. On
+the other hand (see [6, Lemma 2.25])
+ρm =
+√
+2π vol(Pm−1)
+vol(Pm) ,
+hence ρnρn−1 · · · ρ1 = (2π)
+n
+2
+vol(Pn). We have thus verified (4.17).
+□
+References
+[1] Mart´ın Avenda˜no. The number of roots of a lacunary bivariate polynomial on a line. Journal of Symbolic
+Computation, 44(9):1280–1284, 2009.
+[2] D. N. Bernstein. The number of roots of a system of equations. Funkcional. Anal. i Priloˇzen., 9(3):1–4, 1975.
+[3] Fr´ed´eric Bihan and Boulos El-Hilany. A sharp bound on the number of real intersection points of a sparse
+plane curve with a line. Journal of Symbolic Computation, 81:88–96, 2017.
+[4] Fr´ed´eric Bihan and Frank Sottile. New fewnomial upper bounds from Gale dual polynomial systems. Mosc.
+Math. J., 7(3):387–407, 573, 2007.
+[5] Ir´en´ee Briquel and Peter B¨urgisser. The real tau-conjecture is true on average. Random Structures Algo-
+rithms, 57(2):279–303, 2020.
+[6] Peter B¨urgisser and Felipe Cucker. Condition, volume 349 of Grundlehren der Mathematischen Wis-
+senschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2013. The geometry
+of numerical algorithms.
+[7] Peter B¨urgisser, Felipe Cucker, and Pierre Lairez. Rigid continuation paths ii. structured systems. Preprint
+arXiv:2010.10997, 2021.
+[8] Peter B¨urgisser, Alperen A. Erg¨ur, and Josu´e Tonelli-Cueto. On the number of real zeros of random fewno-
+mials. SIAM J. Appl. Algebra Geom., 3(4):721–732, 2019.
+[9] Peter B¨urgisser and Antonio Lerario. Probabilistic Schubert calculus. J. Reine Angew. Math., 760:1–58,
+2020.
+[10] Ren´e Descartes. La G´eom´etrie. Librairie Scientifique A. Hermann, 1886. Digital reproduction of 2008 by
+Project Gutenberg (Ebook number: 26400).
+[11] Mathias Drton, Bernd Sturmfels, and Seth Sullivant. Lectures on algebraic statistics, volume 39 of Oberwol-
+fach Seminars. Birkh¨auser Verlag, Basel, 2009.
+[12] Alan Edelman and Eric Kostlan. How many zeros of a random polynomial are real? Bull. Amer. Math. Soc.
+(N.S.), 32(1):1–37, 1995.
+[13] Jacques Faraut and Adam Kor´anyi. Analysis on symmetric cones. Oxford Mathematical Monographs. The
+Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications.
+[14] Osman G¨uler. Barrier functions in interior point methods. Math. Oper. Res., 21(4):860–885, 1996.
+[15] Bertrand Haas. A simple counterexample to Kouchnirenko’s conjecture. Beitr¨age Algebra Geom., 43(1):1–8,
+2002.
+[16] Fritz Horn and Roy Jackson. General mass action kinetics. Arch. Rational Mech. Anal., 47:81–116, 1972.
+[17] Ralph Howard. The kinematic formula in Riemannian homogeneous spaces. Mem. Amer. Math. Soc.,
+106(509):vi+69, 1993.
+[18] Gorav Jindal, Anurag Pandey, Himanshu Shukla, and Charilaos Zisopoulos. How many zeros of a random
+sparse polynomial are real? In ISSAC’20—Proceedings of the 45th International Symposium on Symbolic
+and Algebraic Computation, pages 273–280. ACM, New York, [2020] ©2020.
+[19] Askold G. Khovanski˘ı. A class of systems of transcendental equations. Dokl. Akad. Nauk SSSR, 255(4):804–
+807, 1980.
+[20] Askold G. Khovanski˘ı. Fewnomials, volume 88 of Translations of Mathematical Monographs. American Math-
+ematical Society, Providence, RI, 1991.
+[21] Pascal Koiran. Shallow circuits with high-powered inputs. Proc. Second Symposium on Innovations in Com-
+puter Science, ICS, 2011.
+[22] Pascal Koiran, Natacha Portier, and S´ebastien Tavenas. On the intersection of a sparse curve and a low-degree
+curve: a polynomial version of the lost theorem. Discrete Comput. Geom., 53(1):48–63, 2015.
+
+16
+PETER B¨URGISSER
+[23] Pascal Koiran, Natacha Portier, and S´ebastien Tavenas. A Wronskian approach to the real τ-conjecture. J.
+Symbolic Comput., 68(part 2):195–214, 2015.
+[24] Anatoli G. Kushnirenko. Poly`edres de Newton et nombres de Milnor. Invent. Math., 32(1):1–31, 1976.
+[25] L´eo Mathis. The Handbook of Zonoid Calculus. PhD thesis, SISSA, 2022.
+[26] Yurii Nesterov and Arkadii Nemirovskii. Interior-point polynomial algorithms in convex programming, vol-
+ume 13 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM),
+Philadelphia, PA, 1994.
+[27] Kaitlyn Phillipson and J. Maurice Rojas. Fewnomial systems with many roots, and an adelic tau conjecture.
+In Proceedings of Bellairs workshop on tropical and non-Archimedean geometry (May 6-13, 2011, Barbados),
+Contemporary Mathematics, volume 605, pages 45–71, 2014.
+[28] Rolf Schneider. Convex bodies: the Brunn-Minkowski theory, volume 151 of Encyclopedia of Mathematics
+and its Applications. Cambridge University Press, Cambridge, expanded edition, 2014.
+[29] Frank Sottile. Real solutions to equations from geometry, volume 57 of University Lecture Series. American
+Mathematical Society, Providence, RI, 2011.
+[30] G¨unter M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag,
+New York, 1995.
+Institute of Mathematics, Technische Universit¨at Berlin
+Email address: pbuerg@math.tu-berlin.de
+
diff --git a/29AyT4oBgHgl3EQfb_di/content/tmp_files/load_file.txt b/29AyT4oBgHgl3EQfb_di/content/tmp_files/load_file.txt
new file mode 100644
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+++ b/29AyT4oBgHgl3EQfb_di/content/tmp_files/load_file.txt
@@ -0,0 +1,1470 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf,len=1469
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='00273v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='PR]  31 Dec 2022  REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS  PETER B¨URGISSER  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Consider a system f1(x) = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , fn(x) = 0 of n random real polynomials in n  variables, where each fi has a prescribed set of exponent vectors described by a set Ai ⊆ Zn  of cardinality ti, whose convex hull is denoted Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Assuming that the coefficients of the fi are  independent standard Gaussian, we prove that the expected number of zeros of the random  system in the positive orthant is at most (2π)− n  2 f0 (t1 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' (tn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Here f0 denotes the  number of vertices of the Minkowski sum P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We also derive a better bound in  the unmixed case where all supports Ai are equal, improving upon [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' All arguments equally  work for real exponent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Introduction  In many applications, we want to understand or find the (positive) real solutions of a system  of multivariate polynomial equations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=', see [11, 16, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Bezout’s theorem, which bounds  the number complex zeros in terms of degrees, usually highly overestimates the number of real  zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This can be already seen from Descartes’ rule of signs [10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' 42], which implies that  a real univariate polynomial with t terms has at most t − 1 positive zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' In 1980, Khovan-  skii [19] obtained a far reaching generalization of Descartes’ rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' He showed that the number  of nondegenerate1 positive solutions of a system f1(x) = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , fn(x) = 0 of n real polynomial  equations in n variables is bounded only in terms of n and the number t of distinct exponent  vectors occurring in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This result in fact allows for any real exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Following  Kushnirenko, one speaks of fewnomial systems, with the idea that the number t of terms is small,  see [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Understanding the complex zeros of fewnomial systems is much simpler: the famous BKK-  Theorem [2, 24] states that for given finite supports A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An ⊆ Zn and Laurent polynomials  fi(x) = �  a∈A ci(a)xa1  1 · · · xan  n  with generic complex coefficients ci(a), the number of complex  solutions in (C×)n of a corresponding system f1(x) = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , fn(x) = 0 is given by n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' times the  mixed volume of the Newton polytopes P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , Pn, where Pi is defined as the convex hull of Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Note that the number of real zeros has little to do with the metric properties of Pi: indeed,  replacing Ai by a multiple mAi amounts to substituting xi by xm  i (m > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Clearly, this does not  change the number of positive real zeros of a fewnomial system, however Pi has been replaced  by mPi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  The bound on the number of real zeros obtained by Khovanskii is exponential in the number t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  It is widely conjectured that this bound is far from optimal: in fact it is conjectured [27] that for  fixed n, the number of nondegenerate positive solutions of a fewnomial system with t exponent  vectors is bounded by a polynomial in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Quite surprisingly, this question is open even for n = 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' 60D05, 14P99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' fewnomials, random polynomials, real algebraic geometry, sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Supported by the ERC under the European Union’s Horizon 2020 research and innovation programme (grant  agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' 787840).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  1i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=', the Jacobian of the system does not vanish at the zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  1  2  PETER B¨URGISSER  For results in special cases, we refer to [4, 1, 29, 22, 23, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Moreover, there is a very interesting  connection to complexity theory [21, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Given this state of affairs of real fewnomial theory, a possible way to advance is to ask  what happens in generic situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This can be made formal by considering random fewnomial  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Fix supports A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An ⊆ Zn of cardinality t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , tn, respectively, and consider a  system of n random polynomials fi(x) as above, but now the coefficients ci(a) are assumed  to be independent standard Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let us denote by N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) the expectation of the  number of nondegenerate positive real zeros of such system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Actually, we work in more generality,  allowing any subsets Ai of Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' There is a large literature on the real zeros of  random polynomials: we refer to [8] for references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  In [8] it was proven that N(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , A) ≤ 21−n�t  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The main result of the present paper is an  extension of this to the mixed case, where the fewnomials may have have different supports Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Our bound depends on the combinatorial structure of the Minkowski sum P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='+ Pn through  the number of its vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We remark that our proof is quite different from the one in [8], which  was quite indirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' If Pi denotes the convex hull of the finite nonempty set Ai ⊆ Rn of cardinality ti,  for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , n, then  N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) ≤ (2π)− n  2 f0 (t1 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' (tn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Here f0 denotes the number of vertices of the Minkowski sum P := P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  The bound in this theorem looks similarly to the one in a conjecture attributed to Kush-  nirenko, which states that the number of positive nondegenerate zeros is always bounded by  (t1 − 1) · · · (tn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' However, this was disproved in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2  In the unmixed situation, where all supports are equal to A, it is well known [12] that the  expected number of positive zeros can be expressed by the volume of the image of the Veronese  like map Rn  >0 → P(RA) sending x to [xa]a∈A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This is a consequence of the kinematic formula  for real projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' In the mixed situation, there is no such simple characterization: one  has to work with the more complicated kinematic formula for products of projective spaces  (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2) that we derive from [17, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' After passing to exponential coordinates w = log x,  we bound the resulting integral over Rn with a strategy inspired by the theory of toric varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  The normal fan of the polytope P affords a decomposition of Rn into the normal cones C at  the vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The resulting integral over C can be bounded in terms of the characteristic  function of the dual cone of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Finally, an explicit a priori bound on the characteristic function  (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4) completes the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The univariate case and a conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The univariate case (n = 1) was settled by  Jindal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' They showed that for any subset S ⊆ R of cardinality t, we have  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1)  N(S) ≤ 2  π  √t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Moreover, they constructed a sequence St ⊆ Z of supports of cardinality t with N(St) ≥ c  √  t for  some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Consider for t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , tn ≥ 1 the supports A1 := St1 × 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An :=  0 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × 0 × Stn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' These supports describe a system of n equations, where the ith equation  depends on xi only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Therefore, N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) = N(St1) · · · N(Stn), which with the above leads  to the lower bound  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2)  N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) ≥ cn√t1 · · · tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  2This conjecture was never published by Kushnirenko and apparently, he did not believe in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS  3  We complement this by showing that for any A = S1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='×Sn in product form, the expectation  N(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , A) can be expressed in terms of the N(Si) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' If A = S1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × Sn for finite Si ⊆ R, then  N(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , A) =  πn  vol(Pn) N(S1) · · · N(Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We conjecture that the lower bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2) is optimal in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let Ai ⊆ Rn be finite nonempty sets of cardinality ti with convex hull Pi, for  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We denote by f0 the number of vertices of P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then  N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) ≤ κ(f0)√t1 · · · tn  for some function κ : N → N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' In particular, N(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , A) ≤ κ(f0) t  n  2 for A ⊆ Rn of cardinality t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  In the special case A = S1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × Sn, by combining (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1) with Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2, we obtain  N(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , A)vol(Pn) ≤ 2n√  t, which is much smaller than what Conjecture 1 predicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Improvement in unmixed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We can improve the dependence on n in the bound of  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1 in the case where all supports are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For A ⊆ Rn of cardinality t ≥ 1 with convex hull P and f0 vertices, we have  N(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , A) ≤  1  vol(Pn) f0  �t−1  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  To compare this with the bound N(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , A) ≤ 21−n�t  n  �  from [8], note that vol(Pn)−1 =  Γ( n+1  2 )π− n+1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Hence, for n → ∞, the new bound goes asymptotically faster to zero than the  old one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' However, the bound in [8] also holds for nonstandard centered Gaussian coefficients  ci(a) ∼ N(0, σi(a)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Location of zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We finish with a result on the typical location of the zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' It is well  known that for random real polynomials, the positive reals zeros x tend to accumulate around 1:  see [12] for the dense and [18] for the sparse case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This means that w = log x accumulates  around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We generalize this to multivariate systems as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Fix a finite supports A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An ⊆ Rn and consider a random system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3) with  independent standard Gaussian coefficients ci(a) for the stretched supports mAi, where m ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Fix ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then the probability that the system has a zero w ∈ Rn with ∥w∥ > ε goes to zero, as  m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Preliminaries  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' A metric property of charts of real projective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Consider the real projective  space Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We shall identify the tangent space T[y]Pm at a point [y] := [y0 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' : ym] with Ry⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  The standard Riemannian metric on Pm is defined by ⟨v, w⟩[y] := ∥y∥−2⟨v, w⟩ for v, w ∈ Ry⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We denote by Py the orthogonal projection onto Ry⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Consider the affine chart (Pm)y0̸=0 → Rm, which maps [y0 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' : ym] to y−1  0 (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , ym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Its  inverse is given by  π: Rm → (Pm)y0̸=0, (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , ym) �→ [1 : y1 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' : ym].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  By [6, Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='8], the derivative of π at y′ := (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , yn) satisfies Dy′π = ∥π(y′)∥−1Py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' From  this we conclude for the spectral norm  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1)  ∥Dy′π∥ ≤ ∥π(y′)∥−1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  4  PETER B¨URGISSER  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' On the quantity σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The relative position of two subspaces of a Euclidean vector space E  can be quantified by a volume like quantity, which is crucial in the study of integral geometry  in homogeneous spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' see [17] and [9, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' To define this quantity, note first that there is an  induced inner product on the exterior algebra Λ(E) given by (see [9, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1)])  ⟨v1 ∧ · · · ∧ vk, w1 ∧ · · · ∧ wk⟩ = det(⟨vi, wj⟩)1≤i,j≤k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  More concretely, ∥v1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ vn∥ = | det[v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , vn]|, where [v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , vn] denotes the matrix with  columns vi ∈ E = Rn  Let V, W be linear subspaces of E of complementary dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We define (see [9, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3)])  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2)  σ(V, W) := ∥v1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ vk ∧ w1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ wm∥ ∈ [0, 1],  where v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , vk and w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , wm are orthonormal bases of V and W, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Clearly,  σ(V, W) = σ(W, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The extreme cases are: σ(V, W) = 0 iff V ∩ W ̸= 0 and σ(V, W) = 1 iff v  and W are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We have σ(V ⊥, W ⊥) = | det p |, if p: V ⊥ → W denotes the restriction of the  orthogonal projection E → W to V ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Moreover, σ(V, W) = σ(V ⊥, W ⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , νm be an orthonormal basis of V ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We decompose νi = ν′  i + ν′′  i according  to E = W ⊕ W ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then p(νi) = ν′  i and | det p | = ∥ν′  1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ ν′  m∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' If ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , ωk denotes an  orthonormal basis of W ⊥, we have  σ(V ⊥, W ⊥) = ∥ν1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ νm ∧ ω1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ ωk∥ = ∥ν′  1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ ν′  m ∧ ω1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ ωk∥ = ∥ν′  1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ ν′  m∥,  the last equality holding since the span of the ν′  i equals W, which is orthogonal to the span of  the wj, which is W ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This proves σ(V ⊥, W ⊥) = | det p |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  For the second assertion, we use that | det p | = | det q |, where q: W ⊥ → V denotes the  restriction of the orthogonal projection E → V to W ⊥, see [7, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  Clearly, the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2) can be extended to more than two subspaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' see [9, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' But if  W = W1 ⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='⊕Wn is an orthogonal decomposition, we can reduce to the case of two subspaces:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3)  σ(V, W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , Wn) = σ(V, W1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Wn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  This is a consequence of [9, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Characteristic function of convex cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We prove here an priori upper bound on the  characteristic function of a convex cone, which is a key ingredient in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  A convex cone C ⊆ Rn is called proper if it is n-dimensional and pointed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=', contained in a  halfspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' It is well known that a convex C ⊆ Rn is proper iff its dual cone  C∗ := {x ∈ Rn | ∀y ∈ C ⟨x, y⟩ ≥ 0}  is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let g ∈ GL(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then K := g(C) is a proper cone and gT (K∗) = C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We denote  by int(C) the interior of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We assign to a proper cone C ⊆ Rn the function  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4)  vC : int(C∗) → R>0, vC(x) :=  �  C  e−⟨x,y⟩ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  One calls vC the characteristic function (or Koszul-Vinberg characteristic) of C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' It is a useful  analytic tool for investigating convex cones, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=', see [13, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3] and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=', Rn  >0 is self dual and  vRn  >0(x) = (x1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' · xn)−1 for x ∈ Rn  >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS  5  The homogeneity property vC(tx) = t−nvC(x) for t > 0, x ∈ int(C∗) is immediate to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Moreover, the transformation formula implies the following invariance property: if g ∈ GL(n, R)  and K := g(C), then gT (K∗) = C∗ and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5)  vK(z) = | det g| vC(gT z)  for z ∈ int(K∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The function log vC is strictly convex and essentially equals Nesterov and Ne-  mirowski’s universal self-concordant barrier function [26, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5], see [14] for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  The following result is well known, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=', see [14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We provide the simple proof for  the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For x ∈ int(C∗) we have  vC(x) = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' vol  �  y ∈ C | ⟨x, y⟩ ≤ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We fix x ∈ int(C∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For t ≥ 0 we define the n − 1-dimensional slice  Ct :=  �  y ∈ C | ⟨x, y⟩ = t∥x∥  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  By Fubini, we get  vC(x) =  �  C  e−⟨x,y⟩ dy =  � ∞  0  voln−1(Ct)e−t∥x∥dt = voln−1(C1)  � ∞  0  tn−1e−t∥x∥dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Note that  � ∞  0  tn−1e−t∥x∥dt =  1  ∥x∥n  � ∞  0  sn−1e−sds =  1  ∥x∥n Γ(n) = (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  ∥x∥n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Moreover, we have  voln−1(C1) = n voln  �  y ∈ C | ⟨x, y⟩ ≤ ∥x|  �  = n ∥x∥n voln  �  y ∈ C | ⟨x, y⟩ ≤ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  It follows that  vC(x) = n ∥x∥n voln  �  y ∈ C | ⟨x, y⟩ ≤ 1  � (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  ∥x∥n  = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' voln  �  y ∈ C | ⟨x, y⟩ ≤ 1  �  ,  completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  The following technical result is essential for the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let C ⊆ Rn be a proper cone and b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then we have  ∥b1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ bn∥ · vC(b1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + bn) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  This bound is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We denote by cone(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn) ⊆ C∗ the convex cone generated by b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Without  loss of generality, we may assume that b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn ∈ C∗ is a basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let b∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , b∗  n denote its  dual basis, that is ⟨b∗  i , bj⟩ = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' In matrix terminology, [b∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , b∗  n]T [b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn] = In, hence  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='6)  det[b∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , b∗  n] det[b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn] = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Moreover, the definition of the dual basis implies that cone(b∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , b∗  n) is the dual cone of  cone(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Therefore, by duality, we get  C ⊆ cone(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn)∗ = cone(b∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , b∗  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Put d := b1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + bn and let y ∈ C such that ⟨d, y⟩ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Since C ⊆ cone(b∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , b∗  n), we can  write y = �  i tib∗  i with ti ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Moreover �  i ti = ⟨d, y⟩ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Thus we have shown the inclusion  {y ∈ C | ⟨d, y⟩ ≤ 1} ⊆ conv{0, b∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , b∗  n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  6  PETER B¨URGISSER  This implies the inequality of volumes  voln{y ∈ C | ⟨d, y⟩ ≤ 1} ≤ volnconv{0, b∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , b∗  n} = 1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='| det[b∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , b∗  n]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Multiplying with n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' | det[b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn]|, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='6) and taking into account Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3, the assertion  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  The optimality is attained for C = Rn  + and bi = diei with di > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Indeed, we have  | det[b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn] |vC(d) = d1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' · dn (d1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' · dn)−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Vertices and normal fan of sums of polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We recall here some basic facts about  polytopes and their normal fans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' see [30, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Let P ⊆ Rn be a full-dimensional polytope and v be a vertex of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The cone Pv of P at v is  defined as the convex cone generated by P − v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' It is a proper cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The dual cone of Pv, also  called the inner normal cone of P at v, is defined as  P ∗  v := {y ∈ Rn | ∀x ∈ P ⟨x, y⟩ ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  The cone P ∗  v is also proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The union over all P ∗  v equals Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Moreover, for v1 ̸= v2, we have  dim(P ∗  v1 ∩ P ∗  v2) < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' In fact, the P ∗  v are the n-dimensional cones of the normal fan of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We will need the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , Pn be polytopes in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' There is an injective map  Vert(P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Pn) → Vert(P1) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Vert(Pn), v �→ (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , vn)  satisfying v = v1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Moreover, if we denote by Πi the cone of Pi at the vertex vi, then  Π := Π1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Πn is the cone of P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Pn at the vertex v1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' In particular,  Π∗ = Π∗  1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∩ Π∗  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For the following, see [28, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' To a nonzero weight ω ∈ Rn we assign the face of Pi,  given by  F(Pi, ω) :=  �  w ∈ Rn | ⟨w, ω⟩ = min  w′∈Pi⟨w′, ω⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We have (see [28, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5])  F(P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Pn, ω) = F(P1, ω) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + F(Pn, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Suppose that F(P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Pn, ω) = {v} is a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then all F(Pi, ω) = {vi} are vertices  and v = v1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  It is easy to see that the vi are uniquely determined by v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  The map  v �→ (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , vn) is as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The remaining assertions are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Random intersections in products of projective spaces  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The kinematic formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We specialize here the general kinematic formula for homo-  geneous spaces from [9, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2] to the case of products of real projective spaces (Theo-  rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For this purpose, we define the average scaling factor and we explain how to bound it  in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Consider the product Ω := Pm1 × · · · × Pmn of real projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The product G :=  O(m1 +1)×· · ·×O(mn +1) of orthogonal groups acts transitively on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' So Ω is a homogeneous  space and we have an induced transitive action of G on the tangent bundle of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We focus on  the special hypersurfaces H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , Hn of Ω of the following shape  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1)  H1 := Pm1−1 × Pm2 × · · · × Pmn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , Hn := Pm1 × Pm2 · · · × Pmn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  They are determined upon selecting hyperplanes Pmi−1 in Pmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Our goal is to investigate the  average cardinality of the intersection Z ∩H1 ∩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='∩Hn of an n-dimensional smooth submanifold  REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS  7  Z ⊆ Ω with random Hi corresponding to independently chosen uniform random hyperplanes  in Pmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Fix a distinguished point ω ∈ Ω and denote by K the stabilizer group of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=', take  ωi = [1 : 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' : 0] for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Notice that we have an induced action of K on the tangent space  T := TωΩ, which we can identify with the standard action of K = O(m1) × · · · × O(mn) on  T = Rm1 × · · · × Rmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This induces an action of K on the Grassmann manifold Gr(d, T ) of  linear subspaces of T with codimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Note that this action is transitive if n = 1, but not  for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We assign to an n-dimensional smooth submanifold Z ⊆ Ω a map  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2)  Z → Gr(n, T )/K, z �→ KgNpZ  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For given p ∈ Z choose any g ∈ G such that gp = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The induced action of g maps  the tangent space TpΩ to TωΩ = T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This transports the normal subspace NpZ ⊆ TpΩ of Z at p  to gNpZ ⊆ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Note that the K-orbit of the subspace gNpZ does not depend on the choice of g,  which shows that the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We call the submanifold Z cohomogeneous if the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2) is constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' see [9, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1] and [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  For instance, a product Z = L1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × Ln of lines Li in Pmi is cohomogeneous: indeed, the  map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2) sends any point p ∈ Z to the K-orbit of R × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The average scaling factor function of the n-dimensional submanifold Z of  Pm1 × · · · × Pmn is the function ¯σZ : Z → [0, 1] defined at p ∈ Z by  ¯σZ(p) := E Liσ(gNpZ, L1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × Ln),  where g ∈ G satisfies gp = ω0, and the expectation is taken over uniformly random lines Li in  T = Rm1 × · · · × Rmn (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2) for the definition of σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Note that due to the averaging over the K-orbit, the choice of g is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The above  definition is consistent with the one in [9, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1], since  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3)  σ(gNpZ, L1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × Ln) = σ(gNpZ, L1 × 0 × · · · × 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , 0 × · · · × 0 × Ln)  by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' indeed note that the n lines L1 × 0 × · · · × 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' are pairwise orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We introduce the notation  ρn := E ∥x∥ =  √  2 Γ( n+1  2 )  Γ( n  2 )  ≤ √n  for standard Gaussian x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We note that by [6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='25],  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4)  vol(Pmi−1)  vol(Pmi)  =  1  √π  Γ( mi+1  2  )  Γ( mi  2 )  =  1  √  2π ρmi,  We can now explicitly state the kinematic formula for products of real projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For any n-dimensional submanifold Z of Pm1 × · · · × Pmn, we have  E g∈G#(Z ∩ g1H1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∩ gnHn) = (2π)− n  2 ρm1 · · · ρmn  �  Z  ¯σZ dZ,  where the hypersurfaces Hi are defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' If σK : Z × H1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × Hn → [0, 1] denotes the average scaling function from [9, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1],  then [9, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2] states that  E g∈G#(Z ∩ g1H1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∩ gnHn) =  1  vol(Ω)n  �  Z×H1×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='×Hn  σK d(Z × H1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × Hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  8  PETER B¨URGISSER  By K-invariance and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3), we have σK(z, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , yn) = ¯σZ(z) for all z ∈ Z and yi ∈ Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Therefore,  E g∈G#(Z ∩ g1H1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∩ gnHn) = vol(H1) · · · vol(Hn)  vol(Ω)n  �  Z  ¯σZ dZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Finally, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4) gives  vol(H1) · · · vol(Hn)  vol(Ω)n  =  n  �  i=1  vol(Pmi−1)  vol(Pmi)  = (2π)− n  2 ρm1 · · · ρmn,  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' A product Z = L1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × Ln of lines Li is cohomogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2 implies  that ¯σZ = (2/π)n/2(ρm1 · · · ρmn)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We shall focus on submanifolds Z arising as the image of an injective smooth map  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5)  ψ: U → Pm1 × · · · × Pmn, ψ(x) := (ψ1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , ψn(x)),  where the ψi : U → Pmi are smooth maps defined on an open subset U ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let us denote by  Jψ(x) :=  �  det((Dxψ)T Dxψ)  the absolute Jacobian of ψ at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The transformation formula implies that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='6)  �  Z  ¯σZ dZ =  �  U  ¯σZ(ψ(x))Jψ(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We next analyze the integrand on the right-hand side more closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let x ∈ U and put Ti := Tψi(x)Pmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Let λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , λn be independent standard  Gaussian linear forms on Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This defines the random linear forms λi ◦ Dxψi on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then  ρm1 · · · ρmn ¯σZ(ψ(x))Jψ(x) = E λ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',λn ∥(λ1 ◦ Dxψ1) ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ (λn ◦ Dxψn)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' To simplify notation, we assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' that p = ψ(x) is the distinguished point ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We also identify Ti with Rmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For ui ∈ Ti with ∥ui∥ = 1 consider the line Li = Rui and the  orthogonal projection pi : Ti → Li, which is is given by pi(w) = µi(w)ui with the linear form on Ti  defined by µi(w) := ⟨w, ui⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Thus the orthogonal projection pL : T1 × · · · × Tn → L1 × · · · × Ln  is described by µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This implies that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='7)  | det(pL ◦ Dxψ)| = ∥(µ1 ◦ Dxψ1) ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ (µn ◦ Dxψn)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  On the other hand, according to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1, we have  σ(L1 × · · · × Ln, NpZ) = | det p′  L|,  where p′  L : TpZ → L1 × · · · × Ln denotes the restriction of pL to TpZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Applying the determinant  to the composition of Dxψ with p′  L, we get  Jψ(x) | det p′  L| = | det(pL ◦ Dxψ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  By averaging over random lines Li, we deduce from the definition of ¯σZ and the above that  Jψ(x)¯σZ(p) = Jψ(x) E Liσ(NpZ, L1 × · · · × Ln) = Jψ(x) E Li| det p′  L| = E Li| det(pL ◦ Dψ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Finally, a standard Gaussian linear form on Ti is obtained as λi = riµi with independent random  variables ri and ui, where ui is uniformly random in the unit sphere of Ti and r2  i is χ2-distributed  REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS  9  with mi degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Thus E ri = ρmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Altogether, we obtain, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='7),  ρm1 · · · ρmnJψ(x)¯σZ(p) = ρm1 · · · ρmnE | det(pL ◦ Dψ)|  = ρm1 · · · ρmnE ∥(µ1 ◦ Dxψ1) ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ (µn ◦ Dxψn)∥  = E ∥(λ1 ◦ Dxψ1) ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ (λn ◦ Dxψn)∥ ,  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Bounding the average scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' In order to bound the quantity in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4,  we use affine charts for the product of projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Denote by yi0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , yimi coordinates  for Pmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Fix 0 ≤ ri ≤ mi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , n, and consider the inverse of the affine chart πiri : Rmi →  (Pmi)yiri̸=0, see Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We describe the maps ψi from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5) in these charts by smooth  functions defined on open subsets of Rn,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='8)  ϕiri : Rn ⊇ Uiri → Rmi,  satisfying ψi := πiri ◦ ϕiri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' In order to simplify notation, we assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='og.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ri = 0 and write  πi := πi0, ϕi := ϕi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' In these charts, the combined map ψ of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5) is represented by a map  ϕ: U → Rm1 × · · · × Rmn, ϕ(x) = (ϕ1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , ϕn(x))  defined on some open subset U ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We view the derivative M(x) := Dxϕ as a matrix of  format (m1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + mn) × n with blocks Mi(x) := Dxϕi ∈ Rmi×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For 1 ≤ ji ≤ mi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , n,  we denote by M(x)j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',jn the n × n submatrix of M(x) obtained by selecting in the ith block  the jith row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let x ∈ U be such that [yi] := ψi(x) ∈ (Pmi)y0̸=0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then  ρm1 · · · ρmn ¯σZ(ψ(x))Jψ(x) ≤  �  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',jn  | det M(x)j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',jn|,  where the sum is over the n-tuples (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , jn) ∈ [m1] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × [mn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' From ψi = πi ◦ ϕi we get Dψi = Dπi ◦ Dϕi, where we drop arguments for notational  simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let λi : Ti → R be a linear form on Ti = Tψi(x)Pmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then, defining wi := λi ◦ Dπi,  λi ◦ Dψi = λi ◦ Dπi ◦ Dϕi = wi ◦ Dϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  If we identify λi ◦ Dxψi with a vector in Rn and wi with a vector in Rmi, then we have the  matrix product of formats n × �  i mi and �  i mi × n,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='9)  R(x) :=  \uf8ee  \uf8ef\uf8f0  (λ1 ◦ Dxψ1)T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  (λn ◦ Dxψ1)T  \uf8f9  \uf8fa\uf8fb =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  wT  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' 0  0  wT  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' 0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  0  0  wT  n  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb ·  \uf8ee  \uf8ef\uf8f0  M1(x)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Mn(x)  \uf8f9  \uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4 tells us that  ρm1 · · · ρmn ¯σZ(ψ(x))Jψ(x) = E λi| det R(x))|,  where the expectation is over independent standard Gaussian λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Note that the resuling random  vector wi := λi ◦ Dπi is not standard Gaussian anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' However ∥Dπi∥ ≤ 1 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1) and  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='6 below implies that E w2  ij ≤ 1 for the jth component wij of wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  From Binet-Cauchy, we obtain from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='9)  (det R(x))2 =  �  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',jn  w2  1j1 · · · w2  njn(det M(x)j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',jn)2,  10  PETER B¨URGISSER  where the sum is over all (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , jn) ∈ [m1] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × [mn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Taking expectations yields  E w(det R(x))2 ≤  �  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',jn  (det M(x)j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',jn)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We conclude that  E w| det R(x))| ≤  �  E w(det R(x))2� 1  2 ≤  �  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',jn  | det M(x)j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',jn|,  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let A ∈ Rp×m with ∥A∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' If y ∈ Rp is standard Gaussian, then the random  variable z := yA satisfies E |zj|2 ≤ 1 for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' From zj = �  i yiaij we get z2  j = �  i,k yiykaijakj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Hence E z2  j = �  i a2  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Finally, �  i a2  ij =  ∥A(ej)∥2 ≤ ∥A∥2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Mixed random fewnomial systems  The goal here is to provide the proofs of the assertions in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Let us first  introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We assign to a real valued function c: A → R on a finite nonempty subset A ⊆ Rn the real  analytic function FA,c : Rn → R  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1)  FA,c(w) :=  �  a∈A  c(a)e⟨a,w⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  In the special case where A consists of integer vectors, FA,c arises from the Laurent polynomial  fA,c(x) = �  a∈A c(a)xa by a substitution: FA,c(w) = fA,c(ew).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Generally, we have the following  equivariance property: for g ∈ GL(n, R) and b ∈ Rn,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2)  FA+b,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='c(w) = e⟨b,w⟩FA,c(w),  Fg(A),g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='c(w) = FA,c(gT w),  where (g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='c)(a) := c(g−1a) and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='c(a) := c(a − b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Suppose now we have n such analytic functions encoded by ci : Ai → R, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Throughout, we denote by ti the cardinality of Ai and by Pi its convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We are interested  in the number N of nondegenerate zeros w ∈ Rn of the system  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3)  FA1,c1(w) = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , FAn,cn(w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Our goal is to study the expected number of nondegenerate zeros for random coefficient func-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' More specifically, we denote by N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) the expectation of N, when all the coeffi-  cients ci(a), for i ∈ [n] and ai ∈ Ai, are independent standard Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Clearly, N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An)  is invariant under permutations of the Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Also, N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) = 0 if ti = 1 for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' More-  over, we have N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) = 0 if dim(P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Pn) < n, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2) implies the following invariance properties  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4)  N(A1 + b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An + bn) = N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An),  N(g(A1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , g(An)) = N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An),  where b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn ∈ Rn and g ∈ GL(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Our main result is Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1 stated in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Note that it gives the correct  answer N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) = 0 if ti = 1 for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' In the case t1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' = tn = 2, the Pi are segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' If they are linearly independent,  P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Pn is a parallelepiped with 2n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Thus, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1 gives N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) ≤  (2/π)  n  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This can be easily verified directly as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Suppose Ai = {ai, bi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We claim that  N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) = 2−n if b1 − a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn − an are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For showing this, by the  REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS  11  invariance properties (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4), it suffices to consider the case where Ai = {0, ei}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3) amounts  to the system ci(0) + ci(ei)ewi = 0, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , n, which has a solution iff ci(0)ci(ei) < 0, for  all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This happens with probability 2−n, hence indeed N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) = 2−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let us look at a special instance of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' To the given finite  nonempty subsets A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An ⊆ Rn, we assign the maps  ψi : Rn  >0 → P(RAi) ≃ Pmi, ψi(x) := [xai]ai∈Ai,  where mi := #Ai − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Recall that Pi denotes the convex hull of Ai and put P := P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We consider the combined map  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5)  ψ: Rn  >0 → Pm1 × · · · × Pmn, ψ(x) := (ψ1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , ψn(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The map ψ is injective iff P is n-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Moreover, if P is not n-dimensional,  then rankDxψ < n for all x ∈ Rn  >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Assume ψ(exp(w)) = ψ(exp(w′)) for w ̸= w′ ∈ Rn Then there are ci ∈ R such that for  all ai ∈ Ai we have ⟨ai, w − w′⟩ = ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Hence, ⟨x, w − w′⟩ = ci for all xi ∈ Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' It follows that  ⟨x, w − w′⟩ = c1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + cn for all x ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Hence dim P < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Conversely, assume there is a nonzero w ∈ Rn and c ∈ R such that ⟨x, w⟩ = c for all x ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Then there are ci ∈ R such that ⟨xi, w⟩ = ci for all xi ∈ Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' It follows that for any x ∈ Rn  >0 and  any s ∈ R we have  ψi(eswx) = [(esw)aixai]ai∈Ai = [es⟨ai,w⟩xai]a∈Ai = [escixai]a∈Ai = ψi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Hence ψ is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Moreover, w is in the kernel of the derivative of ψi at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  We denote by Z the image of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then we can write  N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) = E g∈G#(Z ∩ g1H1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∩ gnHn),  where the hypersurfaces Hi are defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='6), this can be expressed  as  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='6)  N(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , An) = (2π)− n  2 ρm1 · · · ρmn  �  Rn  >  (¯σZ ◦ ψ)Jψ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We make the coordinate change Rn → Rn  >0, (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , wn) �→ x = (e−w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , e−wn), which has  the absolute Jacobian x1 · · · xn, and obtain (slightly abusing notation)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='7)  �  Rn  >  (¯σZ ◦ ψ)Jψ dx =  �  Rn x1 · · · xn(¯σZ ◦ ψ)Jψ dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Recall from Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4 that each vertex v of P defines the inner normal cone Cv := P ∗  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We can write  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='8)  Rn =  �  v  Cv  as the union over the vertices v of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Moreover, we know that dim(Cv ∩ Cv′) < n for different  vertices v, v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Therefore, we can rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='7) as the sum  �  v  �  Cv  x1 · · · xn(¯σZ ◦ ψ)Jψ dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  over the f0 many vertices v of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Fix now a vertex v of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5, there are vertices vi of Pi, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , n,  satisfying v = v1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Note that ai ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  12  PETER B¨URGISSER  We define the map ϕi : Rn  >0 → RAi\\{vi} by  ϕi(x) = (xai−vi)ai∈Ai\\{vi} ∈ RAi\\{vi} ≃ Rmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Note that ϕi expresses ψi in the affine chart given by P(RAi)yvi ̸=0 → Rai∈Ai\\{vi}, which maps  [yai]ai∈Ai to y−1  vi (yai)ai∈Ai\\{vi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' So we are in the setting of Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2 and ϕi is an instance  of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The rows of the matrix M(x) := Dxϕ are labeled by the disjoint union A1⊔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='⊔An and  M(x) has n columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For any n-tuple (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , an) with ai ∈ Ai \\{vi}, we denote by M(x)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an  the n×n submatrix of M(x), obtained by selecting from M(x) the rows numbered by a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5 to bound  ρm1 · · · ρmn  �  Cv  x1 · · · xn(¯σZ ◦ ψ)Jψ dw ≤  �  a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an  �  Cv  x1 · · · xn| det M(x)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an| dw,  where the sum runs over all tuples (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , an) with ai ∈ Ai \\ {vi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' So there are m1 · · · mn many  summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1, it is sufficient to show that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='9)  �  Cv  x1 · · · xn| det M(x)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an| dw ≤ 1  for each vertex v and each selection (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  The component (row) of the derivative Dxϕi corresponding to ai ∈ Ai \\ {vi} is given by  (Dxϕi)ai = xai−vi(ai − vi)diag(x−1  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , x−1  n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Hence the n × n-submatrix M(x)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an of M(x) is given by  M(x)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an = diag(xa1−v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , xan−vn)  \uf8ee  \uf8ef\uf8f0  a1 − v1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  an − vn  \uf8f9  \uf8fa\uf8fb diag(x−1  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , x−1  n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Therefore, setting bi := ai − vi, we get  x1 · · · xn det(M(x)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an) = xb1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='+bn det[b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Let us write Πi for the cone of Pi at the vertex vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' By definition, bi ∈ Π∗  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5,  Π := Π1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='+Πn equals the cone of the polytope P = P1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='+Pn at the vertex v = v1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='+vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Hence bi ∈ Π∗  i ⊆ Π∗  1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∩ Π∗  n = Π∗ = Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We can therefore rewrite the left-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='9) as  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='10)  �  Cv  x1 · · · xn| det M(x)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an| dw =  �  Cv  e−⟨b1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='+bn,w⟩| det[b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , bn]| dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4, this is at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This shows claim (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='9) and finishes the proof of Theo-  rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For finite Si ⊆ R, put A := S1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × Sn, and consider the  maps  ψi : R>0 → P(RSi), xi �→ [xai  i ]ai∈Si,  ψ: Rn  >0 → P(RA), x �→ [xa]a∈A  with images Zi and Z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The kinematic formula for real projective space gives (see  [9, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3])  N(Si) = vol(Zi)  vol(P1),  N(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , A) = vol(Z)  vol(Pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  The key insight is that Z is obtained as the image of Z1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × Zn under the Segre embedding  P(RS1) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' × P(RSn) → P(RS1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ⊗ RSn) ≃ P(RA),  REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS  13  which is well known to be isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Therefore, vol(Z) = vol(Z1) · · · vol(Zn), and this completes  the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Given is a finite subset A ⊆ Rn with convex hull P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2  we can can w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' assume that dim P = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Consider the injective map  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='11)  ψ: Rn  >0 → P(RA), ψ(x) := [xa]a∈A  with image Z ⊆ P(RA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The kinematic formula for real projective space [9, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3] is con-  siderably simpler than the one in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2, since O(m) acts transitively on the Grassmann  manifolds Gr(k, Rm): we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='12)  N(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , A) = vol(Z)  vol(Pn) =  1  vol(Pn)  �  Rn  >0  Jψ(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We now proceed as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We make the coordinate change x = e−w  and decompose the resulting integral according to the decomposition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='8) of Rn into the full  dimensional cones Cv corresponding to vertices v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Thus  �  Rn  >0  Jψ(x) dx =  �  Rn x1 · · · xnJψ(x) dw =  �  Cv  �  Cv  x1 · · · xnJψ(x) dw  For a fixed vertex v of P, we consider the map ϕ: Rn  >0 → RA\\{v} defined by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='13)  ϕ(x) = (xa−v)a∈A\\{v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Then we have ψ(x) = π(ϕ(x)), where π is the inverse of the chart P(RA)yv̸=0 → RA\\{v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' It is  easy to verify that Jψ(x) ≤ Jϕ(x) using ∥Dϕ(x)π∥ ≤ 1, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Le us view M(x) := Dxϕ as a matrix whose rows are labelled by elements of A \\ {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' and  denote by M(x)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an the submatrix of M(x) obtained by selecting the rows labelled by the ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Binet-Cauchy implies that  Jϕ(x)2 = det(M(x)T M(x)) =  �  a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an  (det M(x)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an)2,  with the sum running over all n-element subsets {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , an} of A\\{v}, of which there are  �t−1  n  �  many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' This implies Jϕ(x) ≤ �  a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an | det M(x)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We have arrived at  �  Cv  x1 · · · xnJψ(x) dw ≤  �  a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an  �  Cv  x1 · · · xn| det M(x)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',an| dw ≤  �t − 1  n  �  ,  where the right-hand inequality follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4 as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The key observation is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Define for ε > 0  Dε := {x ∈ Rn | ∥x∥ ≥ ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let C ⊆ Rn be a proper cone, d ∈ int(C∗), and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then  lim  m→∞ mn  �  C∩Bε  e−m⟨d,w⟩ dw = 0  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Since ∩m≥1Dmε = ∅, basic integration theory implies  lim  m→∞  �  C∩Dmε  e−⟨d,u⟩ du = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Making the change of variables u = mw shows the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  14  PETER B¨URGISSER  We now observe the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Let U ⊆ Rn  >0 be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Analogously as for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='6), one shows  that  (2π)− n  2 ρm1 · · · ρmn  �  U  x1 · · · xn(¯σZ ◦ ψ)Jψ dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  equals the expected number of nondegenerate zeros in U of the random system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We follow the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Note that stretching the support does not change the  Newton polytopes Pi and P = P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Fix a vertex v of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5,  there are vertices vi of Pi, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , n, satisfying v = v1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' + vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Tracing the proof  of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='1, one sees that it is sufficient to show that (compare (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='10)) for any selection  a1 ∈ A1 \\ {v1}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , an ∈ An \\ {vn}, the vectors bi = ai − vi satisfy  lim  m→∞  �  Cv  e−m⟨b1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='+bn,w⟩| det[mb1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , mbn]| dw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  However, this is a consequence of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Additional comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' It is instructive to see how (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='12) directly follows from the more  general kinematic formula in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Consider the injective map ψ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='11) with image  Z ⊆ P(RA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We use ψ to define the map  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='14)  ψd : Rn  >0 → (P(RA))n, x �→ (ψ(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , ψ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  The image Zd = {(y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , y) | y ∈ Z} ⊆ (Pm)n of ψd is the diagonal embedding of Z in the  product of projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='6) we have  N(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , A) = (2π)− n  2 ρn  m  �  Rn  >0  (¯σZd ◦ ψd)Jψd dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Via Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4 below, we indeed conclude that  N(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , A) =  1  vol(Pn)  �  Rn  >0  Jψ dx = vol(Z)  vol(Pn),  which is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' For x ∈ Rn  >0 we have  ρn  m ¯σZd(ψd(x))Jψd(x) =  (2π)  n  2  vol(Pn)Jψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='4 applied to the map ψd from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='14) gives  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='15)  ρn  m ¯σZd(ψd(x))Jψd(x) = E λ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=',λn ∥(λ1 ◦ Dxψ) ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ∧ (λn ◦ Dxψ)∥  where the λi are standard Gaussian linear forms on Tψ(x)Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Take an isometry Tψ(x)Pm ≃ Rm,  view λi ∈ Rm as a vector, and view ∆ := Dxψ as a matrix in Rm×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We note that Jψ(x) =  �  det(∆T ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='15) can be written as the expectation E λi| det R(x)|,  with the matrix  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='16)  R(x) :=  \uf8ee  \uf8ef\uf8f0  λT  1 ◦ Dxψ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  λT  n ◦ Dxψ  \uf8f9  \uf8fa\uf8fb =  \uf8ee  \uf8ef\uf8f0  λT  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  λT  n  \uf8f9  \uf8fa\uf8fb · ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  We thus need to prove that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='17)  E λi| det R(x)| =  (2π)  n  2  vol(Pn)  �  det(∆T ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  REAL ZEROS OF MIXED RANDOM FEWNOMIAL SYSTEMS  15  In order to show this, by the singular value decomposition, we may assume that ∆ =  �  D  0  �  ,  where D = diag(σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' , σn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Note that  �  det(∆T ∆) = σ1 · · · σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='16) can be written as  R(x) = ΛD, where Λ ∈ Rn×n is a standard Gaussian square matrix and we get E Λ| det(R(x))| =  σ1 · · · σn E w| det Λ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' It is well known that E Λ| det Λ| = ρnρn−1 · · · ρ1, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=', see [6, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' On  the other hand (see [6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='25])  ρm =  √  2π vol(Pm−1)  vol(Pm) ,  hence ρnρn−1 · · · ρ1 = (2π)  n  2  vol(Pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' We have thus verified (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  □  References  [1] Mart´ın Avenda˜no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The number of roots of a lacunary bivariate polynomial on a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Journal of Symbolic  Computation, 44(9):1280–1284, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' Funkcional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' A sharp bound on the number of real intersection points of a sparse  plane curve with a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' New fewnomial upper bounds from Gale dual polynomial systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Mosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' The real tau-conjecture is true on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Random Structures Algo-  rithms, 57(2):279–303, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' Condition, volume 349 of Grundlehren der Mathematischen Wis-  senschaften [Fundamental Principles of Mathematical Sciences].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Springer, Heidelberg, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The geometry  of numerical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  [7] Peter B¨urgisser, Felipe Cucker, and Pierre Lairez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Rigid continuation paths ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' structured systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Preprint  arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='10997, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  [8] Peter B¨urgisser, Alperen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Erg¨ur, and Josu´e Tonelli-Cueto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' On the number of real zeros of random fewno-  mials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Algebra Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content='  [9] Peter B¨urgisser and Antonio Lerario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Probabilistic Schubert calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' La G´eom´etrie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' Digital reproduction of 2008 by  Project Gutenberg (Ebook number: 26400).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  [11] Mathias Drton, Bernd Sturmfels, and Seth Sullivant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Lectures on algebraic statistics, volume 39 of Oberwol-  fach Seminars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Birkh¨auser Verlag, Basel, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  [12] Alan Edelman and Eric Kostlan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' Analysis on symmetric cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Oxford Mathematical Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' The  Clarendon Press, Oxford University Press, New York, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Oxford Science Publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  [14] Osman G¨uler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Barrier functions in interior point methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content='  [15] Bertrand Haas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' A simple counterexample to Kouchnirenko’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Beitr¨age Algebra Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' General mass action kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' The kinematic formula in Riemannian homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' How many zeros of a random  sparse polynomial are real?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' In ISSAC’20—Proceedings of the 45th International Symposium on Symbolic  and Algebraic Computation, pages 273–280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' ACM, New York, [2020] ©2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' Khovanski˘ı.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content=' Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Nauk SSSR, 255(4):804–  807, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  [20] Askold G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Khovanski˘ı.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Fewnomials, volume 88 of Translations of Mathematical Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' American Math-  ematical Society, Providence, RI, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  [21] Pascal Koiran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Shallow circuits with high-powered inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Second Symposium on Innovations in Com-  puter Science, ICS, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content='  [22] Pascal Koiran, Natacha Portier, and S´ebastien Tavenas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' On the intersection of a sparse curve and a low-degree  curve: a polynomial version of the lost theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Discrete Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content='  16  PETER B¨URGISSER  [23] Pascal Koiran, Natacha Portier, and S´ebastien Tavenas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+page_content='  Institute of Mathematics, Technische Universit¨at Berlin  Email address: pbuerg@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQfb_di/content/2301.00273v1.pdf'}
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+ARcode: HPC Application Recognition Through
+Image-encoded Monitoring Data
+1st Jie Li
+Department of Computer Science
+Texas Tech University
+Lubbock, TX, USA
+jie.li@ttu.edu
+2nd Brandon Cook
+National Energy Research Scientific Computing Center
+Lawrence Berkeley National Laboratory
+Berkeley, CA, USA
+bgcook@lbl.gov
+3rd Yong Chen
+Department of Computer Science
+Texas Tech University
+Lubbock, TX, USA
+yong.chen@ttu.edu
+Abstract—Knowing HPC applications of jobs and analyzing
+their performance behavior play important roles in system
+management and optimizations. The existing approaches detect
+and identify HPC applications through machine learning models.
+However, these approaches rely heavily on the manually extracted
+features from resource utilization data to achieve high prediction
+accuracy. In this study, we propose an innovative application
+recognition method, ARcode, which encodes job monitoring
+data into images and leverages the automatic feature learning
+capability of convolutional neural networks to detect and identify
+applications. Our extensive evaluations based on the dataset
+collected from a large-scale production HPC system show that
+ARcode outperforms the state-of-the-art methodology by up
+to 18.87% in terms of accuracy at high confidence thresholds.
+For some specific applications (BerkeleyGW and e3sm), ARcode
+outperforms by over 20% at a confidence threshold of 0.8.
+Index Terms—High Performance Computing, Application De-
+tection, Deep Learning, Convolutional Neural Network
+I. INTRODUCTION
+As HPC systems are approaching the exaFLOP era, the scale
+and complexity of HPC systems have increased significantly
+over the past few years. Administrators need to understand
+not only the performance of the hardware system, but also the
+typical applications and their characteristics, such as how they
+use the computing resources and how they have been executed
+before [1]–[5]. With the increase of computation capability, the
+resource contention and energy consumption increase as well.
+To improve HPC system efficiency, it is imperative to under-
+stand the characteristics of applications and to guide better
+resource-aware scheduling policies based on the knowledge
+of resource requirements of applications [6]–[8]. Moreover,
+emergent misbehaviour is becoming more prevalent due to the
+large scale and high utilization [9]. For system administrators
+striving to guarantee optimal system performance, detecting
+anomalies and potential errors of applications is an essential
+task [10]–[12].
+An online detection system that is capable of identifying
+applications in real-time, with little or no human intervention,
+would be a boon to system management. However, this is a
+daunting task. Large-scale HPC systems are generally shared
+by a variety of users from different domains. In addition to
+traditional large-scale simulation applications (e.g., molecu-
+lar dynamics, quantum chemistry applications, and climate
+simulations), emerging Machine Learning (ML) and Artificial
+Intelligence (AI) applications have become an increasingly
+critical part of the workloads on HPC systems. Frequently
+used applications and libraries are usually pre-installed on the
+system by system administrators. However, administrators do
+not necessarily possess the knowledge about the executions
+and characteristics of each application. Users could build,
+compile and name their own applications that are not shared
+with others. In addition, if users do not provide application
+information in job submission scripts, it is difficult to know
+what applications these jobs belong to. These various use
+cases and the imprecise mapping of job names to actual
+applications make it difficult to identify applications using
+naive approaches. As an example, we examined the application
+names derived from the job submission script on Cori and
+found that about 42.4% of the names were incorrect.
+Advanced methods for detecting and recognizing applica-
+tions can be divided into static analysis of binaries and/or
+scripts, which can be performed without running a job, and dy-
+namic analysis of system logs and performance metrics, which
+implies analysis during or after job execution. Early works
+explored static analysis of binaries to determine the semantic
+similarity between two applications [13], [14]. However, The
+complexity of HPC systems has reached a point where static
+analysis of the binaries used to run and maintain the detection
+system is no longer feasible. This approach is invasive to
+users’ data and requires a dedicated binaries collection and
+management infrastructure, which is not always of interest for
+system administrators. Moreover, even though the binaries are
+available, it does not perform well if the same application is
+compiled by a different compiler toolchain or optimization
+level [15]–[17].
+Collecting and analyzing system logs and performance
+metrics are critical to combat performance crisis, and they
+are prevalent in HPC systems, although the approach may
+vary. Recently, there has been a growing research interest
+in automatic detection that relies on extensive performance
+metrics and employs ML techniques to identify applica-
+tions [18]–[22]. A representative approach proposed by Ates
+et al. explored building supervised ML models with statistic
+features of monitoring metrics to classify applications [19].
+The classification model relies on thousands of statistical
+arXiv:2301.08612v1  [cs.DC]  20 Jan 2023
+
+features extracted from hundreds of time-series monitoring
+metrics to achieve high prediction accuracy. A major weakness
+of this approach is the high-latency responses of the detection
+model, and the statistical features are only accurate for repre-
+senting the application after the job is finished. In addition, the
+performance of feature-based models is highly dependent on
+feature engineering; using different features has the potential
+to deviate the classification performance [23].
+In this study, we extend the line of performance metrics
+based approaches and propose an innovative method called
+ARcode (stands for Application Recognition code). It is
+an application recognition method utilizing images encoded
+from performance monitoring metrics. Specifically, we lever-
+age monitoring metrics collected from HPC systems (§II-A)
+and encode time-series data to two-dimensional images to
+represent the resource usage patterns of HPC executions (or
+job signatures for simplicity, discussed in §III-B). We then
+build a dataset labelled with application names and train a
+Convolutional Neural Network (CNN) to build the classifi-
+cation model (§III-C). The contributions of this study are
+summarized below:
+• Contrary to other studies where datasets are generated
+from benchmarks and proxy applications, our dataset is
+built from real applications with different input data,
+resource allocations and run times, which well reflects the
+complex real scenarios. Specifically, we collect monitor-
+ing data from a production system and build a dataset of
+performance metrics of twelve popular HPC applications
+where the application names are labelled.
+• Our innovative methodology encodes time-series mon-
+itoring data into two-dimensional images, where the
+performance metrics are creatively represented in a much
+smaller size compared to the original data without losing
+important metric variations. The encoded job signatures
+can be used not only for application classification and
+detection, but also to inspire methods for predicting and
+estimating the resource usage of applications.
+• We use the CNN techniques and train the CNN model
+with the job signatures. The job signatures are generated
+from the performance monitoring data, thus do not in-
+volve collecting and analyzing users’ private data. The
+CNN model, on the other hand, does not require manual
+features engineering, making it easier to be tuned and
+adopted by any HPC sites.
+Through extensive experiments, we find that ARcode
+achieved a competitive classification performance in most
+cases and outperformed by up to 18.87% at high confidence
+thresholds compared to the state-of-the-art methods. When
+detecting some specific applications (e.g., BerkeleyGW and
+e3sm) with a confidence threshold of 0.8, ARcode is better
+than the state-of-the-art methods by over 20% in terms of ac-
+curacy. Meanwhile, ARcode retains the temporal information
+of the monitoring data and is able to recognize running jobs.
+This capability is not available in any state-of-the-art methods.
+The details of all these experiment evaluations are discussed
+Figure. 1: Workflow of LDMS on Cori
+in
+§IV. The ARcode model and dataset used in this study
+can be found in a separate submission of artifacts.
+II. BACKGROUND
+In this section, we briefly introduce the Cori system and
+how we collect job monitoring data. Then, we describe the
+workflow of the monitoring infrastructure being used and
+present the available job-level monitoring metrics.
+A. The Cori System
+Cori1 is a Cray XC40 system at National Energy Research
+Scientific Computing Center (NERSC). It consists of 2,388
+Intel Xeon “Haswell” processor nodes and 9,688 Intel Xeon
+Phi “Knight’s Landing” (KNL) nodes interconnected on Cray
+Aries High-Speed Network, which provides a peak perfor-
+mance of about 30 petaflops. Additionally, Cori is equipped
+with a large scratch Luster File System that provides 432
+GB/s of performance with a capacity of 28.5 petabytes to
+the compute nodes. Cori also has the Cray DataWarp based
+Burst Buffer, offering a 1.8 petabytes burst buffer storage with
+1.7 TB/s in peak bandwidth performance [24], [25]. With the
+mission of accelerating the pace of scientific discovery through
+HPC and data analysis, workloads running on the Cori system
+cover a wide range of scientific disciplines, including lattice
+QCD, materials science, climate research, high energy physics,
+astrophysics, and more.
+B. Monitoring Workflow
+In this study, we utilize the monitoring metrics collected
+from the CPU nodes of the NERSC Cori system, where the in-
+1https://docs.nersc.gov/systems/cori/
+
+Compute Node
+Compute Node
+LDMS Sampler Daemon 
+LDMS Sampler Daemon 
+Haswel
+Aggregation Node
+LDMS Aggregator Daemon
+Parguet
+Large Memory Node
+2TB of MemoryTABLE I: Selected Monitoring Metrics
+Sampler
+Metrics
+Derived Metrics
+Description
+cray aries sampler
+power
+power
+Node Power Consumption
+syspapi
+PAPI TOT INS
+IPC (PAPI TOT INS/PAPI TOT TOT)
+Instruction Per Cycle
+PAPI TOT TOT
+meminfo
+MemTotal
+Mem (MemTotal - MemFree)
+Memory Used
+MemFree
+Figure. 2: Overview of ARcode Design. ARcode encodes the time-series monitoring data into job signatures. In the offline
+training phase, ARcode trains the CNN from a labelled job signature dataset. In the running recognition phase, the CNN
+model is used to detect the encoded job signature. The CNN model is modified so that it can identify novel applications.
+band monitoring tool, LDMS is used [26]. The detailed discus-
+sion of how LDMS works on Cori is out of scope of this paper.
+In Figure 1, we illustrate its workflow for generating job-level
+performance metrics. The workflow includes the following
+steps: 1 LDMS samplers on Haswell and KNL nodes (two
+partitions in Cori) collect in-band metrics at a pre-configured
+frequency; 2 the monitoring data are then sent to aggregation
+nodes, where metrics collected from the same sampler are
+stored in CSV files under the same folder. Each CSV contains
+metrics from multiple nodes and the corresponding metrics for
+a job may span multiple CSV files. To improve the usability of
+the monitoring data, the NERSC LDMS [27], an LDMS data
+processing tool, takes care of the post-processing of CSV files.
+3 NERSC LDMS gets job IDs from Slurm sacct and joins
+the job IDs with CSV files, and 4 submits these files to large
+memory nodes for post-processing, where the same metrics of
+the same job are extracted. 5 The post-processed job-level
+metrics are saved in parquet files by metric samplers for future
+analysis. Compared to raw CSV files, job-level parquet files
+significantly improve the availability of monitoring data and
+the efficiency of querying job performance metrics.
+C. Available Metrics
+The available metrics collected through LDMS on HPC
+systems depend on the configurations and available samplers
+on the hardware platform. As routine tasks of monitoring
+the health of the Cori system, 34 different samplers collect
+metrics related to I/O, network, CPU counters, memory usage,
+power consumption, etc. The total number of metrics collected
+through the sampler varies from 12 to 3,016, depending
+on the sampler. The granularity of collected metrics is one
+second, generating approximately 400MB of monitoring data
+per second on a system wide basis.
+From the perspective of application recognition, it is
+unattractive and impractical to include all of these extensive
+monitoring metrics in one model because processing large
+amounts of time-series data is compute-intensive and time-
+consuming. On the other hand, we envision that the proposed
+methodology should be easily adopted by other HPC systems
+and the selected metrics should be common even when using
+different monitoring infrastructures. Therefore, we select five
+of these metrics and derive three representative metrics for
+creating job signatures. These three metrics are the power
+consumption of the compute node, instruction per cycle (IPC),
+and memory used as shown in Table I. These metrics are
+expected to be available through the monitoring infrastructure
+on a variety of HPC architectures.
+III. METRICS ENCODING AND CLASSIFICATION
+In this section, we first discuss design considerations and
+provide an overview of ARcode design. We then present
+the details of encoding monitoring data, including resampling
+job metrics, converting 1D time-series data into 2D images,
+and encoding multiple traces into a single image. Lastly, we
+introduce the CNN architecture for classifying the encoded
+images.
+A. Design Consideration and Overview
+As discussed in §I, existing solutions either perform a static
+analysis of binaries and/or scripts or leverage machine learning
+methods to extract key features out of extensive monitoring
+data to build application prediction and classification models.
+
+Known App
+App1
+App2
+Known App 2
+Known App n
+App 3
+App 4
+Unknown Apps
+App n(a) Original Power Consumption Trace
+(b) Downsampled Trace by Aggregating
+(c) Original Power Consumption Trace
+(d) Upsampled Trace by Padding
+Figure. 3: Resampling traces by aggregating and padding based on the original trace length and the predefined length. Figure
+a is the trace longer than the predefined length 128, in which the aggregating function is applied to downsample the trace.
+Figure c is a trace shorter than 128, in which the padding is used to upsample the trace. Figure b and d show the corresponding
+resampled traces, respectively.
+Our approach, ARcode, is similar to the monitoring data
+based approaches but with two design considerations: features
+learned without human intervention and data retaining tempo-
+ral information.
+First, current state-of-art application detection models, such
+as Taxonomist [19], extract statistical summaries from raw
+monitoring data to create a feature vector for machine learning
+models, where the classification performance is highly de-
+pendent on the quality of the manually constructed features.
+To improve the usability, one of our considerations is that
+the features of the monitoring traces can be learned without
+human intervention. Second, although statistical features are
+spatially efficient and lightweight when building detection
+models, they lose the temporal information of time-series data,
+thus limiting their use case. To improve the extensibility,
+the other consideration of our model is to retain temporal
+information. So it is able to use part of the monitoring data in
+detection, classification, and prediction of the resource usage
+of applications.
+With these considerations in mind, our proposed approach,
+ARcode, encodes entire time-series monitoring data of jobs
+into unified-size images and leverages deep learning tech-
+niques to learn features. The encoded image, which we named
+as job signature, is a representation of monitoring traces that
+preserve temporal performance behavior of the job.
+As shown in Figure 2, ARcode has two main components:
+the monitoring data encoding and the CNN model. The
+first component, the monitoring data encoding component,
+performs a series of operations on the raw monitoring data.
+It creates job signatures encoding the time-series data and
+represents the jobs as images. The second component is a CNN
+model that is customized to learn features from job signatures
+and to classify these job signatures. ARcode operates in two
+phases. The first phase is the offline training phase, where
+the CNN model is trained from a labelled job signature
+dataset. The training phase can be enhanced with transfer
+learning [28], where the convolutional layers can be transferred
+from a trained job detection model and only the layers that
+make predictions need to be trained. The runtime recognition
+phase is where ARcode operates to detect and to predict the
+applications by job signatures.
+Classification model should not be limited to classify
+known applications already seen in the training phase. To
+make ARcode practically useful, we introduce confidence
+thresholds to help identify applications. When the prediction
+probability exceeds the defined threshold, ARcode labels
+the job with the application name; otherwise, it marks the
+observation as unknown, indicating the job is likely to be a
+new application.
+B. Construction of Job Signature
+1) Resampling Traces: To construct job signatures that can
+be measured for the similarity between pairs of signatures, it
+requires the time series traces to have equal length. Given a
+trace, we resample it with a predefined length l. For a trace T
+of length n, we create T ′ by sampling the data points from T
+
+275.00
+Power Consumption (W)
+250.00
+225.00
+200.00
+75.00
+150.00
+125.00
+14000
+0
+2000
+4000
+6000
+8000
+10000
+12000
+Timesteps220.00
+Power Consumption (W)
+210.00
+200.00
+190.00
+180.00
+170.00
+20
+60
+80
+100
+120
+0
+40
+Timesteps300.00
+Power Consumption (W)
+250.00
+200.00
+.50.00
+100.00
+50.00
+0
+10
+20
+30
+40
+50
+Timesteps300.00
+Power Consumption (W)
+250.00
+200.00
+.50.00
+100.00
+50.00
+0
+20
+40
+60
+80
+100
+120
+Timesteps(a) Polar Coordinate of the Normalized Trace
+(b) Gramian Angular Summation (Left) and Difference (Right) Fields
+Figure. 4: Steps of Gramian Angular Field Conversion. Each encoded image has a resolution of 128 × 128.
+so that n′ = l, where n′ is the length of the sampled trace T ′.
+Note that the predefined length is usually set to a relatively
+small value to reduce the compute time when calculating the
+similarities.
+Considering most jobs on HPC run for hours or even days,
+their corresponding monitoring traces are usually longer than
+the predefined length, i.e, n > l. In this case, we downsample
+the traces. For each set of ⌊n/l⌋ data points, we apply one of
+the functions, such as mean, max, min, median, to calculate the
+aggregated value. Assuming we set the predefined length to be
+120, to resample the power consumption trace of a 10-minute
+job (i.e., the trace has 600 timesteps in total), every 5 data
+points should be aggregated. Figure 3a and Figure 3b illustrate
+the procedure of resampling the power consumption traces of
+a job using the mean value. The original trace contains more
+than 14,000 timesteps, while after resampling, the trace length
+becomes 128 timesteps.
+In case that n < l, i.e., the length of a job metric trace is less
+than the predefined length l, we upsample the original traces.
+Specifically, we add ⌊l/n⌋ paddings between consecutive data
+points and fill with the previous value. Figure 3c and Figure 3d
+show the traces before and after upsampling. After resampling,
+all traces have the length of l, irrespective of the duration of
+the job. The corresponding resampled traces T ′ will be used
+for further processing.
+2) Converting 1D time series to 2D images: As shown
+in Figure 3, the monitoring traces and the corresponding
+resampled traces are univariate time series. To take advantage
+of the feature learning in deep learning architectures, we
+convert 1D time series to 2D images.
+We utilize the Gramian Angular Field (GAF) to transform
+time series into images [29]. Specifically, the time series data
+is first normalized or scaled into the range of [−1, 1]. The
+normalized time series data is then represented in a polar
+coordinate instead of the typical Cartesian coordinate. A Gram
+Matrix like operation is applied on the resulting angles to
+construct 2D images.
+Given a time series trace T = {t1, t2, ..., tn} of n times-
+tamps, we rescale T to have the interval [−1, 1] by the equation
+below:
+˜ti = (ti − max(T)) + (ti − min(T))
+max(T) − min(T)
+(1)
+The value of the time series and its corresponding timestamp
+need to be accounted for so that no information is lost. These
+two quantities are expressed with the angle and the radius
+in polar coordinates, respectively. Mathematically, the angle
+is computed by arccos(˜ti), which lies within [0, π], and the
+radius variable is calculated by i/n, which is in [0, 1]. The
+point can be expressed in polar coordinates (φi, ri), where:
+�
+φi = arccos(˜ti),
+−1 ≤ ˜ti ≤ 1
+ri = i
+n,
+0 ≤ i ≤ n
+(2)
+The encoding function is a composition of bijective func-
+tions, producing one and only one result in the polar coordinate
+system. In addition, as opposed to Cartesian coordinates,
+polar coordinates preserve temporal dependency through the
+r coordinate. An example of a trace represented in polar
+coordinates is shown in Figure 4a, which is transformed from
+the normalized trace of Figure 3d.
+The temporal correlations between each pair of data points
+(ti, tj) are computed by considering the trigonometric sum-
+mation (cos(φi + φj)) or subtraction (cos(φi − φj)), leading
+to the Gramian Matrix called Gramian Angular Summation
+Field (GASF) or Gramian Angular Difference Field (GADF),
+respectively. The GASF is defined as follows:
+GASF =
+�
+����
+cos(φ1 + φ1)
+. . .
+cos(φ1 + φn)
+cos(φ2 + φ1)
+. . .
+cos(φ2 + φn)
+...
+...
+...
+cos(φn + φ1)
+. . .
+cos(φn + φn)
+�
+����
+(3)
+Through the GASF or GADF conversion, the diagonal Gi,i
+contains the original value of the scaled time series, while
+Gi,j represents the relative correlation by superposition of
+directions with respect to time interval |i − j|. Other details
+on time series encoding can be found from [29]. In Figure 4b,
+
+90°
+135°
+45°
+20.00
+180°
+0°
+0.25
+0.50
+0.75
+L.00
+225°
+315°
+270°60
+0
+20
+40
+80
+100
+120
+0
+20
+40
+60
+80
+100
+120
+1.00
+0.00
+-1.00
+120
+0.75
+100
+0.50
+80
+0.25
+0.00
+60
+-0.25
+40
+-0.50
+20
+-0.75
+0
+1.00
+.00
+0
+1we illustrate the encoded 2D images with GASF and GADF
+for the trace in Figure 4a. To keep it concise, we use GASF
+as the 1D time series conversion algorithm and refer to GASF
+as GAF in the following discussion.
+3) Encoding Multiple Time Series Traces: The procedure
+presented in §III-B2 only converts one monitoring trace to
+a 2D image. Since we have multiple monitoring traces cor-
+responding to the same job and we do not want to lose the
+correlation between each pair of traces. The question becomes
+how we can encode multiple time series to a single image such
+that a deep learning architecture can understand. To solve this
+problem, we are inspired by the concept of RGB channels,
+where each RGB channel emphasizes different aspects of the
+original image. Similarly, the power consumption trace, IPC
+trace, and memory used trace can be considered as the R
+channel, G channel, and B channel of the encoded image,
+respectively.
+The output of GAF conversion is nothing but a 2D matrix
+of floating-point numbers that fall in [−1, 1]. To visualize the
+job signature, we rescale the GAFs of the above-mentioned
+three traces to be in [0, 255] by using the below equation:
+�
+GAF = ⌊GAF + 1
+2
+∗ 255⌋
+(4)
+The �
+GAF contains the pixel value of the gray scale image.
+When combining �
+GAF power, �
+GAF ipc, �
+GAF mem as three
+channels of RGB, we create a single color image. Figure 5
+shows the GAFs of the power, IPC and memory usage of a
+job and its corresponding encoded job signature. It is worth
+noting that rescaling GAF to the range [0, 255] is only for
+the purpose of visualizing the job signature while the original
+GAF falling in [−1, 1] can be directly fed in the Convelutional
+Neural Network.
+It is important to note that the channel-like encoding
+methodology is not limited to three channels. Each time series
+trace is converted to a l ×l matrix by the procedure presented
+in §III-B2. An encoded job signature of three channels is
+a l × l × 3 matrix. Encoding one more metric is simply
+adding another dimension in the matrix. More formally, a job
+signature of c metrics is a l × l × c matrix. Even though it
+is not straightforward to be visualized when c is larger than
+3, the CNN model can “understand” and analyze the high-
+dimensional matrix.
+C. Classification Model
+Using the methodology presented in §III-B, the monitoring
+traces of jobs can be represented in images (i.e., job sig-
+natures). Therefore, detecting and identifying applications of
+jobs become an image recognition problem. In this subsec-
+tion, we first introduce the CNN, a deep learning technique
+that has been widely used in image classification problems
+and achieved promising results in many domains. Then, we
+present the CNN architecture that is specifically customized
+for classifying job signatures.
+1) Deep learning using CNN: The performance of con-
+ventional machine learning techniques depends heavily on
+data representation, which requires lots of efforts to design
+preprocessing pipeline and feature engineering. Such feature
+engineering is labor intensive and lacks the ability to extract
+discriminative information from the data [30]. Deep learning,
+on the other hand, explores the possibility to feed raw data to
+the algorithm and automatically discover the features needed
+for detection or classification. The key concept of deep learn-
+ing is to transform the representation at a lower level into a
+representation at a higher and more abstract level; and with
+composition of such transformations, complex functions can
+be learned [31]. As a widely-used deep learning technique,
+CNNs have been successful in image classification problems.
+2) Customized CNN Architecture for Job Signatures: The
+CNN architecture for classifying job signatures is designed as
+shown in Figure 6, which contains the following customized
+layers and parameters:
+1. Input layer for job signature: The inputs are job
+signatures of encoded multiple time series traces, as presented
+in §III-B. The resolution of the job signature is 128 × 128
+and the number of channels equals to the number of traces
+encoded in the job signature (i.e., three in our dataset). Note
+that for other cases that job signatures encode more than three
+monitoring traces, the input shape should be set accordingly.
+2. Layers for feature extraction: In our model, we have
+three layers of CNN to learn features from the job signature,
+each of which contains one convolution layer to extract
+features and one pooling layer to reduce the spatial dimension
+of the convoluted image. The convolution layer extracts image
+features by convolving the job signature with a set of kernels
+and produces one feature map for each kernel. We apply
+the activation function ReLU on the convoluted features. It
+replaces negative values with zero and keeps the positive value.
+The output of ReLU will be the input for the pooling layer.
+The first convolution layer learns 32 kernels, and the sec-
+ond and third convolution layers learn 64 and 128 kernels,
+respectively. We set the kernels to be size of 3×3 and use the
+same padding in the convolution to maintain the dimension of
+output as input. All three pooling layers perform max pooling
+that returns the maximum value from the portion of the image
+covered by a window of size of 2 × 2.
+3. Layers for classification: After going through the feature
+extraction layers, the customized CNN model flattens the
+final output and feeds it to a regular neural network for
+classification. The flattened vector is connected to a fully
+connected layer, through which non-linear combinations of
+the features can be learned. After passing through the fully
+connected layer, we use the softmax activation function in the
+last layer to get the probabilities of the input job signature
+being in a particular class.
+To prevent the CNN model from overfitting and improve the
+generalization of the CNN model, we add several dropout lay-
+ers in the model to randomly disable neurons during training,
+as shown in the dashed red parallelograms in Figure 6. In our
+
+Figure. 5: Encoding Multiple Time Series Traces into a Job Signature with a Resolution of 128 × 128.
+Figure. 6: Customized CNN Architecture for Classifying Job Signatures.
+experiments, the dropout rate before the flatten layer and in
+the fully-connected layer is set as 30% and 70%, respectively.
+IV. EXPERIMENTAL EVALUATION
+In this section, we introduce the experimental evaluation
+of our proposed classification model on a real-world dataset
+collected from the Cori system. Particularly, we first introduce
+the dataset built based on the proposed encoding methodology
+and then, we compare the performance of our model with the
+baseline methods in terms of accuracy.
+A. Dataset
+To the best of our knowledge, there are no publicly released
+monitoring traces with labelled application information col-
+lected from production HPC systems. Ates el al. [32] published
+a dataset containing the monitoring data of benchmarks and
+proxy applications. Google published its cluster traces that do
+not have application information [33]. Therefore, in order to
+train the CNN model for detecting applications running in
+production HPC systems, we built our own dataset.
+On Cori system, some other research groups had imple-
+mented the job metadata management service, where the
+metadata of jobs running on Cori are stored and managed
+in a MySQL database. The job metadata has a field named
+‘application name’ which is derived from the job submission
+script. However, based on our analysis, about 42.4% of derived
+names do not reflect the application accurately. Many of them
+have names like ‘test’, ‘bugs’, ‘b.sh’, etc. Therefore, to build
+a reliable labelled dataset, we select jobs that have accurate
+application names through the job metadata service. In addi-
+tion, since Cori system has the KNL partition and the Haswell
+partition, the monitoring metrics of the same application (even
+with same configurations and same input files) running on
+different architectures could potentially have large variations.
+To avoid discrepancy caused by the architecture, we only focus
+on the jobs running on the same architecture and we select
+KNL jobs as our dataset. Besides, we discard short jobs that
+run for less than 60 seconds since they are likely for testing
+purposes.
+The
+job
+IDs
+of
+the
+selected
+jobs
+are
+used
+with
+NERSC LDMS [27] to obtain the corresponding monitoring
+traces. Considering that a job may have multiple steps and
+that job steps may correspond to different applications, we
+treat the job steps of a job separately. In addition, since a job
+may use multiple nodes (nodes are exclusively used by the
+job on Cori), we aggregate the monitoring metrics from all
+involved nodes and use the average time-series traces to build
+the job signature.
+We select 23,665 jobs from 12 different applications to build
+the job signature dataset after eliminating those jobs that do
+not have monitoring metrics (due to data collection errors or
+historical traces are purged) and balancing the number of jobs
+of each application as much as possible. Details are listed in
+Table II. The numbers of allocated nodes for a job vary from
+1 to a maximum of 2,048. The BerkeleyGW, Espresso and
+
+Footunee Lonin:
+CNIN Ieyr 
+ JAI NING
+CNIN IyEr
+128 × 129 x3
+FT X 
+84 X 04 × B4
+[Poad mps
+32 量 $2 x 128
+Potpd mp
+L
+口
+口
+口0
+20
+40
+60
+80
+100
+120
+20
+40
+60
+80
+100
+120
+20
+40
+60
+80
+100
+120
+0.90 卡
+0.13 
+185.00
+0.12
+182.50
+120 -
+120
+120
+20 
+100
+100
+100 
+40
+80 
+80
+80 -
+60
+09
+60
+80 -
+40 
+40
+40 -
+100
+20 -
+M
+120
+ol
+01
+0
+20
+40
+60
+80
+100
+120
+GAF of Power (Channel R)
+GAF of IPC (Channel G)
+GAF of Mem (Channel B)
+Job SignatureTABLE II: Dataset of Job Signatures
+Application
+# of Jobs
+# of Nodes*
+Runtime(s)*
+Description
+BerkeleyGW
+1899
+1 / 2048
+60 / 172867
+For quasiparticle excitations and optical properties of materials.
+Espresso
+2000
+1 / 2048
+61 / 172858
+For electronic-structure calculations and materials modeling at the nanoscale.
+Gromacs
+1937
+1 / 64
+61 / 172836
+For simulations of proteins, lipids and nucleic acids.
+LAMMPS
+1999
+1 / 64
+60 / 172866
+For molecular dynamics with a focus on materials modeling.
+NWChem
+1992
+1 / 384
+71 / 169209
+For computational chemistry.
+VASP
+2000
+1 / 128
+69 / 172864
+For “ab-initio” quantum-mechanical molecular dynamics (MD) simulations.
+WRF
+1860
+1 / 256
+72 / 174608
+for atmospheric research and operational forecasting applications.
+aims
+2000
+1 / 192
+71 / 172959
+For “ab-initio” molecular simulations.
+chroma
+1996
+2 / 137
+488 / 147447
+For lattice Quantum Chromodynamics calculations (LQCD).
+cp2k
+1993
+1 / 128
+60 / 174608
+For quantum chemistry and solid state physics.
+e3sm
+1989
+1 / 2048
+60 / 173402
+For earth system modeling, simulation and prediction.
+su3
+2000
+1 / 36
+65 / 20277
+For lattice Quantum Chromodynamics calculations (LQCD)
+*The values on the left and right side of ‘/’ indicate the minimum and maximum values of the corresponding characteristics of the jobs.
+s3sm have extremely large-scale jobs where all cores of 2,048
+nodes are used, corresponding to 131,072 physical cores. The
+selected jobs also cover a large range of runtime variations.
+The shortest job, as defined earlier, runs for only 60 seconds.
+The longest job, limited by the Quality of Service of the KNL
+partition, runs for 48 hours. The dataset can be found in a
+separate submission of artifacts.
+It is worth mentioning that, it is up to users to set the resam-
+ling length l and define the resolution based on their computing
+capability. Nevertheless, the CNN model for images of larger
+resolution usually requires more complex architecture and
+the time for fine tuning and training the parameters of the
+architecture becomes high. In our experiments, we set the
+resampling length to be 128 and use a resolution of 128×128
+for job signatures.
+B. Baseline Methods
+To compare the performance of our proposed CNN model
+with state-of-the-art methods for detecting HPC jobs, we
+examine several other performance metrics based approaches.
+The Power Signature [18] and Taxonomist [19] are two repre-
+sentative methods that use statistic features of the time series
+monitoring data to build the classification model. The statistics
+include the minimum, maximum, mean, standard deviation,
+skew, kurtosis and the 5th, 25th, 50th, 75th, 95th percentiles.
+Using the same monitoring metrics as ARcode, namely power
+consumption, IPC and memory usage of the selected jobs,
+we extract statistical features and build several classifiers as
+baselines. These classifiers are Random Forest (RF), linear
+Support Vector Classifier (Liner-SVC), and non-linear Support
+Vector Classifier (SVC). In addition, we include a Random
+Forest model that uses only statistical features extracted from
+the power consumption trace. All of these models use default
+hyperparameters.
+C. Experiment Setup
+The ARcode and baseline models are trained and evaluated
+on a Cori GPU node, where 2 Skylake CPUs, 8 NVIDIA Tesla
+V100 GPUs and 384 GB DDR4 memory are provided. The
+proposed CNN is implemented using the TensorFlow Keras
+library in a Python 3.9 environment. The baseline models are
+implemented in Python leveraging scikit-learn library.
+Figure. 7: Loss (top) and accuracy (bottom) of the training
+and validation data. The vertical dashed line indicates the
+determined epoch of 50.
+We divide the dataset into 60% for training, 20% for
+validation, and 20% for testing. To mitigate overfitting and to
+increase the generalization of ARcode model, we determine
+the optimal epoch number by examining the loss and accuracy
+trends on training and validation data. As shown in Figure 7,
+after epoch reaches to 50, the training loss gets lower than
+validation loss while the accuracy of validation does not
+improve. Therefore, we set epoch to 50. While evaluating the
+classification performance on each application, we use ten-fold
+stratified cross validation to divide the dataset into ten disjoint
+partitions and evaluate model performance with each partition.
+We test the classification performance on different con-
+fidence thresholds. For each baseline classifier, we use the
+one-vs.-rest version of that classifier such that it produces a
+set of real-valued prediction scores for its decision instead
+of a labeled class. In ARcode model, the softmax function
+assigns probabilities of classes in each prediction. We compare
+the prediction probabilities with the confidence thresholds to
+determine the predicted class. For the prediction score si of
+class ci, i ∈ {1, ..., k}, the classifier assigns the label of the
+class by the following expression:
+�
+ci, where si = max(s1, ..., sk),
+if si ≥ threshold
+unknown,
+otherwise
+(5)
+In other words, when the maximum value of the prediction
+
+Training
+4.00
+Validation
+3.00
+S
+LoSS
+2.00
+1.00
+0.80 -
+Accuracy
+0.60
+0.40
+-
+40
+60
+120
+0
+20
+80
+100
+EpochsFigure. 8: Accuracy of ARcode and baseline classifiers at
+different confidence thresholds. The vertical dashed line in-
+dicates the confidence threshold (0.4) above which ARcode
+outperforms all other classifiers.
+Figure. 9: Accuracy of classifiers on each application at a
+confidence threshold of 0.8.
+scores is larger than the threshold, the class with the largest
+score is the predicted class. Otherwise, the classifier labels the
+observation with unknown. When the confidence threshold is
+0, it is the same with the vanilla multi-class classifier.
+D. Experimental Results
+We evaluate the capability of ARcode in identifying appli-
+cations in this subsection. First, we evaluate the classification
+performance on different confidence thresholds and examine
+the performance on each application. Then, we assess the
+performance of identifying applications that have not been
+trained with the models. In addition, since the job signatures
+encode temporal information of the monitoring metrics, we use
+a subset of the monitoring data to construct partial signatures
+to evaluate the ARcode model.
+1) Classification Performance: Figure 8 depicts the accu-
+racy of ARcode and baseline models on different confidence
+thresholds. As we can see from the figure, the Linear-SVC
+model has the worst performance at all confidence thresholds.
+When the confidence threshold is below 0.4, Random Forest
+outperforms all other classifiers, with the highest accuracy
+of 93.81% at a confidence threshold of 0. ARcode follows
+closely with an accuracy of 89.67%. When the confidence
+threshold is above 0.4, the accuracy of Random Forest de-
+creases and ARcode has the best performance. At a threshold
+of 1.0, the SVC model performs a little better than Random
+Figure. 10: Accuracy of ARcode and baseline classifiers on
+detecting novel applications at different confidence thresholds.
+ARcode achieved the highest accuracy when the confidence
+threshold is greater than 0.8.
+Forest, but it is still 18.87% worse than ARcode. At a
+confidence threshold of 0, the accuracy of Random Forest
+on power features is close to ARcode , but it decreases
+significantly as the confidence threshold increases. In sum-
+mary, the accuracy of all classifier decreases with increasing
+confidence thresholds, but the slowest decrease rate is observed
+for ARcode.
+We further examine the classification performance of
+ARcode and Random Forest on each application, as shown
+in Figure 9, where we use a confidence threshold of 0.8
+in the experiment. From this figure, we have the following
+observations. First, the accuracy of ARcode and Random
+Forest are very close in most cases. When detecting Gromacs,
+LAMMPS and cp2k, Random Forest performs slightly bet-
+ter. Second, for some applications such as BerkeleyGW and
+e3sm, ARcode achieves a significantly better accuracy. In
+BerkeleyGW classification, the accuracy of Random Forest is
+62.06% while ARcode achieves 83.94% accuracy. ARcode
+is also 20.21% better than Random Forest in detecting e3sm.
+2) Classification on Novel Applications: Considering that
+in HPC environments, ‘novel’ applications are more prevalent
+than those trained with the classifier, it is appropriate to
+define an ‘unknown’ class for all these novel applications. To
+be practically useful, the classification system must classify
+both known and unknown novel applications. We evaluate
+this capability by removing one application from the training
+set while keeping the testing set untouched. If the removed
+application in the testing set is predicted to be unknown, we
+mark it as a correct prediction.
+The classification performance for novel applications is
+depicted in Figure 10. From this figure, we can observe
+that Random Forest has the best prediction accuracy when
+the confidence threshold is below 0.8, while ARcode beats
+Random Forest once the confidence threshold is greater than
+0.8. ARcode, similar to its performance in known application
+detection, has relatively stable prediction accuracy across all
+confidence thresholds. However, the accuracy of Random
+Forest drops significantly in large confidence thresholds.
+
+100%
+80%
+60%
+Accuracy
+40%
+ARcode (CNN)
+Random Forest
+20%
+SVC
+LinearisvC
+Random Forest on Poweri Features!
+0%
+0.2
+0.0
+0.1
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Confidence ThresholdARcode(CNN)
+Random Forest
+100%
+80%
+Accuracy
+60%
+40%
+20%
+0%
+aims
+WRF
+BerkeleyGW Espresso Gromacs LAMMPS NWCHEM
+VASP
+chroma
+cp2k
+e3sm
+su3
+Applications100%
+80%
+60%
+Accuracy
+40%
+ARcode (CNN)
+Random Forest
+20%
+SVC
+Linearisvc
+Random Forest on Poweri Features
+0%
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Confidence ThresholdFigure. 11: Accuracy of ARcode on each application using
+different channels of the job signature.
+Figure. 12: Accuracy of ARcode on each application using
+partial job signature. The partial job signatures are all built
+from the metrics collected since the startup of jobs.
+3) Classification using Single Channel: In our experiment,
+ARcode encodes three channels of monitoring data in the
+job signature. It is worth knowing whether each channel plays
+the same importance in classification and whether we can use
+one channel for detection while still achieving high accuracy.
+To answer these questions, we train CNN models with each
+of these channels individually and analyze their classification
+performance. The results are shown in Figure 11.
+From the figure we can see that using any of the chan-
+nels gives a competitive accuracy when detecting Espresso,
+LAMMPS, WRF, and chroma compared to using all channels.
+For NWCHEM, the classification accuracy using the memory
+channel is significantly better than using the power or IPC
+channel. We can also see that all channels are important for
+detecting both Gromacs and cp2k with an accuracy improve-
+ment of 26.36% and 19.66%, respectively, compared to using
+only one of the channels. The memory channel is the most
+representative of these channels. In most applications, using
+the memory channel in detection has the closest accuracy to
+using all channels.
+4) Classification using Partial Signatures: The job sig-
+nature retains the temporal information of the time-series
+monitoring data, and the job signature generated before the
+end of the job still holds some of the attributes of the full
+job signature. We use the first 25%, 50% and 75% of the
+time-series data to create partial job signatures and compare
+the detection performance with the full job signature (i.e,
+a job signature built from 100% of the monitoring data).
+This experiment is performed to evaluate the usability of the
+ARcode model for detecting running applications, which is
+not available in the statistics-based models.
+The results are shown in Figure 12. It is not surprising that
+the larger the percentage of encoded data, the better the detec-
+tion performance achieved by ARcode. When encoded with
+25% of the monitoring data, ARcode achieves a relatively
+high accuracy of 77.27% in detecting WRF, but only 17.27%
+and 12.88% in detecting cp2k and Espresso, respectively.
+When detecting NWCHEM, the full job signature brings
+89.79% accuracy; however, the 75% partial job signature
+gives only 46.73% accuracy. The improvement from encoding
+more data varies from application to application. For example,
+encoding 25% more data brings an average improvement of
+24.67% for Espresso, but only 4.65% for WRF.
+5) Sensitivity to the resampling length: The performance
+evaluation described above is based on job signatures with
+a resolution of 128 × 128, i.e., the performance metrics are
+resampled to a length of 128. It is valuable to have knowledge
+of the sensitivity of the classification accuracy in terms of the
+resampling length such that an appropriate resolution can be
+chosen to achieve a balance between accuracy and training
+overhead. To explore this, we further build two job signature
+datasets with resampling lengths of 32 and 64, respectively,
+and use the same CNN architecture to train these models.
+Figure 13 depicts the accuracy (solid lines) and training
+time (horizontal dashed lines) of ARcode using different
+resampling lengths. Note that the training time is constant
+across different thresholds for the same resampling length, as
+illustrated by horizontal dashed lines. This is because training
+time does not vary according to confidence thresholds, which
+are set during the prediction of applications. As illustrated
+from the figure, the training times vary significantly among
+these three resampling lengths. It is over 185 seconds for
+the resampling length of 128. For signatures with resampling
+lengths of 64 and 32, it is 80 and 65 seconds, respectively.
+Additionally, we can observe from the figure that the higher
+resolution it is, the higher prediction accuracy. While the
+accuracy improvement from the resampling length of 64 to
+128 is not significant (only by 0.5% on average), the accuracy
+improvement from the resampling length of 32 to 64 is 3.0%
+on average.
+This result suggests that 64 is the optimal resampling length
+for building job signatures in our experiment, considering the
+trade-off between accuracy and training time. Its prediction
+accuracy is close to that of the highest resolution signature, but
+only uses 43.2% of the training time of the latter. Furthermore,
+the data size of the job signature generated with a resolution
+of 64 × 64 is only 25% of that with 128 × 128.
+V. RELATED WORKS
+Significant amount of work has been reported in the lit-
+erature on detecting and classifying applications. Identifying
+similarities and differences among binary executables was
+explored in [13]–[16]. These approaches are limited by the
+
+ZZIPC
+Memory
+Power
+All
+100%
+80%
+Accuracy
+60%
+40%
+20%
+0%
+WRF
+BerkeleyGWEspresso Gromacs LAMMPS NWCHEM 
+VASP
+aims
+chroma
+cp2k
+e3sm
+su3
+Applications25%
+75%
+100%
+50%
+Z
+100%
+80%
+Accuracy
+60%
+40%
+20%
+0%
+WRF
+BerkeleyGWEspresso Gromacs LAMMPS NWCHEM 
+VASP
+aims
+chroma
+cp2k
+e3sm
+su3
+ApplicationsFigure. 13: Accuracy and training time of ARcode using dif-
+ferent resampling lengths. The horizontal dashed lines indicate
+the training time.
+fact that, they cannot differentiate the same source program
+compiled by different compiler toolchains or optimization
+levels. To overcome this limitation, Blanket Execution [17]
+presented a binary differencing algorithm that compares the
+side effects of functions during executation, which is based
+on the insight that similar codes have semantically similar
+execution behavior.
+However, the binary based approach cannot be applied in the
+HPC environment, where it is impractical to conduct binary
+differencing among hundreds of thousands of executables. In
+addition, obtaining and managing binaries of users’ applica-
+tions are not always possible for HPC researchers. Instead
+of using binaries, Yamamoto et al. [34] used job scripts as
+job information to classify applications with text classification
+techniques. This approach, however, also requires dedicated
+collection and management infrastructures, which are not as
+prevalent as monitoring infrastructures for performance metric.
+Another line of work aims at characterizing HPC appli-
+cations through system logs. Liu et al. extracted features
+from combined logs of multiple subsystems to represent
+applications and build a machine learning model based on
+the eXtreme Gradient Boosting (XGB) algorithm to identify
+HPC applications [35]. DeMasi et al. collected and extracted
+features from Integrated Performance Monitoring (IPM) per-
+formance logs to fingerprint HPC codes [36]. Log analysis is
+more common in characterizing subsystem and user behavior,
+related works can be found in [37]–[40].
+Monitoring data based application detection has been ex-
+plored in [18]–[22]. As a early quantitative study of power
+consumption of HPC workloads, Combs et al. [18] studied
+the applicability of classifying applications through power
+consumption traces. Ramos et al. [21] relied on performance
+counters to model, fingerprint and clustering applications. Zou
+et al. [20] explored detecting illicit applications in GPU-
+accelerated HPC workloads. They used performance counters,
+data movement behavior and resources utilization traces to
+train machine learning models. Taxonomist [19], proposed
+by Ates et al., used over 700 system metrics and a time
+window spanning the whole execution to extract statistical
+features. They enhanced the classification model such that
+unknown applications can be detected. EFD [22] created
+key-values pairs that link execution fingerprints of system
+metrics to application and input size information to implement
+application recognition. Except for using measurements of
+system metrics, they also used the metrics name, node ID and
+time interval to create fingerprints.
+These studies, however, are based on the datasets built from
+the monitoring data of benchmarks and proxy applications,
+where a relative small range of configurations and input size
+are shown in the applications. On the contrary, the model
+and experiment results of ARcode are developed and tested
+on datasets collected from real applications in a large-scale
+production system. In addition, the models of these studies rely
+on the manually defined features extracted from time series
+monitoring data and some studies utilize the input information
+to enhance the detection model. In this work, ARcode allevi-
+ates the effort of features engineering and takes advantages of
+the capability of features learning in CNN model to detect and
+classify encoded monitoring data. Moreover, ARcode retains
+the temporal information of time-series performance metrics,
+enabling detecting applications before jobs are finished. This
+capability bridges a major gap in related works.
+VI. CONCLUSION
+Existing application detection methodologies on HPC sys-
+tems either relies on the data that are not prevalent or require
+intensive effort of feature engineering to build high accuracy
+models. In this study, we aim to provide a solution that can
+be easily used and adopted by any HPC site. We have intro-
+duced ARcode, an application recognition framework which
+is effective and extensible with the following characteristics:
+1) ARcode uses three most common monitoring metrics (i.e.,
+the power consumption of the compute node, instruction per
+cycle, and memory usage) available through the monitoring
+infrastructure on diverse HPC architectures. 2) ARcode allevi-
+ates the effort of feature engineering by leveraging the feature
+learning capability of CNN models. 3) ARcode’s channel-like
+encoding method allows easy encoding of additional metrics
+in job signatures. 4) ARcode encodes job monitoring data into
+images, which translates the application recognition problem
+into an image classification problem. Unlike the statistics-
+based approaches where the temporal information of monitor-
+ing data are lost, the job signature encodes metric variations
+over the runtime. Therefore, the job signature generated from
+part of monitoring data can still be used in detection, but with
+the sacrifice of accuracy.
+Although our evaluation is performed on job signatures
+generated from three common monitoring metrics, HPC re-
+searchers and system administrators can select other represen-
+tative metrics and build job signatures to detect and classify
+applications with specific characteristics such as CPU intensive
+applications and I/O intensive applications. In addition, we
+have seen lots of successful cases of applying image recogni-
+tion in the medical and automobile industries. Their experience
+in training high-accuracy models can be used in our model to
+further improve the recognition accuracy. Moreover, encoding
+
+100%
+128
+32
+64
+32
+64
+128
+200
+(accuracy)
+(training time)
+90%
+180
+Time (
+80%
+Accuracy
+140
+ining
+70%
+120
+Trai
+-100
+60%
+-80
+50%
+60
+0.0
+0.1
+0.2
+0.3
+0.6
+0.7
+0.8
+0.9
+1.0
+0.4
+0.5
+Confidence Thresholdmonitoring data in job signatures offers a new perspective
+of exploring and analyzing the performance monitoring data.
+In our future work, we will further explore the use of job
+signatures to predict resource usage, detect anomalies, and
+identify malicious applications.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf,len=915
+page_content='ARcode: HPC Application Recognition Through  Image-encoded Monitoring Data  1st Jie Li  Department of Computer Science  Texas Tech University  Lubbock, TX, USA  jie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='li@ttu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='edu  2nd Brandon Cook  National Energy Research Scientific Computing Center  Lawrence Berkeley National Laboratory  Berkeley, CA, USA  bgcook@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='gov  3rd Yong Chen  Department of Computer Science  Texas Tech University  Lubbock, TX, USA  yong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='chen@ttu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='edu  Abstract—Knowing HPC applications of jobs and analyzing  their performance behavior play important roles in system  management and optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The existing approaches detect  and identify HPC applications through machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  However, these approaches rely heavily on the manually extracted  features from resource utilization data to achieve high prediction  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In this study, we propose an innovative application  recognition method, ARcode, which encodes job monitoring  data into images and leverages the automatic feature learning  capability of convolutional neural networks to detect and identify  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Our extensive evaluations based on the dataset  collected from a large-scale production HPC system show that  ARcode outperforms the state-of-the-art methodology by up  to 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='87% in terms of accuracy at high confidence thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  For some specific applications (BerkeleyGW and e3sm), ARcode  outperforms by over 20% at a confidence threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Index Terms—High Performance Computing, Application De-  tection, Deep Learning, Convolutional Neural Network  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' INTRODUCTION  As HPC systems are approaching the exaFLOP era, the scale  and complexity of HPC systems have increased significantly  over the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Administrators need to understand  not only the performance of the hardware system, but also the  typical applications and their characteristics, such as how they  use the computing resources and how they have been executed  before [1]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' With the increase of computation capability, the  resource contention and energy consumption increase as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  To improve HPC system efficiency, it is imperative to under-  stand the characteristics of applications and to guide better  resource-aware scheduling policies based on the knowledge  of resource requirements of applications [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Moreover,  emergent misbehaviour is becoming more prevalent due to the  large scale and high utilization [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' For system administrators  striving to guarantee optimal system performance, detecting  anomalies and potential errors of applications is an essential  task [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  An online detection system that is capable of identifying  applications in real-time, with little or no human intervention,  would be a boon to system management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' However, this is a  daunting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Large-scale HPC systems are generally shared  by a variety of users from different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In addition to  traditional large-scale simulation applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=', molecu-  lar dynamics, quantum chemistry applications, and climate  simulations), emerging Machine Learning (ML) and Artificial  Intelligence (AI) applications have become an increasingly  critical part of the workloads on HPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Frequently  used applications and libraries are usually pre-installed on the  system by system administrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' However, administrators do  not necessarily possess the knowledge about the executions  and characteristics of each application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Users could build,  compile and name their own applications that are not shared  with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In addition, if users do not provide application  information in job submission scripts, it is difficult to know  what applications these jobs belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' These various use  cases and the imprecise mapping of job names to actual  applications make it difficult to identify applications using  naive approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' As an example, we examined the application  names derived from the job submission script on Cori and  found that about 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='4% of the names were incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Advanced methods for detecting and recognizing applica-  tions can be divided into static analysis of binaries and/or  scripts, which can be performed without running a job, and dy-  namic analysis of system logs and performance metrics, which  implies analysis during or after job execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Early works  explored static analysis of binaries to determine the semantic  similarity between two applications [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' However, The  complexity of HPC systems has reached a point where static  analysis of the binaries used to run and maintain the detection  system is no longer feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' This approach is invasive to  users’ data and requires a dedicated binaries collection and  management infrastructure, which is not always of interest for  system administrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Moreover, even though the binaries are  available, it does not perform well if the same application is  compiled by a different compiler toolchain or optimization  level [15]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Collecting and analyzing system logs and performance  metrics are critical to combat performance crisis, and they  are prevalent in HPC systems, although the approach may  vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Recently, there has been a growing research interest  in automatic detection that relies on extensive performance  metrics and employs ML techniques to identify applica-  tions [18]–[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' A representative approach proposed by Ates  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' explored building supervised ML models with statistic  features of monitoring metrics to classify applications [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  The classification model relies on thousands of statistical  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='08612v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='DC]  20 Jan 2023  features extracted from hundreds of time-series monitoring  metrics to achieve high prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' A major weakness  of this approach is the high-latency responses of the detection  model, and the statistical features are only accurate for repre-  senting the application after the job is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In addition, the  performance of feature-based models is highly dependent on  feature engineering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' using different features has the potential  to deviate the classification performance [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  In this study, we extend the line of performance metrics  based approaches and propose an innovative method called  ARcode (stands for Application Recognition code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' It is  an application recognition method utilizing images encoded  from performance monitoring metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Specifically, we lever-  age monitoring metrics collected from HPC systems (§II-A)  and encode time-series data to two-dimensional images to  represent the resource usage patterns of HPC executions (or  job signatures for simplicity, discussed in §III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' We then  build a dataset labelled with application names and train a  Convolutional Neural Network (CNN) to build the classifi-  cation model (§III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The contributions of this study are  summarized below:  Contrary to other studies where datasets are generated  from benchmarks and proxy applications, our dataset is  built from real applications with different input data,  resource allocations and run times, which well reflects the  complex real scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Specifically, we collect monitor-  ing data from a production system and build a dataset of  performance metrics of twelve popular HPC applications  where the application names are labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Our innovative methodology encodes time-series mon-  itoring data into two-dimensional images, where the  performance metrics are creatively represented in a much  smaller size compared to the original data without losing  important metric variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The encoded job signatures  can be used not only for application classification and  detection, but also to inspire methods for predicting and  estimating the resource usage of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  We use the CNN techniques and train the CNN model  with the job signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The job signatures are generated  from the performance monitoring data, thus do not in-  volve collecting and analyzing users’ private data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The  CNN model, on the other hand, does not require manual  features engineering, making it easier to be tuned and  adopted by any HPC sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Through extensive experiments, we find that ARcode  achieved a competitive classification performance in most  cases and outperformed by up to 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='87% at high confidence  thresholds compared to the state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' When  detecting some specific applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=', BerkeleyGW and  e3sm) with a confidence threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='8, ARcode is better  than the state-of-the-art methods by over 20% in terms of ac-  curacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Meanwhile, ARcode retains the temporal information  of the monitoring data and is able to recognize running jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  This capability is not available in any state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  The details of all these experiment evaluations are discussed  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 1: Workflow of LDMS on Cori  in  §IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The ARcode model and dataset used in this study  can be found in a separate submission of artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' BACKGROUND  In this section, we briefly introduce the Cori system and  how we collect job monitoring data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Then, we describe the  workflow of the monitoring infrastructure being used and  present the available job-level monitoring metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The Cori System  Cori1 is a Cray XC40 system at National Energy Research  Scientific Computing Center (NERSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' It consists of 2,388  Intel Xeon “Haswell” processor nodes and 9,688 Intel Xeon  Phi “Knight’s Landing” (KNL) nodes interconnected on Cray  Aries High-Speed Network, which provides a peak perfor-  mance of about 30 petaflops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Additionally, Cori is equipped  with a large scratch Luster File System that provides 432  GB/s of performance with a capacity of 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='5 petabytes to  the compute nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Cori also has the Cray DataWarp based  Burst Buffer, offering a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='8 petabytes burst buffer storage with  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='7 TB/s in peak bandwidth performance [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' With the  mission of accelerating the pace of scientific discovery through  HPC and data analysis, workloads running on the Cori system  cover a wide range of scientific disciplines, including lattice  QCD, materials science, climate research, high energy physics,  astrophysics, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Monitoring Workflow  In this study, we utilize the monitoring metrics collected  from the CPU nodes of the NERSC Cori system, where the in-  1https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='nersc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='gov/systems/cori/  Compute Node  Compute Node  LDMS Sampler Daemon  LDMS Sampler Daemon  Haswel  Aggregation Node  LDMS Aggregator Daemon  Parguet  Large Memory Node  2TB of MemoryTABLE I: Selected Monitoring Metrics  Sampler  Metrics  Derived Metrics  Description  cray aries sampler  power  power  Node Power Consumption  syspapi  PAPI TOT INS  IPC (PAPI TOT INS/PAPI TOT TOT)  Instruction Per Cycle  PAPI TOT TOT  meminfo  MemTotal  Mem (MemTotal - MemFree)  Memory Used  MemFree  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 2: Overview of ARcode Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' ARcode encodes the time-series monitoring data into job signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In the offline  training phase, ARcode trains the CNN from a labelled job signature dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In the running recognition phase, the CNN  model is used to detect the encoded job signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The CNN model is modified so that it can identify novel applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  band monitoring tool, LDMS is used [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The detailed discus-  sion of how LDMS works on Cori is out of scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  In Figure 1, we illustrate its workflow for generating job-level  performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The workflow includes the following  steps: 1 LDMS samplers on Haswell and KNL nodes (two  partitions in Cori) collect in-band metrics at a pre-configured  frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 2 the monitoring data are then sent to aggregation  nodes, where metrics collected from the same sampler are  stored in CSV files under the same folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Each CSV contains  metrics from multiple nodes and the corresponding metrics for  a job may span multiple CSV files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' To improve the usability of  the monitoring data, the NERSC LDMS [27], an LDMS data  processing tool, takes care of the post-processing of CSV files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  3 NERSC LDMS gets job IDs from Slurm sacct and joins  the job IDs with CSV files, and 4 submits these files to large  memory nodes for post-processing, where the same metrics of  the same job are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 5 The post-processed job-level  metrics are saved in parquet files by metric samplers for future  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Compared to raw CSV files, job-level parquet files  significantly improve the availability of monitoring data and  the efficiency of querying job performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Available Metrics  The available metrics collected through LDMS on HPC  systems depend on the configurations and available samplers  on the hardware platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' As routine tasks of monitoring  the health of the Cori system, 34 different samplers collect  metrics related to I/O, network, CPU counters, memory usage,  power consumption, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The total number of metrics collected  through the sampler varies from 12 to 3,016, depending  on the sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The granularity of collected metrics is one  second, generating approximately 400MB of monitoring data  per second on a system wide basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  From the perspective of application recognition, it is  unattractive and impractical to include all of these extensive  monitoring metrics in one model because processing large  amounts of time-series data is compute-intensive and time-  consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' On the other hand, we envision that the proposed  methodology should be easily adopted by other HPC systems  and the selected metrics should be common even when using  different monitoring infrastructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Therefore, we select five  of these metrics and derive three representative metrics for  creating job signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' These three metrics are the power  consumption of the compute node, instruction per cycle (IPC),  and memory used as shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' These metrics are  expected to be available through the monitoring infrastructure  on a variety of HPC architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' METRICS ENCODING AND CLASSIFICATION  In this section, we first discuss design considerations and  provide an overview of ARcode design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' We then present  the details of encoding monitoring data, including resampling  job metrics, converting 1D time-series data into 2D images,  and encoding multiple traces into a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Lastly, we  introduce the CNN architecture for classifying the encoded  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Design Consideration and Overview  As discussed in §I, existing solutions either perform a static  analysis of binaries and/or scripts or leverage machine learning  methods to extract key features out of extensive monitoring  data to build application prediction and classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Known App  App1  App2  Known App 2  Known App n  App 3  App 4  Unknown Apps  App n(a) Original Power Consumption Trace  (b) Downsampled Trace by Aggregating  (c) Original Power Consumption Trace  (d) Upsampled Trace by Padding  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 3: Resampling traces by aggregating and padding based on the original trace length and the predefined length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Figure  a is the trace longer than the predefined length 128, in which the aggregating function is applied to downsample the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Figure c is a trace shorter than 128, in which the padding is used to upsample the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Figure b and d show the corresponding  resampled traces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Our approach, ARcode, is similar to the monitoring data  based approaches but with two design considerations: features  learned without human intervention and data retaining tempo-  ral information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  First, current state-of-art application detection models, such  as Taxonomist [19], extract statistical summaries from raw  monitoring data to create a feature vector for machine learning  models, where the classification performance is highly de-  pendent on the quality of the manually constructed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  To improve the usability, one of our considerations is that  the features of the monitoring traces can be learned without  human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Second, although statistical features are  spatially efficient and lightweight when building detection  models, they lose the temporal information of time-series data,  thus limiting their use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' To improve the extensibility,  the other consideration of our model is to retain temporal  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' So it is able to use part of the monitoring data in  detection, classification, and prediction of the resource usage  of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  With these considerations in mind, our proposed approach,  ARcode, encodes entire time-series monitoring data of jobs  into unified-size images and leverages deep learning tech-  niques to learn features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The encoded image, which we named  as job signature, is a representation of monitoring traces that  preserve temporal performance behavior of the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  As shown in Figure 2, ARcode has two main components:  the monitoring data encoding and the CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The  first component, the monitoring data encoding component,  performs a series of operations on the raw monitoring data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  It creates job signatures encoding the time-series data and  represents the jobs as images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The second component is a CNN  model that is customized to learn features from job signatures  and to classify these job signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' ARcode operates in two  phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The first phase is the offline training phase, where  the CNN model is trained from a labelled job signature  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The training phase can be enhanced with transfer  learning [28], where the convolutional layers can be transferred  from a trained job detection model and only the layers that  make predictions need to be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The runtime recognition  phase is where ARcode operates to detect and to predict the  applications by job signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Classification model should not be limited to classify  known applications already seen in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' To  make ARcode practically useful, we introduce confidence  thresholds to help identify applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' When the prediction  probability exceeds the defined threshold, ARcode labels  the job with the application name;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' otherwise, it marks the  observation as unknown, indicating the job is likely to be a  new application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Construction of Job Signature  1) Resampling Traces: To construct job signatures that can  be measured for the similarity between pairs of signatures, it  requires the time series traces to have equal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Given a  trace, we resample it with a predefined length l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' For a trace T  of length n, we create T ′ by sampling the data points from T  275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  Power Consumption (W)  250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  14000  0  2000  4000  6000  8000  10000  12000  Timesteps220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  Power Consumption (W)  210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  20  60  80  100  120  0  40  Timesteps300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  Power Consumption (W)  250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  0  10  20  30  40  50  Timesteps300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  Power Consumption (W)  250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  0  20  40  60  80  100  120  Timesteps(a) Polar Coordinate of the Normalized Trace  (b) Gramian Angular Summation (Left) and Difference (Right) Fields  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 4: Steps of Gramian Angular Field Conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Each encoded image has a resolution of 128 × 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  so that n′ = l, where n′ is the length of the sampled trace T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Note that the predefined length is usually set to a relatively  small value to reduce the compute time when calculating the  similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Considering most jobs on HPC run for hours or even days,  their corresponding monitoring traces are usually longer than  the predefined length, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='e, n > l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In this case, we downsample  the traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' For each set of ⌊n/l⌋ data points, we apply one of  the functions, such as mean, max, min, median, to calculate the  aggregated value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Assuming we set the predefined length to be  120, to resample the power consumption trace of a 10-minute  job (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=', the trace has 600 timesteps in total), every 5 data  points should be aggregated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Figure 3a and Figure 3b illustrate  the procedure of resampling the power consumption traces of  a job using the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The original trace contains more  than 14,000 timesteps, while after resampling, the trace length  becomes 128 timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  In case that n < l, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=', the length of a job metric trace is less  than the predefined length l, we upsample the original traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Specifically, we add ⌊l/n⌋ paddings between consecutive data  points and fill with the previous value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Figure 3c and Figure 3d  show the traces before and after upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' After resampling,  all traces have the length of l, irrespective of the duration of  the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The corresponding resampled traces T ′ will be used  for further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  2) Converting 1D time series to 2D images: As shown  in Figure 3, the monitoring traces and the corresponding  resampled traces are univariate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' To take advantage  of the feature learning in deep learning architectures, we  convert 1D time series to 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  We utilize the Gramian Angular Field (GAF) to transform  time series into images [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Specifically, the time series data  is first normalized or scaled into the range of [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The  normalized time series data is then represented in a polar  coordinate instead of the typical Cartesian coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' A Gram  Matrix like operation is applied on the resulting angles to  construct 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Given a time series trace T = {t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=', tn} of n times-  tamps, we rescale T to have the interval [−1, 1] by the equation  below:  ˜ti = (ti − max(T)) + (ti − min(T))  max(T) − min(T)  (1)  The value of the time series and its corresponding timestamp  need to be accounted for so that no information is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' These  two quantities are expressed with the angle and the radius  in polar coordinates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Mathematically, the angle  is computed by arccos(˜ti), which lies within [0, π], and the  radius variable is calculated by i/n, which is in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The  point can be expressed in polar coordinates (φi, ri), where:  �  φi = arccos(˜ti),  −1 ≤ ˜ti ≤ 1  ri = i  n,  0 ≤ i ≤ n  (2)  The encoding function is a composition of bijective func-  tions, producing one and only one result in the polar coordinate  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In addition, as opposed to Cartesian coordinates,  polar coordinates preserve temporal dependency through the  r coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' An example of a trace represented in polar  coordinates is shown in Figure 4a, which is transformed from  the normalized trace of Figure 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  The temporal correlations between each pair of data points  (ti, tj) are computed by considering the trigonometric sum-  mation (cos(φi + φj)) or subtraction (cos(φi − φj)), leading  to the Gramian Matrix called Gramian Angular Summation  Field (GASF) or Gramian Angular Difference Field (GADF),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The GASF is defined as follows:  GASF =  �  ����  cos(φ1 + φ1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  cos(φ1 + φn)  cos(φ2 + φ1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  cos(φ2 + φn)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  cos(φn + φ1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  cos(φn + φn)  �  ����  (3)  Through the GASF or GADF conversion, the diagonal Gi,i  contains the original value of the scaled time series, while  Gi,j represents the relative correlation by superposition of  directions with respect to time interval |i − j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Other details  on time series encoding can be found from [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In Figure 4b,  90°  135°  45°  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  180°  0°  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='75  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  225°  315°  270°60  0  20  40  80  100  120  0  20  40  60  80  100  120  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  120  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='75  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='50  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='25  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='50  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='75  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  0  1we illustrate the encoded 2D images with GASF and GADF  for the trace in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' To keep it concise, we use GASF  as the 1D time series conversion algorithm and refer to GASF  as GAF in the following discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  3) Encoding Multiple Time Series Traces: The procedure  presented in §III-B2 only converts one monitoring trace to  a 2D image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Since we have multiple monitoring traces cor-  responding to the same job and we do not want to lose the  correlation between each pair of traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The question becomes  how we can encode multiple time series to a single image such  that a deep learning architecture can understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' To solve this  problem, we are inspired by the concept of RGB channels,  where each RGB channel emphasizes different aspects of the  original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Similarly, the power consumption trace, IPC  trace, and memory used trace can be considered as the R  channel, G channel, and B channel of the encoded image,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  The output of GAF conversion is nothing but a 2D matrix  of floating-point numbers that fall in [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' To visualize the  job signature, we rescale the GAFs of the above-mentioned  three traces to be in [0, 255] by using the below equation:  �  GAF = ⌊GAF + 1  2  ∗ 255⌋  (4)  The �  GAF contains the pixel value of the gray scale image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  When combining �  GAF power, �  GAF ipc, �  GAF mem as three  channels of RGB, we create a single color image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Figure 5  shows the GAFs of the power, IPC and memory usage of a  job and its corresponding encoded job signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' It is worth  noting that rescaling GAF to the range [0, 255] is only for  the purpose of visualizing the job signature while the original  GAF falling in [−1, 1] can be directly fed in the Convelutional  Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  It is important to note that the channel-like encoding  methodology is not limited to three channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Each time series  trace is converted to a l ×l matrix by the procedure presented  in §III-B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' An encoded job signature of three channels is  a l × l × 3 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Encoding one more metric is simply  adding another dimension in the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' More formally, a job  signature of c metrics is a l × l × c matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Even though it  is not straightforward to be visualized when c is larger than  3, the CNN model can “understand” and analyze the high-  dimensional matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Classification Model  Using the methodology presented in §III-B, the monitoring  traces of jobs can be represented in images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=', job sig-  natures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Therefore, detecting and identifying applications of  jobs become an image recognition problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In this subsec-  tion, we first introduce the CNN, a deep learning technique  that has been widely used in image classification problems  and achieved promising results in many domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Then, we  present the CNN architecture that is specifically customized  for classifying job signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  1) Deep learning using CNN: The performance of con-  ventional machine learning techniques depends heavily on  data representation, which requires lots of efforts to design  preprocessing pipeline and feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Such feature  engineering is labor intensive and lacks the ability to extract  discriminative information from the data [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Deep learning,  on the other hand, explores the possibility to feed raw data to  the algorithm and automatically discover the features needed  for detection or classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The key concept of deep learn-  ing is to transform the representation at a lower level into a  representation at a higher and more abstract level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' and with  composition of such transformations, complex functions can  be learned [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' As a widely-used deep learning technique,  CNNs have been successful in image classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  2) Customized CNN Architecture for Job Signatures: The  CNN architecture for classifying job signatures is designed as  shown in Figure 6, which contains the following customized  layers and parameters:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Input layer for job signature: The inputs are job  signatures of encoded multiple time series traces, as presented  in §III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The resolution of the job signature is 128 × 128  and the number of channels equals to the number of traces  encoded in the job signature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=', three in our dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Note  that for other cases that job signatures encode more than three  monitoring traces, the input shape should be set accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Layers for feature extraction: In our model, we have  three layers of CNN to learn features from the job signature,  each of which contains one convolution layer to extract  features and one pooling layer to reduce the spatial dimension  of the convoluted image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The convolution layer extracts image  features by convolving the job signature with a set of kernels  and produces one feature map for each kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' We apply  the activation function ReLU on the convoluted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' It  replaces negative values with zero and keeps the positive value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  The output of ReLU will be the input for the pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  The first convolution layer learns 32 kernels, and the sec-  ond and third convolution layers learn 64 and 128 kernels,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' We set the kernels to be size of 3×3 and use the  same padding in the convolution to maintain the dimension of  output as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' All three pooling layers perform max pooling  that returns the maximum value from the portion of the image  covered by a window of size of 2 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Layers for classification: After going through the feature  extraction layers, the customized CNN model flattens the  final output and feeds it to a regular neural network for  classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The flattened vector is connected to a fully  connected layer, through which non-linear combinations of  the features can be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' After passing through the fully  connected layer, we use the softmax activation function in the  last layer to get the probabilities of the input job signature  being in a particular class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  To prevent the CNN model from overfitting and improve the  generalization of the CNN model, we add several dropout lay-  ers in the model to randomly disable neurons during training,  as shown in the dashed red parallelograms in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In our  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 5: Encoding Multiple Time Series Traces into a Job Signature with a Resolution of 128 × 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 6: Customized CNN Architecture for Classifying Job Signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  experiments, the dropout rate before the flatten layer and in  the fully-connected layer is set as 30% and 70%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' EXPERIMENTAL EVALUATION  In this section, we introduce the experimental evaluation  of our proposed classification model on a real-world dataset  collected from the Cori system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Particularly, we first introduce  the dataset built based on the proposed encoding methodology  and then, we compare the performance of our model with the  baseline methods in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Dataset  To the best of our knowledge, there are no publicly released  monitoring traces with labelled application information col-  lected from production HPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Ates el al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' [32] published  a dataset containing the monitoring data of benchmarks and  proxy applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Google published its cluster traces that do  not have application information [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Therefore, in order to  train the CNN model for detecting applications running in  production HPC systems, we built our own dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  On Cori system, some other research groups had imple-  mented the job metadata management service, where the  metadata of jobs running on Cori are stored and managed  in a MySQL database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The job metadata has a field named  ‘application name’ which is derived from the job submission  script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' However, based on our analysis, about 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='4% of derived  names do not reflect the application accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Many of them  have names like ‘test’, ‘bugs’, ‘b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='sh’, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Therefore, to build  a reliable labelled dataset, we select jobs that have accurate  application names through the job metadata service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In addi-  tion, since Cori system has the KNL partition and the Haswell  partition, the monitoring metrics of the same application (even  with same configurations and same input files) running on  different architectures could potentially have large variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  To avoid discrepancy caused by the architecture, we only focus  on the jobs running on the same architecture and we select  KNL jobs as our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Besides, we discard short jobs that  run for less than 60 seconds since they are likely for testing  purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  The  job  IDs  of  the  selected  jobs  are  used  with  NERSC LDMS [27] to obtain the corresponding monitoring  traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Considering that a job may have multiple steps and  that job steps may correspond to different applications, we  treat the job steps of a job separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In addition, since a job  may use multiple nodes (nodes are exclusively used by the  job on Cori), we aggregate the monitoring metrics from all  involved nodes and use the average time-series traces to build  the job signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  We select 23,665 jobs from 12 different applications to build  the job signature dataset after eliminating those jobs that do  not have monitoring metrics (due to data collection errors or  historical traces are purged) and balancing the number of jobs  of each application as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Details are listed in  Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The numbers of allocated nodes for a job vary from  1 to a maximum of 2,048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The BerkeleyGW, Espresso and  Footunee Lonin:  CNIN Ieyr  JAI NING  CNIN IyEr  128 × 129 x3  FT X  84 X 04 × B4  [Poad mps  32 量 $2 x 128  Potpd mp  L  口  口  口0  20  40  60  80  100  120  20  40  60  80  100  120  20  40  60  80  100  120  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='90 卡  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='13  185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='12  182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='50  120 -  120  120  20  100  100  100  40  80  80  80 -  60  09  60  80 -  40  40  40 -  100  20 -  M  120  ol  01  0  20  40  60  80  100  120  GAF of Power (Channel R)  GAF of IPC (Channel G)  GAF of Mem (Channel B)  Job SignatureTABLE II: Dataset of Job Signatures  Application  # of Jobs  # of Nodes*  Runtime(s)*  Description  BerkeleyGW  1899  1 / 2048  60 / 172867  For quasiparticle excitations and optical properties of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Espresso  2000  1 / 2048  61 / 172858  For electronic-structure calculations and materials modeling at the nanoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Gromacs  1937  1 / 64  61 / 172836  For simulations of proteins, lipids and nucleic acids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  LAMMPS  1999  1 / 64  60 / 172866  For molecular dynamics with a focus on materials modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  NWChem  1992  1 / 384  71 / 169209  For computational chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  VASP  2000  1 / 128  69 / 172864  For “ab-initio” quantum-mechanical molecular dynamics (MD) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  WRF  1860  1 / 256  72 / 174608  for atmospheric research and operational forecasting applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  aims  2000  1 / 192  71 / 172959  For “ab-initio” molecular simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  chroma  1996  2 / 137  488 / 147447  For lattice Quantum Chromodynamics calculations (LQCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  cp2k  1993  1 / 128  60 / 174608  For quantum chemistry and solid state physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  e3sm  1989  1 / 2048  60 / 173402  For earth system modeling, simulation and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  su3  2000  1 / 36  65 / 20277  For lattice Quantum Chromodynamics calculations (LQCD)  The values on the left and right side of ‘/’ indicate the minimum and maximum values of the corresponding characteristics of the jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  s3sm have extremely large-scale jobs where all cores of 2,048  nodes are used, corresponding to 131,072 physical cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The  selected jobs also cover a large range of runtime variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  The shortest job, as defined earlier, runs for only 60 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  The longest job, limited by the Quality of Service of the KNL  partition, runs for 48 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The dataset can be found in a  separate submission of artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  It is worth mentioning that, it is up to users to set the resam-  ling length l and define the resolution based on their computing  capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Nevertheless, the CNN model for images of larger  resolution usually requires more complex architecture and  the time for fine tuning and training the parameters of the  architecture becomes high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In our experiments, we set the  resampling length to be 128 and use a resolution of 128×128  for job signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Baseline Methods  To compare the performance of our proposed CNN model  with state-of-the-art methods for detecting HPC jobs, we  examine several other performance metrics based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  The Power Signature [18] and Taxonomist [19] are two repre-  sentative methods that use statistic features of the time series  monitoring data to build the classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The statistics  include the minimum, maximum, mean, standard deviation,  skew, kurtosis and the 5th, 25th, 50th, 75th, 95th percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Using the same monitoring metrics as ARcode, namely power  consumption, IPC and memory usage of the selected jobs,  we extract statistical features and build several classifiers as  baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' These classifiers are Random Forest (RF), linear  Support Vector Classifier (Liner-SVC), and non-linear Support  Vector Classifier (SVC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In addition, we include a Random  Forest model that uses only statistical features extracted from  the power consumption trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' All of these models use default  hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Experiment Setup  The ARcode and baseline models are trained and evaluated  on a Cori GPU node, where 2 Skylake CPUs, 8 NVIDIA Tesla  V100 GPUs and 384 GB DDR4 memory are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The  proposed CNN is implemented using the TensorFlow Keras  library in a Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='9 environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The baseline models are  implemented in Python leveraging scikit-learn library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 7: Loss (top) and accuracy (bottom) of the training  and validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The vertical dashed line indicates the  determined epoch of 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  We divide the dataset into 60% for training, 20% for  validation, and 20% for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' To mitigate overfitting and to  increase the generalization of ARcode model, we determine  the optimal epoch number by examining the loss and accuracy  trends on training and validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' As shown in Figure 7,  after epoch reaches to 50, the training loss gets lower than  validation loss while the accuracy of validation does not  improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Therefore, we set epoch to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' While evaluating the  classification performance on each application, we use ten-fold  stratified cross validation to divide the dataset into ten disjoint  partitions and evaluate model performance with each partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  We test the classification performance on different con-  fidence thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' For each baseline classifier, we use the  one-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='-rest version of that classifier such that it produces a  set of real-valued prediction scores for its decision instead  of a labeled class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In ARcode model, the softmax function  assigns probabilities of classes in each prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' We compare  the prediction probabilities with the confidence thresholds to  determine the predicted class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' For the prediction score si of  class ci, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=', k}, the classifier assigns the label of the  class by the following expression:  �  ci, where si = max(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=', sk),  if si ≥ threshold  unknown,  otherwise  (5)  In other words, when the maximum value of the prediction  Training  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  Validation  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  S  LoSS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='80 -  Accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='40 40  60  120  0  20  80  100  EpochsFigure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 8: Accuracy of ARcode and baseline classifiers at  different confidence thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The vertical dashed line in-  dicates the confidence threshold (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='4) above which ARcode  outperforms all other classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 9: Accuracy of classifiers on each application at a  confidence threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  scores is larger than the threshold, the class with the largest  score is the predicted class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Otherwise, the classifier labels the  observation with unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' When the confidence threshold is  0, it is the same with the vanilla multi-class classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Experimental Results  We evaluate the capability of ARcode in identifying appli-  cations in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' First, we evaluate the classification  performance on different confidence thresholds and examine  the performance on each application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Then, we assess the  performance of identifying applications that have not been  trained with the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In addition, since the job signatures  encode temporal information of the monitoring metrics, we use  a subset of the monitoring data to construct partial signatures  to evaluate the ARcode model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  1) Classification Performance: Figure 8 depicts the accu-  racy of ARcode and baseline models on different confidence  thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' As we can see from the figure, the Linear-SVC  model has the worst performance at all confidence thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  When the confidence threshold is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='4, Random Forest  outperforms all other classifiers, with the highest accuracy  of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='81% at a confidence threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' ARcode follows  closely with an accuracy of 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='67%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' When the confidence  threshold is above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='4, the accuracy of Random Forest de-  creases and ARcode has the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' At a threshold  of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='0, the SVC model performs a little better than Random  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 10: Accuracy of ARcode and baseline classifiers on  detecting novel applications at different confidence thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  ARcode achieved the highest accuracy when the confidence  threshold is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Forest, but it is still 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='87% worse than ARcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' At a  confidence threshold of 0, the accuracy of Random Forest  on power features is close to ARcode , but it decreases  significantly as the confidence threshold increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In sum-  mary, the accuracy of all classifier decreases with increasing  confidence thresholds, but the slowest decrease rate is observed  for ARcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  We further examine the classification performance of  ARcode and Random Forest on each application, as shown  in Figure 9, where we use a confidence threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='8  in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' From this figure, we have the following  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' First, the accuracy of ARcode and Random  Forest are very close in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' When detecting Gromacs,  LAMMPS and cp2k, Random Forest performs slightly bet-  ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Second, for some applications such as BerkeleyGW and  e3sm, ARcode achieves a significantly better accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In  BerkeleyGW classification, the accuracy of Random Forest is  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='06% while ARcode achieves 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='94% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' ARcode  is also 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='21% better than Random Forest in detecting e3sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  2) Classification on Novel Applications: Considering that  in HPC environments, ‘novel’ applications are more prevalent  than those trained with the classifier, it is appropriate to  define an ‘unknown’ class for all these novel applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' To  be practically useful, the classification system must classify  both known and unknown novel applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' We evaluate  this capability by removing one application from the training  set while keeping the testing set untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' If the removed  application in the testing set is predicted to be unknown, we  mark it as a correct prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  The classification performance for novel applications is  depicted in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' From this figure, we can observe  that Random Forest has the best prediction accuracy when  the confidence threshold is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='8, while ARcode beats  Random Forest once the confidence threshold is greater than  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' ARcode, similar to its performance in known application  detection, has relatively stable prediction accuracy across all  confidence thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' However, the accuracy of Random  Forest drops significantly in large confidence thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  100%  80%  60%  Accuracy  40%  ARcode (CNN)  Random Forest  20%  SVC  LinearisvC  Random Forest on Poweri Features!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  0%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='0  Confidence ThresholdARcode(CNN)  Random Forest  100%  80%  Accuracy  60%  40%  20%  0%  aims  WRF  BerkeleyGW Espresso Gromacs LAMMPS NWCHEM  VASP  chroma  cp2k  e3sm  su3  Applications100%  80%  60%  Accuracy  40%  ARcode (CNN)  Random Forest  20%  SVC  Linearisvc  Random Forest on Poweri Features  0%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='0  Confidence ThresholdFigure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 11: Accuracy of ARcode on each application using  different channels of the job signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 12: Accuracy of ARcode on each application using  partial job signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The partial job signatures are all built  from the metrics collected since the startup of jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  3) Classification using Single Channel: In our experiment,  ARcode encodes three channels of monitoring data in the  job signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' It is worth knowing whether each channel plays  the same importance in classification and whether we can use  one channel for detection while still achieving high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  To answer these questions, we train CNN models with each  of these channels individually and analyze their classification  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The results are shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  From the figure we can see that using any of the chan-  nels gives a competitive accuracy when detecting Espresso,  LAMMPS, WRF, and chroma compared to using all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  For NWCHEM, the classification accuracy using the memory  channel is significantly better than using the power or IPC  channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' We can also see that all channels are important for  detecting both Gromacs and cp2k with an accuracy improve-  ment of 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='36% and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='66%, respectively, compared to using  only one of the channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The memory channel is the most  representative of these channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In most applications, using  the memory channel in detection has the closest accuracy to  using all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  4) Classification using Partial Signatures: The job sig-  nature retains the temporal information of the time-series  monitoring data, and the job signature generated before the  end of the job still holds some of the attributes of the full  job signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' We use the first 25%, 50% and 75% of the  time-series data to create partial job signatures and compare  the detection performance with the full job signature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='e,  a job signature built from 100% of the monitoring data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  This experiment is performed to evaluate the usability of the  ARcode model for detecting running applications, which is  not available in the statistics-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  The results are shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' It is not surprising that  the larger the percentage of encoded data, the better the detec-  tion performance achieved by ARcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' When encoded with  25% of the monitoring data, ARcode achieves a relatively  high accuracy of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='27% in detecting WRF, but only 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='27%  and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='88% in detecting cp2k and Espresso, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  When detecting NWCHEM, the full job signature brings  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='79% accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' however, the 75% partial job signature  gives only 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='73% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The improvement from encoding  more data varies from application to application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' For example,  encoding 25% more data brings an average improvement of  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='67% for Espresso, but only 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='65% for WRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  5) Sensitivity to the resampling length: The performance  evaluation described above is based on job signatures with  a resolution of 128 × 128, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=', the performance metrics are  resampled to a length of 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' It is valuable to have knowledge  of the sensitivity of the classification accuracy in terms of the  resampling length such that an appropriate resolution can be  chosen to achieve a balance between accuracy and training  overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' To explore this, we further build two job signature  datasets with resampling lengths of 32 and 64, respectively,  and use the same CNN architecture to train these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Figure 13 depicts the accuracy (solid lines) and training  time (horizontal dashed lines) of ARcode using different  resampling lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Note that the training time is constant  across different thresholds for the same resampling length, as  illustrated by horizontal dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' This is because training  time does not vary according to confidence thresholds, which  are set during the prediction of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' As illustrated  from the figure, the training times vary significantly among  these three resampling lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' It is over 185 seconds for  the resampling length of 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' For signatures with resampling  lengths of 64 and 32, it is 80 and 65 seconds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Additionally, we can observe from the figure that the higher  resolution it is, the higher prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' While the  accuracy improvement from the resampling length of 64 to  128 is not significant (only by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='5% on average), the accuracy  improvement from the resampling length of 32 to 64 is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='0%  on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  This result suggests that 64 is the optimal resampling length  for building job signatures in our experiment, considering the  trade-off between accuracy and training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Its prediction  accuracy is close to that of the highest resolution signature, but  only uses 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='2% of the training time of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Furthermore,  the data size of the job signature generated with a resolution  of 64 × 64 is only 25% of that with 128 × 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' RELATED WORKS  Significant amount of work has been reported in the lit-  erature on detecting and classifying applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Identifying  similarities and differences among binary executables was  explored in [13]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' These approaches are limited by the  ZZIPC  Memory  Power  All  100%  80%  Accuracy  60%  40%  20%  0%  WRF  BerkeleyGWEspresso Gromacs LAMMPS NWCHEM  VASP  aims  chroma  cp2k  e3sm  su3  Applications25%  75%  100%  50%  Z  100%  80%  Accuracy  60%  40%  20%  0%  WRF  BerkeleyGWEspresso Gromacs LAMMPS NWCHEM  VASP  aims  chroma  cp2k  e3sm  su3  ApplicationsFigure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 13: Accuracy and training time of ARcode using dif-  ferent resampling lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' The horizontal dashed lines indicate  the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  fact that, they cannot differentiate the same source program  compiled by different compiler toolchains or optimization  levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' To overcome this limitation, Blanket Execution [17]  presented a binary differencing algorithm that compares the  side effects of functions during executation, which is based  on the insight that similar codes have semantically similar  execution behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  However, the binary based approach cannot be applied in the  HPC environment, where it is impractical to conduct binary  differencing among hundreds of thousands of executables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In  addition, obtaining and managing binaries of users’ applica-  tions are not always possible for HPC researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Instead  of using binaries, Yamamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' [34] used job scripts as  job information to classify applications with text classification  techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' This approach, however, also requires dedicated  collection and management infrastructures, which are not as  prevalent as monitoring infrastructures for performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Another line of work aims at characterizing HPC appli-  cations through system logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' extracted features  from combined logs of multiple subsystems to represent  applications and build a machine learning model based on  the eXtreme Gradient Boosting (XGB) algorithm to identify  HPC applications [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' DeMasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' collected and extracted  features from Integrated Performance Monitoring (IPM) per-  formance logs to fingerprint HPC codes [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Log analysis is  more common in characterizing subsystem and user behavior,  related works can be found in [37]–[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Monitoring data based application detection has been ex-  plored in [18]–[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' As a early quantitative study of power  consumption of HPC workloads, Combs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' [18] studied  the applicability of classifying applications through power  consumption traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Ramos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' [21] relied on performance  counters to model, fingerprint and clustering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Zou  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' [20] explored detecting illicit applications in GPU-  accelerated HPC workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' They used performance counters,  data movement behavior and resources utilization traces to  train machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Taxonomist [19], proposed  by Ates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=', used over 700 system metrics and a time  window spanning the whole execution to extract statistical  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' They enhanced the classification model such that  unknown applications can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' EFD [22] created  key-values pairs that link execution fingerprints of system  metrics to application and input size information to implement  application recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Except for using measurements of  system metrics, they also used the metrics name, node ID and  time interval to create fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  These studies, however, are based on the datasets built from  the monitoring data of benchmarks and proxy applications,  where a relative small range of configurations and input size  are shown in the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' On the contrary, the model  and experiment results of ARcode are developed and tested  on datasets collected from real applications in a large-scale  production system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In addition, the models of these studies rely  on the manually defined features extracted from time series  monitoring data and some studies utilize the input information  to enhance the detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In this work, ARcode allevi-  ates the effort of features engineering and takes advantages of  the capability of features learning in CNN model to detect and  classify encoded monitoring data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Moreover, ARcode retains  the temporal information of time-series performance metrics,  enabling detecting applications before jobs are finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' This  capability bridges a major gap in related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' CONCLUSION  Existing application detection methodologies on HPC sys-  tems either relies on the data that are not prevalent or require  intensive effort of feature engineering to build high accuracy  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In this study, we aim to provide a solution that can  be easily used and adopted by any HPC site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' We have intro-  duced ARcode, an application recognition framework which  is effective and extensible with the following characteristics:  1) ARcode uses three most common monitoring metrics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=',  the power consumption of the compute node, instruction per  cycle, and memory usage) available through the monitoring  infrastructure on diverse HPC architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 2) ARcode allevi-  ates the effort of feature engineering by leveraging the feature  learning capability of CNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 3) ARcode’s channel-like  encoding method allows easy encoding of additional metrics  in job signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' 4) ARcode encodes job monitoring data into  images, which translates the application recognition problem  into an image classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Unlike the statistics-  based approaches where the temporal information of monitor-  ing data are lost, the job signature encodes metric variations  over the runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Therefore, the job signature generated from  part of monitoring data can still be used in detection, but with  the sacrifice of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  Although our evaluation is performed on job signatures  generated from three common monitoring metrics, HPC re-  searchers and system administrators can select other represen-  tative metrics and build job signatures to detect and classify  applications with specific characteristics such as CPU intensive  applications and I/O intensive applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' In addition, we  have seen lots of successful cases of applying image recogni-  tion in the medical and automobile industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Their experience  in training high-accuracy models can be used in our model to  further improve the recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content=' Moreover, encoding  100%  128  32  64  32  64  128  200  (accuracy)  (training time)  90%  180  Time (  80%  Accuracy  140  ining  70%  120  Trai  100  60%  80  50%  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='5  Confidence Thresholdmonitoring data in job signatures offers a new perspective  of exploring and analyzing the performance monitoring data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
+page_content='  In our future work, we will further explore the use of job  signatures to predict resource usage, detect anomalies, and  identify malicious applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf'}
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diff --git a/6dFJT4oBgHgl3EQflizf/content/tmp_files/2301.11584v1.pdf.txt b/6dFJT4oBgHgl3EQflizf/content/tmp_files/2301.11584v1.pdf.txt
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+Robust variance-regularized risk minimization
+with concomitant scaling
+Matthew J. Holland
+Osaka University
+Abstract
+Under losses which are potentially heavy-tailed, we consider the task of minimizing sums
+of the loss mean and standard deviation, without trying to accurately estimate the variance.
+By modifying a technique for variance-free robust mean estimation to fit our problem
+setting, we derive a simple learning procedure which can be easily combined with standard
+gradient-based solvers to be used in traditional machine learning workflows. Empirically,
+we verify that our proposed approach, despite its simplicity, performs as well or better
+than even the best-performing candidates derived from alternative criteria such as CVaR
+or DRO risks on a variety of datasets.
+Contents
+1
+Introduction
+2
+2
+Background
+3
+2.1
+Robust mean estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+2.2
+Good-enough ancillary scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.3
+A bridge between two problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2.4
+Overview of contributions and limitations . . . . . . . . . . . . . . . . . . . . .
+6
+3
+Theory
+7
+3.1
+Links to the mean-SD objective . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+3.2
+Guiding the optimal threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.3
+Deriving an algorithm using finite-sample theory . . . . . . . . . . . . . . . . .
+8
+3.4
+Stationary points of mean-variance . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+3.5
+Comparison with dual form of DRO risk . . . . . . . . . . . . . . . . . . . . . .
+11
+4
+Empirical analysis
+11
+4.1
+Simulated noisy classification on the plane . . . . . . . . . . . . . . . . . . . . .
+12
+4.2
+Classification on real datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+A Technical appendix
+18
+A.1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+A.2 Convexity and smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+B Additional proofs
+20
+C Empirical test appendix
+27
+1
+arXiv:2301.11584v1  [stat.ML]  27 Jan 2023
+
+1
+Introduction
+Traditionally, the “textbook definition” of a statistical machine learning problem is formulated
+in terms of making decisions which minimize the expected value of a random loss [9, 27, 35].
+More precisely, the traditional setup has us minimize Eµ L(h) with respect to a decision h,
+where we denote random losses as L(h) ..= ℓ(h; Z), with a random data point Z ∼ µ, and ℓ(·)
+is a loss function assigning real values to (decision, data) pairs. This problem class is very
+general in that it covers a wide range of learning problems both supervised and unsupervised,
+but it is limited in the sense that it only aspired to be optimal on average, with no guarantees
+for other aspects of performance such as loss deviations, resilience to worst-case examples and
+distribution shift, sub-population disparity, and class-balanced error. While it is sometimes
+possible to account for these issues by modifying the base loss function ℓ (e.g., logit-adjusted
+softmax cross-entropy for balanced error [26]), there is a growing literature looking at prin-
+cipled, systematic modifications to the “risk,” i.e., a non-random numerical property of the
+distribution of L(h) to be optimized in h, leaving the base loss ℓ(·) fixed. Some prominent
+examples are weighted sums of loss quantiles [25], distributionally robust optimization (DRO)
+risk [13], conditional value-at-risk (CVaR) [7], tilted risk [20], and more general optimized
+certainty equivalent (OCE) risks [19], among others. It is well-known that many risks can be
+expressed in terms of location-deviation sums, with the canonical example being a weighted
+sum of the loss mean and standard deviation (or variance) [31, §2]. We refer the reader to some
+recent surveys [14, 16, 32] for more general background on developments in learning criteria.
+In this work, the criterion of interest is the mean loss regularized by standard deviation
+(SD), when losses are allowed to be heavy-tailed. More formally, we allow for heavy tails in
+the sense that all we assume is that the second moment Eµ|L(h)|2 is finite, and the ultimate
+objective of interest is the mean-SD criterion
+MSµ(h; λ) ..= Eµ L(h) +
+�
+λ Vµ L(h)
+(1)
+with loss variance denoted by Vµ L ..= Eµ(L − Eµ L)2, and weighting parameter λ ≥ 0. This
+mean-SD objective (1) and its mean-variance counterpart have a long history in the literature
+on decision making under uncertainty, including the influential work of Markowitz [23] on
+optimal portfolio selection. In the context of machine learning, it is well-known that one can
+obtain “fast rate” bounds on the expected loss when variance is small (see [11, §1]), though the
+problem of actually ensuring that loss deviations are sufficiently small is an entirely separate
+matter. In this direction, Maurer and Pontil [24] bound the (population) expected loss using a
+weighted sum of the sample mean and standard deviation. Their “sample variance penalized”
+objective is convenient to compute and can be used to guarantee fast rates in theory, but a
+lack of convexity makes it hard to minimize in practice. A convex approximation is developed
+by Duchi and Namkoong [11], who show that a sub-class of (empirical) DRO risks can be used
+to approximate the sample mean-SD objective, again yielding fast rates when the (population)
+variance is small enough. The critical limitation to this approach is poor guarantees under
+heavy-tailed losses; while we gain in terms of convexity, the empirical DRO risk of [11] is at least
+as sensitive to outliers as the naive empirical objective (i.e., directly minimizing the sample
+mean and SD), which is already known to result in highly sub-optimal performance guarantees
+under heavy tails [5, 10, 15]. Recent work by Zhai et al. [36] studies a natural strategy for
+robustifying the DRO objective (called DORO), which discards a specified fraction of the
+largest losses. While the impact of outliers can be reduced under the right setting of DORO,
+their approach is limited to non-negative losses, and the impact that such one-sided trimming
+has on the resulting mean-SD sum, our ultimate object of interest, is unknown.
+2
+
+With this context in mind, in this paper we propose a new approach to robustly minimize
+the objective (1) under heavy-tailed losses, without a priori knowledge of anything but the
+fact that variance is finite. Our key technique is based on extending a convex program of Sun
+[34] from one-dimensional mean estimation to our mean-variance objective MSµ(h; λ) under
+general losses. After some motivating background points in §2.1–§2.2, we describe our basic
+approach and summarize our contributions in §2.3–§2.4. Theoretical analysis comes in §3, and
+based upon formal properties of the proposed objective function, we derive a general-purpose
+procedure summarized in Algorithm 1, and tested empirically in §4.
+Our main finding is
+that the simple algorithm we derive works remarkably well on both simulated and real-world
+datasets without any fine-tuning, despite sacrificing the convexity enjoyed by procedures based
+on criteria such as CVaR and DRO. Software and notebooks to reproduce all results in this
+paper are provided in an online repository.1
+2
+Background
+Before we describe our proposed approach to the mean-SD task described in §1, we start with
+a much simpler problem, namely the task of robust mean estimation. This will allow us to
+highlight key technical points from the literature which provide both conceptual and technical
+context for our proposal. Key points from the existing literature are introduced in §2.1–§2.2,
+and building upon this we introduce our method in §2.3–§2.4.
+2.1
+Robust mean estimation
+Let X be a random variable. For the moment, our goal will be to construct an accurate empirical
+estimate of the mean Eµ X, assuming only that the variance Vµ X = Eµ X2 − (Eµ X)2 is both
+defined and finite. We assume access to an independent and identically distributed (IID) sample
+X1, . . . , Xn. Since higher-order moments may be infinite, the tails of X may be “heavy” and
+decidedly non-Gaussian, causing problems for the usual empirical mean. This problem setting
+is now very well-understood; see Lugosi and Mendelson [22] for an authoritative reference.
+One very well-known approach is to use M-estimators [18], namely to design an estimator
+An ≈ Eµ X satisfying
+An ∈ arg min
+a∈R
+b
+n
+n
+�
+i=1
+ρ
+�Xi − a
+b
+�
+(2)
+where ρ: R → R+ is a function that is approximately quadratic near zero, but grows more
+slowly in the limit, i.e., large deviations are penalized in a sub-quadratic manner, where “large”
+is relative to the scaling parameter b > 0, used to control bias. When ρ(·) is convex, differen-
+tiable, and the solution set is non-empty, the condition (2) is equivalent to
+1
+n
+n
+�
+i=1
+ρ′
+�Xi − An
+b
+�
+= 0
+(3)
+and when the derivative ρ′(·) is bounded such that
+− log(1 + −x + γx2) ≤ ρ′(x) ≤ log(1 + x + γx2),
+x ∈ R
+(4)
+for some constant 0 < γ < ∞, then the analytical approach of Catoni [6] tells us that when b2
+scales with Vµ X/n, the deviations |An − Eµ X| enjoy sub-Gaussian tails, namely upper bounds
+1https://github.com/feedbackward/bdd-mv
+3
+
+-2
+0
+2
+2
+0
+2
+-2
+0
+2
+′
+-2
+0
+2
+′′
+Figure 1: From left to right, we plot the graphs of ρ(·), ρ′(·), and ρ′′(·) with ρ as in (6). In the middle plot,
+the dotted curves represent the upper (blue) and lower (dark pink) bounds in (4) with γ = 1.
+of the order O(
+�
+log(1/δ) Vµ X/n) with probability at least 1 − δ. Under these weak assump-
+tions, such guarantees are essentially optimal [10]. While a very important result, for practical
+purposes, the need for knowledge of Vµ X is a significant limitation, since without finite higher-
+order moments, it is not plausible to obtain variance estimates with analogous sub-Gaussian
+guarantees (e.g., impossibility results of [10]). There do exist other robust estimators such
+as median-of-means [22, §2.1] which do not require variance information, and this illustrates
+the fact that knowledge of the variance is sufficient, although not necessary, for sub-Gaussian
+mean estimation under heavy tails.
+2.2
+Good-enough ancillary scaling
+Since sub-Gaussian estimates of the variance Vµ X are not possible under our weak assump-
+tions, it is natural to ask whether there exists a middle-ground, namely whether or not it is
+possible to construct a (data-driven) procedure for setting the scale b > 0 in (2) which is “good
+enough” in the sense that the resulting An is sub-Gaussian, even though the scale itself cannot
+be. An initial (affirmative) answer to this question was given in recent work by Sun [34], whose
+basic idea we briefly review here, with some slight re-formulation for readability and additional
+generality.
+Essentially, the underlying idea in [34] is to utilize the convexity of ρ in (2), and to solve
+for both a ∈ R and b > 0 simultaneously, while penalizing b in such a way as to encourage
+scaling which is “good enough” as mentioned. More precisely, the empirical objective
+�Sn(a, b) ..= βb + b
+n
+n
+�
+i=1
+ρ
+�Xi − a
+b
+�
+(5)
+plays a central role, where 0 < β < 1 is a parameter we can control, and ρ is fixed as
+ρ(x) =
+�
+x2 + 1 − 1,
+x ∈ R
+(6)
+which is differentiable, and satisfies the Catoni condition (4) with γ = 1 (see Figure 1). If we
+fix b > 0, then the solution sets (in a) of both �Sn(a, b) and b × �Sn(a, b) are identical, and it
+should be noted that the re-scaled map x �→ b2ρ(x/b) = b
+√
+x2 + b2 − b2 closely approximates
+x �→ x2/2 as b grows large (Figure 2), and is well-known as the “pseudo Huber” or “smooth
+Huber” function, where b acts as a smoothing parameter.2
+When considering the joint objective �Sn(a, b), from the computational side, one important
+fact is that this function is convex on R × (0, ∞) (see Lemma 8). From the statistical side of
+2Barron [2, §1] gives a summary of this and related functions from the perspective of loss function design.
+This is not the only smoothed variant of the classic Huber function [17], see for example Rey [29, §6.4.4].
+4
+
+1
+0
+1
+0.0
+0.2
+0.4
+x
+b2 (x/b)
+0.1
+2.0
+b value
+Figure 2: Graphs of the smooth Huber function, with ρ as in (6), over a range of smoothing parameters. For
+visual comparison, the graph of x �→ x2/2 is plotted with a thick dashed green curve.
+things, the solutions
+(An, Bn) ∈ arg min
+a∈R,b>0
+�Sn(a, b)
+(7)
+are such that under certain regularity conditions, the deviations |An − Eµ X| are nearly optimal
+(sub-Gaussian, up to poly-logarithmic factors) [34, §3.3].3 The corresponding Bn of course
+cannot give us sub-Gaussian estimates of the variance under such weak assumptions, but it
+does scale in a desirable way [34, §3.2], and when bias is mitigated by setting β sufficiently small
+given the sample size n, the resulting Bn is good enough to provide such guarantees for An,
+which is the ultimate goal anyways. By taking on a slightly more difficult optimization problem,
+it is possible to get away with not having prior knowledge or sub-Gaussian estimates of the
+variance. We use this basic insight as a stepping stone to our approach for learning algorithms
+charged with selecting a decision h such that the loss L(h) has a small mean-variance.
+2.3
+A bridge between two problems
+To develop our proposal, we now return to the more general learning setup, where the test
+data is a random vector Z ∼ µ, test loss is L(h) ..= ℓ(h; Z), and we have n IID training
+points Z1, . . . , Zn yielding losses Li(·) ..= ℓ(·; Zi), i ∈ [n]. If our goal was to simply minimize
+the traditional risk Eµ L(h) over h ∈ H under heavy-tailed losses, then in principle we could
+extend the approach of §2.2 to robustly estimate the test risk using
+(An(h), Bn(h)) ∈ arg min
+a∈R,b>0
+�
+βb + b
+n
+n
+�
+i=1
+ρ
+�Li(h) − a
+b
+��
+(8)
+and design a learning algorithm using (8) as follows:
+Hn ∈ arg min
+h∈H
+An(h).
+(9)
+Under some regularity conditions, the machinery of Brownlees et al. [5] could then be combined
+with pointwise concentration inequalities in [34] to control the tails of Eµ L(Hn) under just finite
+loss variance. Our goal however is not to minimize the expected loss, but rather the mean-SD
+sum (1). Furthermore, the bi-level program inherent in (9) is not computationally congenial
+from the perspective of large-scale machine learning tasks. To ease the computational burden
+3Strictly speaking, the objective used in [34] is �Sn(a, b)/β, but all key results easily translate to our setup.
+5
+
+while at the same time building a bridge between these two problems, we consider a new
+objective function taking the form
+�Cn(h; a, b) ..= αa + βb + λb
+n
+n
+�
+i=1
+ρ
+�Li(h) − a
+b
+�
+(10)
+with parameters α ≥ 0 and β ≥ 0. We call (10) the modified Sun-Huber objective, since ρ is
+fixed as (6), and this form plays a special role in our analysis. Compared with that of (9),
+this objective is a simple function of h, and gradient-based minimizers can be easily applied
+assuming the underlying loss ℓ(·) is sufficiently smooth. On the other hand, it is “biased” in
+the sense that it penalizes not just the loss location (whenever α > 0), but the loss scale as well
+(whenever β > 0). Intuitively, some kind of deviation-driven “bias” is precisely what we need
+from the standpoint of minimizing the mean-SD objective MSµ(h; λ), but it is not immediately
+clear how this objective relates to �Cn(h; a, b), and it is equally unclear if we can just plug this
+new objective into standard machine learning workflows (e.g., using stochastic gradient-based
+optimizers) and achieve the desired effect without a prohibitive amount of manual tuning.
+2.4
+Overview of contributions and limitations
+With our basic idea described and some key questions raised, we summarize the central points
+that characterize the rest of this paper, and also highlight the limitations of this work. Broadly
+speaking, the new proposal here is a class of empirical “risk” minimizers, namely any learning
+algorithm which minimizes the new empirical objective (10). More explicitly, this refers to all
+procedures which returns a triplet satisfying
+(Hn, An, Bn) ∈
+arg min
+h∈H,a∈R,b>0
+�Cn(h; a, b)
+(11)
+where H denotes a set of feasible decisions, and we note that each element of this class is
+characterized by the settings of α, β, and λ used to define �Cn. In analogy with the strat-
+egy employed in §2.2, we do not expect An and Bn to provide sub-Gaussian estimates; we
+simply hope that these estimates are good enough to ensure the mean-SD is smaller and/or
+better-behaved when compared to standard benchmarks such as mean-based empirical risk
+minimization (ERM) and DRO-based algorithms. Theoretically, we are interested in identify-
+ing links between the proposed objective �Cn and loss properties such as Eµ L(h) and Vµ L(h),
+with particular emphasis on how the settings of α, β, and λ influence such links.
+Our main theory-driven contribution is the derivation of a principled approach to determine
+�Cn (i.e., set α and β), before seeing any training data, in such a way that we can balance between
+“biased but robust” ρ-based deviations and “unbiased but outlier-sensitive” squared deviations
+that arise in the loss variance. Details are in §3.1–§3.3, and a concise procedure is summarized
+in Algorithm 1. We do not, however, consider the behavior of MSµ(Hn; λ) for a particular
+implementation of (11) (e.g., SGD) from a theoretical viewpoint; the implementation is left
+abstract. This is where the empirical analysis of §4 comes in. We provide evidence using
+simulated and real data that our procedure can be quite useful, even using a rudimentary
+implementation where we wrap base loss objects and naively pass them to standard stochastic
+gradient-based learning routines, with no manual tweaking of parameters.
+6
+
+3
+Theory
+3.1
+Links to the mean-SD objective
+We would like to make the connection between the proposed objective (10) and the ultimate
+objective (1) a bit more transparent. To do this, we will make use of the population version of
+�Cn, denoted henceforth by Cµ and defined as
+Cµ(h; a, b) ..= αa + βb + λb Eµ ρ
+�L(h) − a
+b
+�
+.
+(12)
+Let us fix the decision h and threshold a, paying close attention to the optimal value of the
+scale b, denoted here by bµ(h, a). More explicitly, consider any positive real number satisfying
+bµ(h, a) ∈ arg min
+b>0
+Cµ(h; a, b).
+(13)
+While it is not explicit in our notation, the optimal scale in (13) depends critically on the
+value of β. Intuitively, a smaller value of β leads to a weaker penalty for taking b large, thus
+encouraging a larger value of bµ(h, a). In fact, one can show that viewing bµ(h, a) as a function
+of the parameter β, in the limit we have (proof in §B)
+lim
+β→0+ bµ(h, a) = ∞.
+(14)
+Combining this with the fact that
+lim
+b→∞ b Eµ ρ
+�L(h) − a
+b
+�
+= 0
+(15)
+also holds (proof in §B), by re-scaling to avoid trivial limits we can obtain a result which
+sharply bounds the proposed learning criterion at the optimal scale using the square root of
+quadratic deviations, thereby establishing a clear link to the desired mean-SD objective (1).
+Proposition 1. Let H be such that Eµ|L(h)|2 < ∞ for each h ∈ H. If we set α = α(β) such
+that α(β)/√β → �α ∈ [0, ∞) as β → 0+, then in this limit, with appropriate re-scaling the
+scale-optimized learning criteria can be bounded above and below as
+�αa + (1/2)
+�
+λ Eµ(L(h) − a)2 ≤ lim
+β→0+ min
+b>0
+Cµ(h; a, b)
+√β
+≤ �αa + 4
+�
+λ Eµ(L(h) − a)2
+for any choice of threshold a ∈ R and weight α ≥ 0.
+In the special case where a = Eµ L(h) and �α > 0, we naturally recover mean-SD sums akin to
+those studied in an ERM framework by Maurer and Pontil [24] and those bounded from above
+using convex surrogates by Duchi and Namkoong [11].
+Of course in practice, we will only ever be working with fixed values of β, and the entire
+point of introducing new criteria (namely �Cn and Cµ) was to give us some control over how
+sensitive our objective is to loss tails. The following result makes the nature of this control
+(through β) more transparent.
+Proposition 2. Let H and L(h) be as stated in Proposition 1. Letting bµ(h, a) be as specified
+in (13), we define a Bernoulli random variable
+I(h; a) ..= I {|L(h) − a| ≤ bµ(h, a)}
+7
+
+for any choice of h ∈ H and a ∈ R. The optimal scale can then be bounded by
+λ
+4β Eµ I(h; a)(L(h) − a)2 ≤ b2
+µ(h, a) ≤ λ
+2β Eµ(L(h) − a)2
+for any choice of 0 < β < λ and a ∈ R.
+While it is difficult to pin down exactly how bµ(h, a) changes as a function of β, Proposition 2
+clearly shows us the appealing property that optimal scale induced by the proposed objective
+function essentially falls between the (tail-sensitive) quadratic deviations and a (tail-insensitive)
+truncated variant, with the truncation threshold loosening as β shrinks.
+3.2
+Guiding the optimal threshold
+Since the preceding Propositions 1–2 both hold for any choice of threshold a ∈ R, they clearly
+hold when both a and b are optimal, i.e., when a and b are set as
+(aµ(h), bµ(h)) ∈ arg min
+a∈R,b>0
+Cµ(h; a, b).
+(16)
+In particular, using first-order conditions, the inclusion (16) is equivalent to the following two
+equalities holding at once:
+Eµ
+�
+�
+L(h) − aµ(h)
+�
+(L(h) − aµ(h))2 + b2µ(h)
+�
+� = α,
+Eµ
+�
+�
+bµ(h)
+�
+(L(h) − aµ(h))2 + b2µ(h)
+�
+� = 1 − β/λ.
+(17)
+Given the context of our analysis in §3.1, let us consider the effect of taking β towards zero.
+For any non-trivial random loss, the second equality asks that bµ(h) grow without bound as
+β → 0+, while |aµ(h)| must be either bounded or grow slower than bµ(h). On the other hand,
+if α is too large (i.e., α > 1) then the first equality will be impossible to satisfy. In addition to
+taking 0 < α < 1, note that if we multiply both sides of the first equality in (17) by bµ(h) and
+apply Proposition 2, then under this optimality condition we must have
+Eµ
+�
+�
+�
+�
+L(h) − aµ(h)
+�
+(L(h)−aµ(h)
+bµ(h)
+)2 + 1
+�
+�
+�
+� ≤ α
+�
+λ
+2β Eµ(L(h) − aµ(h))2.
+(18)
+With this inequality in place, we adopt the following strategy: encourage the optimal location
+to converge as aµ(h) → Eµ L(h) when β → 0+. Since λ > 0 is assumed to be fixed in advance,
+the only way to ensure this using (18) is to set α = α(β) such that
+lim
+β→0+
+α(β)
+√β = 0.
+(19)
+While (19) gives us a rather clear condition for determining α given β, we still do not have a
+principled setting for β. This point will be treated in the following sub-section.
+3.3
+Deriving an algorithm using finite-sample theory
+To complement the preceding analysis and discussion centered around the population objective
+(12), we now return to the empirical objective function �Cn(h; a, b) introduced in (10). We
+maintain the running assumption that the training data Z1, . . . , Zn are an IID sample from µ,
+8
+
+and thus the losses Li(h), i = 1, . . . , n are independent given any fixed h. With h and b > 0
+fixed for the moment, we will now take a closer look at the optimal (empirical) threshold that
+arises from this objective function, namely any random variable An(h, b) satisfying
+An(h, b) ∈ arg min
+a∈R
+�Cn(h; a, b).
+(20)
+Using the property (4) of the smooth Huber-like function ρ, we can demonstrate how data-
+driven thresholds satisfying (20) are concentrated at a point near the expected loss, where α
+and b play a key role in how close this point is to the mean.
+Proposition 3 (Concentration at a shifted location). Taking 0 ≤ α < 1, b > 0, and 0 < δ < 1,
+with large enough n it is always possible to satisfy the condition
+4α
+λ ≤ 4
+�Vµ L(h)
+b2
++ log(2/δ)
+n
+�
+≤ 1 − 4α
+λ ,
+and when this condition is satisfied, the data-driven threshold An(h, b) in (20) satisfies
+����An(h, b) −
+�
+Eµ L(h) − 2α
+λ b
+����� ≤ 2
+�Vµ L(h)
+b
++ b log(2/δ)
+n
+�
+with probability no less than 1 − δ.
+This result can be seen as an extension of [34, Prop. 3.1] for the function (5) used in mean
+estimation to our generalized learning problem, although we use a different proof strategy
+which does not require strong convexity of �Cn (with respect to a).
+With Proposition 3 established, conventional wisdom might incline one to pursue a O(1/√n)
+rate in the upper bound; in this case, setting β ∝ 1/n is a natural strategy since Proposition
+2 tells us that for the population objective, the optimal setting of b scales with
+�
+λ/β. While
+this is natural from the perspective of tight concentration bounds for An(h, b), we argue that a
+different strategy is more appropriate when we actually consider how (Hn, An, Bn) will behave
+in the full joint optimization (11). The most obvious reason for this is that the joint objective
+lacks convexity and smoothness, as the following result summarizes.
+Proposition 4 (Joint objective is non-convex and non-smooth). Even when H is a compact
+convex set and the base loss function ℓ(·; Z) is convex, the mapping (h, a, b) �→ �Cn is not convex
+in general, and is non-smooth in the sense that its gradient is not Lipschitz continuous on
+H × R × (0, ∞).
+In consideration of Proposition 4, standard complexity results for typical optimizers such as
+stochastic gradient descent to achieve a ε-stationary point are on the order of O(ε−4); see
+Davis and Drusvyatskiy [8] for example.4
+With this in mind, setting β ∝ 1/n to achieve
+O(ε−2) sample complexity for error bounds of An(h, b) seems superfluous if in the end the
+dominant complexity for solving the ultimate problem (11) will be of the order O(ε−4). As
+such, in order to match this rate, the more natural strategy is to set β ∝ 1/√n, or more
+precisely to set
+β = β0
+√n
+(21)
+where β0 > 0 is a constant used to ensure 0 < β < λ. This, coupled with α(β) = β to satisfy
+(19) from the previous sub-section, is our proposed setting to determine (α, β) (and thus �Cn)
+using just knowledge of n, and without having observed any data points. This procedure is
+summarized in Algorithm 1, and will be the subject of empirical analysis later in §4.
+4Even if the objective were smooth, the same rates are typical; see for example Ghadimi and Lan [12].
+9
+
+Algorithm 1 Modified Sun-Huber
+Inputs: data Z1, . . . , Zn and parameter λ > 0.
+Set: β = β0/√n, with β0 such that 0 < β < λ.
+{Based on (21).}
+Set: α = β.
+{Satisfies (19).}
+Minimize: �Cn(h; a, b) in (h, a, b) using α and β as above.
+3.4
+Stationary points of mean-variance
+Having established links between the proposed objective and the mean-SD objective, we next
+consider the mean-variance objective
+MVµ(h) ..= Eµ L(h) + Vµ L(h).
+(22)
+This quantity can be expressed as the minimum value of a convex function, namely we have
+MVµ(h) = min
+a∈R
+�
+a + Eµ(L(h) − a)2 + 1
+2
+�
+= aMV(h) + Eµ(L(h) − aMV(h))2 + 1
+2
+(23)
+where on the right-most side we have set aMV(h) ..= Eµ L(h)−1. Assuming the underlying loss
+is differentiable, the gradient with respect to h can be written as
+MV′
+µ(h) = Eµ L′(h) + Eµ L(h) L′(h) − Eµ L(h) Eµ L′(h)
+= Eµ L′(h) + Eµ (L(h) − Eµ L(h)) L′(h)
+= Eµ (L(h) − (Eµ L(h) − 1)) L′(h)
+which implies a stationarity condition of
+MV′
+µ(h) = 0 ⇐⇒ Eµ (L(h) − (Eµ L(h) − 1)) L′(h) = 0.
+(24)
+Similarly, the partial derivative of the learning criterion (12) taken with respect to h is
+∂
+∂h Cµ(h; a, b) = Eµ
+�
+L(h) − a
+�
+(L(h) − a)2 + b2
+�
+L′(h)
+and thus multiplying both sides by b > 0, we obtain a simple stationarity condition of
+∂
+∂h Cµ(h; a, b) = 0 ⇐⇒ Eµ
+�
+�
+L(h) − a
+�
+(L(h)−a
+b
+)2 + 1
+�
+� L′(h) = 0.
+(25)
+With the right threshold setting, obviously the two conditions become very similar as b grows
+large. The following result makes this precise.
+Proposition 5. Let loss function ℓ and data distribution µ be such that the random vector
+L(h) L′(h) is integrable and has a norm with finite mean, i.e., Eµ∥L(h) L′(h)∥ < ∞ for some
+choice of h ∈ H. Then, for any a ∈ R, defining
+f(h; a) ..= lim
+b→∞ b ∂
+∂h Cµ(h; a, b)
+(26)
+the stationary points of the mean-variance objective are related to those of the proposed objective
+(12) through the following equivalence:
+f(h; aMV(h)) = 0 ⇐⇒
+∂
+∂h MVµ(h) = 0
+where MVµ(h) is as defined in (22).
+10
+
+1.0
+0.5
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+Figure 3: Graph of the Legendre transform ρ∗ as given in (28) over (−1, 1).
+3.5
+Comparison with dual form of DRO risk
+Some readers may notice that the proposed (population) objective (12) looks quite similar to
+the dual form of DRO risks:
+DROµ(h; β) ..=
+inf
+a∈R,b>0
+�
+a + βb + b Eµ φ∗
+�L(h) − a
+b
+��
+(27)
+where φ∗ is the Legendre-Fenchel convex conjugate φ∗(x) ..= supu∈R[xu − φ(u)] induced by a
+function φ : R → R, assumed to be convex and lower semi-continuous, with φ(1) = 0 and
+φ(x) = ∞ whenever x < 0 (cf. [33, §3.2]). Given this similarity, one might ask whether or not
+some form of DRO risk can be reverse engineered from our proposed objective. Taking up this
+point briefly, we first note that the conjugate of ρ given by (6) is
+ρ∗(x) ..= sup
+u∈R
+[xu − ρ(u)] = sup
+u∈R
+�
+xu −
+�
+u2 + 1 + 1
+�
+.
+From the non-negative nature of ρ, clearly ρ∗(0) = −ρ(0) = 0. For x ̸= 0, note that taking the
+derivative of concave function u �→ xu − ρ(u) and setting it to zero, we obtain the first-order
+optimality conditions
+u
+√
+u2 + 1 = x ⇐⇒
+sign(x)
+�
+1 + 1/u2 = x ⇐⇒
+1
+x2 = 1 + 1/u2 ⇐⇒ u =
+sign(x)
+�
+1/x2 − 1.
+Plugging this solution in whenever |x| < 1 and doing a bit of algebra readily yields the simple
+closed-form expression
+ρ∗(x) =
+�
+�
+�
+x2
+√
+1−x2 + 1 −
+1
+√
+1−x2 ,
+if 0 ≤ |x| < 1
+∞,
+else.
+(28)
+As can be readily observed from both (28) and Figure 3, this function does not satisfy any
+of the requirements placed on φ except convexity, and thus despite the similar form, the non-
+monotonic nature of ρ is in sharp contrast with monotonicity of typical cases of φ∗ that arise
+in the DRO literature (e.g. [3, §3]), and does not readily imply a “primal” DRO objective that
+can be recovered using ρ∗.
+4
+Empirical analysis
+Our investigation in the previous section led us to Algorithm 1, giving us a principled and
+precise strategy to construct the objective function �Cn, but leaving the actual minimization
+procedure abstract.
+Here we make this concrete by implementing a simple gradient-based
+minimizer of this objective, and comparing this procedure with natural benchmarks from the
+literature.
+11
+
+4.1
+Simulated noisy classification on the plane
+As a simplified and controlled setting to start with, we generate random data points on the
+plane which are mostly linearly separable, save for a single distant outlier (Figure 4). Before
+we consider off-sample generalization, here we focus simply on the training loss distribution
+properties as a function of algorithm iterations.
+Experiment setup
+We generate n = 100 training data points using two Gaussian distri-
+butions on the plane to represent two classes, with each class having the same number of
+points.
+We choose a single point uniformly at random, and perturb it by multiplying the
+scalar -10. We compare our proposed procedure (denoted “Modified Sun-Huber”) with three
+alternatives: traditional mean-based empirical risk minimization (denoted “Vanilla ERM”),
+conditional value-at-risk (CVaR) [7], and the well-studied χ2-DRO risk [11, 13]. In light of
+Algorithm 1, we set λ = log(n)/√n > β = β0/√n, and try a variety of β0 values just for
+reference. For all the aforementioned methods, we set the base loss ℓ(·) to the usual binary
+logistic loss (linear model), and run (batch) gradient descent on the empirical risk objectives
+implied by each of these methods (see §C for details), with a fixed step size of 0.01 over 15,000
+iterations. Alternative settings of step size and iteration number were not tested. All methods
+are initialized at the same point, shown in Figure 4.
+Results and discussion
+In Figure 5, we show the empirical mean-SD trajectories for the
+base loss, over algorithm iterations (log10 scale), for each method of interest. Using our no-
+tation, this is the sample version of MSµ(h; λ) in (1), with λ = 1 fixed. All methods besides
+vanilla ERM have multiple settings that were tested, and the results for each are distinguished
+using curves of different color. Our method tests different values of β0, CVaR tests different
+quantile levels, and DRO tests different robustness levels (details in §C). Since Vanilla ERM
+is designed to optimize the average loss, it is perhaps not surprising that it fails in terms of
+the mean-SD objective. On the other hand, the proposed method (for any choice of β0) is as
+good or better than all the competing methods. As a basic sanity check, in Figure 6 we also
+consider the error rate (average zero-one loss) and model norm trajectories over iterations for
+each method. For each method, we plot just one trajectory, namely the one achieving the best
+final error rate. While our method is not designed to minimize the average loss and typical
+surrogate theory does not apply, we find that the error rate is surprisingly good, albeit with
+slower convergence than the other methods. Note also how the error rate for CVaR matches
+that of Vanilla ERM; this is in fact the CVaR setting with the worst final mean-SD value. On
+the other hand, the proposed method performs well from both perspectives at once.
+4.2
+Classification on real datasets
+We proceed to experiments using real-world datasets, some of which are orders of magnitude
+larger than the simple setup given in §4.1, and which include multi-class classification tasks.
+Experiment setup
+We make use of four well-known datasets, all available from online
+repositories: adult,5 australian,6 cifar10,7 and fashion_mnist.8 For multi-class datasets,
+we extend the binary logistic loss to the usual multi-class logistic regression loss under a linear
+5https://archive.ics.uci.edu/ml/datasets/Adult
+6https://archive.ics.uci.edu/ml/datasets/statlog+(australian+credit+approval)
+7https://www.cs.toronto.edu/~kriz/cifar.html
+8https://github.com/zalandoresearch/fashion-mnist
+12
+
+model, with one linear model for each class. Features for all datasets are normalized to [0, 1],
+with one-hot representations of categorical features. The learning algorithms being compared
+here are the same as described in §4.1, except that now we implement each method using
+mini-batch stochastic gradient descent (batch size 32), and do 30 epochs (i.e., 30 passes over
+the training data). In addition, our proposed “Modified Sun-Huber” method performs almost
+identically for the range of β0 values tested in §4.1, and thus we have simply fixed β0 = 0.9, so
+there is only one trajectory curve this time. On the other hand, we now try a range of step sizes
+for each method, choosing the best step size in terms of average (base) loss value on validation
+data for each method. We run five independent trials, and for each trial we randomly re-shuffle
+the dataset, taking 80% for training, 10% for validation (used to select step sizes), and 10%
+for final testing.
+Results and discussion
+Our main results are shown in Figure 7, where once again we
+plot the trajectory of the mean-SD objective, but this time computed on test data, and given
+as a function of epoch number, rather than individual iterations.
+Since there are multiple
+trials, the curves drawn represent averages taken over trials, and the lightly shaded region
+above/below each curve shows standard deviation over trials. Perhaps surprisingly, the very
+simple implementation of our proposed Algorithm 1 (fixed step size, no regularization) works
+remarkably well on a number of datasets. From the perspective of mean-SD minimization,
+for three our of four datasets, the proposed method is far better than Vanilla ERM, and as
+good or better than even the best settings of CVaR and DRO viewed after the fact. Regarding
+the sub-standard performance observed on fashion_mnist, detailed analysis shows that more
+fine-tuned settings of α and β can readily bring the method up to par; the non-convex and non-
+smooth nature of �Cn naturally means that some tasks will require more careful settings than
+are captured by our Algorithm 1, and indeed will take explicit account of the optimizer to be
+used. We leave both the theoretical grounding and empirical testing of such optimizer-aligned
+mean-SD minimizers for future work.
+13
+
+Figure 4: 2D classification example from §4.1. The red line represents the initial value used by each method.
+102
+1
+2
+Vanilla ERM
+102
+Modified Sun-Huber
+102
+CVaR
+102
+2-DRO risk
+0.1
+0.9
+0.0
+0.8
+0.0
+0.3
+Mean + SD (2D classification with outliers)
+Figure 5: Trajectory of the (empirical) mean-SD objective (1) over iterations. Colors correspond to different
+choices from each class: β0 for Modified Sun-Huber, quantile level for CVaR, and constraint level for DRO.
+101
+103
+0.0
+0.2
+0.4
+Error rate
+101
+103
+0
+2
+4
+Norm
+Trajectory with best final error rate
+Vanilla ERM
+Modified Sun-Huber
+CVaR
+2-DRO risk
+Figure 6:
+From each method class, we show the classification error rate and Euclidean norm trajectories
+corresponding to the setting that achieved the best error rate after the final iteration.
+14
+
+0
+20
+0.7
+0.8
+Vanilla ERM
+0
+20
+Modified Sun-Huber
+0
+20
+CVaR risk
+0
+20
+2-DRO risk
+Mean + SD (dataset: adult)
+0
+20
+0.8
+1.0
+Vanilla ERM
+0
+20
+Modified Sun-Huber
+0
+20
+CVaR risk
+0
+20
+2-DRO risk
+Mean + SD (dataset: australian)
+0
+20
+2.5
+3.0
+3.5
+Vanilla ERM
+0
+20
+Modified Sun-Huber
+0
+20
+CVaR risk
+0
+20
+2-DRO risk
+Mean + SD (dataset: cifar10)
+0
+20
+1.5
+2.0
+2.5
+Vanilla ERM
+0
+20
+Modified Sun-Huber
+0
+20
+CVaR risk
+0
+20
+2-DRO risk
+Mean + SD (dataset: fashion_mnist)
+Figure 7: Mean-SD trajectories on real-world datasets as described in §4.2, given as a function of epochs and
+averaged over multiple independent trials. Coloring for CVaR and DRO is analogous to that of Figure 5.
+15
+
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+volume 139 of Proceedings of Machine Learning Research, pages 12345–12355.
+A
+Technical appendix
+A.1
+Basic facts
+Assuming ρ is defined as in (6), let us consider the function
+f(x, a, b) ..= αa + βb + bρ
+�x − a
+b
+�
+(29)
+= αa + βb +
+�
+(x − a)2 + b2 − b
+(30)
+= αa +
+�
+(x − a)2 + b2 − (1 − β)b.
+(31)
+The partial derivatives are as follows.
+∂xf(x, a, b) =
+x − a
+�
+(x − a)2 + b2
+(32)
+∂af(x, a, b) = α −
+x − a
+�
+(x − a)2 + b2
+(33)
+∂bf(x, a, b) =
+b
+�
+(x − a)2 + b2 − (1 − β)
+(34)
+The corresponding second derivatives are as follows.
+∂2
+xf(x, a, b) =
+1
+�
+(x − a)2 + b2 −
+(x − a)2
+((x − a)2 + b2)3/2 =
+b2
+((x − a)2 + b2)3/2
+(35)
+∂2
+af(x, a, b) =
+1
+�
+(x − a)2 + b2 −
+(x − a)2
+((x − a)2 + b2)3/2 =
+b2
+((x − a)2 + b2)3/2
+(36)
+∂2
+b f(x, a, b) =
+1
+�
+(x − a)2 + b2 −
+b2
+((x − a)2 + b2)3/2 =
+(x − a)2
+((x − a)2 + b2)3/2
+(37)
+The remaining elements of the Hessian of f(x, a, b) follow easily, given as follows.
+∂a∂xf(x, a, b) =
+−1
+�
+(x − a)2 + b2 +
+(x − a)2
+((x − a)2 + b2)3/2 =
+−b2
+((x − a)2 + b2)3/2
+(38)
+∂b∂xf(x, a, b) =
+−b(x − a)
+((x − a)2 + b2)3/2
+(39)
+∂b∂af(x, a, b) =
+b(x − a)
+((x − a)2 + b2)3/2
+(40)
+Lemma 6 (Useful inequalities).
+1
+1 + x ≤ 1 − x
+2,
+0 ≤ x ≤ 1.
+(41)
+(1 + x)c ≥ 1 + cx,
+x ≥ −1, c ∈ R \ (0, 1).
+(42)
+18
+
+A.2
+Convexity and smoothness
+Lemma 7. The map x �→ 1/√1 + x is convex on [0, ∞).
+Lemma 8 (Properties of partial objective). With ρ as in (6) and β ≥ 0, the function
+(x, b) �→ βb + bρ
+�x
+b
+�
+is convex and (1 + max{1 − β, β})-Lipschitz (in ∥·∥1) on R × (0, ∞), but its gradient is not
+(globally) Lipschitz, and thus the function is not smooth.9
+Proof of Lemma 8. For notational convenience, setting 0 < β < 1, let us denote
+g(x, b) ..= βb + bρ(x/b),
+x ∈ R, b > 0
+with ρ as in (6). From the partial derivatives (32) and (34), it is clear that we have
+−1 ≤ ∂xg(x, b) ≤ 1,
+−(1 − β) ≤ ∂bg(x, b) ≤ β
+when evaluated at any choice of x ∈ R and b > 0. It follows that the gradient norm can be
+bounded as
+∥∇g(x, b)∥1 ≤ 1 + max{(1 − β), β}
+and thus g(·) is Lipschitz continuous in ∥·∥1 (and also ∥·∥2).10
+Next, let us denote the Hessian of g(·) evaluated at (x, b) by H. Basic calculus gives us the
+simple form
+H ..=
+1
+(x2 + b2)3/2
+�
+b2
+−xb
+−xb
+x2
+�
+and for any pair of real values u = (u1, u2), we have
+⟨Hu, u⟩ =
+1
+(x2 + b2)3/2 (u1b − u2x)2 ≥ 0.
+(43)
+Since this holds for any choice of x ∈ R and b > 0, the Hessian is thus positive semi-definite,
+implying that g(·) is (jointly) convex [28, Thm. 2.1.4].
+On the other hand, the function g(·) is not smooth. To see this, first note that having
+chosen any u such that ∥u∥ ≤ 1, we have that the (operator) norm is bounded below as
+∥H∥ = sup
+∥u′∥≤1
+�
+sup
+∥u′′∥≤1
+⟨Hu′, u′′⟩
+�
+≥ ⟨Hu, u⟩.
+Then, as a concrete example, consider setting x = b, with u = (u1, u2) such that u1 ̸= u2.
+Recalling the lower bound (43), we have
+∥H∥ ≥
+b2
+(2b2)3/2 (u1 − u2)2 = (u1 − u2)2
+(
+√
+2)3b
+→ ∞
+in the limit as b → 0+.
+As such, the gradient of g(·) cannot be Lipschitz continuous on
+R × (0, ∞), and thus g(·) is not smooth [28, Thm. 2.1.6].
+9We prove that the Hessian’s norm is unbounded, which implies (via Nesterov [28, Thm. 2.1.6]) that the
+convex function of interest cannot be smooth.
+10That bounded gradients imply Lipschitz continuity is a general fact on linear spaces [21, §7.3, Prop. 2].
+19
+
+B
+Additional proofs
+Proof of Proposition 1. To begin, note that the function
+b �→ b Eµ ρ
+�L(h) − a
+b
+�
+= Eµ
+��
+(L(h) − a)2 + b2 − b
+�
+(44)
+is monotonic (non-increasing) on (0, ∞) (follows clearly from (34)). We will use this property
+moving forward. Recalling the upper and lower bounds of Proposition 2, we re-write them as
+clo(β)
+β
+≤ b2
+µ(h, a) ≤ chi
+β
+(45)
+using the shorthand notation
+clo(β) ..= λ
+4 Eµ I(h; a)(L(h) − a)2
+chi ..= λ
+2 Eµ(L(h) − a)2
+and noting that while chi is free of β, clo(β) depends on β through the definition of I(h; a).
+Fixing 0 < β < λ for now and recalling the form of Cµ in (12), the preceding bounds (45) and
+monotonicity of (44) can be used to obtain a lower bound of the form
+min
+b>0 Cµ(h; a, b) ≥ αa +
+�
+βclo(β) + λ√chi Eµ
+�
+�
+�
+(L(h) − a)2
+chi
++ 1
+β −
+�
+1
+β
+�
+� .
+(46)
+Using the fact (14) and applying dominated convergence [1, Thm. 1.6.9], in the limit we have
+lim
+β→0+ clo(β) = λ
+4 Eµ(L(h) − a)2.
+Dividing both sides of (46) by √β, setting α = α(β) as in the proposition statement, and
+taking the limit as β → 0+, we obtain
+lim
+β→0+ min
+b>0
+Cµ(h; a, b)
+√β
+≥ �αa +
+�
+λ
+4 Eµ(L(h) − a)2 + λ Eµ(L(h) − a)2
+2√chi
+= �αa +
+�
+λ
+4 Eµ(L(h) − a)2 +
+�
+λ
+2 Eµ(L(h) − a)2
+= �αa +
+�1
+2 + 1
+√
+2
+� �
+λ Eµ(L(h) − a)2.
+The first inequality uses the fact that for any c > 0, we have
+√
+cx + x2 − x → c/2 as x → ∞,
+and also uses dominated convergence. The remaining equalities just follow from plugging in
+the definition of chi and cleaning up terms. This proves the desired lower bound.
+As for the upper bound of interest, a perfectly analogous argument can be applied. Using
+Proposition 2 again and taking β small enough that
+clo(β) ≥ chi/4
+(47)
+20
+
+holds (always possible), we can obtain upper bounds of the form
+min
+b>0 Cµ(h; a, b) ≤ αa +
+�
+βchi + λ
+�
+clo(β) Eµ
+��
+(L(h) − a)2
+clo(β)
++ 1
+β −
+�
+1
+β
+�
+≤ αa +
+�
+βchi + λ√chi Eµ
+�
+�
+�
+4(L(h) − a)2
+chi
++ 1
+β −
+�
+1
+β
+�
+�
+(48)
+noting that the latter inequality (48) follows from using (47) as well as clo(β) ≤ chi. As with
+the lower bound argument in the preceding paragraph, we set α = α(β), divide both sides by
+√β, and take the limit as β → 0+. This results in
+lim
+β→0+ min
+b>0
+Cµ(h; a, b)
+√β
+≤ �αa +
+�
+λ
+2 Eµ(L(h) − a)2 + 2λ Eµ(L(h) − a)2
+√chi
+= �αa +
+�
+λ
+2 Eµ(L(h) − a)2 + 2
+�
+2λ Eµ(L(h) − a)2
+= �αa +
+�
+2
+√
+2 + 1
+√
+2
+� �
+λ Eµ(L(h) − a)2
+which gives us the desired upper bound. The bounds given in the proposition statement are
+slightly looser, but more readable.
+Proof of Proposition 2. We adapt key elements of the scale control used by Sun [34, §2] to our
+setting. We start by looking at first-order conditions for optimality of b > 0. First, note that
+∂
+∂b Cµ(h; a, b) = β + λ ∂
+∂b
+�
+Eµ
+�
+(L(h) − a)2 + b2 − b
+�
+= β + λ Eµ
+�
+b
+�
+(L(h) − a)2 + b2
+�
+− λ.
+As such, it follows that
+Eµ
+�
+b
+�
+(L(h) − a)2 + b2
+�
+= 1 − β/λ
+(49)
+is equivalent to ∂b Cµ(h; a, b) = 0. Obviously, the left-hand side of (49) is non-negative for all
+b ≥ 0 and bounded above by 1 for all b ≥ 0, a ∈ R, and h ∈ H. Thus (49) can only hold for
+0 ≤ β ≤ λ. Using convexity (Lemma 7) and Jensen’s inequality [1, Thm. 6.3.5], we have
+Eµ
+�
+b
+�
+(L(h) − a)2 + b2
+�
+= Eµ
+�
+�
+1
+�
+(L(h)−a
+b
+)2 + 1
+�
+� ≥
+�
+�
+1
+�
+Eµ(L(h)−a
+b
+)2 + 1
+�
+�
+and thus whenever (49) holds, we know that
+(1 − β/λ)2 ≥
+1
+Eµ(L(h)−a
+b
+)2 + 1
+must also hold. Re-arranging terms, we see that this implies
+b2 ≤ (1 − β/λ)2 Eµ(L(h) − a)2
+1 − (1 − β/λ)2
+.
+21
+
+For readability, set η ..= β/λ, and note that since
+(1 − η)2
+1 − (1 − η)2 = (1 − η)2
+2η − η2 =
+(1 − η)2
+2η(1 − η/2) ≤ (1 − η)2
+2η(1 − η) ≤ 1
+2η
+we can obtain the cleaner (but looser) upper bound
+b2 ≤ Eµ(L(h) − a)2
+2η
+= λ
+2β Eµ(L(h) − a)2
+for any choice of 0 < β ≤ λ and a ∈ R. Since the first-order condition (49) is necessary for
+optimality [28, Thm. 1.2.1], it follows that
+b2
+µ(h, a) ≤ Eµ(L(h) − a)2
+2β
+(50)
+which is the desired upper bound.
+Considering a lower bound next, note first that using the concavity of x �→ √x on R+,
+another application of Jensen’s inequality gives us
+Eµ
+�
+b
+�
+(L(h) − a)2 + b2
+�
+= Eµ
+�
+b2
+(L(h) − a)2 + b2 ≤
+�
+�
+�
+�Eµ
+�
+1
+(L(h)−a
+b
+)2 + 1
+�
+.
+(51)
+Using the inequality 1/(x + 1) ≤ 1 − x/2 for all 0 ≤ x ≤ 1 ((41) in Lemma 6), this suggests
+a natural event to use as a condition.
+More precisely, writing E ..= I {|L(h) − a| ≤ b} for
+readability, note that we have
+1
+(L(h)−a
+b
+)2 + 1
+=
+1 − E
+(L(h)−a
+b
+)2 + 1
++
+E
+(L(h)−a
+b
+)2 + 1
+≤
+1 − E
+(L(h)−a
+b
+)2 + 1
++ E
+�
+1 − 1
+2
+�L(h) − a
+b
+�2�
+=
+�
+1 − E
+(L(h)−a
+b
+)2 + 1
++ E
+�
+�
+��
+�
+≤1
+−E
+2
+�L(h) − a
+b
+�2
+.
+Taking expectation and utilizing (51), whenever (49) holds, we have
+1 − β/λ = Eµ
+�
+b
+�
+(L(h) − a)2 + b2
+�
+≤
+�
+1 − Eµ
+E
+2
+�L(h) − a
+b
+�2
+.
+(52)
+With this established, note that via helper inequality (42), for any β ≤ λ we have
+(1 − β/λ)2 ≥ 1 − 2β/λ
+and thus in light of (52), we may conclude that
+1 − 2β/λ ≤ 1 − Eµ
+E
+2
+�L(h) − a
+b
+�2
+which implies
+λ
+4β Eµ E(h; a, b)(L(h) − a)2 ≤ b2
+22
+
+noting that we have written E(h; a, b) to emphasize the dependence on h, a, and b. Once again
+since the first-order condition (49) is necessary for optimality, we may conclude that
+λ
+4β Eµ E(h; a, bµ(h, a))(L(h) − a)2 ≤ b2
+µ(h, a)
+(53)
+which is the remaining desired inequality.
+Proof of the limit (14). Recall from the proof of Proposition 2 the first-order optimality con-
+dition (49), which is satisfied by any solution bµ(h, a) given by (13), i.e., we have
+Eµ
+�
+�
+bµ(h, a)
+�
+(L(h) − a)2 + (bµ(h, a))2
+�
+� = 1 − β/λ
+(54)
+for any 0 < β ≤ λ. Defining g(β) ..= 1 − β/λ and taking any 0 < β2 < β1 ≤ λ, clearly we have
+g(β1) < g(β2) and thus using the equality (54), we must have that bµ(h, a; β2) ≥ bµ(h, a; β1),
+otherwise it would result in a contradiction of (54). Using this monotonicity, clearly
+E(h; a, bµ(h, a; β1)) ≤ E(h; a, bµ(h, a; β2))
+and thus
+Eµ E(h; a, bµ(h, a; β1))(L(h) − a)2 ≤ E(h; a, bµ(h, a; β2))(L(h) − a)2.
+Applying this to the lower bound in Proposition 2, we have
+lim inf
+β→0+ b2
+µ(h, a) ≥ lim
+β→0+
+λ
+4β Eµ I(h; a)(L(h) − a)2 = ∞
+as desired.
+Proof of the limit (15). Note that we can easily bound the random variable of interest as
+0 ≤ bρ
+�L(h) − a
+b
+�
+=
+�
+(L(h) − a)2 + b2 − b ≤ |L(h) − a|
+(55)
+for any choice of 0 < b < ∞. Some straightforward calculus shows that
+lim
+b→∞ bρ
+�L(h) − a
+b
+�
+= 0
+in a pointwise sense. Since the upper bound in (55) is µ-integrable by assumption, a simple
+application of dominated convergence [1, Thm. 1.6.9] yields
+lim
+b→∞ b Eµ ρ
+�L(h) − a
+b
+�
+= Eµ
+�
+lim
+b→∞ bρ
+�L(h) − a
+b
+��
+= 0
+as desired.
+23
+
+Proof of Proposition 3. From condition (20), since any solution must also be a stationary point
+[28, Thm. 1.2.1], we know that An ..= An(h, b) must satisfy the first-order condition
+λ
+n
+n
+�
+i=1
+ρ′
+�Li(h) − An
+b
+�
+= α
+which is equivalent to
+b
+n
+n
+�
+i=1
+ρ′
+�Li(h) − An
+b
+�
+= α
+λb.
+(56)
+Next we make use of the argument developed by Catoni [6, §2]. First note that fixing any
+a ∈ R and b > 0, we have
+E exp
+� n
+�
+i=1
+ρ′
+�Li(h) − a
+b
+��
+= E
+� n
+�
+i=1
+exp
+�
+ρ′
+�Li(h) − a
+b
+���
+=
+n
+�
+i=1
+Ei exp
+�
+ρ′
+�Li(h) − a
+b
+��
+≤
+n
+�
+i=1
+Ei
+�
+1 + Li(h) − a
+b
++ γ
+b2 (Li(h) − a)2
+�
+=
+�Eµ L(h) − a
+b
++ γ
+b2 Eµ(L(h) − a)2
+�n
+≤ exp
+�n
+b (Eµ L(h) − a) + nγ
+b2 Eµ(L(h) − a)2
+�
+.
+(57)
+The second equality above follows from the independence of the training data, and the first
+inequality uses the upper bound in (4), which is satisfied by ρ given in (6) with γ = 1, though
+we leave γ as is to illustrate how more general results are obtained. The third equality just uses
+the fact that the training data is an IID sample from µ, and the final inequality culminating
+in (57) just uses the bound 1 + x ≤ exp(x). Using Markov’s inequality and taking 0 < δ < 1,
+it is straightforward to show that (57) implies a 1 − δ event (over the draw of Z1, . . . , Zn) in
+which we have
+n
+�
+i=1
+ρ′
+�Li(h) − a
+b
+�
+≤ n
+b (Eµ L(h) − a) + nγ
+b2 Eµ(L(h) − a)2 + log(1/δ).
+Multiplying both sides by b/n, on the same “good” event, we have
+b
+n
+n
+�
+i=1
+ρ′
+�Li(h) − a
+b
+�
+≤ Eµ L(h) − a + γ
+b Eµ(L(h) − a)2 + b log(1/δ)
+n
+= Eµ L(h) − a + γ
+b
+�
+Vµ L(h) + (Eµ L(h) − a)2�
++ b log(1/δ)
+n
+(58)
+where (58) follows from expanding the quadratic term and doing some algebra.
+With the
+equality (56) in mind, subtracting a constant from both sides of (58), note that we equivalently
+have
+b
+n
+n
+�
+i=1
+ρ′
+�Li(h) − a
+b
+�
+− α
+λb ≤ p(a)
+(59)
+24
+
+where we have defined
+p(a) ..= Eµ L(h) − a + γ
+b
+�
+Vµ L(h) + (Eµ L(h) − a)2�
++ b log(1/δ)
+n
+− α
+λb
+(60)
+for readability. Note that p(·) in (60) is a polynomial of degree 2, and can be written as
+p(a) = ua2 + va + w
+(61)
+with coefficients defined as
+u ..= γ
+b
+v ..= (−1)
+�
+1 + 2γ Eµ L(h)
+b
+�
+w ..= Eµ L(h) + γ
+b Eµ|L(h)|2 + b log(1/δ)
+n
+− α
+λb.
+This polynomial has real roots whenever v2 − 4uw ≥ 0, and some algebra shows that this is
+equivalent to
+0 ≤ D ≤ 1, where D ..= 4
+��γ
+b
+�2
+Vµ L(h) + γ log(1/δ)
+n
+− γα
+λ
+�
+.
+(62)
+Assuming this holds, denoting by a+ the smallest root of p(·), i.e., the smallest of satisfying
+p(a+) = 0, the critical fact of interest to us is that An ≤ a+ on the good event of (59). This is
+valid due to two facts: first, the left-hand side of (59) is a decreasing function of a; second, due
+to (56), we know that An is a root of the left-hand side of (59). With this key fact in hand,
+using the quadratic formula we have
+An ≤ a+
+= Eµ L(h) + b
+2γ
+�
+1 −
+√
+1 − D
+�
+= Eµ L(h) + b
+2γ
+�
+1 −
+√
+1 − D
+� �
+1 +
+√
+1 − D
+�
+�
+1 +
+√
+1 − D
+�
+= Eµ L(h) + b
+2γ
+D
+�
+1 +
+√
+1 − D
+�
+≤ Eµ L(h) + b
+2γ D.
+Taking the two ends of this inequality chain together and expanding D, we have
+An ≤ Eµ L(h) − 2(α/λ)b + 2
+�γ
+b Vµ L(h) + b log(1/δ)
+n
+�
+(63)
+with probability no less than 1 − δ, assuming that n, b, and α are such that 0 ≤ D ≤ 1 holds.
+This gives us the desired upper bound.
+To obtain a lower bound, a perfectly analogous argument can be applied. First, using the
+lower bound in (4) and the fact that ρ′(−x) = −ρ′(x), we know that
+ρ′
+�a − Li(h)
+b
+�
+≤ log
+�
+1 + a − Li(h)
+b
++ γ
+b2 (a − Li(h))2
+�
+(64)
+25
+
+for any a ∈ R, b > 0, and i ∈ [n]. Plugging this inequality (64) into an argument analogous to
+the chain of inequalities that led to (58) earlier, it is clear that again on an event of probability
+no less than 1 − δ, we have
+b
+n
+n
+�
+i=1
+ρ′
+�a − Li(h)
+b
+�
+≤ a − Eµ L(h) + γ
+b
+�
+Vµ L(h) + (Eµ L(h) − a)2�
++ b log(1/δ)
+n
+.
+(65)
+Once again the upper bound we can bound this using a polynomial of degree 2, namely
+b
+n
+n
+�
+i=1
+ρ′
+�a − Li(h)
+b
+�
++ α
+λb ≤ q(a)
+(66)
+where we have defined
+q(a) ..= a − Eµ L(h) + γ
+b
+�
+Vµ L(h) + (Eµ L(h) − a)2�
++ b log(1/δ)
+n
++ α
+λb.
+(67)
+Now, since An is a root of the left-hand side of (66) viewed as a function of a, and this function
+is monotonically increasing, it is evident that denoting the largest root of q(·) (when it exists)
+by a−, we have An ≥ a−, a lower bound in contrast to the An ≤ a+ upper bound used earlier.
+For completeness, we write this polynomial as
+q(a) = u′a2 + v′a + w′
+(68)
+with coefficients
+u′ ..= γ
+b
+v′ ..=
+�
+1 − 2γ Eµ L(h)
+b
+�
+w′ ..= (−1) Eµ L(h) + γ
+b Eµ|L(h)|2 + b log(1/δ)
+n
++ α
+λb.
+We have two real roots whenever
+1 ≥ D′ ..= 4
+��γ
+b
+�2
+Vµ L(h) + γ log(1/δ)
+n
++ γα
+λ
+�
+(69)
+holds, and thus we obtain a high probability lower bound on An as follows:
+An ≥ a−
+= Eµ L(h) − b
+2γ
+�
+1 −
+√
+1 − D′
+�
+≥ Eµ L(h) − b
+2γ D′.
+Expanding D′ gives us the lower bound
+An ≥ Eµ L(h) − 2(α/λ)b − 2
+�γ
+b Vµ L(h) + b log(1/δ)
+n
+�
+(70)
+with probability no less than 1 − δ, as desired.
+26
+
+Let us conclude this proof by organizing the technical assumptions. First of all, for the
+two quadratics used in the preceding bounds, we require both (62) and (69) to hold. It is
+straightforward to verify that having these conditions both hold is equivalent to the following:
+4γα
+λ
+≤ 4
+��γ
+b
+�2
+Vµ L(h) + γ log(1/δ)
+n
+�
+≤ 1 − 4γα
+λ .
+(71)
+As such, whenever α, δ, and b are such that (71) holds, using a union bound, it follows that
+with probability no less than 1 − 2δ, we have a bound on
+|An − (Eµ L(h) − 2(α/λ)b)| ≤ 2
+�γ
+b Vµ L(h) + b log(1/δ)
+n
+�
+as desired. The proposition statement takes a cleaner form since we have γ = 1.
+Proof of Proposition 4. The lack of convexity follows from the fact that the composition of
+two convex functions need not be convex when the outermost function is non-monotonic (see
+for example Boyd and Vandenberghe [4, Ch. 3]), and the lack of smoothness follows a fortiori
+from Lemma 8.
+C
+Empirical test appendix
+For reference, we give the population versions of the CVaR and χ2-DRO criteria used in the
+empirical tests of §4. First, it is well known (see Rockafellar and Uryasev [30]) that CVaR at
+quantile level ξ can be represented as
+CVaRµ(h; ξ) = inf
+a∈R
+�
+a +
+1
+1 − ξ Eµ (L(h) − a)+
+�
+(72)
+where (x)+ ..= max{0, x}. Similarly, DRO risk based on the Cressie-Read family of divergence
+functions is formulated (for any c > 1) using
+DROµ(h; η) = inf
+a∈R
+�
+a + (1 + c(c − 1)η)1/c �
+Eµ (L(h) − a)c∗
++
+�1/c∗�
+(73)
+where c∗ ..= c/(c − 1), and χ2-DRO is the special case where c = 2 [11, 13, 36]. The different
+“robustness levels” mentioned in §4 correspond to a re-parameterized quantity �η ∈ (0, 1),
+related to η by the equality η = (1/(1 − �η) − 1)/2. Just as our �Cn(h; a, b) is solved jointly in
+(h, a, b), our empirical tests minimize the empirical versions of (72) and (73) jointly in (h, a).
+27
+
diff --git a/6dFJT4oBgHgl3EQflizf/content/tmp_files/load_file.txt b/6dFJT4oBgHgl3EQflizf/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..61c3dec3d72f86752b51156ee5334adff66eb924
--- /dev/null
+++ b/6dFJT4oBgHgl3EQflizf/content/tmp_files/load_file.txt
@@ -0,0 +1,1131 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf,len=1130
+page_content='Robust variance-regularized risk minimization  with concomitant scaling  Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Holland  Osaka University  Abstract  Under losses which are potentially heavy-tailed, we consider the task of minimizing sums  of the loss mean and standard deviation, without trying to accurately estimate the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  By modifying a technique for variance-free robust mean estimation to fit our problem  setting, we derive a simple learning procedure which can be easily combined with standard  gradient-based solvers to be used in traditional machine learning workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Empirically,  we verify that our proposed approach, despite its simplicity, performs as well or better  than even the best-performing candidates derived from alternative criteria such as CVaR  or DRO risks on a variety of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Contents  1  Introduction  2  2  Background  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1  Robust mean estimation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
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+page_content='  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2  Good-enough ancillary scaling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
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+page_content='1  Simulated noisy classification on the plane .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
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+page_content='  18  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2 Convexity and smoothness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
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+page_content='  19  B Additional proofs  20  C Empirical test appendix  27  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='11584v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='ML]  27 Jan 2023  1  Introduction  Traditionally, the “textbook definition” of a statistical machine learning problem is formulated  in terms of making decisions which minimize the expected value of a random loss [9, 27, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  More precisely, the traditional setup has us minimize Eµ L(h) with respect to a decision h,  where we denote random losses as L(h) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Z), with a random data point Z ∼ µ, and ℓ(·)  is a loss function assigning real values to (decision, data) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' This problem class is very  general in that it covers a wide range of learning problems both supervised and unsupervised,  but it is limited in the sense that it only aspired to be optimal on average, with no guarantees  for other aspects of performance such as loss deviations, resilience to worst-case examples and  distribution shift, sub-population disparity, and class-balanced error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' While it is sometimes  possible to account for these issues by modifying the base loss function ℓ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', logit-adjusted  softmax cross-entropy for balanced error [26]), there is a growing literature looking at prin-  cipled, systematic modifications to the “risk,” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', a non-random numerical property of the  distribution of L(h) to be optimized in h, leaving the base loss ℓ(·) fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Some prominent  examples are weighted sums of loss quantiles [25], distributionally robust optimization (DRO)  risk [13], conditional value-at-risk (CVaR) [7], tilted risk [20], and more general optimized  certainty equivalent (OCE) risks [19], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' It is well-known that many risks can be  expressed in terms of location-deviation sums, with the canonical example being a weighted  sum of the loss mean and standard deviation (or variance) [31, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We refer the reader to some  recent surveys [14, 16, 32] for more general background on developments in learning criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  In this work, the criterion of interest is the mean loss regularized by standard deviation  (SD), when losses are allowed to be heavy-tailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' More formally, we allow for heavy tails in  the sense that all we assume is that the second moment Eµ|L(h)|2 is finite, and the ultimate  objective of interest is the mean-SD criterion  MSµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= Eµ L(h) +  �  λ Vµ L(h)  (1)  with loss variance denoted by Vµ L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= Eµ(L − Eµ L)2, and weighting parameter λ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' This  mean-SD objective (1) and its mean-variance counterpart have a long history in the literature  on decision making under uncertainty, including the influential work of Markowitz [23] on  optimal portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' In the context of machine learning, it is well-known that one can  obtain “fast rate” bounds on the expected loss when variance is small (see [11, §1]), though the  problem of actually ensuring that loss deviations are sufficiently small is an entirely separate  matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' In this direction, Maurer and Pontil [24] bound the (population) expected loss using a  weighted sum of the sample mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Their “sample variance penalized”  objective is convenient to compute and can be used to guarantee fast rates in theory, but a  lack of convexity makes it hard to minimize in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' A convex approximation is developed  by Duchi and Namkoong [11], who show that a sub-class of (empirical) DRO risks can be used  to approximate the sample mean-SD objective, again yielding fast rates when the (population)  variance is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The critical limitation to this approach is poor guarantees under  heavy-tailed losses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' while we gain in terms of convexity, the empirical DRO risk of [11] is at least  as sensitive to outliers as the naive empirical objective (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', directly minimizing the sample  mean and SD), which is already known to result in highly sub-optimal performance guarantees  under heavy tails [5, 10, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Recent work by Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' [36] studies a natural strategy for  robustifying the DRO objective (called DORO), which discards a specified fraction of the  largest losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' While the impact of outliers can be reduced under the right setting of DORO,  their approach is limited to non-negative losses, and the impact that such one-sided trimming  has on the resulting mean-SD sum, our ultimate object of interest, is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  2  With this context in mind, in this paper we propose a new approach to robustly minimize  the objective (1) under heavy-tailed losses, without a priori knowledge of anything but the  fact that variance is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Our key technique is based on extending a convex program of Sun  [34] from one-dimensional mean estimation to our mean-variance objective MSµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' λ) under  general losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' After some motivating background points in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1–§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2, we describe our basic  approach and summarize our contributions in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='3–§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Theoretical analysis comes in §3, and  based upon formal properties of the proposed objective function, we derive a general-purpose  procedure summarized in Algorithm 1, and tested empirically in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Our main finding is  that the simple algorithm we derive works remarkably well on both simulated and real-world  datasets without any fine-tuning, despite sacrificing the convexity enjoyed by procedures based  on criteria such as CVaR and DRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Software and notebooks to reproduce all results in this  paper are provided in an online repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1  2  Background  Before we describe our proposed approach to the mean-SD task described in §1, we start with  a much simpler problem, namely the task of robust mean estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' This will allow us to  highlight key technical points from the literature which provide both conceptual and technical  context for our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Key points from the existing literature are introduced in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1–§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2,  and building upon this we introduce our method in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='3–§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1  Robust mean estimation  Let X be a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' For the moment, our goal will be to construct an accurate empirical  estimate of the mean Eµ X, assuming only that the variance Vµ X = Eµ X2 − (Eµ X)2 is both  defined and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We assume access to an independent and identically distributed (IID) sample  X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Since higher-order moments may be infinite, the tails of X may be “heavy” and  decidedly non-Gaussian, causing problems for the usual empirical mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' This problem setting  is now very well-understood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' see Lugosi and Mendelson [22] for an authoritative reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  One very well-known approach is to use M-estimators [18], namely to design an estimator  An ≈ Eµ X satisfying  An ∈ arg min  a∈R  b  n  n  �  i=1  ρ  �Xi − a  b  �  (2)  where ρ: R → R+ is a function that is approximately quadratic near zero, but grows more  slowly in the limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', large deviations are penalized in a sub-quadratic manner, where “large”  is relative to the scaling parameter b > 0, used to control bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' When ρ(·) is convex, differen-  tiable, and the solution set is non-empty, the condition (2) is equivalent to  1  n  n  �  i=1  ρ′  �Xi − An  b  �  = 0  (3)  and when the derivative ρ′(·) is bounded such that  − log(1 + −x + γx2) ≤ ρ′(x) ≤ log(1 + x + γx2),  x ∈ R  (4)  for some constant 0 < γ < ∞, then the analytical approach of Catoni [6] tells us that when b2  scales with Vµ X/n, the deviations |An − Eµ X| enjoy sub-Gaussian tails, namely upper bounds  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='com/feedbackward/bdd-mv  3  2  0  2  2  0  2  2  0  2  ′  2  0  2  ′′  Figure 1: From left to right, we plot the graphs of ρ(·), ρ′(·), and ρ′′(·) with ρ as in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' In the middle plot,  the dotted curves represent the upper (blue) and lower (dark pink) bounds in (4) with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  of the order O(  �  log(1/δ) Vµ X/n) with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Under these weak assump-  tions, such guarantees are essentially optimal [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' While a very important result, for practical  purposes, the need for knowledge of Vµ X is a significant limitation, since without finite higher-  order moments, it is not plausible to obtain variance estimates with analogous sub-Gaussian  guarantees (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', impossibility results of [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' There do exist other robust estimators such  as median-of-means [22, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1] which do not require variance information, and this illustrates  the fact that knowledge of the variance is sufficient, although not necessary, for sub-Gaussian  mean estimation under heavy tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2  Good-enough ancillary scaling  Since sub-Gaussian estimates of the variance Vµ X are not possible under our weak assump-  tions, it is natural to ask whether there exists a middle-ground, namely whether or not it is  possible to construct a (data-driven) procedure for setting the scale b > 0 in (2) which is “good  enough” in the sense that the resulting An is sub-Gaussian, even though the scale itself cannot  be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' An initial (affirmative) answer to this question was given in recent work by Sun [34], whose  basic idea we briefly review here, with some slight re-formulation for readability and additional  generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Essentially, the underlying idea in [34] is to utilize the convexity of ρ in (2), and to solve  for both a ∈ R and b > 0 simultaneously, while penalizing b in such a way as to encourage  scaling which is “good enough” as mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' More precisely, the empirical objective  �Sn(a, b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= βb + b  n  n  �  i=1  ρ  �Xi − a  b  �  (5)  plays a central role, where 0 < β < 1 is a parameter we can control, and ρ is fixed as  ρ(x) =  �  x2 + 1 − 1,  x ∈ R  (6)  which is differentiable, and satisfies the Catoni condition (4) with γ = 1 (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' If we  fix b > 0, then the solution sets (in a) of both �Sn(a, b) and b × �Sn(a, b) are identical, and it  should be noted that the re-scaled map x �→ b2ρ(x/b) = b  √  x2 + b2 − b2 closely approximates  x �→ x2/2 as b grows large (Figure 2), and is well-known as the “pseudo Huber” or “smooth  Huber” function, where b acts as a smoothing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2  When considering the joint objective �Sn(a, b), from the computational side, one important  fact is that this function is convex on R × (0, ∞) (see Lemma 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' From the statistical side of  2Barron [2, §1] gives a summary of this and related functions from the perspective of loss function design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  This is not the only smoothed variant of the classic Huber function [17], see for example Rey [29, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  4  1  0  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='4  x  b2 (x/b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  b value  Figure 2: Graphs of the smooth Huber function, with ρ as in (6), over a range of smoothing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' For  visual comparison, the graph of x �→ x2/2 is plotted with a thick dashed green curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  things, the solutions  (An, Bn) ∈ arg min  a∈R,b>0  �Sn(a, b)  (7)  are such that under certain regularity conditions, the deviations |An − Eµ X| are nearly optimal  (sub-Gaussian, up to poly-logarithmic factors) [34, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='3 The corresponding Bn of course  cannot give us sub-Gaussian estimates of the variance under such weak assumptions, but it  does scale in a desirable way [34, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2], and when bias is mitigated by setting β sufficiently small  given the sample size n, the resulting Bn is good enough to provide such guarantees for An,  which is the ultimate goal anyways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' By taking on a slightly more difficult optimization problem,  it is possible to get away with not having prior knowledge or sub-Gaussian estimates of the  variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We use this basic insight as a stepping stone to our approach for learning algorithms  charged with selecting a decision h such that the loss L(h) has a small mean-variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='3  A bridge between two problems  To develop our proposal, we now return to the more general learning setup, where the test  data is a random vector Z ∼ µ, test loss is L(h) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Z), and we have n IID training  points Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' , Zn yielding losses Li(·) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= ℓ(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Zi), i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' If our goal was to simply minimize  the traditional risk Eµ L(h) over h ∈ H under heavy-tailed losses, then in principle we could  extend the approach of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2 to robustly estimate the test risk using  (An(h), Bn(h)) ∈ arg min  a∈R,b>0  �  βb + b  n  n  �  i=1  ρ  �Li(h) − a  b  ��  (8)  and design a learning algorithm using (8) as follows:  Hn ∈ arg min  h∈H  An(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (9)  Under some regularity conditions, the machinery of Brownlees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' [5] could then be combined  with pointwise concentration inequalities in [34] to control the tails of Eµ L(Hn) under just finite  loss variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Our goal however is not to minimize the expected loss, but rather the mean-SD  sum (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Furthermore, the bi-level program inherent in (9) is not computationally congenial  from the perspective of large-scale machine learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' To ease the computational burden  3Strictly speaking, the objective used in [34] is �Sn(a, b)/β, but all key results easily translate to our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  5  while at the same time building a bridge between these two problems, we consider a new  objective function taking the form  �Cn(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= αa + βb + λb  n  n  �  i=1  ρ  �Li(h) − a  b  �  (10)  with parameters α ≥ 0 and β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We call (10) the modified Sun-Huber objective, since ρ is  fixed as (6), and this form plays a special role in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Compared with that of (9),  this objective is a simple function of h, and gradient-based minimizers can be easily applied  assuming the underlying loss ℓ(·) is sufficiently smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' On the other hand, it is “biased” in  the sense that it penalizes not just the loss location (whenever α > 0), but the loss scale as well  (whenever β > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Intuitively, some kind of deviation-driven “bias” is precisely what we need  from the standpoint of minimizing the mean-SD objective MSµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' λ), but it is not immediately  clear how this objective relates to �Cn(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b), and it is equally unclear if we can just plug this  new objective into standard machine learning workflows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', using stochastic gradient-based  optimizers) and achieve the desired effect without a prohibitive amount of manual tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='4  Overview of contributions and limitations  With our basic idea described and some key questions raised, we summarize the central points  that characterize the rest of this paper, and also highlight the limitations of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Broadly  speaking, the new proposal here is a class of empirical “risk” minimizers, namely any learning  algorithm which minimizes the new empirical objective (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' More explicitly, this refers to all  procedures which returns a triplet satisfying  (Hn, An, Bn) ∈  arg min  h∈H,a∈R,b>0  �Cn(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b)  (11)  where H denotes a set of feasible decisions, and we note that each element of this class is  characterized by the settings of α, β, and λ used to define �Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' In analogy with the strat-  egy employed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2, we do not expect An and Bn to provide sub-Gaussian estimates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' we  simply hope that these estimates are good enough to ensure the mean-SD is smaller and/or  better-behaved when compared to standard benchmarks such as mean-based empirical risk  minimization (ERM) and DRO-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Theoretically, we are interested in identify-  ing links between the proposed objective �Cn and loss properties such as Eµ L(h) and Vµ L(h),  with particular emphasis on how the settings of α, β, and λ influence such links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Our main theory-driven contribution is the derivation of a principled approach to determine  �Cn (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', set α and β), before seeing any training data, in such a way that we can balance between  “biased but robust” ρ-based deviations and “unbiased but outlier-sensitive” squared deviations  that arise in the loss variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Details are in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1–§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='3, and a concise procedure is summarized  in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We do not, however, consider the behavior of MSµ(Hn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' λ) for a particular  implementation of (11) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', SGD) from a theoretical viewpoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' the implementation is left  abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' This is where the empirical analysis of §4 comes in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We provide evidence using  simulated and real data that our procedure can be quite useful, even using a rudimentary  implementation where we wrap base loss objects and naively pass them to standard stochastic  gradient-based learning routines, with no manual tweaking of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  6  3  Theory  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1  Links to the mean-SD objective  We would like to make the connection between the proposed objective (10) and the ultimate  objective (1) a bit more transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' To do this, we will make use of the population version of  �Cn, denoted henceforth by Cµ and defined as  Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= αa + βb + λb Eµ ρ  �L(h) − a  b  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (12)  Let us fix the decision h and threshold a, paying close attention to the optimal value of the  scale b, denoted here by bµ(h, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' More explicitly, consider any positive real number satisfying  bµ(h, a) ∈ arg min  b>0  Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (13)  While it is not explicit in our notation, the optimal scale in (13) depends critically on the  value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Intuitively, a smaller value of β leads to a weaker penalty for taking b large, thus  encouraging a larger value of bµ(h, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' In fact, one can show that viewing bµ(h, a) as a function  of the parameter β, in the limit we have (proof in §B)  lim  β→0+ bµ(h, a) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (14)  Combining this with the fact that  lim  b→∞ b Eµ ρ  �L(h) − a  b  �  = 0  (15)  also holds (proof in §B), by re-scaling to avoid trivial limits we can obtain a result which  sharply bounds the proposed learning criterion at the optimal scale using the square root of  quadratic deviations, thereby establishing a clear link to the desired mean-SD objective (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Let H be such that Eµ|L(h)|2 < ∞ for each h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' If we set α = α(β) such  that α(β)/√β → �α ∈ [0, ∞) as β → 0+, then in this limit, with appropriate re-scaling the  scale-optimized learning criteria can be bounded above and below as  �αa + (1/2)  �  λ Eµ(L(h) − a)2 ≤ lim  β→0+ min  b>0  Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b)  √β  ≤ �αa + 4  �  λ Eµ(L(h) − a)2  for any choice of threshold a ∈ R and weight α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  In the special case where a = Eµ L(h) and �α > 0, we naturally recover mean-SD sums akin to  those studied in an ERM framework by Maurer and Pontil [24] and those bounded from above  using convex surrogates by Duchi and Namkoong [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Of course in practice, we will only ever be working with fixed values of β, and the entire  point of introducing new criteria (namely �Cn and Cµ) was to give us some control over how  sensitive our objective is to loss tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The following result makes the nature of this control  (through β) more transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Let H and L(h) be as stated in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Letting bµ(h, a) be as specified  in (13), we define a Bernoulli random variable  I(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= I {|L(h) − a| ≤ bµ(h, a)}  7  for any choice of h ∈ H and a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The optimal scale can then be bounded by  λ  4β Eµ I(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a)(L(h) − a)2 ≤ b2  µ(h, a) ≤ λ  2β Eµ(L(h) − a)2  for any choice of 0 < β < λ and a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  While it is difficult to pin down exactly how bµ(h, a) changes as a function of β, Proposition 2  clearly shows us the appealing property that optimal scale induced by the proposed objective  function essentially falls between the (tail-sensitive) quadratic deviations and a (tail-insensitive)  truncated variant, with the truncation threshold loosening as β shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2  Guiding the optimal threshold  Since the preceding Propositions 1–2 both hold for any choice of threshold a ∈ R, they clearly  hold when both a and b are optimal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', when a and b are set as  (aµ(h), bµ(h)) ∈ arg min  a∈R,b>0  Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (16)  In particular, using first-order conditions, the inclusion (16) is equivalent to the following two  equalities holding at once:  Eµ  �  �  L(h) − aµ(h)  �  (L(h) − aµ(h))2 + b2µ(h)  �  � = α,  Eµ  �  �  bµ(h)  �  (L(h) − aµ(h))2 + b2µ(h)  �  � = 1 − β/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (17)  Given the context of our analysis in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1, let us consider the effect of taking β towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  For any non-trivial random loss, the second equality asks that bµ(h) grow without bound as  β → 0+, while |aµ(h)| must be either bounded or grow slower than bµ(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' On the other hand,  if α is too large (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', α > 1) then the first equality will be impossible to satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' In addition to  taking 0 < α < 1, note that if we multiply both sides of the first equality in (17) by bµ(h) and  apply Proposition 2, then under this optimality condition we must have  Eµ  �  �  �  �  L(h) − aµ(h)  �  (L(h)−aµ(h)  bµ(h)  )2 + 1  �  �  �  � ≤ α  �  λ  2β Eµ(L(h) − aµ(h))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (18)  With this inequality in place, we adopt the following strategy: encourage the optimal location  to converge as aµ(h) → Eµ L(h) when β → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Since λ > 0 is assumed to be fixed in advance,  the only way to ensure this using (18) is to set α = α(β) such that  lim  β→0+  α(β)  √β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (19)  While (19) gives us a rather clear condition for determining α given β, we still do not have a  principled setting for β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' This point will be treated in the following sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='3  Deriving an algorithm using finite-sample theory  To complement the preceding analysis and discussion centered around the population objective  (12), we now return to the empirical objective function �Cn(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b) introduced in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We  maintain the running assumption that the training data Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' , Zn are an IID sample from µ,  8  and thus the losses Li(h), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' , n are independent given any fixed h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' With h and b > 0  fixed for the moment, we will now take a closer look at the optimal (empirical) threshold that  arises from this objective function, namely any random variable An(h, b) satisfying  An(h, b) ∈ arg min  a∈R  �Cn(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (20)  Using the property (4) of the smooth Huber-like function ρ, we can demonstrate how data-  driven thresholds satisfying (20) are concentrated at a point near the expected loss, where α  and b play a key role in how close this point is to the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Proposition 3 (Concentration at a shifted location).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Taking 0 ≤ α < 1, b > 0, and 0 < δ < 1,  with large enough n it is always possible to satisfy the condition  4α  λ ≤ 4  �Vµ L(h)  b2  + log(2/δ)  n  �  ≤ 1 − 4α  λ ,  and when this condition is satisfied, the data-driven threshold An(h, b) in (20) satisfies  ����An(h, b) −  �  Eµ L(h) − 2α  λ b  ����� ≤ 2  �Vµ L(h)  b  + b log(2/δ)  n  �  with probability no less than 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  This result can be seen as an extension of [34, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1] for the function (5) used in mean  estimation to our generalized learning problem, although we use a different proof strategy  which does not require strong convexity of �Cn (with respect to a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  With Proposition 3 established, conventional wisdom might incline one to pursue a O(1/√n)  rate in the upper bound;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' in this case, setting β ∝ 1/n is a natural strategy since Proposition  2 tells us that for the population objective, the optimal setting of b scales with  �  λ/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' While  this is natural from the perspective of tight concentration bounds for An(h, b), we argue that a  different strategy is more appropriate when we actually consider how (Hn, An, Bn) will behave  in the full joint optimization (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The most obvious reason for this is that the joint objective  lacks convexity and smoothness, as the following result summarizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Proposition 4 (Joint objective is non-convex and non-smooth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Even when H is a compact  convex set and the base loss function ℓ(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Z) is convex, the mapping (h, a, b) �→ �Cn is not convex  in general, and is non-smooth in the sense that its gradient is not Lipschitz continuous on  H × R × (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  In consideration of Proposition 4, standard complexity results for typical optimizers such as  stochastic gradient descent to achieve a ε-stationary point are on the order of O(ε−4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' see  Davis and Drusvyatskiy [8] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='4  With this in mind, setting β ∝ 1/n to achieve  O(ε−2) sample complexity for error bounds of An(h, b) seems superfluous if in the end the  dominant complexity for solving the ultimate problem (11) will be of the order O(ε−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' As  such, in order to match this rate, the more natural strategy is to set β ∝ 1/√n, or more  precisely to set  β = β0  √n  (21)  where β0 > 0 is a constant used to ensure 0 < β < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' This, coupled with α(β) = β to satisfy  (19) from the previous sub-section, is our proposed setting to determine (α, β) (and thus �Cn)  using just knowledge of n, and without having observed any data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' This procedure is  summarized in Algorithm 1, and will be the subject of empirical analysis later in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  4Even if the objective were smooth, the same rates are typical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' see for example Ghadimi and Lan [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  9  Algorithm 1 Modified Sun-Huber  Inputs: data Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' , Zn and parameter λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Set: β = β0/√n, with β0 such that 0 < β < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  {Based on (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='}  Set: α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  {Satisfies (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='}  Minimize: �Cn(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b) in (h, a, b) using α and β as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='4  Stationary points of mean-variance  Having established links between the proposed objective and the mean-SD objective, we next  consider the mean-variance objective  MVµ(h) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= Eµ L(h) + Vµ L(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (22)  This quantity can be expressed as the minimum value of a convex function, namely we have  MVµ(h) = min  a∈R  �  a + Eµ(L(h) − a)2 + 1  2  �  = aMV(h) + Eµ(L(h) − aMV(h))2 + 1  2  (23)  where on the right-most side we have set aMV(h) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= Eµ L(h)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Assuming the underlying loss  is differentiable, the gradient with respect to h can be written as  MV′  µ(h) = Eµ L′(h) + Eµ L(h) L′(h) − Eµ L(h) Eµ L′(h)  = Eµ L′(h) + Eµ (L(h) − Eµ L(h)) L′(h)  = Eµ (L(h) − (Eµ L(h) − 1)) L′(h)  which implies a stationarity condition of  MV′  µ(h) = 0 ⇐⇒ Eµ (L(h) − (Eµ L(h) − 1)) L′(h) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (24)  Similarly, the partial derivative of the learning criterion (12) taken with respect to h is  ∂  ∂h Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b) = Eµ  �  L(h) − a  �  (L(h) − a)2 + b2  �  L′(h)  and thus multiplying both sides by b > 0, we obtain a simple stationarity condition of  ∂  ∂h Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b) = 0 ⇐⇒ Eµ  �  �  L(h) − a  �  (L(h)−a  b  )2 + 1  �  � L′(h) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (25)  With the right threshold setting, obviously the two conditions become very similar as b grows  large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The following result makes this precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Let loss function ℓ and data distribution µ be such that the random vector  L(h) L′(h) is integrable and has a norm with finite mean, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', Eµ∥L(h) L′(h)∥ < ∞ for some  choice of h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Then, for any a ∈ R, defining  f(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= lim  b→∞ b ∂  ∂h Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b)  (26)  the stationary points of the mean-variance objective are related to those of the proposed objective  (12) through the following equivalence:  f(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' aMV(h)) = 0 ⇐⇒  ∂  ∂h MVµ(h) = 0  where MVµ(h) is as defined in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  Figure 3: Graph of the Legendre transform ρ∗ as given in (28) over (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='5  Comparison with dual form of DRO risk  Some readers may notice that the proposed (population) objective (12) looks quite similar to  the dual form of DRO risks:  DROµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.=  inf  a∈R,b>0  �  a + βb + b Eµ φ∗  �L(h) − a  b  ��  (27)  where φ∗ is the Legendre-Fenchel convex conjugate φ∗(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= supu∈R[xu − φ(u)] induced by a  function φ : R → R, assumed to be convex and lower semi-continuous, with φ(1) = 0 and  φ(x) = ∞ whenever x < 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' [33, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Given this similarity, one might ask whether or not  some form of DRO risk can be reverse engineered from our proposed objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Taking up this  point briefly, we first note that the conjugate of ρ given by (6) is  ρ∗(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= sup  u∈R  [xu − ρ(u)] = sup  u∈R  �  xu −  �  u2 + 1 + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  From the non-negative nature of ρ, clearly ρ∗(0) = −ρ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' For x ̸= 0, note that taking the  derivative of concave function u �→ xu − ρ(u) and setting it to zero, we obtain the first-order  optimality conditions  u  √  u2 + 1 = x ⇐⇒  sign(x)  �  1 + 1/u2 = x ⇐⇒  1  x2 = 1 + 1/u2 ⇐⇒ u =  sign(x)  �  1/x2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Plugging this solution in whenever |x| < 1 and doing a bit of algebra readily yields the simple  closed-form expression  ρ∗(x) =  �  �  �  x2  √  1−x2 + 1 −  1  √  1−x2 ,  if 0 ≤ |x| < 1  ∞,  else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (28)  As can be readily observed from both (28) and Figure 3, this function does not satisfy any  of the requirements placed on φ except convexity, and thus despite the similar form, the non-  monotonic nature of ρ is in sharp contrast with monotonicity of typical cases of φ∗ that arise  in the DRO literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' [3, §3]), and does not readily imply a “primal” DRO objective that  can be recovered using ρ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  4  Empirical analysis  Our investigation in the previous section led us to Algorithm 1, giving us a principled and  precise strategy to construct the objective function �Cn, but leaving the actual minimization  procedure abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Here we make this concrete by implementing a simple gradient-based  minimizer of this objective, and comparing this procedure with natural benchmarks from the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1  Simulated noisy classification on the plane  As a simplified and controlled setting to start with, we generate random data points on the  plane which are mostly linearly separable, save for a single distant outlier (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Before  we consider off-sample generalization, here we focus simply on the training loss distribution  properties as a function of algorithm iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Experiment setup  We generate n = 100 training data points using two Gaussian distri-  butions on the plane to represent two classes, with each class having the same number of  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  We choose a single point uniformly at random, and perturb it by multiplying the  scalar -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We compare our proposed procedure (denoted “Modified Sun-Huber”) with three  alternatives: traditional mean-based empirical risk minimization (denoted “Vanilla ERM”),  conditional value-at-risk (CVaR) [7], and the well-studied χ2-DRO risk [11, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' In light of  Algorithm 1, we set λ = log(n)/√n > β = β0/√n, and try a variety of β0 values just for  reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' For all the aforementioned methods, we set the base loss ℓ(·) to the usual binary  logistic loss (linear model), and run (batch) gradient descent on the empirical risk objectives  implied by each of these methods (see §C for details), with a fixed step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='01 over 15,000  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Alternative settings of step size and iteration number were not tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' All methods  are initialized at the same point, shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Results and discussion  In Figure 5, we show the empirical mean-SD trajectories for the  base loss, over algorithm iterations (log10 scale), for each method of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Using our no-  tation, this is the sample version of MSµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' λ) in (1), with λ = 1 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' All methods besides  vanilla ERM have multiple settings that were tested, and the results for each are distinguished  using curves of different color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Our method tests different values of β0, CVaR tests different  quantile levels, and DRO tests different robustness levels (details in §C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Since Vanilla ERM  is designed to optimize the average loss, it is perhaps not surprising that it fails in terms of  the mean-SD objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' On the other hand, the proposed method (for any choice of β0) is as  good or better than all the competing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' As a basic sanity check, in Figure 6 we also  consider the error rate (average zero-one loss) and model norm trajectories over iterations for  each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' For each method, we plot just one trajectory, namely the one achieving the best  final error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' While our method is not designed to minimize the average loss and typical  surrogate theory does not apply, we find that the error rate is surprisingly good, albeit with  slower convergence than the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Note also how the error rate for CVaR matches  that of Vanilla ERM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' this is in fact the CVaR setting with the worst final mean-SD value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' On  the other hand, the proposed method performs well from both perspectives at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2  Classification on real datasets  We proceed to experiments using real-world datasets, some of which are orders of magnitude  larger than the simple setup given in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1, and which include multi-class classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Experiment setup  We make use of four well-known datasets, all available from online  repositories: adult,5 australian,6 cifar10,7 and fashion_mnist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='8 For multi-class datasets,  we extend the binary logistic loss to the usual multi-class logistic regression loss under a linear  5https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='edu/ml/datasets/Adult  6https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='edu/ml/datasets/statlog+(australian+credit+approval)  7https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='toronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='edu/~kriz/cifar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='html  8https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='com/zalandoresearch/fashion-mnist  12  model, with one linear model for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Features for all datasets are normalized to [0, 1],  with one-hot representations of categorical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The learning algorithms being compared  here are the same as described in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1, except that now we implement each method using  mini-batch stochastic gradient descent (batch size 32), and do 30 epochs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', 30 passes over  the training data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' In addition, our proposed “Modified Sun-Huber” method performs almost  identically for the range of β0 values tested in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1, and thus we have simply fixed β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='9, so  there is only one trajectory curve this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' On the other hand, we now try a range of step sizes  for each method, choosing the best step size in terms of average (base) loss value on validation  data for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We run five independent trials, and for each trial we randomly re-shuffle  the dataset, taking 80% for training, 10% for validation (used to select step sizes), and 10%  for final testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Results and discussion  Our main results are shown in Figure 7, where once again we  plot the trajectory of the mean-SD objective, but this time computed on test data, and given  as a function of epoch number, rather than individual iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Since there are multiple  trials, the curves drawn represent averages taken over trials, and the lightly shaded region  above/below each curve shows standard deviation over trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Perhaps surprisingly, the very  simple implementation of our proposed Algorithm 1 (fixed step size, no regularization) works  remarkably well on a number of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' From the perspective of mean-SD minimization,  for three our of four datasets, the proposed method is far better than Vanilla ERM, and as  good or better than even the best settings of CVaR and DRO viewed after the fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Regarding  the sub-standard performance observed on fashion_mnist, detailed analysis shows that more  fine-tuned settings of α and β can readily bring the method up to par;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' the non-convex and non-  smooth nature of �Cn naturally means that some tasks will require more careful settings than  are captured by our Algorithm 1, and indeed will take explicit account of the optimizer to be  used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We leave both the theoretical grounding and empirical testing of such optimizer-aligned  mean-SD minimizers for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  13  Figure 4: 2D classification example from §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The red line represents the initial value used by each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  102  1  2  Vanilla ERM  102  Modified Sun-Huber  102  CVaR  102  2-DRO risk  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='3  Mean + SD (2D classification with outliers)  Figure 5: Trajectory of the (empirical) mean-SD objective (1) over iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Colors correspond to different  choices from each class: β0 for Modified Sun-Huber, quantile level for CVaR, and constraint level for DRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  101  103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='4  Error rate  101  103  0  2  4  Norm  Trajectory with best final error rate  Vanilla ERM  Modified Sun-Huber  CVaR  2-DRO risk  Figure 6:  From each method class, we show the classification error rate and Euclidean norm trajectories  corresponding to the setting that achieved the best error rate after the final iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  14  0  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='8  Vanilla ERM  0  20  Modified Sun-Huber  0  20  CVaR risk  0  20  2-DRO risk  Mean + SD (dataset: adult)  0  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  Vanilla ERM  0  20  Modified Sun-Huber  0  20  CVaR risk  0  20  2-DRO risk  Mean + SD (dataset: australian)  0  20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='5  Vanilla ERM  0  20  Modified Sun-Huber  0  20  CVaR risk  0  20  2-DRO risk  Mean + SD (dataset: cifar10)  0  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='5  Vanilla ERM  0  20  Modified Sun-Huber  0  20  CVaR risk  0  20  2-DRO risk  Mean + SD (dataset: fashion_mnist)  Figure 7: Mean-SD trajectories on real-world datasets as described in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2, given as a function of epochs and  averaged over multiple independent trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Coloring for CVaR and DRO is analogous to that of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
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+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Risk-adaptive approaches to learning and decision making: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  arXiv preprint arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='00856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  [33] Shapiro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Distributionally robust stochastic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' SIAM Journal on  Optimization, 27(4):2258–2275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  [34] Sun, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Do we need to estimate the variance in robust mean estimation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' arXiv  preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='00118v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  [35] Vapnik, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The Nature of Statistical Learning Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Statistics for Engineering  and Information Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Springer, 2nd edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  17  [36] Zhai, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', Dan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', Kolter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', and Ravikumar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' DORO: Distributional and  outlier robust optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' In 38th International Conference on Machine Learning (ICML),  volume 139 of Proceedings of Machine Learning Research, pages 12345–12355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  A  Technical appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1  Basic facts  Assuming ρ is defined as in (6), let us consider the function  f(x, a, b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= αa + βb + bρ  �x − a  b  �  (29)  = αa + βb +  �  (x − a)2 + b2 − b  (30)  = αa +  �  (x − a)2 + b2 − (1 − β)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (31)  The partial derivatives are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  ∂xf(x, a, b) =  x − a  �  (x − a)2 + b2  (32)  ∂af(x, a, b) = α −  x − a  �  (x − a)2 + b2  (33)  ∂bf(x, a, b) =  b  �  (x − a)2 + b2 − (1 − β)  (34)  The corresponding second derivatives are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  ∂2  xf(x, a, b) =  1  �  (x − a)2 + b2 −  (x − a)2  ((x − a)2 + b2)3/2 =  b2  ((x − a)2 + b2)3/2  (35)  ∂2  af(x, a, b) =  1  �  (x − a)2 + b2 −  (x − a)2  ((x − a)2 + b2)3/2 =  b2  ((x − a)2 + b2)3/2  (36)  ∂2  b f(x, a, b) =  1  �  (x − a)2 + b2 −  b2  ((x − a)2 + b2)3/2 =  (x − a)2  ((x − a)2 + b2)3/2  (37)  The remaining elements of the Hessian of f(x, a, b) follow easily, given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  ∂a∂xf(x, a, b) =  −1  �  (x − a)2 + b2 +  (x − a)2  ((x − a)2 + b2)3/2 =  −b2  ((x − a)2 + b2)3/2  (38)  ∂b∂xf(x, a, b) =  −b(x − a)  ((x − a)2 + b2)3/2  (39)  ∂b∂af(x, a, b) =  b(x − a)  ((x − a)2 + b2)3/2  (40)  Lemma 6 (Useful inequalities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  1  1 + x ≤ 1 − x  2,  0 ≤ x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (41)  (1 + x)c ≥ 1 + cx,  x ≥ −1, c ∈ R \\ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (42)  18  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2  Convexity and smoothness  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The map x �→ 1/√1 + x is convex on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Lemma 8 (Properties of partial objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' With ρ as in (6) and β ≥ 0, the function  (x, b) �→ βb + bρ  �x  b  �  is convex and (1 + max{1 − β, β})-Lipschitz (in ∥·∥1) on R × (0, ∞), but its gradient is not  (globally) Lipschitz, and thus the function is not smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='9  Proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' For notational convenience, setting 0 < β < 1, let us denote  g(x, b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= βb + bρ(x/b),  x ∈ R, b > 0  with ρ as in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' From the partial derivatives (32) and (34), it is clear that we have  −1 ≤ ∂xg(x, b) ≤ 1,  −(1 − β) ≤ ∂bg(x, b) ≤ β  when evaluated at any choice of x ∈ R and b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' It follows that the gradient norm can be  bounded as  ∥∇g(x, b)∥1 ≤ 1 + max{(1 − β), β}  and thus g(·) is Lipschitz continuous in ∥·∥1 (and also ∥·∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='10  Next, let us denote the Hessian of g(·) evaluated at (x, b) by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Basic calculus gives us the  simple form  H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.=  1  (x2 + b2)3/2  �  b2  −xb  −xb  x2  �  and for any pair of real values u = (u1, u2), we have  ⟨Hu, u⟩ =  1  (x2 + b2)3/2 (u1b − u2x)2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (43)  Since this holds for any choice of x ∈ R and b > 0, the Hessian is thus positive semi-definite,  implying that g(·) is (jointly) convex [28, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  On the other hand, the function g(·) is not smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' To see this, first note that having  chosen any u such that ∥u∥ ≤ 1, we have that the (operator) norm is bounded below as  ∥H∥ = sup  ∥u′∥≤1  �  sup  ∥u′′∥≤1  ⟨Hu′, u′′⟩  �  ≥ ⟨Hu, u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Then, as a concrete example, consider setting x = b, with u = (u1, u2) such that u1 ̸= u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Recalling the lower bound (43), we have  ∥H∥ ≥  b2  (2b2)3/2 (u1 − u2)2 = (u1 − u2)2  (  √  2)3b  → ∞  in the limit as b → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  As such, the gradient of g(·) cannot be Lipschitz continuous on  R × (0, ∞), and thus g(·) is not smooth [28, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  9We prove that the Hessian’s norm is unbounded, which implies (via Nesterov [28, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='6]) that the  convex function of interest cannot be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  10That bounded gradients imply Lipschitz continuity is a general fact on linear spaces [21, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='3, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  19  B  Additional proofs  Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' To begin, note that the function  b �→ b Eµ ρ  �L(h) − a  b  �  = Eµ  ��  (L(h) − a)2 + b2 − b  �  (44)  is monotonic (non-increasing) on (0, ∞) (follows clearly from (34)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We will use this property  moving forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Recalling the upper and lower bounds of Proposition 2, we re-write them as  clo(β)  β  ≤ b2  µ(h, a) ≤ chi  β  (45)  using the shorthand notation  clo(β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= λ  4 Eµ I(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a)(L(h) − a)2  chi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= λ  2 Eµ(L(h) − a)2  and noting that while chi is free of β, clo(β) depends on β through the definition of I(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Fixing 0 < β < λ for now and recalling the form of Cµ in (12), the preceding bounds (45) and  monotonicity of (44) can be used to obtain a lower bound of the form  min  b>0 Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b) ≥ αa +  �  βclo(β) + λ√chi Eµ  �  �  �  (L(h) − a)2  chi  + 1  β −  �  1  β  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (46)  Using the fact (14) and applying dominated convergence [1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='9], in the limit we have  lim  β→0+ clo(β) = λ  4 Eµ(L(h) − a)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Dividing both sides of (46) by √β, setting α = α(β) as in the proposition statement, and  taking the limit as β → 0+, we obtain  lim  β→0+ min  b>0  Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b)  √β  ≥ �αa +  �  λ  4 Eµ(L(h) − a)2 + λ Eµ(L(h) − a)2  2√chi  = �αa +  �  λ  4 Eµ(L(h) − a)2 +  �  λ  2 Eµ(L(h) − a)2  = �αa +  �1  2 + 1  √  2  � �  λ Eµ(L(h) − a)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  The first inequality uses the fact that for any c > 0, we have  √  cx + x2 − x → c/2 as x → ∞,  and also uses dominated convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The remaining equalities just follow from plugging in  the definition of chi and cleaning up terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' This proves the desired lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  As for the upper bound of interest, a perfectly analogous argument can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Using  Proposition 2 again and taking β small enough that  clo(β) ≥ chi/4  (47)  20  holds (always possible), we can obtain upper bounds of the form  min  b>0 Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b) ≤ αa +  �  βchi + λ  �  clo(β) Eµ  ��  (L(h) − a)2  clo(β)  + 1  β −  �  1  β  �  ≤ αa +  �  βchi + λ√chi Eµ  �  �  �  4(L(h) − a)2  chi  + 1  β −  �  1  β  �  �  (48)  noting that the latter inequality (48) follows from using (47) as well as clo(β) ≤ chi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' As with  the lower bound argument in the preceding paragraph, we set α = α(β), divide both sides by  √β, and take the limit as β → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' This results in  lim  β→0+ min  b>0  Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b)  √β  ≤ �αa +  �  λ  2 Eµ(L(h) − a)2 + 2λ Eµ(L(h) − a)2  √chi  = �αa +  �  λ  2 Eµ(L(h) − a)2 + 2  �  2λ Eµ(L(h) − a)2  = �αa +  �  2  √  2 + 1  √  2  � �  λ Eµ(L(h) − a)2  which gives us the desired upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The bounds given in the proposition statement are  slightly looser, but more readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We adapt key elements of the scale control used by Sun [34, §2] to our  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' We start by looking at first-order conditions for optimality of b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' First, note that  ∂  ∂b Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b) = β + λ ∂  ∂b  �  Eµ  �  (L(h) − a)2 + b2 − b  �  = β + λ Eµ  �  b  �  (L(h) − a)2 + b2  �  − λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  As such, it follows that  Eµ  �  b  �  (L(h) − a)2 + b2  �  = 1 − β/λ  (49)  is equivalent to ∂b Cµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Obviously, the left-hand side of (49) is non-negative for all  b ≥ 0 and bounded above by 1 for all b ≥ 0, a ∈ R, and h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Thus (49) can only hold for  0 ≤ β ≤ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Using convexity (Lemma 7) and Jensen’s inequality [1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='5], we have  Eµ  �  b  �  (L(h) − a)2 + b2  �  = Eµ  �  �  1  �  (L(h)−a  b  )2 + 1  �  � ≥  �  �  1  �  Eµ(L(h)−a  b  )2 + 1  �  �  and thus whenever (49) holds, we know that  (1 − β/λ)2 ≥  1  Eµ(L(h)−a  b  )2 + 1  must also hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Re-arranging terms, we see that this implies  b2 ≤ (1 − β/λ)2 Eµ(L(h) − a)2  1 − (1 − β/λ)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  21  For readability, set η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= β/λ, and note that since  (1 − η)2  1 − (1 − η)2 = (1 − η)2  2η − η2 =  (1 − η)2  2η(1 − η/2) ≤ (1 − η)2  2η(1 − η) ≤ 1  2η  we can obtain the cleaner (but looser) upper bound  b2 ≤ Eµ(L(h) − a)2  2η  = λ  2β Eµ(L(h) − a)2  for any choice of 0 < β ≤ λ and a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Since the first-order condition (49) is necessary for  optimality [28, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1], it follows that  b2  µ(h, a) ≤ Eµ(L(h) − a)2  2β  (50)  which is the desired upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Considering a lower bound next, note first that using the concavity of x �→ √x on R+,  another application of Jensen’s inequality gives us  Eµ  �  b  �  (L(h) − a)2 + b2  �  = Eµ  �  b2  (L(h) − a)2 + b2 ≤  �  �  �  �Eµ  �  1  (L(h)−a  b  )2 + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (51)  Using the inequality 1/(x + 1) ≤ 1 − x/2 for all 0 ≤ x ≤ 1 ((41) in Lemma 6), this suggests  a natural event to use as a condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  More precisely, writing E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= I {|L(h) − a| ≤ b} for  readability, note that we have  1  (L(h)−a  b  )2 + 1  =  1 − E  (L(h)−a  b  )2 + 1  +  E  (L(h)−a  b  )2 + 1  ≤  1 − E  (L(h)−a  b  )2 + 1  + E  �  1 − 1  2  �L(h) − a  b  �2�  =  �  1 − E  (L(h)−a  b  )2 + 1  + E  �  �  ��  �  ≤1  −E  2  �L(h) − a  b  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Taking expectation and utilizing (51), whenever (49) holds, we have  1 − β/λ = Eµ  �  b  �  (L(h) − a)2 + b2  �  ≤  �  1 − Eµ  E  2  �L(h) − a  b  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (52)  With this established, note that via helper inequality (42), for any β ≤ λ we have  (1 − β/λ)2 ≥ 1 − 2β/λ  and thus in light of (52), we may conclude that  1 − 2β/λ ≤ 1 − Eµ  E  2  �L(h) − a  b  �2  which implies  λ  4β Eµ E(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b)(L(h) − a)2 ≤ b2  22  noting that we have written E(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b) to emphasize the dependence on h, a, and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Once again  since the first-order condition (49) is necessary for optimality, we may conclude that  λ  4β Eµ E(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, bµ(h, a))(L(h) − a)2 ≤ b2  µ(h, a)  (53)  which is the remaining desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Proof of the limit (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Recall from the proof of Proposition 2 the first-order optimality con-  dition (49), which is satisfied by any solution bµ(h, a) given by (13), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', we have  Eµ  �  �  bµ(h, a)  �  (L(h) − a)2 + (bµ(h, a))2  �  � = 1 − β/λ  (54)  for any 0 < β ≤ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Defining g(β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= 1 − β/λ and taking any 0 < β2 < β1 ≤ λ, clearly we have  g(β1) < g(β2) and thus using the equality (54), we must have that bµ(h, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' β2) ≥ bµ(h, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' β1),  otherwise it would result in a contradiction of (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Using this monotonicity, clearly  E(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, bµ(h, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' β1)) ≤ E(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, bµ(h, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' β2))  and thus  Eµ E(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, bµ(h, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' β1))(L(h) − a)2 ≤ E(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, bµ(h, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' β2))(L(h) − a)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Applying this to the lower bound in Proposition 2, we have  lim inf  β→0+ b2  µ(h, a) ≥ lim  β→0+  λ  4β Eµ I(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a)(L(h) − a)2 = ∞  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Proof of the limit (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Note that we can easily bound the random variable of interest as  0 ≤ bρ  �L(h) − a  b  �  =  �  (L(h) − a)2 + b2 − b ≤ |L(h) − a|  (55)  for any choice of 0 < b < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Some straightforward calculus shows that  lim  b→∞ bρ  �L(h) − a  b  �  = 0  in a pointwise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Since the upper bound in (55) is µ-integrable by assumption, a simple  application of dominated convergence [1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='9] yields  lim  b→∞ b Eµ ρ  �L(h) − a  b  �  = Eµ  �  lim  b→∞ bρ  �L(h) − a  b  ��  = 0  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  23  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' From condition (20), since any solution must also be a stationary point  [28, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='1], we know that An .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= An(h, b) must satisfy the first-order condition  λ  n  n  �  i=1  ρ′  �Li(h) − An  b  �  = α  which is equivalent to  b  n  n  �  i=1  ρ′  �Li(h) − An  b  �  = α  λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (56)  Next we make use of the argument developed by Catoni [6, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' First note that fixing any  a ∈ R and b > 0, we have  E exp  � n  �  i=1  ρ′  �Li(h) − a  b  ��  = E  � n  �  i=1  exp  �  ρ′  �Li(h) − a  b  ���  =  n  �  i=1  Ei exp  �  ρ′  �Li(h) − a  b  ��  ≤  n  �  i=1  Ei  �  1 + Li(h) − a  b  + γ  b2 (Li(h) − a)2  �  =  �Eµ L(h) − a  b  + γ  b2 Eµ(L(h) − a)2  �n  ≤ exp  �n  b (Eµ L(h) − a) + nγ  b2 Eµ(L(h) − a)2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (57)  The second equality above follows from the independence of the training data, and the first  inequality uses the upper bound in (4), which is satisfied by ρ given in (6) with γ = 1, though  we leave γ as is to illustrate how more general results are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The third equality just uses  the fact that the training data is an IID sample from µ, and the final inequality culminating  in (57) just uses the bound 1 + x ≤ exp(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Using Markov’s inequality and taking 0 < δ < 1,  it is straightforward to show that (57) implies a 1 − δ event (over the draw of Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' , Zn) in  which we have  n  �  i=1  ρ′  �Li(h) − a  b  �  ≤ n  b (Eµ L(h) − a) + nγ  b2 Eµ(L(h) − a)2 + log(1/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Multiplying both sides by b/n, on the same “good” event, we have  b  n  n  �  i=1  ρ′  �Li(h) − a  b  �  ≤ Eµ L(h) − a + γ  b Eµ(L(h) − a)2 + b log(1/δ)  n  = Eµ L(h) − a + γ  b  �  Vµ L(h) + (Eµ L(h) − a)2�  + b log(1/δ)  n  (58)  where (58) follows from expanding the quadratic term and doing some algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  With the  equality (56) in mind, subtracting a constant from both sides of (58), note that we equivalently  have  b  n  n  �  i=1  ρ′  �Li(h) − a  b  �  − α  λb ≤ p(a)  (59)  24  where we have defined  p(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= Eµ L(h) − a + γ  b  �  Vµ L(h) + (Eµ L(h) − a)2�  + b log(1/δ)  n  − α  λb  (60)  for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Note that p(·) in (60) is a polynomial of degree 2, and can be written as  p(a) = ua2 + va + w  (61)  with coefficients defined as  u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= γ  b  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= (−1)  �  1 + 2γ Eµ L(h)  b  �  w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= Eµ L(h) + γ  b Eµ|L(h)|2 + b log(1/δ)  n  − α  λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  This polynomial has real roots whenever v2 − 4uw ≥ 0, and some algebra shows that this is  equivalent to  0 ≤ D ≤ 1, where D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= 4  ��γ  b  �2  Vµ L(h) + γ log(1/δ)  n  − γα  λ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (62)  Assuming this holds, denoting by a+ the smallest root of p(·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=', the smallest of satisfying  p(a+) = 0, the critical fact of interest to us is that An ≤ a+ on the good event of (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' This is  valid due to two facts: first, the left-hand side of (59) is a decreasing function of a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' second, due  to (56), we know that An is a root of the left-hand side of (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' With this key fact in hand,  using the quadratic formula we have  An ≤ a+  = Eµ L(h) + b  2γ  �  1 −  √  1 − D  �  = Eµ L(h) + b  2γ  �  1 −  √  1 − D  � �  1 +  √  1 − D  �  �  1 +  √  1 − D  �  = Eµ L(h) + b  2γ  D  �  1 +  √  1 − D  �  ≤ Eµ L(h) + b  2γ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Taking the two ends of this inequality chain together and expanding D, we have  An ≤ Eµ L(h) − 2(α/λ)b + 2  �γ  b Vµ L(h) + b log(1/δ)  n  �  (63)  with probability no less than 1 − δ, assuming that n, b, and α are such that 0 ≤ D ≤ 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  This gives us the desired upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  To obtain a lower bound, a perfectly analogous argument can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' First, using the  lower bound in (4) and the fact that ρ′(−x) = −ρ′(x), we know that  ρ′  �a − Li(h)  b  �  ≤ log  �  1 + a − Li(h)  b  + γ  b2 (a − Li(h))2  �  (64)  25  for any a ∈ R, b > 0, and i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Plugging this inequality (64) into an argument analogous to  the chain of inequalities that led to (58) earlier, it is clear that again on an event of probability  no less than 1 − δ, we have  b  n  n  �  i=1  ρ′  �a − Li(h)  b  �  ≤ a − Eµ L(h) + γ  b  �  Vµ L(h) + (Eµ L(h) − a)2�  + b log(1/δ)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (65)  Once again the upper bound we can bound this using a polynomial of degree 2, namely  b  n  n  �  i=1  ρ′  �a − Li(h)  b  �  + α  λb ≤ q(a)  (66)  where we have defined  q(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= a − Eµ L(h) + γ  b  �  Vµ L(h) + (Eµ L(h) − a)2�  + b log(1/δ)  n  + α  λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (67)  Now, since An is a root of the left-hand side of (66) viewed as a function of a, and this function  is monotonically increasing, it is evident that denoting the largest root of q(·) (when it exists)  by a−, we have An ≥ a−, a lower bound in contrast to the An ≤ a+ upper bound used earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  For completeness, we write this polynomial as  q(a) = u′a2 + v′a + w′  (68)  with coefficients  u′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= γ  b  v′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.=  �  1 − 2γ Eµ L(h)  b  �  w′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= (−1) Eµ L(h) + γ  b Eµ|L(h)|2 + b log(1/δ)  n  + α  λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  We have two real roots whenever  1 ≥ D′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= 4  ��γ  b  �2  Vµ L(h) + γ log(1/δ)  n  + γα  λ  �  (69)  holds, and thus we obtain a high probability lower bound on An as follows:  An ≥ a−  = Eµ L(h) − b  2γ  �  1 −  √  1 − D′  �  ≥ Eµ L(h) − b  2γ D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Expanding D′ gives us the lower bound  An ≥ Eµ L(h) − 2(α/λ)b − 2  �γ  b Vµ L(h) + b log(1/δ)  n  �  (70)  with probability no less than 1 − δ, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  26  Let us conclude this proof by organizing the technical assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' First of all, for the  two quadratics used in the preceding bounds, we require both (62) and (69) to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' It is  straightforward to verify that having these conditions both hold is equivalent to the following:  4γα  λ  ≤ 4  ��γ  b  �2  Vµ L(h) + γ log(1/δ)  n  �  ≤ 1 − 4γα  λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  (71)  As such, whenever α, δ, and b are such that (71) holds, using a union bound, it follows that  with probability no less than 1 − 2δ, we have a bound on  |An − (Eµ L(h) − 2(α/λ)b)| ≤ 2  �γ  b Vµ L(h) + b log(1/δ)  n  �  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The proposition statement takes a cleaner form since we have γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The lack of convexity follows from the fact that the composition of  two convex functions need not be convex when the outermost function is non-monotonic (see  for example Boyd and Vandenberghe [4, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' 3]), and the lack of smoothness follows a fortiori  from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  C  Empirical test appendix  For reference, we give the population versions of the CVaR and χ2-DRO criteria used in the  empirical tests of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' First, it is well known (see Rockafellar and Uryasev [30]) that CVaR at  quantile level ξ can be represented as  CVaRµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' ξ) = inf  a∈R  �  a +  1  1 − ξ Eµ (L(h) − a)+  �  (72)  where (x)+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= max{0, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Similarly, DRO risk based on the Cressie-Read family of divergence  functions is formulated (for any c > 1) using  DROµ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' η) = inf  a∈R  �  a + (1 + c(c − 1)η)1/c �  Eµ (L(h) − a)c∗  +  �1/c∗�  (73)  where c∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='.= c/(c − 1), and χ2-DRO is the special case where c = 2 [11, 13, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' The different  “robustness levels” mentioned in §4 correspond to a re-parameterized quantity �η ∈ (0, 1),  related to η by the equality η = (1/(1 − �η) − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' Just as our �Cn(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content=' a, b) is solved jointly in  (h, a, b), our empirical tests minimize the empirical versions of (72) and (73) jointly in (h, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
+page_content='  27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFJT4oBgHgl3EQflizf/content/2301.11584v1.pdf'}
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+arXiv:2301.11593v1  [math.AG]  27 Jan 2023
+THE DONOVAN–WEMYSS CONJECTURE VIA THE
+TRIANGULATED AUSLANDER–IYAMA CORRESPONDENCE
+GUSTAVO JASSO, BERNHARD KELLER, AND FERNANDO MURO
+Abstract. We provide an outline of the proof of the Donovan–Wemyss Con-
+jecture in the context of the Homological Minimal Model Program for three-
+folds. The proof relies on results of August, of Hua and the second-named au-
+thor, Wemyss, and on the Triangulated Auslander–Iyama Correspondence—a
+recent result by the first- and third-named authors.
+Contents
+Introduction
+1
+1.
+Preliminaries
+2
+2.
+The Derived Donovan–Wemyss Conjecture
+7
+3.
+Uniqueness of the 2Z-derived contraction algebra
+8
+4.
+Concluding remarks
+20
+References
+23
+Introduction
+We work over the field C of complex numbers. A compound Du Val (=cDV)
+singularity is a complete local hypersurface
+R ∼= C�x, y, z, t�
+(f + tg) ,
+where C�x, y, z�/(f) is a Kleinian surface singularity and g ∈ C�x, y, z, t� is arbi-
+trary. Introduced by Reid in the early 1980s [Rei83], cDV singularities form an
+important class of three-dimensional singularities in birational geometry and play
+a significant role in the Minimal Model Program (MMP) for threefolds [KM98,
+Sec. 5.3] as well as in the Homological MMP [Wem18].
+We refer the reader
+to [Aug19, Ch. 1] and [Wem21] for introductions to the subject.
+This note is concerned with the following geometric situation: Let R be an iso-
+lated cDV singularity and p: X → Spec(R) a crepant resolution, that is p is a
+proper birational map with smooth source such that the pullback of the dualising
+sheaf of Spec(R) along f is the dualising sheaf of X. It follows that Spec(R) has
+a unique singular point m and the (reduced) exceptional fibre p−1(m) = �n
+i=1 Ci is
+a union of curves, with Ci ∼= P1
+C [VdB04, Lemma 3.4.1]. To these data, Donovan
+and Wemyss [DW16, DW19] associate a (basic, connected) finite-dimensional alge-
+bra Λcon = Λcon(p), the contraction algebra of p, which represents the functor of
+‘simultaneous non-commutative deformations’ of the reduced exceptional fibre. By
+construction, Λcon is a Cn-augmented algebra, and hence in particular determines
+the number n of irreducible components of the exceptional fibre. The contraction
+2020 Mathematics Subject Classification. Primary 14E30; Secondary 13D03.
+Key words and phrases. Minimal model program; compound Du Val singularity; crepant res-
+olution; contraction algebra; Hochschild cohomology; Auslander correspondence.
+1
+
+2
+G. JASSO, B. KELLER, AND F. MURO
+algebra encodes a surprising amount of information stemming from the given geo-
+metric setup. For example, when p contracts a single curve, the contraction algebra
+recovers known invariants such as Reid’s width [Rei83] and the Gopakumar–Vafa
+invariants [Kat08], see [Tod15]. Neither the dimension nor the Gabriel quiver of
+contraction algebras suffice for differentiating cDV singularities [DW16, Table 2].
+In fact, it is well-known that there are continuous families of pairwise non isomor-
+phic cDV singularities (that is ‘cDV singularities have moduli’). Notwithstanding,
+at the risk of stating the obvious, let us point out that the contraction algebra is
+equiped with crucial data in the form of the multiplication law and that this law
+is essential in recovering the above mentioned invariants. Equipped with their al-
+gebra structure, contraction algebras distinguish between non-isomorphic isolated
+cDV singularities that admit a crepant resolution in all known examples. These
+considerations motivate the following remarkable conjecture.
+Conjecture A (Donovan and Wemyss [DW16]). Let R1 and R2 be isolated cDV
+singularities with crepant resolutions
+p1 : X1 → Spec(R1)
+and
+p2 : X2 → Spec(R2).
+Then, the contraction algebras Λcon(p1) and Λcon(p2) are derived equivalent if and
+only if there is an isomorphism of algebras R1 ∼= R2.
+The original conjecture was formulated only in the case of single-curve contrac-
+tions; algebraically, this corresponds to the case where the contraction algebras are
+local and thus derived equivalence reduces to mere isomorphism of algebras since
+contraction algebras are basic, see [Zim14, Prop. 6.7.4] for example. In the above
+form, which allows for contracting multiple curves, the conjecture appeared in print
+in [Aug20, Conj. 1.3].
+That the contraction algebras of a given isolated cDV singularity are derived
+equivalent follows by combining results from Wemyss [Wem18] and Dugas [Dug15].
+In this note we provide an outline of the proof of the remaining part of Conjecture A.
+This proof first appeared in the appendix to [JM22] written by the second-named
+author where it is explained how the conjecture follows by combining previous
+results of August [Aug20] and [HK18] with the Triangulated Auslander–Iyama Co-
+rrespondence—the main result in [JM22]. For the sake of concreteness, we restrict
+ourselves to the specific context of the conjecture, with the understanding that
+most concepts and results that are presented here are but special cases of a much
+more general theory that the reader can find in the original sources. We hope that
+this sacrifice in generality makes the proof of the conjecture accessible to a broader
+readership.
+Acknowledgements. The authors thank Michael Wemyss and Zheng Hua for in-
+teresting conversations and email exchanges; in addition, they thank M. W. for sug-
+gesting the terminology ‘2Z-derived contraction algebra’ used in these proceedings.
+The third-named author is algo grateful to Matt Booth for answering questions.
+Financial
+support. F.
+M.
+was
+partially
+supported
+by
+grants
+PID2020-
+117971GB-C21
+funded
+by
+MCIN/AEI/10.13039/501100011033,
+US-1263032
+(US/JUNTA/FEDER, UE), and P20 01109 (JUNTA/FEDER, UE).
+1. Preliminaries
+In this section we collect preliminary definitions and results that are needed in
+our proof of Conjecture A. We use freely the theories of differential graded cate-
+gories [Kel94, Kel06] and A∞-categories [LH03]. We denote the derived category
+of an algebra or, more generally, a DG algebra A by D(A); the perfect derived
+
+THE DONOVAN–WEMYSS CONJECTURE
+3
+category of A, that is the full subcategory of D(A) spanned by its compact objects,
+is denoted by Dc(A). All (DG) modules are right (DG) modules.
+1.1. 2Z-cluster tilting objects. Let T be a triangulated category whose underly-
+ing additive category is Krull–Schmidt and has finite-dimensional morphism spaces.
+Definition 1.1.1 ([IY08, GKO13]). A basic1 object T ∈ T is 2-cluster tilting if the
+following conditions hold:
+(1) The object T is rigid: T(T, T [1]) = 0.
+(2) For each object X ∈ T there exists a triangle T1 → T0 → X → T1[1] with
+T0, T1 ∈ add(T ), where add(T ) is the smallest additive subcategory of T
+containing T that is closed under direct summands.
+We say that T is 2Z-cluster tilting if it is 2-cluster tilting and T ∼= T [2].
+Remark 1.1.2. Clearly, if T ∈ T is a 2- or 2Z-cluster tilting object, then T generates
+T as a triangulated category with split idempotents (which is to say that T is a
+classical generator of T). In particular, if the triangulated category T is algebraic2
+then there exists a DG algebra A and an equivalence of triangulated categories
+T
+∼
+−→ Dc(A),
+T �−→ A.
+Remark 1.1.3. Given a basic 2-cluster tilting object T ∈ T, one can produce a new
+such object by a procedure called mutation that, in a nutshell, replaces a single
+indecomposable direct summand of T by a new one, see [IY08] for a precise defini-
+tion. This process, which can be iterated, is an important reason for introducing
+2-cluster tilting objects into the framework of the Homological MMP, see [Wem18]
+and compare with Section 1.3.
+Remark 1.1.4. In general, 2Z-cluster tilting objects are not invariant under mu-
+tation (however, see [HI11, Sec. 4]). In the context of the Homological MMP this
+problem does not occur since the notions of 2- and 2Z-cluster tilting object coincide,
+see Section 1.2.
+1.2. Maximal Cohen–Macaulay modules and singularity categories. Let
+R be an isolated cDV singularity and
+CM(R) := {M ∈ mod(R)| depth(M) = dim(R)}
+be the category of maximal Cohen–Macaulay R-modules [Yos90, LW12]. The cate-
+gory CM(R) is a Frobenius exact category and, therefore, the stable category CM(R)
+has an induced structure of a triangulated category; moreover, there is a canonical
+equivalence of triangulated categories
+CM(R)
+∼
+−→ Dsg(R) := Db(mod(R))/ Kb(proj(R)),
+where Dsg(R) is the singularity category of R, see [Buc21] for details. We record
+the following facts for later use:
+• [Yos90, Prop. 1.18] Since R is complete local, CM(R) ≃ Dsg(R) is a Krull–
+Schmidt category with finite-dimensional morphism spaces.
+• [Aus78] Since R is 3-dimensional, CM(R) ≃ Dsg(R) is a 2-Calabi–Yau tri-
+angulated category [Kon98, Kel08], that is there is a natural isomorphism
+DHom(X, Y )
+≃
+−→ Hom(Y, X[2]),
+X, Y ∈ CM(R) ≃ Dsg(R),
+where V �→ DV denotes the passage to the C-linear dual.
+1An object in a Krull–Schmidt additive category is basic if all of its indecomposable direct
+summands have multiplicity one.
+2A triangulated category is algebraic if it is equivalent—as a triangulated category—to the
+stable category of a Frobenius exact category [Kel94].
+
+4
+G. JASSO, B. KELLER, AND F. MURO
+• [Eis80] Since R is a hypersurface, CM(R) ≃ Dsg(R) is a 2-periodic triangu-
+lated category, that is there is an isomorphism of exact functors [2] ∼= 1.
+In particular, the notions of 2- and 2Z-cluster tilting object coincide in this
+context.
+• The endomorphism algebra of any basic object X in CM(R) ≃ Dsg(R) is
+a symmetric algebra.
+This is an immediate consequence of the natural
+isomorphisms
+Hom(X, X) ∼= D Hom(X, X[2]) ∼= D Hom(X, X) .
+We also consider the DG category Dsg(R)dg, which is defined as the DG quo-
+tient [Kel99, Dri04] of the canonical DG enhancements of the triangulated cate-
+gories Db(mod(R)) and Kb(proj(R)). By construction,
+H0(Dsg(R)dg) = Dsg(R).
+1.3. Contraction algebras via 2Z-cluster tilting objects. Let R be an isolated
+cDV singularity and p: X → Spec(R) a crepant resolution. As explained in the
+introduction, to this geometric setup Donovan and Wemyss associate a basic finite-
+dimensional algebra Λcon = Λcon(p). We recall an alternative construction of the
+algebra Λcon that is more adapted to the methods we utilise in this note. Given p
+as above, a theorem of Van den Bergh [VdB04] furnishes a tilting bundle
+OX ⊕ N = OX ⊕ N(p) ∈ coh(X)
+and Wemyss proves [Wem18] that there is an isomorphism of algebras
+Λcon ∼= EndR(N)
+between the contraction algebra of p and the stable endomorphism algebra of
+N := H0(N) ∈ CM(R). Remarkably, when viewed as an object of the triangulated
+category CM(R), the R-module N is a 2Z-cluster tilting object. Conversely, given
+a 2Z-cluster tilting object T ∈ CM(R), there exists a crepant resolution of Spec(R)
+whose associated contraction algebra is isomorphic to EndR(T ). We summarise the
+previous discussion in the following theorem.3
+Theorem 1.3.1 ([Wem18]). Let R be an isolated cDV singularity and assume
+that Spec(R) admits a crepant resolution. Then, the contraction algebras of R are
+precisely the endomorphism algebras of 2Z-cluster tilting objects in the triangulated
+category CM(R) ≃ Dsg(R).
+The following theorem of August reduces Conjecture A from a derived equiv-
+alence to an isomorphism problem.
+The proof leverages the characterisation of
+contraction algebras provided by Theorem 1.3.1.
+Theorem 1.3.2 ([Aug20, Thm. 1.4]). Let R be an isolated cDV singularity and
+assume that Spec(R) admits a crepant resolution. The contraction algebras of R
+form a single and complete derived equivalence class of basic algebras.
+Corollary 1.3.3. Let R1 and R2 be isolated cDV singularities with crepant reso-
+lutions
+p1 : X1 → Spec(R1)
+and
+p2 : X2 → Spec(R2)
+and corresponding contraction algebras Λ1 = Λcon(p1) and Λ2 = Λcon(p2). If the
+algebras Λ1 and Λ2 are derived equivalent, then there exists a contraction algebra
+Λ of R2 such that Λ ∼= Λ1.
+3Wemyss proves even more: Up to isomorphism on both sides, crepant resolutions of R cor-
+respond bijectively to (basic) 2Z-cluster tilting objects in Dsg(R). In particular, the number of
+isomorphism classes of 2Z-cluster tilting objects in Dsg(R) if finite, for the number of minimal
+models of Spec(R) is finite [KM87].
+
+THE DONOVAN–WEMYSS CONJECTURE
+5
+1.4. Hochschild cohomology. Let A be a graded algebra.
+The bigraded
+Hochschild cochain complex has components
+Cp,q(A, A) := HomC
+�
+A⊗p, A(q)
+�
+,
+p ≥ 0, q ∈ Z,
+where V �→ V (1) is the (vertical) degree shift of graded vector spaces, equipped
+with the Hochschild differential x �→ ∂(x) of bidegree (1, 0), see [Mur20] for the
+precise definition, related structure (described below) and sign conventions. The
+first degree is called horizontal or Hochschild degree and the second is the vertical
+or internal degree; the sum of both is the total degree and we denote it by |x| (we
+also use this notation for the degree of an element in a singly-graded vector space).
+The component of total degree n of the Hochschild complex is
+�
+p+q=n
+Cp,q(A, A) .
+The Hochschild complex is equipped with a brace algebra structure, consisting
+of operations called braces (which we do not describe explicitly here)
+C•,∗(A, A)⊗n+1
+−→ C•,∗(A, A) ,
+x0 ⊗ x1 ⊗ · · · ⊗ xn �−→ x0{x1, . . . , xn},
+that are defined for all n ≥ 1, and satisfy the brace relation
+x{y1, . . . , yp}{z1, . . . , zq}
+=
+�
+0≤i1≤j1≤···≤ip≤jp≤q
+(−1)ǫ{z1, . . . , zi1, y1{zi1+1, . . . , zj1}, zj1+1, . . .
+. . . , zip, yq{zip+1, . . . , zjp}, zjp+1, . . . , zq}.
+Above, ǫ reflects the Koszul sign rule with respect to the total degree shifted by
+−1. Brace operations have horizontal degree −n and vertical degree 0 and the
+n-th brace operation vanishes when x0 has horizontal degree < n. The Hochschild
+complex is a bigraded associative algebra equipped with the cup-product
+x · y = (−1)|x|−1m2{x, y},
+where m2 ∈ C2,0(A, A) is, up to a sign, the multiplication in the graded algebra A.
+The Hochschild complex also has the structure of a (horizontally-shifted) graded
+Lie algebra, with Lie bracket
+[x, y] = x{y} − (−1)(|x|−1)(|y|−1)y{x}
+of horizontal degree −1 and vertical degree 0. The Hochschild differential is given
+by
+∂(x) = [m2, x]
+and satisfies the corresponding Leibniz rules with respect to the previous associative
+product and Lie bracket.
+The relation between brace operations, the Hochschild differential and the cup
+product are encoded in the following straightforward consequences of the brace
+relation.
+
+6
+G. JASSO, B. KELLER, AND F. MURO
+Lemma 1.4.1. The following formula holds for all n ≥ 1:
+∂(x0{x1, . . . , xn}) = ∂(x0){x1, . . . , xn}
++
+n
+�
+i=1
+(−1)
+�i−1
+j=0 |xj|−ix0{x1, . . . , ∂(xi), . . . , xn}
++ (−1)|x0|−1+|x0||x1|x1 · x0{x2, . . . , xn−1}
++
+n−1
+�
+i=1
+(−1)
+�i
+j=0 |xi|−i−1x0{x1, . . . , xi · xi+1, . . . , xn}
++ (−1)
+�n−1
+j=0 |xj|−n−1x0{x1, . . . , xn−1} · xn.
+Lemma 1.4.2. The following formula holds for all n ≥ 1:
+(x · y){z1, . . . , zn} =
+n
+�
+i=0
+(−1)|y| �i
+j=1(|zi|−1)x{z1, . . . , zi} · y{zi+1, . . . , zn}.
+Lemma 1.4.1 for n = 1 proves that the induced associative product in Hochschild
+cohomology
+HH•,∗(A, A) = H•,∗(C•,∗(A, A))
+(the cohomology of the Hochschild complex) is graded commutative with respect to
+the total degree. This products satisfies the following compatibility relation with
+the (horizontally shifted) Lie algebra structure,
+[x, y · z] = [x, y] · z + (−1)(|x|−1)|y|y · [x, z],
+and hence Hochschild cohomology is a Gernstenhaber algebra. For this we use both
+Lemmas 1.4.1 and 1.4.2, for n = 2 and n = 1 respectively. The Hochschild complex
+C•,∗(A, M) and Hochschild cohomology HH•,∗(A, M) are defined, more generally,
+for M an A-bimodule, but it does not have any multiplicative structure in this
+general case. For the existence of a graded associative algebra structure it suffices
+that M be an associative algebra in A-bimodules, see also [Mur22, Sec. 1].
+1.5. Minimal A∞-algebras. We now describe minimal A∞-algebras and their
+morphisms in terms of the Hochschild complex. A minimal A∞-algebra structure
+on a graded algebra A is a Hochschild cochain
+m = (0, 0, 0, m3, . . . , mn, . . . ) ∈ C•,∗(A, A)
+of total degree 2 such that the Maurer–Cartan equation
+(1.5.1)
+∂(m) + m{m} = ∂(m) + 1
+2[m, m] = 0
+is satisfied. The pair (A, m) is also denoted
+(A, m3, . . . , mn, . . . ).
+If g : A′ → A is a graded algebra isomorphism, then
+m ∗ g = (0, 0, 0, g−1m3g⊗3, . . . , g−1mng⊗n, . . . )
+is a minimal A∞-algebra structure on A′. If
+m′ = (0, 0, 0, m′
+3, . . . , m′
+n, . . . ) ∈ C•,∗(A, A)
+is another minimal A∞-algebra structure on A, an A∞-isomorphism with identity
+linear part
+f : (A, m) −→ (A, m′)
+is a Hochschild cochain
+f = (0, 0, f2, f3, . . . , fn, . . . ) ∈ C•,∗(A, A)
+
+THE DONOVAN–WEMYSS CONJECTURE
+7
+of total degree 1 such that the following Hochschild cochain vanishes
+(1.5.2)
+∂(f) + f · f +
+�
+r≥0
+m′{f, r. . ., f} − m − f{m}.
+More generally, an A∞-isomorphism between minimal A∞-algebras
+f : (A, m) −→ (A′, m′)
+consists of an isomorphism of graded algebras
+f1 : A −→ A′
+and a Hochschild cochain
+(0, 0, f2, f3, . . . , fn, . . . ) ∈ C•,∗(A, A′)
+of total degree 1 such that
+(0, 0, f −1
+1 f2, f −1
+1 f3, . . . , f −1
+1 fn, . . . ): (A, m) −→ (A, m′ ∗ f1)
+is an A∞-isomorphism with identity linear part.
+2. The Derived Donovan–Wemyss Conjecture
+In this section we discuss one of the main results in [HK18]—crucial to our proof
+of the Dononvan–Wemyss conjecture—and explain how it implies a derived version
+of the conjecture (Corollary 2.2.4).
+2.1. 2Z-derived contraction algebras. By means of the equivalence of triangu-
+lated categories CM(R) ≃ Dsg(R), the contraction algebra associated to a crepant
+resolution p of an isolated cDV singularity can be promoted to the DG algebra
+Λcon = Λcon(p) := REnd(N)
+given by the derived endomorphism algebra of the corresponding 2Z-cluster tilt-
+ing object N = N(p) ∈ Dsg(R)dg. By construction H0(Λcon) ∼= Λcon and, as a
+consequence of the 2-periodicity of the singularity category of R,
+H•(Λcon) ∼= Λcon[ı±1] = Λcon ⊗C C[ı±1],
+|ı| = −2.
+We refer to the DG algebra Λcon as the 2Z-derived contraction algebra of p. The
+(soft) non-positive truncation Λ≤0
+con = τ ≤0Λcon of Λcon is quasi-isomorphic to the
+derived contraction algebra of p considered for example in [Boo19, Boo21, HK18],
+and there is an isomorphism of graded algebras
+H•(Λ≤0
+con) ∼= Λcon[ı] = Λcon ⊗C C[ı],
+|ı| = −2.
+The 2Z-derived contraction algebra Λcon is a localisation of Λ≤0
+con (see [Boo21,
+Thms. 6.4.6 and 7.2.3] and [HK18, Thm. 4.17]) and Λcon can also be interpreted
+as a non-connective variant of Λ≤0
+con.
+Notice also that, since 2Z-cluster tilting objects are in particular classical gen-
+erators, there is a canonical quasi-equivalence of DG categories
+Dc(Λcon)dg
+∼
+−→ Dsg(R)dg
+that induces an equivalence of triangulated categories
+Dc(Λcon)
+∼
+−→ Dsg(R).
+Although we do not need this fact in the sequel, we mention that there is an
+equivalence of triangulated categories4 [HK18, Lemma 5.12]
+C(Λ≤0
+con) := Dc(Λ≤0
+con)/ Dfd(Λ≤0
+con) ≃ Dsg(R),
+4For a DG algebra A, we denote by Dfd(A) the full subcategory of D(A) spanned by the DG
+A-modules with finite-dimensional total cohomology.
+
+8
+G. JASSO, B. KELLER, AND F. MURO
+that is compatible with the canonical DG enhancements on either side. The cate-
+gory C(Λ≤0
+con) is known as the Amiot cluster category of Λ≤0
+con [Ami07] and, indeed,
+establishing a link between birational geometry and the theory of cluster categories
+was one of the objectives in [HK18].
+2.2. The Derived Donovan–Wemyss Conjecture. The following theorem of
+Hua and the second-named author settles a derived version of Conjecture A.
+Theorem 2.2.1 ([HK18, Thm. 5.9]). Let R = C�x, y, z, t�/(f) be an isolated cDV
+singularity. Then, there is an isomorphism of algebras
+HH0(Dsg(R)dg) ∼=
+C�x, y, z, t�
+(f, ∂xf, ∂yf, ∂zf, ∂tf)
+between the 0-th Hochschild cohomology of the DG category Dsg(R)dg and the Tyu-
+rina algebra of R. In particular, if R′ is a further isolated cDV singularity such
+that the DG categories Dsg(R)dg and Dsg(R′)dg are quasi-equivalent, then there is
+an isomorphism of algebras R ∼= R′.
+Remark 2.2.2. The proof of Theorem 2.2.1 relies on a comparison result [Kel18,
+Kel19] between the singular Hochschild cohomology (=Hochschild–Tate cohomol-
+ogy) of R and the Hochschild cohomology of the DG category Dsg(R)dg.
+The
+appearance of the Tyurina algebra stems from an earlier result of the Buenos Aires
+Cyclic Homology Group [GGRV92, Thm. 3.2.7]. That the Tyurina algebra, together
+with the dimension of R, determines the isomorphism type of the singularity is a
+theorem of Mather and Yau [MY82].
+Remark 2.2.3. In [Dyc11], Dyckerhoff shows that the 0-th Hochschild cohomology
+of Dsg(R)dg—viewed as a Z/2-graded DG category—is isomorphic to the Milnor
+algebra
+C�x, y, z, t�
+(∂xf, ∂yf, ∂zf, ∂tf)
+of the singularity (which does not determine the isomorphism type of the singularity,
+even if one knows the dimension). Thus, in Theorem 2.2.1 it is crucial to consider
+Dsg(R)dg as a Z-graded DG category.
+Corollary 2.2.4 (Derived Donovan–Wemyss Conjecture). Let R1 and R2 be iso-
+lated cDV singularities with crepant resolutions
+p1 : X1 → Spec(R1)
+and
+p2 : X2 → Spec(R2).
+If the 2Z-derived contraction algebras Λcon(p1) and Λcon(p2) are quasi-isomorphic,
+then there is an isomorphism of algebras R1 ∼= R2.
+Proof. Indeed, if the DG algebras Λcon(p1) and Λcon(p2) are quasi-isomorphic, then
+the DG categories
+Dc(Λcon(p1))dg ≃ Dsg(R1)dg
+and
+Dc(Λcon(p2))dg ≃ Dsg(R2)dg
+are quasi-equivalent. Theorem 2.2.1 then implies the existence of an isomorphism
+of algebras R1 ∼= R2.
+□
+3. Uniqueness of the 2Z-derived contraction algebra
+In this section we prove that 2Z-derived contraction algebras are determined
+up to quasi-isomorphism by their zeroth cohomology plus a minimal amount of
+additional algebraic data (see Corollary 3.4.6 for the precise statement). Before
+that, we formulate a closely related result (Theorem 3.1.1) that states that two 2Z-
+derived contraction algebras whose zeroth cohomologies are isomorphic as algebras
+must be quasi-isomorphic, and use this result to prove Conjecture A.
+
+THE DONOVAN–WEMYSS CONJECTURE
+9
+3.1. Proof of the Donovan–Wemyss Conjecture. In view of Corollaries 1.3.3
+and 2.2.4, Conjecture A is an immediate consequence of the following theorem, the
+proof of which is given in Section 3.4.
+Theorem 3.1.1. Let R1 and R2 be isolated cDV singularities with crepant resolu-
+tions
+p1 : X1 → Spec(R1)
+and
+p2 : X2 → Spec(R2).
+If the contraction algebras Λ(p1) and Λ(p2) are isomorphic, then the 2Z-derived
+contraction algebras Λcon(p1) and Λcon(p2) are quasi-isomorphic.
+Remark
+3.1.2.
+Theorem 3.1.1
+is
+a
+special
+case
+of
+[JM22,
+Thm.
+5.1.10],
+see Section 4.2.
+Proof of Conjecture A using Theorem 3.1.1. Let R1 and R2 be isolated cDV sin-
+gularities with crepant resolutions
+p1 : X1 → Spec(R1)
+and
+p2 : X2 → Spec(R2)
+whose corresponding contraction algebras Λcon(p1) and Λcon(p2) are derived equiva-
+lent. In view of Corollaries 1.3.3 and 2.2.4, we may and we will assume that Λcon(p1)
+and Λcon(p2) are isomorphic and hence, by Theorem 3.1.1, the 2Z-derived contrac-
+tion algebras Λcon(p1) and Λcon(p2) are quasi-isomorphic. Finally, Corollary 2.2.4
+yields the desired algebra isomorphism R1 ∼= R2.
+□
+3.2. The restricted universal Massey product. The proof of Theorem 3.1.1
+makes use of an invariant of the 2Z-derived contraction algebra, a certain Hochschild
+cohomology class of bidegree (4, −2) that we call the restricted universal Massey
+product. As we explain below, this invariant is induced by the first possibly non-
+trivial higher operation on a minimal A∞-algebra model of the 2Z-derived contrac-
+tion algebra.
+Setting 3.2.1. Fix an isolated cDV singularity R that admits a crepant resolution
+p: X → Spec(R), and let Λ = Λcon(p) be the corresponding 2Z-derived contraction
+algebra so that H0(Λ) ∼= Λ = Λcon(p) is the contraction algebra defined by Donovan
+and Wemyss. For simplicity, we treat the isomorphism of graded algebras
+H•(Λ) ∼= Λ[ı±1] = Λ ⊗C C[ı±1],
+|ı| = −2,
+as an identification.
+Kadeishvili’s Homotopy Transfer Theorem [Kad82] provides us with a minimal
+A∞-algebra structure, unique up to A∞-isomorphism with identity linear part,
+B = (Λ[ı±1], m3, m4, m5, · · · )
+on the cohomology algebra Λ[ı±1]. Since Λ[ı±1] is concentrated in even degrees and,
+by definition,
+mn : Λ[ı±1]⊗n −→ Λ[ı±1]
+is a morphism of degree 2 − n, we conclude that mn = 0 whenever n is odd. We
+write
+B = (Λ[ı±1], m4, m6, m8, · · · )
+as a way to record this observation. We refer to B as a minimal (A∞-algebra)
+model of the DG algebra Λ and fix it for the rest of the section.
+Remark 3.2.2. The passage from DG to A∞-algebras is a matter of technical
+convenience: The homotopy theories of non-unital DG and of A∞-algebras are
+equivalent, [LV12, Thm. 11.4.8]. In particular, two non-unital DG algebras are
+quasi-isomorphic if and only if their minimal models are A∞-isomorphic [LV12,
+Thms. 11.4.9 and 10.3.10]. Here, we are exclusively interested in unital DG algebras,
+
+10
+G. JASSO, B. KELLER, AND F. MURO
+but this is not a problem since by [Mur14, Prop. 6.2] two unital DG algebras are
+quasi-isomorphic as non-unital DG algebras if and only if they are quasi-isomorphic
+as unital DG algebras.
+Consider now the bigraded Hochschild cochain complex
+Cp,q�
+Λ[ı±1], Λ[ı±1]
+� := HomC
+�
+Λ[ı±1]⊗p, Λ[ı±1](q)
+�
+,
+p ≥ 0, q ∈ Z,
+recalled in Section 1.4. Since m3 = 0, the A∞-equations imply that ∂(m4) = 0
+([LH03, Lemme B.4.1]); hence we obtain a class
+(3.2.3)
+{m4} =
+�
+mΛ
+4
+�
+∈ HH4,−2�
+Λ[ı±1], Λ[ı±1]
+�
+that we call the universal Massey product (of length 4). It follows from the definition
+of A∞-morphism ([LH03, Lemme B.4.2]) that the class {m4} does not depend on
+the choice of minimal model for Λ and hence the universal Massey product can and
+will be regarded as an invariant of the latter DG algebra.
+Consider now the graded-algebra morphism j : Λ ֒→ Λ[ı±1] given by the inclusion
+of the degree 0 part. The morphism j induces a restriction morphism on Hochschild
+cohomology5
+j∗: HH•,∗�
+Λ[ı±1], Λ[ı±1]
+�
+−→ HH•,∗�
+Λ, Λ[ı±1]
+�
+,
+where the space on the right is the Hochschild cohomology of Λ, viewed as a graded
+algebra concentrated in degree 0, with coefficients in the graded Λ-bimodule Λ[ı±1].
+In particular, since the degree −2 component Λ · ı of Λ[ı±1] is isomorphic to the
+diagonal Λ-bimodule, we obtain a class
+(3.2.4)
+j∗{m4} = j∗�
+mΛ
+4
+�
+∈ HH4,−2�
+Λ, Λ[ı±1]
+�
+= HH4(Λ, Λ · ı) = Ext4
+Λe(Λ, Λ) ,
+where Λe = Λ ⊗C Λop is the eveloping algebra of Λ; we call the class j∗{m4}
+the restricted universal Massey product (of length 4) and, as with the unrestricted
+version, we regard it as an invariant of the 2Z-derived contraction algebra Λ. Notice
+also that the previous discussion applies verbatim to any DG algebra A whose
+cohomology is isomorphic to the graded algebra Λ[ı±1], so that we may associate
+to A its restricted universal Massey product j∗�
+mA
+4
+�
+.
+The following theorem is the first main step towards the proof of Theorem 3.1.1.
+Theorem 3.2.5. The restricted universal Massey product j∗{m4}, when viewed as
+an element of the space Ext4
+Λe(Λ, Λ) of Yoneda extensions of Λ-bimodules, can be
+represented by an exact sequence
+0 → Λ → P3 → P2 → P1 → P0 → Λ → 0
+with projective middle terms. In particular, Ω4
+Λe(Λ) ∼= Λ in the stable category of
+Λ-bimodules.
+Proof. The first claim is a special case of [JM22, Cor. 4.5.17]. Indeed, by definition,
+the 2Z-derived contraction algebra Λ is the derived endomorphism algebra of a 2Z-
+cluster tilting object in Dc(Λ) ≃ Dsg(R), which is one of the equivalent conditions
+in loc. cit. The second claim follows immediately from the first.
+□
+Remark 3.2.6. The proof of [JM22, Cor. 4.5.17], and hence that of Theorem 3.2.5, is
+non-trivial. In the special case of the contraction algebra, it is possible that detailed
+knowledge of the first non-trivial higher operation m4 of some minimal model of
+the 2Z-derived contraction algebra allows for establishig the desired property of the
+restricted universal Massey product j∗{m4} directly. The approach taken in [JM22],
+which deals with an abstract and more general situation, rather leverages the fact
+that Λ is the endomorphism algebra of a 2Z-cluster tilting object T ∈ Dsg(R).
+5In fact, the morphism j∗ is surjective, see [JM22, Prop. 4.6.9] and take σ = 1 and d = 2
+(which is even and hence no signs occur in the formulas therein).
+
+THE DONOVAN–WEMYSS CONJECTURE
+11
+The upshot is that the additive closure add(T ) of T has an induced structure of a
+so-called 4-angulated category, that is add(T ) is equipped with a natural class of
+diagrams □GKO, called 4-angles, of the form6
+T1 → T2 → T3 → T4 → T1[2]
+that satisfies axioms analogous to those of triangulated categories [GKO13]. On
+the other hand, an extension of Λ-bimodules
+0 → Λ → P3 → P2 → P1 → P0 → Λ → 0
+with P0, P1, P2 projective-injective (but perhaps not P3) that represents the class
+j∗{m4} ∈ Ext4
+Λe(Λ, Λ) yields a class of 4-angles □j∗{m4} defined in terms of certain
+exactness properties [Ami07, Lin19]; the class □j∗{m4} is a priori not known to
+form a 4-angulation of add(T ). The crux of the argument is then to prove that
+□GKO = □j∗{m4}
+so that the class □j∗{m4} is indeed a 4-angulation of add(T ); this agreement relies
+on a delicate analysis of the relationship between Toda brackets, Massey products
+and the classes □GKO and □j∗{m4}. Finally, in view of the exactness properties
+defining the class □j∗{m4} (now known to be 4-angulation), a theorem of Auslander
+and Reiten [AR91] for detecting projective bimodules implies that P3 must be a
+projective Λ-bimodule, which is what Theorem 3.2.5 claims. The reader is referred
+to [JM22] for details.
+Recall that the contraction algebra is Frobenius (in fact, symmetric). Conse-
+quently, its enveloping algebra is also a Frobenius algebra and we may consider the
+Hochschild–Tate cohomology
+HH•,∗�
+Λ, Λ[ı±1]
+� := Ext•,∗
+Λe
+�
+Λ, Λ[ı±1]
+�
+defined in terms of the extension spaces in the stable category of graded Λ-
+bimodules; thus,
+HH>0,∗�
+Λ, Λ[ı±1]
+�
+= HH>0,∗�
+Λ, Λ[ı±1]
+�
+and there is a surjection
+HH0,∗�
+Λ, Λ[ı±1]
+�
+։ HH0,∗�
+Λ, Λ[ı±1]
+�
+.
+The multiplication on Λ[ı±1] endows HH•,∗�
+Λ, Λ[ı±1]
+�
+with the structure of a bi-
+graded algebra, see [Mur22, Sec. 5] for details.
+Corollary 3.2.7. The restricted universal Massey product j∗{m4}, when viewed
+as an element of the Hochschild–Tate cohomology HH•,∗�
+Λ, Λ[ı±1]
+�
+, is a unit.
+Proof. Immediate from Theorem 3.2.5 and [Mur22, Prop. 5.7 and Rmk. 5.8], which
+characterises the units in HH•,∗�
+Λ, Λ[ı±1]
+�
+of positive Hochschild (=horizontal) de-
+gree.
+□
+Remark 3.2.8. In Corollary 3.2.7 it is essential to pass from Hochschild to
+Hochschild–Tate cohomology in order to have units of positive Hochschild degree.
+6Recall that [2] ∼
+= 1 in Dsg(R).
+
+12
+G. JASSO, B. KELLER, AND F. MURO
+3.3. Hochschild cohomology of the graded contraction algebra. In this
+section we compute the Hochschild cohomology of the graded algebra
+Λcon[ı±1] = Λ[ı±1] = Λ ⊗C C[ı±1],
+|ı| = −2,
+that we call graded contraction algebra, in terms of the Hochschild cohomology of
+the Dononvan–Wemyss contraction algebra Λcon = Λ.
+First, notice that ı lies in the (graded) centre of Λ[ı±1], which is
+Z(Λ[ı±1]) = HH0,∗�
+Λ[ı±1], Λ[ı±1]
+�
+;
+hence
+ı ∈ HH0,−2�
+Λ[ı±1], Λ[ı±1]
+�
+.
+We introduce the fractional Euler derivation
+¯δ ∈ C1,0�
+Λ[ı±1], Λ[ı±1]
+�
+,
+which acts by the formula
+¯δ : a �−→ |a|
+2 a,
+where we observe that |a|
+2 is an integer since Λ[ı±1] is concentrated in even degrees.
+It is a cocycle with cohomology class
+δ ∈ HH1,0�
+Λ[ı±1], Λ[ı±1]
+�
+.
+Proposition 3.3.1. The following statements hold:
+(1) There is an isomorphism of graded commutative algebras
+HH•,∗�
+Λ[ı±1], Λ[ı±1]
+� ∼= HH•(Λ, Λ) [ı±1, δ].
+The graded Lie algebra structure on the right hand side is induced by the
+(usual) Lie algebra structure on HH•,∗(Λ, Λ) by setting
+[ı, HH•(Λ, Λ)] = 0,
+[ı, ı] = 0,
+[δ, HH•(Λ, Λ)] = 0,
+[δ, ı] = −ı.
+(2) There is an isomorphism of graded algebras
+HH•,∗�
+Λ, Λ[ı±1]
+� ∼= HH•(Λ, Λ) [ı±1].
+Moreover, the morphism
+j∗ : HH•,∗�
+Λ[ı±1], Λ[ı±1]
+�
+−→ HH•,∗�
+Λ, Λ[ı±1]
+�
+induced by the inclusion j : Λ ֒→ Λ[ı±1] of the degree 0 part is the apparent
+natural projection with kernel the graded ideal generated by δ.
+(3) There is an isomorphism of graded algebras
+HH•,∗�
+Λ, Λ[ı±1]
+� ∼= HH•(Λ, Λ) [ı±1].
+Furthermore, the comparison map
+HH•,∗�
+Λ, Λ[ı±1]
+�
+−→ HH•,∗�
+Λ, Λ[ı±1]
+�
+is the apparent extension of the comparison map HH•(Λ, Λ) → HH•(Λ, Λ).
+Proof. All of the forthcoming claims follow from the proof of [JM22, Prop. 4.6.9]
+for σ = 1Λ and d = 2.
+(1) The Hochschild complex C•,∗�
+Λ[ı±1], Λ[ı±1]
+�
+contains the subcomplex
+C•,∗
+C[ı±1]
+�
+Λ[ı±1], Λ[ı±1]
+�
+of C[ı±1]-linear cochains; this subcomplex is also an associative subalgebra and a
+Lie subalgebra of the C-linear Hochschild complex.
+The composite
+C•,∗
+C[ı±1]
+�
+Λ[ı±1], Λ[ı±1]
+�
+i֒→ C•,∗�
+Λ[ı±1], Λ[ı±1]
+�
+j∗
+−→ C•,∗�
+Λ, Λ[ı±1]
+�
+,
+
+THE DONOVAN–WEMYSS CONJECTURE
+13
+of the inclusion of the C[ı±1]-linear Hochschild cochains into the C-linear ones with
+the restriction of scalars along the inclusion j : Λ ֒→ Λ[ı±1] of the degree 0 part is
+an isomorphism of DG algebras. The target, unlike the source, does not a priori
+carry any Lie algebra structure. Nevertheless, there is an obvious isomorphism of
+DG algebras
+C•,∗�
+Λ, Λ[ı±1]
+� ∼= C•(Λ, Λ) [ı±1]
+that we regard as an identification, and the composite isomorphism
+C•,∗
+C[ı±1]
+�
+Λ[ı±1], Λ[ı±1]
+� ∼= C•(Λ, Λ) [ı±1]
+is also a Lie algebra map when we regard the target as a Lie algebra extension of
+C•(Λ, Λ) with ı a central element (in the Lie-algebra sense).
+The morphism
+C•,∗�
+Λ[ı±1], Λ[ı±1]
+�
+−→ C•(Λ, Λ) [ı±1, ¯δ],
+x �−→ j∗(x) − ı−1 · j∗([x, ı]) · ¯δ,
+is a quasi-isomorphism of DG algebras with quasi-inverse
+C•(Λ, Λ) [ı±1] ⊕ C•(Λ, Λ) [ı±1] · ¯δ −→ C•,∗�
+Λ[ı±1], Λ[ı±1]
+�
+,
+x + y · ¯δ �−→ i(x) + i(y) · ¯δ.
+(The latter is just a morphism of complexes since ¯δ2 ̸= 0 in the target, it only
+vanishes in cohomology.)
+In fact, the relevant composite equals the identity of
+C•(Λ, Λ) [ı±1, ¯δ].
+The Lie bracket formulas in the statement of the proposition
+follow from the definition of the fractional Euler class, C[ı±1]-linear cochains and
+degree considerations.
+(2) The statement follows easily from the previous computations.
+(3) The statement is consequece of the fact that HH•,∗(Λ, Λ) is obtained from
+HH•,∗(Λ, Λ) by inverting any element of
+HH4(Λ, Λ) = Ext4
+Λe(Λ, Λ)
+representing the 4-periodicty of Λ, and similarly when the coefficients lie in Λ[ı±1].
+□
+Below, we use the isomorphisms in Proposition 3.3.1 as identifications.
+Corollary 3.3.2. Let u ∈ HH4(Λ, Λ) be a unit. There exists a unique Hochschild
+class
+m ∈ HH4,−2�
+Λ[ı±1], Λ[ı±1]
+�
+such that
+j∗(m) = u · ı,
+1
+2[m, m] = 0.
+Proof. The first equation in the statement is equivalent to m being of the form
+(3.3.3)
+m = (u + x · δ) · ı
+for some x ∈ HH3(Λ, Λ). Using the relations in a Gerstenhaber algebra, the second
+equation is equivalent to
+0 = ([u, u] − 2u · x) · ı2 − 2[x, u] · ı2 · δ.
+This means that both summands must vanish. For the first one, this is equivalent
+to
+x = 1
+2u−1[u, u].
+This is takes place in the piece of HH•(Λ, Λ) that agrees with HH•(Λ, Λ), and is
+compatible with the second summand since
+0 = 1
+2[[u, u], u] = [ux, u] = u[x, u] − [u, u]x = u[x, u] − u−1[u, u]2 = u[x, u],
+
+14
+G. JASSO, B. KELLER, AND F. MURO
+so [x, u] = 0. The first step follows from the graded Jacobi identity and we also
+use that [u, u]2 = 0 since [u, u] has odd total degree and Hochschild cohomology is
+graded commutative.
+□
+The following result should be compared with equation (3.3.3); its proof is similar
+to that of Corollary 3.3.2.
+Corollary 3.3.4. Let u ∈ HH4(Λ, Λ) be a unit such that [u, u] = 0. Given
+(x + y · δ) · ıq ∈ HHp,−2q�
+Λ[ı±1], Λ[ı±1]
+�
+with p ≥ 2, x ∈ HHp(Λ, Λ) and y ∈ HHp−1(Λ, Λ), if [u · ı, (x + y · δ) · ıq] = 0 then
+(x + y · δ) · ıq = [u · ı, u−1 · δ · x · ıq−1].
+We obtain the following more precise information on a minimal A∞-model of the
+2Z-derived contraction algebra.
+Proposition 3.3.5. The 2Z-derived contraction algebra has a minimal A∞-model
+(Λ[ı±1], m4, m6, · · · )
+such that mn is C[ı±1]-linear for all n ≥ 4. In particular, {m4} = u · ı for some
+unit u ∈ HH4(Λ, Λ) satisfying [u, u] = 0.
+Proof. The first part follows from [HK18].
+The rest is a direct consecuence of
+Proposition 3.3.1 and the fact that [{m4}, {m4}] = 0, which follows from (1.5.1).
+□
+3.4. Proof of Theorem 3.1.1. The introduction of the restricted universal
+Massey product of Λ is justified by the following result and the subsequent corol-
+lary. Theorem 3.4.1 is an immediate consequence of [JM22, Thm. B], and the latter
+theorem is obtained an application of the obstruction theory for the existence of
+A∞-structures developed by the third-named author in [Mur20]. In this note we
+give a direct proof of Theorem 3.4.1 that leverages our detailed knowledge of the
+relationship between the Hochschild cohomology of the contraction algebra and
+that if its graded variant (see Section 3.3), although part of the techniques used to
+prove [JM22, Thm. B] are utilised in some guise.
+Theorem 3.4.1. Let A be a DG algebra such that H•(A) = Λ[ı±1] as graded
+algebras. If
+j∗�
+mA
+4
+�
+= j∗�
+mΛ
+4
+�
+∈ HH•,∗�
+Λ, Λ[ı±1]
+�
+,
+then A is quasi-isomorphic to the 2Z-derived contraction algebra Λ via a quasi-iso-
+morphism that induces the identity in cohomology.
+Proof. Let
+(Λ[ı±1], m4, m6, . . . )
+be a minimal model for the 2Z-derived contraction algebra as in Proposition 3.3.5,
+and
+(Λ[ı±1], m′
+4, m′
+6, . . . )
+a minimal model for A. Inductively, we will construct an A∞-isomorphism with
+identity linear part
+f = (0, 0, 0, f3, 0, f5, . . . ): (Λ[ı±1], m4, m6, . . . ) −→ (Λ[ı±1], m′
+4, m′
+6, . . . ),
+and this clearly suffices to prove the claim. Notice that, necessarily, f2n = 0 for all
+n ≥ 0 since Λ[ı±1] is concentrated in even degrees.
+We proceed as follows. For all n ≥ 0 we define a Hochschild cochain of total
+degree 1
+f (n) = (0, 0, 0, f (n)
+3
+, 0, . . . , f (n)
+2n+1, 0, . . . )
+
+THE DONOVAN–WEMYSS CONJECTURE
+15
+such that f (n) coincides with f (n−1) up to Hochschild degree 2n − 2 and
+(3.4.2)n
+∂(f (n)) + f (n) · f (n) +
+�
+r≥0
+m′{f (n), r. . ., f (n)} − m − f (n){m}
+vanishes up to Hochschild degree 2n + 2. If we achieve this goal, then we can take
+f = (0, 0, 0, f (3)
+3 , 0, . . . , f (n)
+2n−3, 0, . . . ).
+Indeed, f coincides with f (n) up to Hochschild degree 2n − 2, so (1.5.2) coincides
+with (3.4.2)n up to Hochschild degree 2n − 1. In particular (1.5.2) vanishes up to
+Hochschild degree 2n − 1 for all n ≥ 0. Therefore (1.5.2) fully vanishes, so f is
+indeed an A∞-isomorphism with identity linear part.
+We start with f (0) = 0.
+With this choice, (3.4.2)0 clearly vanishes up to
+Hochschild degree 2.
+Below, when defining f (n) we will only specify f (n)
+2n−1 and f (n)
+2n+1 since in smaller
+Hochschild degrees they are determined by f (n−1) and in higher Hochschild degrees
+they are irrelevant. Moreover, we will also use that (3.4.2)n−1 and (3.4.2)n agree
+(and hence both vanish) up to Hochschild degree 2n − 1.
+Since j∗{m4} = j∗{m′
+4}, then {m4} = {m′
+4} by Corollary 3.3.2, so there exists
+f (1)
+3
+such that
+(3.4.3)
+∂(f (1)
+3 ) + m′
+4 − m4 = 0.
+This proves that (3.4.2)1 vanishes up to Hochschild degree 4.
+Assume we have constructed up to f (n−1) for some n ≥ 2. Let us see how to
+construct f (n). We know by [LH03, Lemme B.4.2] that the Hochschild degree 2n+2
+part of (3.4.2)n−1, that we simply denote by a, is an obstruction cocycle (∂(a) = 0)
+which vanishes in cohomology if and only if there exists f (n)
+2n+1 such that, taking
+f (n)
+2n−1 = f (n−1)
+2n−1 , (3.4.2)n vanishes up to Hochschild degree 2n + 2. Indeed, any
+f (n)
+2n+1 such that a + ∂(f (n)
+2n+1) = 0 would do. We claim that
+(3.4.4)
+[m4, a] + ∂(b − f n−1
+3
+{a})
+vanishes, where b is the Hochschild degree 2n + 4 part of (3.4.2)n−1. We prove this
+claim below. Now, we deduce from Proposition 3.3.5 and Corollary 3.3.4 that there
+exist Hochschild cochains c2n−1 and c2n+1 such that
+a + ∂(c2n+1) + [m4, c2n−1] = 0,
+∂(c2n−1) = 0.
+If we set
+f (n)
+2n−1 = f (n−1)
+2n−1 + c2n−1,
+f (n)
+2n+1 =
+�
+c5 + f (1)
+3 {c3} − 1
+2c3{c3},
+n = 2;
+c2n+1 + f3{c2n−1},
+n > 2;
+we complete the induction step since the Hochschild degree 2n part of (3.4.2)n is,
+∂(c2n−1) = 0,
+and its Hochschild degree 2n + 2 part is, for n = 2,
+a + ∂
+�
+c5 + f (1)
+3 {c3} − 1
+2c3{c3}
+�
++ c3 · c3 + m′
+4{c3} − c3{m4}
+= a + ∂(c5) + ∂
+�
+f (1)
+3
+�
+{c3} + f (1)
+3 {∂(c3)} + m′
+4{c3} − c3{m4}
+= a + ∂(c5) + (m4 − m′
+4){c3} + m′
+4{c3} − c3{m4}
+= a + ∂(c5) + [m4, c3] = 0,
+
+16
+G. JASSO, B. KELLER, AND F. MURO
+where we use that ∂(c3{c3}) = 2c3 · c3 by Lemma 1.4.1, and for n > 2,
+a + ∂
+�
+c2n+1 + f3{c2n−1}
+�
++ m′
+4{c2n−1} − c2n−1{m4}
+= a + ∂(c2n+1) + ∂
+�
+f (n−1)
+3
+�
+{c2n−1} + f (n−1)
+3
+{∂(c2n−1)} + m′
+4{c2n−1} − c2n−1{m4}
+= a + ∂(c2n+1) + (m4 − m′
+4){c2n−1} + m′
+4{c2n−1} − c2n−1{m4}
+= a + ∂(c2n+1) + [m4, c2n−1] = 0.
+We finish the proof with the vanishing of (3.4.4). In what follows, let us write
+Ξ = (3.4.2)n−1 and f = f (n−1), so as not to overload notation. Note that (3.4.4) is
+the Hochschild degree 2n + 5 part of
+(3.4.5)
+[m, Ξ] + ∂(Ξ − f{Ξ}).
+This cochain vanishes in Hochschild degrees < 2n + 5.
+We now start a series of computations. We number most terms for bookkeeping
+purposes. In the first equation we use Lemma 1.4.1,
+∂(Ξ) =
+1
+∂(f) · f −
+2
+f · ∂f +
+3
+�
+r≥0
+∂(m′){f, r. . ., f}
+−
+4
+�
+r≥1
+r
+�
+i=1
+m′{f, i−1
+. . ., ∂(f), r−i
+. . ., f}
+−
+5
+�
+r≥1
+f · m′{f, r−1
+. . ., f} −
+6
+�
+r≥2
+r−1
+�
+i=1
+m′{f, i−1
+. . ., f 2, r−i−1
+. . . , f}
++
+7
+�
+r≥1
+m′{f, r−1
+. . ., f} · f −
+8
+∂(m) −
+9
+∂(f){m} −
+10
+f{∂(m)} +
+11
+f · m −
+12
+m · f
+Since m and m′ are A∞-algebra structures,
+8 = −m{m},
+10 = −f{m{m}},
+3 = −
+�
+r≥0
+m′{m′}{f, r. . ., f}
+= −
+�
+r≥0
+�
+0≤i≤j≤r
+m′{f,
+i. . ., m′{f, j−i
+. . ., f}, r−j
+. . ., f},
+= −
+13
+�
+r≥0
+m′{m′{f, r. . ., f}} −
+14
+�
+r≥1
+�
+0≤i≤j≤r
+j−i<r
+m′{f,
+i. . ., m′{f, j−i
+. . ., f}, r−j
+. . ., f}
+
+THE DONOVAN–WEMYSS CONJECTURE
+17
+Here we also use the brace relation. We also split the following summations in two
+parts,
+4 =
+15
+m′{∂(f)} +
+16
+�
+r≥2
+r
+�
+i=1
+m′{f, i−1
+. . ., ∂(f), r−i
+. . ., f},
+6 =
+17
+m′{f 2} +
+18
+�
+r≥3
+r−1
+�
+i=1
+m′{f, i−1
+. . ., f 2, r−i−1
+. . . , f} .
+Consider the following cochain, that we decompose using the brace relation,
+19
+�
+r≥0
+m′{f, r. . ., f}{m} =
+�
+r≥0
+r
+�
+i=0
+m′{f,
+i. . ., m, r−i
+. . ., f}
++
+�
+r≥1
+r
+�
+i=0
+m′{f, i−1
+. . ., f{m}, r−i
+. . ., f}
+=
+20
+m′{m} +
+21
+�
+r≥1
+r
+�
+i=0
+m′{f,
+i. . ., m, r−i
+. . ., f} +
+22
+m′{f{m}}
++
+23
+�
+r≥2
+r
+�
+i=0
+m′{f, i−1
+. . ., f{m}, r−i
+. . ., f} .
+Consider also the following cochain, which is computed by using Lemma 1.4.2,
+24
+f 2{m} =
+25
+f · f{m} −
+26
+f{m} · f .
+Since Ξ vanishes up to Hochschild degree 2n + 1 and its Hochschild degree 2n + 2
+part is a cocycle, the following cochains vanish up to Hochschild degree 2n + 5,
+�
+r≥2
+r
+�
+i=1
+m′{f, i−1
+. . ., Ξ, r−i
+. . ., f} = 16 + 18 + 14 − 21 − 23 ,
+∂(f){Ξ} + m′{Ξ} − m{Ξ},
+f{∂(Ξ)}.
+Notice that
+Ξ · f = 1 + f 3 + 7 − 12 − 26 ,
+f · Ξ = 2 + f 3 + 5 − 11 − 25 ,
+m′{Ξ} = 15 + 17 + 13 − 20 − 22 ,
+Ξ{m} = 9 + 24 + 19 + 8 + 10 .
+
+18
+G. JASSO, B. KELLER, AND F. MURO
+Using all the above, we obtain the following relations, where ≡ stands for con-
+gruence modulo cochains vanishing in Hochschild degrees < 2n + 5,
+(3.4.5) = m{Ξ} + Ξ{m} + ∂(Ξ) − ∂(f){Ξ} − f{∂(Ξ)} + f · Ξ − Ξ · f
+≡ m{Ξ} −
+�
+9 + 24 + 19 + 8 + 10
+�
++
+�
+1 − 2 + 3 − 4 − 5 − 6 + 7 − 8 − 9 − 10 + 11 − 12
+�
++
+�
+15 + 17 + 13 − 20 − 22
+�
+− m{Ξ} − 0
++
+�
+2 + f 3 + 5 − 11 − 25
+�
+−
+�
+1 + f 3 + 7 − 12 − 26
+�
++
+�
+16 + 18 + 14 − 21 − 23
+�
+= 0.
+This finally concludes the proof.
+□
+Corollary 3.4.6. Let A be a DG algebra such that H•(A) = Λ[ı±1] as graded
+algebras. If the restricted universal Massey product
+j∗�
+mA
+4
+�
+∈ HH•,∗�
+Λ, Λ[ı±1]
+�
+is a unit, then A is quasi-isomorphic to the 2Z-derived contraction algebra Λ.
+Proof. Firstly, we observe that the group of graded-algebra automorphisms of Λ[ı±1]
+acts on the right of HH•,∗�
+Λ, Λ[ı±1]
+�
+by conjugation.
+In particular, the group
+Z(Λ)× of units of the centre of Λ acts on HH•,∗�
+Λ, Λ[ı±1]
+�
+via the graded-algebra
+automorphisms
+gu : x �−→ xui,
+|x| = 2i,
+where u ∈ Z(Λ)×. The induced action on
+HH4,−2�
+Λ, Λ[ı±1]
+� ∼= Ext4
+Λe(Λ, Λ)
+has the following explicit description: Given a unit u ∈ Z(Λ)× and an exact se-
+quence of Λ-bimodules
+η: 0 → Λ
+f−→ X3 → X2 → X1 → X0 → Λ → 0,
+we let [η] · u be the class of the exact sequence
+0 → Λ
+f ′
+−→ X3 → X2 → X1 → X0 → Λ → 0,
+where f ′ := u−1f.
+The above action clearly restricts to the set of units in
+HH•,∗�
+Λ, Λ[ı±1]
+�
+of bidegree (4, −2), since these are precisely the classes that can be
+represented by an exact sequence with projective(-injective) middle terms [Mur22,
+Rmk. 5.8], and is in fact transitive on this set. To prove the latter claim on the
+transitivity of the action we appeal to [Che21, Cor. 2.3], which allows us to lift
+stable bimodule isomorphisms Λ ≃ Λ to honest bimodule isomorphisms Λ ∼= Λ, see
+the proof of [JM22, Prop. 2.2.16].
+Let u ∈ Z(Λ)× be a unit such that
+j∗�
+mA
+4
+�
+· u = j∗�
+mΛ
+4
+�
+∈ HH4,−2�
+Λ, Λ[ı±1]
+�
+.
+Given a minimal model
+(Λ[ı±1], mA
+4 , mA
+6 , mA
+8 , · · · )
+of the DG algebra A we define a new minimal A∞-algebra structure
+(Λ[ı±1], mA
+4 , mA
+6 , mA
+8 , · · · ) = (Λ[ı±1], mA
+4 , mA
+6 , mA
+8 , · · · ) ∗ gu
+with n-ary opearations mA
+n := g−1
+u mA
+4 g⊗n
+u
+(notice that mA
+4 ̸= mA
+4 as soon as u ̸= 1).
+It is easy to see that
+j∗�
+mA
+4
+�
+= j∗�
+mA
+4
+�
+· u = j∗�
+mΛ
+4
+�
+
+THE DONOVAN–WEMYSS CONJECTURE
+19
+and that there is an isomorphism of A∞-algebras
+(Λ[ı±1], mA
+4 , mA
+6 , mA
+8 , · · · ) ⇝ (Λ[ı±1], mA
+4 , mA
+6 , mA
+8 , · · · ).
+Thus, A is quasi-isomorphic to any DG algebra model B of the minimal A∞-
+algebra on the right-hand side, see Remark 3.2.2, and by Theorem 3.4.1 the DG
+algebras B and Λ are quasi-isomorphic. Consequently, the DG algebras A and Λ
+are isomorphic, which is what we needed to prove.
+□
+We are ready to prove Theorem 3.1.1.
+Proof of Theorem 3.1.1. Let R1 and R2 be isolated cDV singularities with crepant
+resolutions
+p1 : X1 → Spec(R1)
+and
+p2 : X2 → Spec(R2)
+whose corresponding contraction algebras Λ1 = Λ(p1) and Λ2 = Λ(p2) are isomor-
+phic. We need to prove that the 2Z-derived contraction algebras Λ1 = Λcon(p1)
+and Λ2 = Λcon(p2) are quasi-isomorphic. Recall that
+H•(Λ1) ∼= Λ1[ı±1]
+and
+H•(Λ2) ∼= Λ2[ı±1],
+|ı| = −2.
+In view of the assumption that Λ1 ∼= Λ2, we obtain a chain of isomorphisms of
+graded algebras
+H•(Λ1) ∼= Λ1[ı±1] ∼= Λ2[ı±1] ∼= H•(Λ2).
+In particular, we may and we will identify minimal models of Λ1 and Λ2 via the
+above isomorphism. For simplicity, let Λ = Λ1 ∼= Λ2. Corollary 3.2.7 shows that
+the restricted universal Massey products
+j∗�
+mΛ1
+4
+�
+∈ HH4,−2�
+Λ, Λ[ı±1]
+�
+and
+j∗�
+mΛ2
+4
+�
+∈ HH4,−2�
+Λ, Λ[ı±1]
+�
+are units in the Hochschild–Tate cohomology HH•,∗�
+Λ, Λ[ı±1]
+�
+. Corollary 3.4.6 im-
+plies that the DG algebras Λ1 and Λ2 are quasi-isomorphic, which is what we
+needed to prove.
+□
+We also have the following important corollary.
+Corollary 3.4.7. Let R be an isolated cDV singularity that admits a crepant res-
+olution. Then, the singularity category Dsg(R) admits a unique DG enhancement
+in the sense of [BK90].
+Proof. Let A be a DG enhancement of Dsg(R). By definition, this means that A
+is a pre-triangulated DG category and there exists an equivalence of triangulated
+categories
+H0(A) ≃ Dsg(R).
+Let T ∈ Dsg(R) be a 2Z-cluster tilting object and A the derived endomorphism alge-
+bra of T computed by means of the DG enhancement A. In particular Dc(A)dg ≃ A
+since T is a classical generator. By definition, H•(A) ∼= H•(Λ) where Λ is the de-
+rived endomorphism algebra of T computed by means of the canonical DG enhance-
+ment of Dsg(R). The proofs of Theorem 3.2.5 and Corollary 3.2.7 only rely on the
+fact that T ∈ Dsg(R) is a 2Z-cluster tilting object, see Remark 3.2.6. Consequently,
+the restricted universal Massey product j∗�
+mA
+4
+�
+is a unit in HH•,∗�
+Λ, Λ[ı±1]
+�
+and,
+by Corollary 3.4.6, the DG algebras A and Λ are quasi-isomorphic. Therefore, the
+DG categories
+A ≃ Dc(A)dg
+and
+Dsg(R)dg
+are quasi-equivalent. This shows that every DG enhancement of Dsg(R) is equiva-
+lent to the canonical DG enhancement and the claim follows.
+□
+
+20
+G. JASSO, B. KELLER, AND F. MURO
+Remark 3.4.8. Corollary 3.4.7 is stronger that Conjecture A, as it shows that the
+DG category Dsg(R)dg is determined by Λcon up to isomorphism in Hmo, the Morita
+category of small DG categories [Tab05].
+4. Concluding remarks
+In this section we collect some observations, some of which follow easily from
+the results in the previous sections.
+4.1. Formality of contraction algebras. Recall that a DG algebra A is formal
+if there is a quasi-isomorphism A ≃ H•(A), where the graded algebra H•(A) is
+viewed as a DG algebra with vanishing differential. We observe that 2Z-derived
+contraction algebras are almost never formal.7
+Theorem 4.1.1. Let R be an isolated cDV singularity with a crepant resolution
+and Λcon a 2Z-derived contraction algebra for R.
+The following statements are
+equivalent:
+(1) The 2Z-derived contraction algebra Λcon is formal.
+(2) There is an isomorphism of algebras Λcon ∼= C.
+(3) There is an isomorphism of algebras
+R ∼= C�x, y, z, t�/(xy − zt),
+so that Spec(R) is the base of the Atiyah flop [Ati58].
+If the above equivalent conditions hold, then there is a quasi-isomorphism
+Λcon ≃ C[ı±1],
+|ı| = −2,
+where the graded algebra C[ı±1] is equipped with the trivial differential.
+Proof. (1)⇒(2) If the 2Z-derived contraction algebra Λcon is formal, then its re-
+stricted universal Massey product j∗{m4} is represented by the trivial sequence in
+HH4,−2�
+Λcon, Λcon[ı±1]
+�
+= Ext4
+Λecon(Λcon, Λcon). In view of Theorem 3.2.5, this can
+only happen if Λcon is projective as a Λcon-bimodule or, equivalently, if the algebra
+Λcon is semisimple. Since contraction algebras are basic and connected, we must
+have an isomorphism of algebras Λcon ∼= C, which is what we needed to prove.
+(2)⇒(1) If Λcon ∼= C, then there is an isomorphism of graded algebras
+H•(Λcon) ∼= Λcon[ı±1] ∼= C[ı±1],
+|ı| = −2.
+It is well-known (and easy to prove using Kadeishvili’s Theorem [Kad88], for exam-
+ple) that the latter graded algebra is intrinsically formal, that is every DG algebra
+with cohomology algebra C[ı±1] is formal. In particular, Λcon is formal.8
+(2)⇔(3) In view of the validity of Conjecture A, it is enough to observe that C is
+indeed isomorphic to the contraction algebra of the Atiyah flop [DW16, Table 2].
+□
+Remark 4.1.2. The (non-)formality of the derived contraction algebra Λ≤0
+con is inves-
+tigated in [Boo19, Ch. 9] where minimal models of this DG algebra are computed
+in various examples, see also [Boo18, Boo21, Boo22].
+7This fact will come as no surprise to the experts, but we did not find a proof of it in the
+literature.
+8Alternatively, one can prove this fact using the results in this note as follows: Since the
+enveloping algebra
+Λe
+con ∼
+= C ⊗C C ∼
+= C
+is semisimple, the Hochschild–Tate cohomology HH•,∗�
+Λcon, Λcon[ı±1]
+�
+vanishes in positive
+Hochschild degrees. Then, by Theorem 3.4.1, the 2Z-derived contraction algebra Λcon is quasi-
+isomorphic to its cohomology algebra H•(Λcon) for the condition on the agreement of the corre-
+sponding universal Massey products is trivially satisfied. Of course, this is essentially the same
+proof as the one using Kadeishvili’s Theorem, which is indeed a special case of [JM22, Thm. B].
+
+THE DONOVAN–WEMYSS CONJECTURE
+21
+Remark 4.1.3. More generally, Kadeishvili’s Theorem [Kad88] can be used to prove
+that the Laurent polynomial algebra K[ı±1] = K ⊗C C[ı±1] is intrinsically formal
+if K is a finite-dimensional algebra of projective dimension at most 2 as a K-
+bimodule [Sai20, Cor. 4.2].
+This K, however, cannot be a contraction algebra
+unless K = C since contraction algebras are basic and connected, and a symmetric
+C-algebra has finite projective dimension as a bimodule over itself if and only if it
+is separable (semisimple).
+4.2. Relationship to the Triangulated Auslander–Iyama Correspondence.
+Let T be a triangulated category whose underlying additive category is Krull–
+Schmidt and has finite-dimensional morphism spaces.
+A basic object T ∈ T is
+d-cluster tilting, d ≥ 1, if the following conditions are satisfied [IY08, Bel15]:
+• The object T is d-rigid: T(T, T [i]) = 0 for all 0 < i < d.
+• For each object X ∈ T there exists a diagram
+Td−2
+· · ·
+T1
+T0
+Td−1
+Xd−2
+· · ·
+X2
+X1
+X
++1
++1
++1
+in which Ti ∈ add(T ), 0 ≤ i < d, the oriented triangles denote exact triangles in
+T and the unoriented triangles commute. The object T is dZ-cluster tilting if it
+is d-cluster tilting and T ∼= T [d]. Theorem 3.1.1 is a special case of the theorem
+below. For a finite-dimensional algebra Λ, we let proj(Λ) be the category of finite-
+dimensional projective Λ-modules. For example, if Λ = T(T, T ), then the Yoneda
+functor
+T ⊇ add(T )
+∼
+−→ proj(Λ),
+X �−→ T(T, X),
+is an equivalence (of additive categories).
+Theorem
+4.2.1
+(Triangulated
+Auslander–Iyama
+Correspondence
+[JM22,
+Thm. 5.1.10]). Let d ≥ 1.
+There is a bijective correspondence between the
+following:
+(1) Quasi-isomorphism classes of DG algebras A that satisfy the following:
+• The algebra H0(A) is a basic finite-dimensional algebra.
+• The free DG A-module A ∈ Dc(A) is a dZ-cluster tilting object.
+(2) Equivalence classes of pairs (Λ, I) consisting of
+• a basic finite-dimensional self-injective algebra Λ and
+• an invertible Λ-bimodule I such that Ωd+2
+Λe (Λ) ∼= I in the stable category
+of Λ-bimodules.
+The correspondence is given by the formula A �→ (H0(A), H−d(A)).
+Remark 4.2.2. The case d = 1 of Theorem 4.2.1 is one way to formulate the main
+result in [Mur22]. Indeed, an object T ∈ T is 1-cluster tilting if and only if it is
+1Z-cluster tilting if and only if add(T ) = T; the latter condition means that T is a
+triangulated category of finite type in the terminology of [Mur22].
+Remark 4.2.3. Let (Λ, I) be a pair as in Theorem 4.2.1(2). Since the algebra Λ is
+assumed to be basic, the map
+Aut(Λ) −→ Pic(Λ),
+σ �−→ [1Λσ]
+from the group of algebra automorphisms of Λ to the Picard group of invertible
+Λ-bimodules induces an isomorphism of groups [Bol84, Prop. 3.8]
+Out(Λ)
+∼
+−→ Pic(Λ),
+[σ] �−→ [1Λσ],
+where Out(Λ) = Aut(Λ)/ Inn(Λ) is the group of outer automorphisms of Λ. In par-
+ticular, there exists σ ∈ Aut(Λ) such that I ∼= 1Λσ as Λ-bimodules. The condition
+
+22
+G. JASSO, B. KELLER, AND F. MURO
+Ωd+2
+Λe (Λ) ≃ 1Λσ expresses the fact that the algebra Λ is twisted (d + 2)-periodic
+with respect to σ. When σ = 1 or, equivalently, I ∼= Λ, the algebra Λ is said to be
+(d+ 2)-periodic. For example, contraction algebras are known to be 4-periodic. We
+refer the reader to [ES08] and the references therein for information on (twisted)
+periodic algebras.
+Theorem 3.1.1
+follows
+from
+the
+injectivity
+of
+the
+correspondence
+in Theorem 4.2.1 with d = 2; the proof of Theorem 3.1.1 outlined in this
+note effectively goes through the proof of the latter in the special case of
+2Z-derived contraction algebras.
+We record the resulting characterisation of
+contraction algebras for the sake of completeness.
+Theorem 4.2.4. Let R be an isolated cDV singularity with a crepant resolution
+p: X → Spec(R).
+Up to quasi-isomorphism, the 2Z-derived contraction algebra
+Λ = Λcon(p) is the unique DG algebra with the following properties:
+(1) Λ ∈ Dc(Λ) is a 2Z-cluster tilting object.
+(2) There is an isomorphism of algebras H0(Λ) ∼= Λ = Λcon(p).
+(3) There is an isomorphism of Λ-bimodules H−2(Λ) ∼= Λ.
+In other words, Λ is determined up to quasi-isomorphism by its image (Λ, Λ) under
+the Triangulated Auslander–Iyama Correspondence.
+Proof. That the 2Z-derived contraction algebra satisfies the first two properties
+follows from Theorem 1.3.1 since, by definition, Λ is the derived endomorphism
+algebra of a 2Z-cluster tilting object in Dsg(R) ≃ Dc(Λ). Therefore Λ belongs to
+the class of DG algebras in Theorem 4.2.1(1) with d = 2. Given that Λ is 4-periodic,
+the third property follows. Moreover, Λ is determined up to quasi-isomorphism by
+its image (Λ, Λ) under the Triangulated Auslander–Iyama Correspondence, which
+is what we needed to prove.
+□
+4.3. Isolated cDV singularities with non-smooth minimal models. Con-
+traction algebras are defined for an arbitrary cDV singularity R that is neither
+isolated nor admits a crepant resolution. However, contraction algebras are finite-
+dimensional if and only if R defines an isolated singularity [DW19, Summary 5.6],
+and this finite-dimensionality is crucial to our approach. On the other hand, R
+admits a crepant resolution if and only if the singularity category Dsg(R) admits a
+2Z-cluster tilting object, see [BIKR08, Thm. 5.4] and the references therein. If, on
+the other hand, the minimal models9 of R are singular, then the contraction algebras
+of R are the endomorphism algebras of maximal rigid objects in Dsg(R) [Wem18],
+that is objects T ∈ Dsg(R) such that
+add(T ) = {X ∈ Dsg(R)| Hom(T ⊕ X, (T ⊕ X)[1]) = 0}.
+It is easy to verify that 2-cluster tilting objects are maximal rigid, but the converse is
+false in general [BIKR08, BMV10]; moreover, if there exists a 2-cluster tilting object
+then every maximal rigid object is also 2-cluster tilting [BIRS09, Thm. II.1.8].10 In
+any case, one may still define the 2Z-derived contraction algebras of R as the derived
+endomorphism algebras of maximal rigid objects in Dsg(R), computed in terms of
+the canonical DG enhancement of the latter triangulated category. Note, however,
+that Theorem 4.2.1 does not cover the case of maximal rigid objects that are not
+9In the context of the MMP in dimension three and higher, minimal models play the role of
+minimal resolutions of surfaces. We do not recall the technical definition in this note, but only
+mention that minimal models are permitted to have ‘mild’ singularities as long as they remain
+‘closer’ to the original space than a smooth resolution (which always exists by a famous theorem
+of Hironaka [Hir64].)
+10The reader should compare this statement with the following geometric fact: If one minimal
+model of Spec(R) is smooth, then all of its minimal models are also smooth [Kol89, Cor. 4.11].
+
+THE DONOVAN–WEMYSS CONJECTURE
+23
+2Z-cluster tilting and hence it cannot be applied to prove a version of Theorem 4.2.4
+for isolated cDV singularities that do not admit a crepant resolution. Finally, we
+mention that the apparent variant of Conjecture A does not hold for isolated cDV
+singularities whose minimal models are not smooth, see [Boo21, Ex 8.4.2] for an
+explicit counterexample.
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+Invent. Math., 211(2):435–521, 2018.
+[Wem21]
+Michael Wemyss. A lockdown survey on cDV singularities, 2021, arXiv:2103.16990
+[math.AG].
+[Yos90]
+Yuji Yoshino. Cohen-Macaulay modules over Cohen-Macaulay rings, volume 146 of
+London Mathematical Society Lecture Note Series. Cambridge University Press, Cam-
+bridge, 1990.
+[Zim14]
+Alexander Zimmermann. Representation theory, volume 19 of Algebra and Applica-
+tions. Springer, Cham, 2014. A homological algebra point of view.
+(G. Jasso) Lund University, Centre for Mathematical Sciences, S¨olvegatan 18A, 22100
+Lund, Sweden
+Email address: gustavo.jasso@math.lu.se
+URL: https://www.maths.lu.se/staff/gustavo-jasso/
+(B. Keller) Universit´e Paris Cit´e, UFR de Math´ematiques, Case 7012, Bˆatiment Sophie
+Germain, 8 place Aur´elie Nemours, 75013 Paris Cedex 13, France
+Email address: bernhard.keller@imj-prg.fr
+URL: https://webusers.imj-prg.fr/~bernhard.keller/
+(F. Muro) Universidad de Sevilla,
+Facultad de Matem´aticas,
+Departamento de
+´Algebra, Calle Tarfia s/n, 41012 Sevilla, Spain
+Email address: fmuro@us.es
+URL: https://personal.us.es/fmuro/
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf,len=1483
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='11593v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='AG]  27 Jan 2023  THE DONOVAN–WEMYSS CONJECTURE VIA THE  TRIANGULATED AUSLANDER–IYAMA CORRESPONDENCE  GUSTAVO JASSO, BERNHARD KELLER, AND FERNANDO MURO  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We provide an outline of the proof of the Donovan–Wemyss Con-  jecture in the context of the Homological Minimal Model Program for three-  folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The proof relies on results of August, of Hua and the second-named au-  thor, Wemyss, and on the Triangulated Auslander–Iyama Correspondence—a  recent result by the first- and third-named authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Contents  Introduction  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Preliminaries  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The Derived Donovan–Wemyss Conjecture  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Uniqueness of the 2Z-derived contraction algebra  8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Concluding remarks  20  References  23  Introduction  We work over the field C of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' A compound Du Val (=cDV)  singularity is a complete local hypersurface  R ∼= C�x, y, z, t�  (f + tg) ,  where C�x, y, z�/(f) is a Kleinian surface singularity and g ∈ C�x, y, z, t� is arbi-  trary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Introduced by Reid in the early 1980s [Rei83], cDV singularities form an  important class of three-dimensional singularities in birational geometry and play  a significant role in the Minimal Model Program (MMP) for threefolds [KM98,  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3] as well as in the Homological MMP [Wem18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  We refer the reader  to [Aug19, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 1] and [Wem21] for introductions to the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  This note is concerned with the following geometric situation: Let R be an iso-  lated cDV singularity and p: X → Spec(R) a crepant resolution, that is p is a  proper birational map with smooth source such that the pullback of the dualising  sheaf of Spec(R) along f is the dualising sheaf of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' It follows that Spec(R) has  a unique singular point m and the (reduced) exceptional fibre p−1(m) = �n  i=1 Ci is  a union of curves, with Ci ∼= P1  C [VdB04, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' To these data, Donovan  and Wemyss [DW16, DW19] associate a (basic, connected) finite-dimensional alge-  bra Λcon = Λcon(p), the contraction algebra of p, which represents the functor of  ‘simultaneous non-commutative deformations’ of the reduced exceptional fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' By  construction, Λcon is a Cn-augmented algebra, and hence in particular determines  the number n of irreducible components of the exceptional fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The contraction  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Primary 14E30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Secondary 13D03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Minimal model program;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' compound Du Val singularity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' crepant res-  olution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' contraction algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Hochschild cohomology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Auslander correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  1  2  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' JASSO, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' KELLER, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' MURO  algebra encodes a surprising amount of information stemming from the given geo-  metric setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' For example, when p contracts a single curve, the contraction algebra  recovers known invariants such as Reid’s width [Rei83] and the Gopakumar–Vafa  invariants [Kat08], see [Tod15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Neither the dimension nor the Gabriel quiver of  contraction algebras suffice for differentiating cDV singularities [DW16, Table 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  In fact, it is well-known that there are continuous families of pairwise non isomor-  phic cDV singularities (that is ‘cDV singularities have moduli’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Notwithstanding,  at the risk of stating the obvious, let us point out that the contraction algebra is  equiped with crucial data in the form of the multiplication law and that this law  is essential in recovering the above mentioned invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Equipped with their al-  gebra structure, contraction algebras distinguish between non-isomorphic isolated  cDV singularities that admit a crepant resolution in all known examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' These  considerations motivate the following remarkable conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Conjecture A (Donovan and Wemyss [DW16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R1 and R2 be isolated cDV  singularities with crepant resolutions  p1 : X1 → Spec(R1)  and  p2 : X2 → Spec(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Then, the contraction algebras Λcon(p1) and Λcon(p2) are derived equivalent if and  only if there is an isomorphism of algebras R1 ∼= R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The original conjecture was formulated only in the case of single-curve contrac-  tions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' algebraically, this corresponds to the case where the contraction algebras are  local and thus derived equivalence reduces to mere isomorphism of algebras since  contraction algebras are basic, see [Zim14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In the above  form, which allows for contracting multiple curves, the conjecture appeared in print  in [Aug20, Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  That the contraction algebras of a given isolated cDV singularity are derived  equivalent follows by combining results from Wemyss [Wem18] and Dugas [Dug15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  In this note we provide an outline of the proof of the remaining part of Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  This proof first appeared in the appendix to [JM22] written by the second-named  author where it is explained how the conjecture follows by combining previous  results of August [Aug20] and [HK18] with the Triangulated Auslander–Iyama Co-  rrespondence—the main result in [JM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' For the sake of concreteness, we restrict  ourselves to the specific context of the conjecture, with the understanding that  most concepts and results that are presented here are but special cases of a much  more general theory that the reader can find in the original sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We hope that  this sacrifice in generality makes the proof of the conjecture accessible to a broader  readership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The authors thank Michael Wemyss and Zheng Hua for in-  teresting conversations and email exchanges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' in addition, they thank M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' for sug-  gesting the terminology ‘2Z-derived contraction algebra’ used in these proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The third-named author is algo grateful to Matt Booth for answering questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Financial  support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  was  partially  supported  by  grants  PID2020-  117971GB-C21  funded  by  MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='13039/501100011033,  US-1263032  (US/JUNTA/FEDER, UE), and P20 01109 (JUNTA/FEDER, UE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Preliminaries  In this section we collect preliminary definitions and results that are needed in  our proof of Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We use freely the theories of differential graded cate-  gories [Kel94, Kel06] and A∞-categories [LH03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We denote the derived category  of an algebra or, more generally, a DG algebra A by D(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' the perfect derived  THE DONOVAN–WEMYSS CONJECTURE  3  category of A, that is the full subcategory of D(A) spanned by its compact objects,  is denoted by Dc(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' All (DG) modules are right (DG) modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 2Z-cluster tilting objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let T be a triangulated category whose underly-  ing additive category is Krull–Schmidt and has finite-dimensional morphism spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 ([IY08, GKO13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' A basic1 object T ∈ T is 2-cluster tilting if the  following conditions hold:  (1) The object T is rigid: T(T, T [1]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (2) For each object X ∈ T there exists a triangle T1 → T0 → X → T1[1] with  T0, T1 ∈ add(T ), where add(T ) is the smallest additive subcategory of T  containing T that is closed under direct summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  We say that T is 2Z-cluster tilting if it is 2-cluster tilting and T ∼= T [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Clearly, if T ∈ T is a 2- or 2Z-cluster tilting object, then T generates  T as a triangulated category with split idempotents (which is to say that T is a  classical generator of T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In particular, if the triangulated category T is algebraic2  then there exists a DG algebra A and an equivalence of triangulated categories  T  ∼  −→ Dc(A),  T �−→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Given a basic 2-cluster tilting object T ∈ T, one can produce a new  such object by a procedure called mutation that, in a nutshell, replaces a single  indecomposable direct summand of T by a new one, see [IY08] for a precise defini-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' This process, which can be iterated, is an important reason for introducing  2-cluster tilting objects into the framework of the Homological MMP, see [Wem18]  and compare with Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In general, 2Z-cluster tilting objects are not invariant under mu-  tation (however, see [HI11, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In the context of the Homological MMP this  problem does not occur since the notions of 2- and 2Z-cluster tilting object coincide,  see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Maximal Cohen–Macaulay modules and singularity categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let  R be an isolated cDV singularity and  CM(R) := {M ∈ mod(R)| depth(M) = dim(R)}  be the category of maximal Cohen–Macaulay R-modules [Yos90, LW12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The cate-  gory CM(R) is a Frobenius exact category and, therefore, the stable category CM(R)  has an induced structure of a triangulated category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' moreover, there is a canonical  equivalence of triangulated categories  CM(R)  ∼  −→ Dsg(R) := Db(mod(R))/ Kb(proj(R)),  where Dsg(R) is the singularity category of R, see [Buc21] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We record  the following facts for later use:  [Yos90, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='18] Since R is complete local, CM(R) ≃ Dsg(R) is a Krull–  Schmidt category with finite-dimensional morphism spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [Aus78] Since R is 3-dimensional, CM(R) ≃ Dsg(R) is a 2-Calabi–Yau tri-  angulated category [Kon98, Kel08], that is there is a natural isomorphism  DHom(X, Y )  ≃  −→ Hom(Y, X[2]),  X, Y ∈ CM(R) ≃ Dsg(R),  where V �→ DV denotes the passage to the C-linear dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  1An object in a Krull–Schmidt additive category is basic if all of its indecomposable direct  summands have multiplicity one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  2A triangulated category is algebraic if it is equivalent—as a triangulated category—to the  stable category of a Frobenius exact category [Kel94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  4  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' JASSO, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' KELLER, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' MURO  [Eis80] Since R is a hypersurface, CM(R) ≃ Dsg(R) is a 2-periodic triangu-  lated category, that is there is an isomorphism of exact functors [2] ∼= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  In particular, the notions of 2- and 2Z-cluster tilting object coincide in this  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The endomorphism algebra of any basic object X in CM(R) ≃ Dsg(R) is  a symmetric algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  This is an immediate consequence of the natural  isomorphisms  Hom(X, X) ∼= D Hom(X, X[2]) ∼= D Hom(X, X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  We also consider the DG category Dsg(R)dg, which is defined as the DG quo-  tient [Kel99, Dri04] of the canonical DG enhancements of the triangulated cate-  gories Db(mod(R)) and Kb(proj(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' By construction,  H0(Dsg(R)dg) = Dsg(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Contraction algebras via 2Z-cluster tilting objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R be an isolated  cDV singularity and p: X → Spec(R) a crepant resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' As explained in the  introduction, to this geometric setup Donovan and Wemyss associate a basic finite-  dimensional algebra Λcon = Λcon(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We recall an alternative construction of the  algebra Λcon that is more adapted to the methods we utilise in this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Given p  as above, a theorem of Van den Bergh [VdB04] furnishes a tilting bundle  OX ⊕ N = OX ⊕ N(p) ∈ coh(X)  and Wemyss proves [Wem18] that there is an isomorphism of algebras  Λcon ∼= EndR(N)  between the contraction algebra of p and the stable endomorphism algebra of  N := H0(N) ∈ CM(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Remarkably, when viewed as an object of the triangulated  category CM(R), the R-module N is a 2Z-cluster tilting object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Conversely, given  a 2Z-cluster tilting object T ∈ CM(R), there exists a crepant resolution of Spec(R)  whose associated contraction algebra is isomorphic to EndR(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We summarise the  previous discussion in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 ([Wem18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R be an isolated cDV singularity and assume  that Spec(R) admits a crepant resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Then, the contraction algebras of R are  precisely the endomorphism algebras of 2Z-cluster tilting objects in the triangulated  category CM(R) ≃ Dsg(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The following theorem of August reduces Conjecture A from a derived equiv-  alence to an isomorphism problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The proof leverages the characterisation of  contraction algebras provided by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2 ([Aug20, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R be an isolated cDV singularity and  assume that Spec(R) admits a crepant resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The contraction algebras of R  form a single and complete derived equivalence class of basic algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R1 and R2 be isolated cDV singularities with crepant reso-  lutions  p1 : X1 → Spec(R1)  and  p2 : X2 → Spec(R2)  and corresponding contraction algebras Λ1 = Λcon(p1) and Λ2 = Λcon(p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' If the  algebras Λ1 and Λ2 are derived equivalent, then there exists a contraction algebra  Λ of R2 such that Λ ∼= Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  3Wemyss proves even more: Up to isomorphism on both sides, crepant resolutions of R cor-  respond bijectively to (basic) 2Z-cluster tilting objects in Dsg(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In particular, the number of  isomorphism classes of 2Z-cluster tilting objects in Dsg(R) if finite, for the number of minimal  models of Spec(R) is finite [KM87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  THE DONOVAN–WEMYSS CONJECTURE  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Hochschild cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let A be a graded algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The bigraded  Hochschild cochain complex has components  Cp,q(A, A) := HomC  �  A⊗p, A(q)  �  ,  p ≥ 0, q ∈ Z,  where V �→ V (1) is the (vertical) degree shift of graded vector spaces, equipped  with the Hochschild differential x �→ ∂(x) of bidegree (1, 0), see [Mur20] for the  precise definition, related structure (described below) and sign conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The  first degree is called horizontal or Hochschild degree and the second is the vertical  or internal degree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' the sum of both is the total degree and we denote it by |x| (we  also use this notation for the degree of an element in a singly-graded vector space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The component of total degree n of the Hochschild complex is  �  p+q=n  Cp,q(A, A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The Hochschild complex is equipped with a brace algebra structure, consisting  of operations called braces (which we do not describe explicitly here)  C•,∗(A, A)⊗n+1  −→ C•,∗(A, A) ,  x0 ⊗ x1 ⊗ · · · ⊗ xn �−→ x0{x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , xn},  that are defined for all n ≥ 1, and satisfy the brace relation  x{y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , yp}{z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , zq}  =  �  0≤i1≤j1≤···≤ip≤jp≤q  (−1)ǫ{z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , zi1, y1{zi1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , zj1}, zj1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , zip, yq{zip+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , zjp}, zjp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , zq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Above, ǫ reflects the Koszul sign rule with respect to the total degree shifted by  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Brace operations have horizontal degree −n and vertical degree 0 and the  n-th brace operation vanishes when x0 has horizontal degree < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The Hochschild  complex is a bigraded associative algebra equipped with the cup-product  x · y = (−1)|x|−1m2{x, y},  where m2 ∈ C2,0(A, A) is, up to a sign, the multiplication in the graded algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The Hochschild complex also has the structure of a (horizontally-shifted) graded  Lie algebra, with Lie bracket  [x, y] = x{y} − (−1)(|x|−1)(|y|−1)y{x}  of horizontal degree −1 and vertical degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The Hochschild differential is given  by  ∂(x) = [m2, x]  and satisfies the corresponding Leibniz rules with respect to the previous associative  product and Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The relation between brace operations, the Hochschild differential and the cup  product are encoded in the following straightforward consequences of the brace  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  6  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' JASSO, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' KELLER, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' MURO  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The following formula holds for all n ≥ 1:  ∂(x0{x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , xn}) = ∂(x0){x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , xn}  +  n  �  i=1  (−1)  �i−1  j=0 |xj|−ix0{x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , ∂(xi), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , xn}  + (−1)|x0|−1+|x0||x1|x1 · x0{x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , xn−1}  +  n−1  �  i=1  (−1)  �i  j=0 |xi|−i−1x0{x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , xi · xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , xn}  + (−1)  �n−1  j=0 |xj|−n−1x0{x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , xn−1} · xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The following formula holds for all n ≥ 1:  (x · y){z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , zn} =  n  �  i=0  (−1)|y| �i  j=1(|zi|−1)x{z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , zi} · y{zi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , zn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 for n = 1 proves that the induced associative product in Hochschild  cohomology  HH•,∗(A, A) = H•,∗(C•,∗(A, A))  (the cohomology of the Hochschild complex) is graded commutative with respect to  the total degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' This products satisfies the following compatibility relation with  the (horizontally shifted) Lie algebra structure,  [x, y · z] = [x, y] · z + (−1)(|x|−1)|y|y · [x, z],  and hence Hochschild cohomology is a Gernstenhaber algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' For this we use both  Lemmas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2, for n = 2 and n = 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The Hochschild complex  C•,∗(A, M) and Hochschild cohomology HH•,∗(A, M) are defined, more generally,  for M an A-bimodule, but it does not have any multiplicative structure in this  general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' For the existence of a graded associative algebra structure it suffices  that M be an associative algebra in A-bimodules, see also [Mur22, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Minimal A∞-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We now describe minimal A∞-algebras and their  morphisms in terms of the Hochschild complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' A minimal A∞-algebra structure  on a graded algebra A is a Hochschild cochain  m = (0, 0, 0, m3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , mn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' ) ∈ C•,∗(A, A)  of total degree 2 such that the Maurer–Cartan equation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1)  ∂(m) + m{m} = ∂(m) + 1  2[m, m] = 0  is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The pair (A, m) is also denoted  (A, m3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , mn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  If g : A′ → A is a graded algebra isomorphism, then  m ∗ g = (0, 0, 0, g−1m3g⊗3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , g−1mng⊗n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' )  is a minimal A∞-algebra structure on A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' If  m′ = (0, 0, 0, m′  3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , m′  n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' ) ∈ C•,∗(A, A)  is another minimal A∞-algebra structure on A, an A∞-isomorphism with identity  linear part  f : (A, m) −→ (A, m′)  is a Hochschild cochain  f = (0, 0, f2, f3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , fn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' ) ∈ C•,∗(A, A)  THE DONOVAN–WEMYSS CONJECTURE  7  of total degree 1 such that the following Hochschild cochain vanishes  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2)  ∂(f) + f · f +  �  r≥0  m′{f, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f} − m − f{m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  More generally, an A∞-isomorphism between minimal A∞-algebras  f : (A, m) −→ (A′, m′)  consists of an isomorphism of graded algebras  f1 : A −→ A′  and a Hochschild cochain  (0, 0, f2, f3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , fn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' ) ∈ C•,∗(A, A′)  of total degree 1 such that  (0, 0, f −1  1 f2, f −1  1 f3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , f −1  1 fn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' ): (A, m) −→ (A, m′ ∗ f1)  is an A∞-isomorphism with identity linear part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The Derived Donovan–Wemyss Conjecture  In this section we discuss one of the main results in [HK18]—crucial to our proof  of the Dononvan–Wemyss conjecture—and explain how it implies a derived version  of the conjecture (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 2Z-derived contraction algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' By means of the equivalence of triangu-  lated categories CM(R) ≃ Dsg(R), the contraction algebra associated to a crepant  resolution p of an isolated cDV singularity can be promoted to the DG algebra  Λcon = Λcon(p) := REnd(N)  given by the derived endomorphism algebra of the corresponding 2Z-cluster tilt-  ing object N = N(p) ∈ Dsg(R)dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' By construction H0(Λcon) ∼= Λcon and, as a  consequence of the 2-periodicity of the singularity category of R,  H•(Λcon) ∼= Λcon[ı±1] = Λcon ⊗C C[ı±1],  |ı| = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  We refer to the DG algebra Λcon as the 2Z-derived contraction algebra of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The  (soft) non-positive truncation Λ≤0  con = τ ≤0Λcon of Λcon is quasi-isomorphic to the  derived contraction algebra of p considered for example in [Boo19, Boo21, HK18],  and there is an isomorphism of graded algebras  H•(Λ≤0  con) ∼= Λcon[ı] = Λcon ⊗C C[ı],  |ı| = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The 2Z-derived contraction algebra Λcon is a localisation of Λ≤0  con (see [Boo21,  Thms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3] and [HK18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='17]) and Λcon can also be interpreted  as a non-connective variant of Λ≤0  con.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Notice also that, since 2Z-cluster tilting objects are in particular classical gen-  erators, there is a canonical quasi-equivalence of DG categories  Dc(Λcon)dg  ∼  −→ Dsg(R)dg  that induces an equivalence of triangulated categories  Dc(Λcon)  ∼  −→ Dsg(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Although we do not need this fact in the sequel, we mention that there is an  equivalence of triangulated categories4 [HK18, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='12]  C(Λ≤0  con) := Dc(Λ≤0  con)/ Dfd(Λ≤0  con) ≃ Dsg(R),  4For a DG algebra A, we denote by Dfd(A) the full subcategory of D(A) spanned by the DG  A-modules with finite-dimensional total cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  8  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' JASSO, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' KELLER, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' MURO  that is compatible with the canonical DG enhancements on either side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The cate-  gory C(Λ≤0  con) is known as the Amiot cluster category of Λ≤0  con [Ami07] and, indeed,  establishing a link between birational geometry and the theory of cluster categories  was one of the objectives in [HK18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The Derived Donovan–Wemyss Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The following theorem of  Hua and the second-named author settles a derived version of Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 ([HK18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R = C�x, y, z, t�/(f) be an isolated cDV  singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Then, there is an isomorphism of algebras  HH0(Dsg(R)dg) ∼=  C�x, y, z, t�  (f, ∂xf, ∂yf, ∂zf, ∂tf)  between the 0-th Hochschild cohomology of the DG category Dsg(R)dg and the Tyu-  rina algebra of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In particular, if R′ is a further isolated cDV singularity such  that the DG categories Dsg(R)dg and Dsg(R′)dg are quasi-equivalent, then there is  an isomorphism of algebras R ∼= R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 relies on a comparison result [Kel18,  Kel19] between the singular Hochschild cohomology (=Hochschild–Tate cohomol-  ogy) of R and the Hochschild cohomology of the DG category Dsg(R)dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The  appearance of the Tyurina algebra stems from an earlier result of the Buenos Aires  Cyclic Homology Group [GGRV92, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' That the Tyurina algebra, together  with the dimension of R, determines the isomorphism type of the singularity is a  theorem of Mather and Yau [MY82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In [Dyc11], Dyckerhoff shows that the 0-th Hochschild cohomology  of Dsg(R)dg—viewed as a Z/2-graded DG category—is isomorphic to the Milnor  algebra  C�x, y, z, t�  (∂xf, ∂yf, ∂zf, ∂tf)  of the singularity (which does not determine the isomorphism type of the singularity,  even if one knows the dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Thus, in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 it is crucial to consider  Dsg(R)dg as a Z-graded DG category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4 (Derived Donovan–Wemyss Conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R1 and R2 be iso-  lated cDV singularities with crepant resolutions  p1 : X1 → Spec(R1)  and  p2 : X2 → Spec(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  If the 2Z-derived contraction algebras Λcon(p1) and Λcon(p2) are quasi-isomorphic,  then there is an isomorphism of algebras R1 ∼= R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Indeed, if the DG algebras Λcon(p1) and Λcon(p2) are quasi-isomorphic, then  the DG categories  Dc(Λcon(p1))dg ≃ Dsg(R1)dg  and  Dc(Λcon(p2))dg ≃ Dsg(R2)dg  are quasi-equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 then implies the existence of an isomorphism  of algebras R1 ∼= R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Uniqueness of the 2Z-derived contraction algebra  In this section we prove that 2Z-derived contraction algebras are determined  up to quasi-isomorphism by their zeroth cohomology plus a minimal amount of  additional algebraic data (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='6 for the precise statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Before  that, we formulate a closely related result (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1) that states that two 2Z-  derived contraction algebras whose zeroth cohomologies are isomorphic as algebras  must be quasi-isomorphic, and use this result to prove Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  THE DONOVAN–WEMYSS CONJECTURE  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Proof of the Donovan–Wemyss Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In view of Corollaries 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4, Conjecture A is an immediate consequence of the following theorem, the  proof of which is given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R1 and R2 be isolated cDV singularities with crepant resolu-  tions  p1 : X1 → Spec(R1)  and  p2 : X2 → Spec(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  If the contraction algebras Λ(p1) and Λ(p2) are isomorphic, then the 2Z-derived  contraction algebras Λcon(p1) and Λcon(p2) are quasi-isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Remark  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1  is  a  special  case  of  [JM22,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='10],  see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof of Conjecture A using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R1 and R2 be isolated cDV sin-  gularities with crepant resolutions  p1 : X1 → Spec(R1)  and  p2 : X2 → Spec(R2)  whose corresponding contraction algebras Λcon(p1) and Λcon(p2) are derived equiva-  lent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In view of Corollaries 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4, we may and we will assume that Λcon(p1)  and Λcon(p2) are isomorphic and hence, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1, the 2Z-derived contrac-  tion algebras Λcon(p1) and Λcon(p2) are quasi-isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Finally, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4  yields the desired algebra isomorphism R1 ∼= R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The restricted universal Massey product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1  makes use of an invariant of the 2Z-derived contraction algebra, a certain Hochschild  cohomology class of bidegree (4, −2) that we call the restricted universal Massey  product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' As we explain below, this invariant is induced by the first possibly non-  trivial higher operation on a minimal A∞-algebra model of the 2Z-derived contrac-  tion algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Setting 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Fix an isolated cDV singularity R that admits a crepant resolution  p: X → Spec(R), and let Λ = Λcon(p) be the corresponding 2Z-derived contraction  algebra so that H0(Λ) ∼= Λ = Λcon(p) is the contraction algebra defined by Donovan  and Wemyss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' For simplicity, we treat the isomorphism of graded algebras  H•(Λ) ∼= Λ[ı±1] = Λ ⊗C C[ı±1],  |ı| = −2,  as an identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Kadeishvili’s Homotopy Transfer Theorem [Kad82] provides us with a minimal  A∞-algebra structure, unique up to A∞-isomorphism with identity linear part,  B = (Λ[ı±1], m3, m4, m5, · · · )  on the cohomology algebra Λ[ı±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Since Λ[ı±1] is concentrated in even degrees and,  by definition,  mn : Λ[ı±1]⊗n −→ Λ[ı±1]  is a morphism of degree 2 − n, we conclude that mn = 0 whenever n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We  write  B = (Λ[ı±1], m4, m6, m8, · · · )  as a way to record this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We refer to B as a minimal (A∞-algebra)  model of the DG algebra Λ and fix it for the rest of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The passage from DG to A∞-algebras is a matter of technical  convenience: The homotopy theories of non-unital DG and of A∞-algebras are  equivalent, [LV12, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In particular, two non-unital DG algebras are  quasi-isomorphic if and only if their minimal models are A∞-isomorphic [LV12,  Thms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Here, we are exclusively interested in unital DG algebras,  10  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' JASSO, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' KELLER, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' MURO  but this is not a problem since by [Mur14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2] two unital DG algebras are  quasi-isomorphic as non-unital DG algebras if and only if they are quasi-isomorphic  as unital DG algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Consider now the bigraded Hochschild cochain complex  Cp,q�  Λ[ı±1], Λ[ı±1]  � := HomC  �  Λ[ı±1]⊗p, Λ[ı±1](q)  �  ,  p ≥ 0, q ∈ Z,  recalled in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Since m3 = 0, the A∞-equations imply that ∂(m4) = 0  ([LH03, Lemme B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' hence we obtain a class  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3)  {m4} =  �  mΛ  4  �  ∈ HH4,−2�  Λ[ı±1], Λ[ı±1]  �  that we call the universal Massey product (of length 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' It follows from the definition  of A∞-morphism ([LH03, Lemme B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2]) that the class {m4} does not depend on  the choice of minimal model for Λ and hence the universal Massey product can and  will be regarded as an invariant of the latter DG algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Consider now the graded-algebra morphism j : Λ ֒→ Λ[ı±1] given by the inclusion  of the degree 0 part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The morphism j induces a restriction morphism on Hochschild  cohomology5  j∗: HH•,∗�  Λ[ı±1], Λ[ı±1]  �  −→ HH•,∗�  Λ, Λ[ı±1]  �  ,  where the space on the right is the Hochschild cohomology of Λ, viewed as a graded  algebra concentrated in degree 0, with coefficients in the graded Λ-bimodule Λ[ı±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  In particular, since the degree −2 component Λ · ı of Λ[ı±1] is isomorphic to the  diagonal Λ-bimodule, we obtain a class  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4)  j∗{m4} = j∗�  mΛ  4  �  ∈ HH4,−2�  Λ, Λ[ı±1]  �  = HH4(Λ, Λ · ı) = Ext4  Λe(Λ, Λ) ,  where Λe = Λ ⊗C Λop is the eveloping algebra of Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' we call the class j∗{m4}  the restricted universal Massey product (of length 4) and, as with the unrestricted  version, we regard it as an invariant of the 2Z-derived contraction algebra Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Notice  also that the previous discussion applies verbatim to any DG algebra A whose  cohomology is isomorphic to the graded algebra Λ[ı±1], so that we may associate  to A its restricted universal Massey product j∗�  mA  4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The following theorem is the first main step towards the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The restricted universal Massey product j∗{m4}, when viewed as  an element of the space Ext4  Λe(Λ, Λ) of Yoneda extensions of Λ-bimodules, can be  represented by an exact sequence  0 → Λ → P3 → P2 → P1 → P0 → Λ → 0  with projective middle terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In particular, Ω4  Λe(Λ) ∼= Λ in the stable category of  Λ-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The first claim is a special case of [JM22, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Indeed, by definition,  the 2Z-derived contraction algebra Λ is the derived endomorphism algebra of a 2Z-  cluster tilting object in Dc(Λ) ≃ Dsg(R), which is one of the equivalent conditions  in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The second claim follows immediately from the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The proof of [JM22, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='17], and hence that of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5, is  non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In the special case of the contraction algebra, it is possible that detailed  knowledge of the first non-trivial higher operation m4 of some minimal model of  the 2Z-derived contraction algebra allows for establishig the desired property of the  restricted universal Massey product j∗{m4} directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The approach taken in [JM22],  which deals with an abstract and more general situation, rather leverages the fact  that Λ is the endomorphism algebra of a 2Z-cluster tilting object T ∈ Dsg(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  5In fact, the morphism j∗ is surjective, see [JM22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='9] and take σ = 1 and d = 2  (which is even and hence no signs occur in the formulas therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  THE DONOVAN–WEMYSS CONJECTURE  11  The upshot is that the additive closure add(T ) of T has an induced structure of a  so-called 4-angulated category, that is add(T ) is equipped with a natural class of  diagrams □GKO, called 4-angles, of the form6  T1 → T2 → T3 → T4 → T1[2]  that satisfies axioms analogous to those of triangulated categories [GKO13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' On  the other hand, an extension of Λ-bimodules  0 → Λ → P3 → P2 → P1 → P0 → Λ → 0  with P0, P1, P2 projective-injective (but perhaps not P3) that represents the class  j∗{m4} ∈ Ext4  Λe(Λ, Λ) yields a class of 4-angles □j∗{m4} defined in terms of certain  exactness properties [Ami07, Lin19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' the class □j∗{m4} is a priori not known to  form a 4-angulation of add(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The crux of the argument is then to prove that  □GKO = □j∗{m4}  so that the class □j∗{m4} is indeed a 4-angulation of add(T );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' this agreement relies  on a delicate analysis of the relationship between Toda brackets, Massey products  and the classes □GKO and □j∗{m4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Finally, in view of the exactness properties  defining the class □j∗{m4} (now known to be 4-angulation), a theorem of Auslander  and Reiten [AR91] for detecting projective bimodules implies that P3 must be a  projective Λ-bimodule, which is what Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5 claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The reader is referred  to [JM22] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Recall that the contraction algebra is Frobenius (in fact, symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Conse-  quently, its enveloping algebra is also a Frobenius algebra and we may consider the  Hochschild–Tate cohomology  HH•,∗�  Λ, Λ[ı±1]  � := Ext•,∗  Λe  �  Λ, Λ[ı±1]  �  defined in terms of the extension spaces in the stable category of graded Λ-  bimodules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' thus,  HH>0,∗�  Λ, Λ[ı±1]  �  = HH>0,∗�  Λ, Λ[ı±1]  �  and there is a surjection  HH0,∗�  Λ, Λ[ı±1]  �  ։ HH0,∗�  Λ, Λ[ı±1]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The multiplication on Λ[ı±1] endows HH•,∗�  Λ, Λ[ı±1]  �  with the structure of a bi-  graded algebra, see [Mur22, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 5] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The restricted universal Massey product j∗{m4}, when viewed  as an element of the Hochschild–Tate cohomology HH•,∗�  Λ, Λ[ı±1]  �  , is a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Immediate from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5 and [Mur22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='7 and Rmk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='8], which  characterises the units in HH•,∗�  Λ, Λ[ı±1]  �  of positive Hochschild (=horizontal) de-  gree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='7 it is essential to pass from Hochschild to  Hochschild–Tate cohomology in order to have units of positive Hochschild degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  6Recall that [2] ∼  = 1 in Dsg(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  12  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' JASSO, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' KELLER, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' MURO  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Hochschild cohomology of the graded contraction algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In this  section we compute the Hochschild cohomology of the graded algebra  Λcon[ı±1] = Λ[ı±1] = Λ ⊗C C[ı±1],  |ı| = −2,  that we call graded contraction algebra, in terms of the Hochschild cohomology of  the Dononvan–Wemyss contraction algebra Λcon = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  First, notice that ı lies in the (graded) centre of Λ[ı±1], which is  Z(Λ[ı±1]) = HH0,∗�  Λ[ı±1], Λ[ı±1]  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  hence  ı ∈ HH0,−2�  Λ[ı±1], Λ[ı±1]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  We introduce the fractional Euler derivation  ¯δ ∈ C1,0�  Λ[ı±1], Λ[ı±1]  �  ,  which acts by the formula  ¯δ : a �−→ |a|  2 a,  where we observe that |a|  2 is an integer since Λ[ı±1] is concentrated in even degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  It is a cocycle with cohomology class  δ ∈ HH1,0�  Λ[ı±1], Λ[ı±1]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The following statements hold:  (1) There is an isomorphism of graded commutative algebras  HH•,∗�  Λ[ı±1], Λ[ı±1]  � ∼= HH•(Λ, Λ) [ı±1, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The graded Lie algebra structure on the right hand side is induced by the  (usual) Lie algebra structure on HH•,∗(Λ, Λ) by setting  [ı, HH•(Λ, Λ)] = 0,  [ı, ı] = 0,  [δ, HH•(Λ, Λ)] = 0,  [δ, ı] = −ı.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (2) There is an isomorphism of graded algebras  HH•,∗�  Λ, Λ[ı±1]  � ∼= HH•(Λ, Λ) [ı±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Moreover, the morphism  j∗ : HH•,∗�  Λ[ı±1], Λ[ı±1]  �  −→ HH•,∗�  Λ, Λ[ı±1]  �  induced by the inclusion j : Λ ֒→ Λ[ı±1] of the degree 0 part is the apparent  natural projection with kernel the graded ideal generated by δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (3) There is an isomorphism of graded algebras  HH•,∗�  Λ, Λ[ı±1]  � ∼= HH•(Λ, Λ) [ı±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Furthermore, the comparison map  HH•,∗�  Λ, Λ[ı±1]  �  −→ HH•,∗�  Λ, Λ[ı±1]  �  is the apparent extension of the comparison map HH•(Λ, Λ) → HH•(Λ, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' All of the forthcoming claims follow from the proof of [JM22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='9]  for σ = 1Λ and d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (1) The Hochschild complex C•,∗�  Λ[ı±1], Λ[ı±1]  �  contains the subcomplex  C•,∗  C[ı±1]  �  Λ[ı±1], Λ[ı±1]  �  of C[ı±1]-linear cochains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' this subcomplex is also an associative subalgebra and a  Lie subalgebra of the C-linear Hochschild complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The composite  C•,∗  C[ı±1]  �  Λ[ı±1], Λ[ı±1]  �  i֒→ C•,∗�  Λ[ı±1], Λ[ı±1]  �  j∗  −→ C•,∗�  Λ, Λ[ı±1]  �  ,  THE DONOVAN–WEMYSS CONJECTURE  13  of the inclusion of the C[ı±1]-linear Hochschild cochains into the C-linear ones with  the restriction of scalars along the inclusion j : Λ ֒→ Λ[ı±1] of the degree 0 part is  an isomorphism of DG algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The target, unlike the source, does not a priori  carry any Lie algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Nevertheless, there is an obvious isomorphism of  DG algebras  C•,∗�  Λ, Λ[ı±1]  � ∼= C•(Λ, Λ) [ı±1]  that we regard as an identification, and the composite isomorphism  C•,∗  C[ı±1]  �  Λ[ı±1], Λ[ı±1]  � ∼= C•(Λ, Λ) [ı±1]  is also a Lie algebra map when we regard the target as a Lie algebra extension of  C•(Λ, Λ) with ı a central element (in the Lie-algebra sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The morphism  C•,∗�  Λ[ı±1], Λ[ı±1]  �  −→ C•(Λ, Λ) [ı±1, ¯δ],  x �−→ j∗(x) − ı−1 · j∗([x, ı]) · ¯δ,  is a quasi-isomorphism of DG algebras with quasi-inverse  C•(Λ, Λ) [ı±1] ⊕ C•(Λ, Λ) [ı±1] · ¯δ −→ C•,∗�  Λ[ı±1], Λ[ı±1]  �  ,  x + y · ¯δ �−→ i(x) + i(y) · ¯δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (The latter is just a morphism of complexes since ¯δ2 ̸= 0 in the target, it only  vanishes in cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=')  In fact, the relevant composite equals the identity of  C•(Λ, Λ) [ı±1, ¯δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The Lie bracket formulas in the statement of the proposition  follow from the definition of the fractional Euler class, C[ı±1]-linear cochains and  degree considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (2) The statement follows easily from the previous computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (3) The statement is consequece of the fact that HH•,∗(Λ, Λ) is obtained from  HH•,∗(Λ, Λ) by inverting any element of  HH4(Λ, Λ) = Ext4  Λe(Λ, Λ)  representing the 4-periodicty of Λ, and similarly when the coefficients lie in Λ[ı±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  Below, we use the isomorphisms in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 as identifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let u ∈ HH4(Λ, Λ) be a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' There exists a unique Hochschild  class  m ∈ HH4,−2�  Λ[ı±1], Λ[ı±1]  �  such that  j∗(m) = u · ı,  1  2[m, m] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The first equation in the statement is equivalent to m being of the form  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3)  m = (u + x · δ) · ı  for some x ∈ HH3(Λ, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Using the relations in a Gerstenhaber algebra, the second  equation is equivalent to  0 = ([u, u] − 2u · x) · ı2 − 2[x, u] · ı2 · δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  This means that both summands must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' For the first one, this is equivalent  to  x = 1  2u−1[u, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  This is takes place in the piece of HH•(Λ, Λ) that agrees with HH•(Λ, Λ), and is  compatible with the second summand since  0 = 1  2[[u, u], u] = [ux, u] = u[x, u] − [u, u]x = u[x, u] − u−1[u, u]2 = u[x, u],  14  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' JASSO, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' KELLER, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' MURO  so [x, u] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The first step follows from the graded Jacobi identity and we also  use that [u, u]2 = 0 since [u, u] has odd total degree and Hochschild cohomology is  graded commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  The following result should be compared with equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' its proof is similar  to that of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let u ∈ HH4(Λ, Λ) be a unit such that [u, u] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Given  (x + y · δ) · ıq ∈ HHp,−2q�  Λ[ı±1], Λ[ı±1]  �  with p ≥ 2, x ∈ HHp(Λ, Λ) and y ∈ HHp−1(Λ, Λ), if [u · ı, (x + y · δ) · ıq] = 0 then  (x + y · δ) · ıq = [u · ı, u−1 · δ · x · ıq−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  We obtain the following more precise information on a minimal A∞-model of the  2Z-derived contraction algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The 2Z-derived contraction algebra has a minimal A∞-model  (Λ[ı±1], m4, m6, · · · )  such that mn is C[ı±1]-linear for all n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In particular, {m4} = u · ı for some  unit u ∈ HH4(Λ, Λ) satisfying [u, u] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The first part follows from [HK18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The rest is a direct consecuence of  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 and the fact that [{m4}, {m4}] = 0, which follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The introduction of the restricted universal  Massey product of Λ is justified by the following result and the subsequent corol-  lary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 is an immediate consequence of [JM22, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' B], and the latter  theorem is obtained an application of the obstruction theory for the existence of  A∞-structures developed by the third-named author in [Mur20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In this note we  give a direct proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 that leverages our detailed knowledge of the  relationship between the Hochschild cohomology of the contraction algebra and  that if its graded variant (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3), although part of the techniques used to  prove [JM22, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' B] are utilised in some guise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let A be a DG algebra such that H•(A) = Λ[ı±1] as graded  algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' If  j∗�  mA  4  �  = j∗�  mΛ  4  �  ∈ HH•,∗�  Λ, Λ[ı±1]  �  ,  then A is quasi-isomorphic to the 2Z-derived contraction algebra Λ via a quasi-iso-  morphism that induces the identity in cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let  (Λ[ı±1], m4, m6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' )  be a minimal model for the 2Z-derived contraction algebra as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5,  and  (Λ[ı±1], m′  4, m′  6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' )  a minimal model for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Inductively, we will construct an A∞-isomorphism with  identity linear part  f = (0, 0, 0, f3, 0, f5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' ): (Λ[ı±1], m4, m6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' ) −→ (Λ[ı±1], m′  4, m′  6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' ),  and this clearly suffices to prove the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Notice that, necessarily, f2n = 0 for all  n ≥ 0 since Λ[ı±1] is concentrated in even degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  We proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' For all n ≥ 0 we define a Hochschild cochain of total  degree 1  f (n) = (0, 0, 0, f (n)  3  , 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , f (n)  2n+1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' )  THE DONOVAN–WEMYSS CONJECTURE  15  such that f (n) coincides with f (n−1) up to Hochschild degree 2n − 2 and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2)n  ∂(f (n)) + f (n) · f (n) +  �  r≥0  m′{f (n), r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f (n)} − m − f (n){m}  vanishes up to Hochschild degree 2n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' If we achieve this goal, then we can take  f = (0, 0, 0, f (3)  3 , 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , f (n)  2n−3, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Indeed, f coincides with f (n) up to Hochschild degree 2n − 2, so (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2) coincides  with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2)n up to Hochschild degree 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In particular (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2) vanishes up to  Hochschild degree 2n − 1 for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Therefore (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2) fully vanishes, so f is  indeed an A∞-isomorphism with identity linear part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  We start with f (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  With this choice, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2)0 clearly vanishes up to  Hochschild degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Below, when defining f (n) we will only specify f (n)  2n−1 and f (n)  2n+1 since in smaller  Hochschild degrees they are determined by f (n−1) and in higher Hochschild degrees  they are irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Moreover, we will also use that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2)n−1 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2)n agree  (and hence both vanish) up to Hochschild degree 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Since j∗{m4} = j∗{m′  4}, then {m4} = {m′  4} by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2, so there exists  f (1)  3  such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3)  ∂(f (1)  3 ) + m′  4 − m4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  This proves that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2)1 vanishes up to Hochschild degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Assume we have constructed up to f (n−1) for some n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let us see how to  construct f (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We know by [LH03, Lemme B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2] that the Hochschild degree 2n+2  part of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2)n−1, that we simply denote by a, is an obstruction cocycle (∂(a) = 0)  which vanishes in cohomology if and only if there exists f (n)  2n+1 such that, taking  f (n)  2n−1 = f (n−1)  2n−1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2)n vanishes up to Hochschild degree 2n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Indeed, any  f (n)  2n+1 such that a + ∂(f (n)  2n+1) = 0 would do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We claim that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4)  [m4, a] + ∂(b − f n−1  3  {a})  vanishes, where b is the Hochschild degree 2n + 4 part of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2)n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We prove this  claim below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Now, we deduce from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4 that there  exist Hochschild cochains c2n−1 and c2n+1 such that  a + ∂(c2n+1) + [m4, c2n−1] = 0,  ∂(c2n−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  If we set  f (n)  2n−1 = f (n−1)  2n−1 + c2n−1,  f (n)  2n+1 =  �  c5 + f (1)  3 {c3} − 1  2c3{c3},  n = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  c2n+1 + f3{c2n−1},  n > 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  we complete the induction step since the Hochschild degree 2n part of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2)n is,  ∂(c2n−1) = 0,  and its Hochschild degree 2n + 2 part is, for n = 2,  a + ∂  �  c5 + f (1)  3 {c3} − 1  2c3{c3}  �  + c3 · c3 + m′  4{c3} − c3{m4}  = a + ∂(c5) + ∂  �  f (1)  3  �  {c3} + f (1)  3 {∂(c3)} + m′  4{c3} − c3{m4}  = a + ∂(c5) + (m4 − m′  4){c3} + m′  4{c3} − c3{m4}  = a + ∂(c5) + [m4, c3] = 0,  16  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' JASSO, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' KELLER, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' MURO  where we use that ∂(c3{c3}) = 2c3 · c3 by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1, and for n > 2,  a + ∂  �  c2n+1 + f3{c2n−1}  �  + m′  4{c2n−1} − c2n−1{m4}  = a + ∂(c2n+1) + ∂  �  f (n−1)  3  �  {c2n−1} + f (n−1)  3  {∂(c2n−1)} + m′  4{c2n−1} − c2n−1{m4}  = a + ∂(c2n+1) + (m4 − m′  4){c2n−1} + m′  4{c2n−1} − c2n−1{m4}  = a + ∂(c2n+1) + [m4, c2n−1] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  We finish the proof with the vanishing of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In what follows, let us write  Ξ = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2)n−1 and f = f (n−1), so as not to overload notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4) is  the Hochschild degree 2n + 5 part of  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5)  [m, Ξ] + ∂(Ξ − f{Ξ}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  This cochain vanishes in Hochschild degrees < 2n + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  We now start a series of computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We number most terms for bookkeeping  purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In the first equation we use Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1,  ∂(Ξ) =  1  ∂(f) · f −  2  f · ∂f +  3  �  r≥0  ∂(m′){f, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f}  −  4  �  r≥1  r  �  i=1  m′{f, i−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', ∂(f), r−i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f}  −  5  �  r≥1  f · m′{f, r−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f} −  6  �  r≥2  r−1  �  i=1  m′{f, i−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f 2, r−i−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , f}  +  7  �  r≥1  m′{f, r−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f} · f −  8  ∂(m) −  9  ∂(f){m} −  10  f{∂(m)} +  11  f · m −  12  m · f  Since m and m′ are A∞-algebra structures,  8 = −m{m},  10 = −f{m{m}},  3 = −  �  r≥0  m′{m′}{f, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f}  = −  �  r≥0  �  0≤i≤j≤r  m′{f,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', m′{f, j−i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f}, r−j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f},  = −  13  �  r≥0  m′{m′{f, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f}} −  14  �  r≥1  �  0≤i≤j≤r  j−i<r  m′{f,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', m′{f, j−i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f}, r−j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f}  THE DONOVAN–WEMYSS CONJECTURE  17  Here we also use the brace relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We also split the following summations in two  parts,  4 =  15  m′{∂(f)} +  16  �  r≥2  r  �  i=1  m′{f, i−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', ∂(f), r−i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f},  6 =  17  m′{f 2} +  18  �  r≥3  r−1  �  i=1  m′{f, i−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f 2, r−i−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' , f} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Consider the following cochain, that we decompose using the brace relation,  19  �  r≥0  m′{f, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f}{m} =  �  r≥0  r  �  i=0  m′{f,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', m, r−i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f}  +  �  r≥1  r  �  i=0  m′{f, i−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f{m}, r−i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f}  =  20  m′{m} +  21  �  r≥1  r  �  i=0  m′{f,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', m, r−i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f} +  22  m′{f{m}}  +  23  �  r≥2  r  �  i=0  m′{f, i−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f{m}, r−i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Consider also the following cochain, which is computed by using Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2,  24  f 2{m} =  25  f · f{m} −  26  f{m} · f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Since Ξ vanishes up to Hochschild degree 2n + 1 and its Hochschild degree 2n + 2  part is a cocycle, the following cochains vanish up to Hochschild degree 2n + 5,  �  r≥2  r  �  i=1  m′{f, i−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', Ξ, r−i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', f} = 16 + 18 + 14 − 21 − 23 ,  ∂(f){Ξ} + m′{Ξ} − m{Ξ},  f{∂(Ξ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Notice that  Ξ · f = 1 + f 3 + 7 − 12 − 26 ,  f · Ξ = 2 + f 3 + 5 − 11 − 25 ,  m′{Ξ} = 15 + 17 + 13 − 20 − 22 ,  Ξ{m} = 9 + 24 + 19 + 8 + 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  18  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' JASSO, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' KELLER, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' MURO  Using all the above, we obtain the following relations, where ≡ stands for con-  gruence modulo cochains vanishing in Hochschild degrees < 2n + 5,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5) = m{Ξ} + Ξ{m} + ∂(Ξ) − ∂(f){Ξ} − f{∂(Ξ)} + f · Ξ − Ξ · f  ≡ m{Ξ} −  �  9 + 24 + 19 + 8 + 10  �  +  �  1 − 2 + 3 − 4 − 5 − 6 + 7 − 8 − 9 − 10 + 11 − 12  �  +  �  15 + 17 + 13 − 20 − 22  �  − m{Ξ} − 0  +  �  2 + f 3 + 5 − 11 − 25  �  −  �  1 + f 3 + 7 − 12 − 26  �  +  �  16 + 18 + 14 − 21 − 23  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  This finally concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let A be a DG algebra such that H•(A) = Λ[ı±1] as graded  algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' If the restricted universal Massey product  j∗�  mA  4  �  ∈ HH•,∗�  Λ, Λ[ı±1]  �  is a unit, then A is quasi-isomorphic to the 2Z-derived contraction algebra Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Firstly, we observe that the group of graded-algebra automorphisms of Λ[ı±1]  acts on the right of HH•,∗�  Λ, Λ[ı±1]  �  by conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  In particular, the group  Z(Λ)× of units of the centre of Λ acts on HH•,∗�  Λ, Λ[ı±1]  �  via the graded-algebra  automorphisms  gu : x �−→ xui,  |x| = 2i,  where u ∈ Z(Λ)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The induced action on  HH4,−2�  Λ, Λ[ı±1]  � ∼= Ext4  Λe(Λ, Λ)  has the following explicit description: Given a unit u ∈ Z(Λ)× and an exact se-  quence of Λ-bimodules  η: 0 → Λ  f−→ X3 → X2 → X1 → X0 → Λ → 0,  we let [η] · u be the class of the exact sequence  0 → Λ  f ′  −→ X3 → X2 → X1 → X0 → Λ → 0,  where f ′ := u−1f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The above action clearly restricts to the set of units in  HH•,∗�  Λ, Λ[ı±1]  �  of bidegree (4, −2), since these are precisely the classes that can be  represented by an exact sequence with projective(-injective) middle terms [Mur22,  Rmk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='8], and is in fact transitive on this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' To prove the latter claim on the  transitivity of the action we appeal to [Che21, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3], which allows us to lift  stable bimodule isomorphisms Λ ≃ Λ to honest bimodule isomorphisms Λ ∼= Λ, see  the proof of [JM22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Let u ∈ Z(Λ)× be a unit such that  j∗�  mA  4  �  u = j∗�  mΛ  4  �  ∈ HH4,−2�  Λ, Λ[ı±1]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Given a minimal model  (Λ[ı±1], mA  4 , mA  6 , mA  8 , · · · )  of the DG algebra A we define a new minimal A∞-algebra structure  (Λ[ı±1], mA  4 , mA  6 , mA  8 , · · · ) = (Λ[ı±1], mA  4 , mA  6 , mA  8 , · · · ) ∗ gu  with n-ary opearations mA  n := g−1  u mA  4 g⊗n  u  (notice that mA  4 ̸= mA  4 as soon as u ̸= 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  It is easy to see that  j∗�  mA  4  �  = j∗�  mA  4  �  u = j∗�  mΛ  4  �  THE DONOVAN–WEMYSS CONJECTURE  19  and that there is an isomorphism of A∞-algebras  (Λ[ı±1], mA  4 , mA  6 , mA  8 , · · · ) ⇝ (Λ[ı±1], mA  4 , mA  6 , mA  8 , · · · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Thus, A is quasi-isomorphic to any DG algebra model B of the minimal A∞-  algebra on the right-hand side, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2, and by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 the DG  algebras B and Λ are quasi-isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Consequently, the DG algebras A and Λ  are isomorphic, which is what we needed to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  We are ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R1 and R2 be isolated cDV singularities with crepant  resolutions  p1 : X1 → Spec(R1)  and  p2 : X2 → Spec(R2)  whose corresponding contraction algebras Λ1 = Λ(p1) and Λ2 = Λ(p2) are isomor-  phic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We need to prove that the 2Z-derived contraction algebras Λ1 = Λcon(p1)  and Λ2 = Λcon(p2) are quasi-isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Recall that  H•(Λ1) ∼= Λ1[ı±1]  and  H•(Λ2) ∼= Λ2[ı±1],  |ı| = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  In view of the assumption that Λ1 ∼= Λ2, we obtain a chain of isomorphisms of  graded algebras  H•(Λ1) ∼= Λ1[ı±1] ∼= Λ2[ı±1] ∼= H•(Λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  In particular, we may and we will identify minimal models of Λ1 and Λ2 via the  above isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' For simplicity, let Λ = Λ1 ∼= Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='7 shows that  the restricted universal Massey products  j∗�  mΛ1  4  �  ∈ HH4,−2�  Λ, Λ[ı±1]  �  and  j∗�  mΛ2  4  �  ∈ HH4,−2�  Λ, Λ[ı±1]  �  are units in the Hochschild–Tate cohomology HH•,∗�  Λ, Λ[ı±1]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='6 im-  plies that the DG algebras Λ1 and Λ2 are quasi-isomorphic, which is what we  needed to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  We also have the following important corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R be an isolated cDV singularity that admits a crepant res-  olution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Then, the singularity category Dsg(R) admits a unique DG enhancement  in the sense of [BK90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let A be a DG enhancement of Dsg(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' By definition, this means that A  is a pre-triangulated DG category and there exists an equivalence of triangulated  categories  H0(A) ≃ Dsg(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Let T ∈ Dsg(R) be a 2Z-cluster tilting object and A the derived endomorphism alge-  bra of T computed by means of the DG enhancement A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In particular Dc(A)dg ≃ A  since T is a classical generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' By definition, H•(A) ∼= H•(Λ) where Λ is the de-  rived endomorphism algebra of T computed by means of the canonical DG enhance-  ment of Dsg(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The proofs of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='7 only rely on the  fact that T ∈ Dsg(R) is a 2Z-cluster tilting object, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Consequently,  the restricted universal Massey product j∗�  mA  4  �  is a unit in HH•,∗�  Λ, Λ[ı±1]  �  and,  by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='6, the DG algebras A and Λ are quasi-isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Therefore, the  DG categories  A ≃ Dc(A)dg  and  Dsg(R)dg  are quasi-equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' This shows that every DG enhancement of Dsg(R) is equiva-  lent to the canonical DG enhancement and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  20  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' JASSO, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' KELLER, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' MURO  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='7 is stronger that Conjecture A, as it shows that the  DG category Dsg(R)dg is determined by Λcon up to isomorphism in Hmo, the Morita  category of small DG categories [Tab05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Concluding remarks  In this section we collect some observations, some of which follow easily from  the results in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Formality of contraction algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Recall that a DG algebra A is formal  if there is a quasi-isomorphism A ≃ H•(A), where the graded algebra H•(A) is  viewed as a DG algebra with vanishing differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We observe that 2Z-derived  contraction algebras are almost never formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='7  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R be an isolated cDV singularity with a crepant resolution  and Λcon a 2Z-derived contraction algebra for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The following statements are  equivalent:  (1) The 2Z-derived contraction algebra Λcon is formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (2) There is an isomorphism of algebras Λcon ∼= C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (3) There is an isomorphism of algebras  R ∼= C�x, y, z, t�/(xy − zt),  so that Spec(R) is the base of the Atiyah flop [Ati58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  If the above equivalent conditions hold, then there is a quasi-isomorphism  Λcon ≃ C[ı±1],  |ı| = −2,  where the graded algebra C[ı±1] is equipped with the trivial differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' (1)⇒(2) If the 2Z-derived contraction algebra Λcon is formal, then its re-  stricted universal Massey product j∗{m4} is represented by the trivial sequence in  HH4,−2�  Λcon, Λcon[ı±1]  �  = Ext4  Λecon(Λcon, Λcon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In view of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='5, this can  only happen if Λcon is projective as a Λcon-bimodule or, equivalently, if the algebra  Λcon is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Since contraction algebras are basic and connected, we must  have an isomorphism of algebras Λcon ∼= C, which is what we needed to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (2)⇒(1) If Λcon ∼= C, then there is an isomorphism of graded algebras  H•(Λcon) ∼= Λcon[ı±1] ∼= C[ı±1],  |ı| = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  It is well-known (and easy to prove using Kadeishvili’s Theorem [Kad88], for exam-  ple) that the latter graded algebra is intrinsically formal, that is every DG algebra  with cohomology algebra C[ı±1] is formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In particular, Λcon is formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='8  (2)⇔(3) In view of the validity of Conjecture A, it is enough to observe that C is  indeed isomorphic to the contraction algebra of the Atiyah flop [DW16, Table 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The (non-)formality of the derived contraction algebra Λ≤0  con is inves-  tigated in [Boo19, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 9] where minimal models of this DG algebra are computed  in various examples, see also [Boo18, Boo21, Boo22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  7This fact will come as no surprise to the experts, but we did not find a proof of it in the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  8Alternatively, one can prove this fact using the results in this note as follows: Since the  enveloping algebra  Λe  con ∼  = C ⊗C C ∼  = C  is semisimple, the Hochschild–Tate cohomology HH•,∗�  Λcon, Λcon[ı±1]  �  vanishes in positive  Hochschild degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Then, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1, the 2Z-derived contraction algebra Λcon is quasi-  isomorphic to its cohomology algebra H•(Λcon) for the condition on the agreement of the corre-  sponding universal Massey products is trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Of course, this is essentially the same  proof as the one using Kadeishvili’s Theorem, which is indeed a special case of [JM22, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  THE DONOVAN–WEMYSS CONJECTURE  21  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' More generally, Kadeishvili’s Theorem [Kad88] can be used to prove  that the Laurent polynomial algebra K[ı±1] = K ⊗C C[ı±1] is intrinsically formal  if K is a finite-dimensional algebra of projective dimension at most 2 as a K-  bimodule [Sai20, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  This K, however, cannot be a contraction algebra  unless K = C since contraction algebras are basic and connected, and a symmetric  C-algebra has finite projective dimension as a bimodule over itself if and only if it  is separable (semisimple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Relationship to the Triangulated Auslander–Iyama Correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Let T be a triangulated category whose underlying additive category is Krull–  Schmidt and has finite-dimensional morphism spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  A basic object T ∈ T is  d-cluster tilting, d ≥ 1, if the following conditions are satisfied [IY08, Bel15]:  The object T is d-rigid: T(T, T [i]) = 0 for all 0 < i < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  For each object X ∈ T there exists a diagram  Td−2  · ·  T1  T0  Td−1  Xd−2  · ·  X2  X1  X  +1  +1  +1  in which Ti ∈ add(T ), 0 ≤ i < d, the oriented triangles denote exact triangles in  T and the unoriented triangles commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The object T is dZ-cluster tilting if it  is d-cluster tilting and T ∼= T [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 is a special case of the theorem  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' For a finite-dimensional algebra Λ, we let proj(Λ) be the category of finite-  dimensional projective Λ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' For example, if Λ = T(T, T ), then the Yoneda  functor  T ⊇ add(T )  ∼  −→ proj(Λ),  X �−→ T(T, X),  is an equivalence (of additive categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1  (Triangulated  Auslander–Iyama  Correspondence  [JM22,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  There is a bijective correspondence between the  following:  (1) Quasi-isomorphism classes of DG algebras A that satisfy the following:  The algebra H0(A) is a basic finite-dimensional algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The free DG A-module A ∈ Dc(A) is a dZ-cluster tilting object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (2) Equivalence classes of pairs (Λ, I) consisting of  a basic finite-dimensional self-injective algebra Λ and  an invertible Λ-bimodule I such that Ωd+2  Λe (Λ) ∼= I in the stable category  of Λ-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  The correspondence is given by the formula A �→ (H0(A), H−d(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The case d = 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 is one way to formulate the main  result in [Mur22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Indeed, an object T ∈ T is 1-cluster tilting if and only if it is  1Z-cluster tilting if and only if add(T ) = T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' the latter condition means that T is a  triangulated category of finite type in the terminology of [Mur22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let (Λ, I) be a pair as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Since the algebra Λ is  assumed to be basic, the map  Aut(Λ) −→ Pic(Λ),  σ �−→ [1Λσ]  from the group of algebra automorphisms of Λ to the Picard group of invertible  Λ-bimodules induces an isomorphism of groups [Bol84, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='8]  Out(Λ)  ∼  −→ Pic(Λ),  [σ] �−→ [1Λσ],  where Out(Λ) = Aut(Λ)/ Inn(Λ) is the group of outer automorphisms of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In par-  ticular, there exists σ ∈ Aut(Λ) such that I ∼= 1Λσ as Λ-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The condition  22  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' JASSO, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' KELLER, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' MURO  Ωd+2  Λe (Λ) ≃ 1Λσ expresses the fact that the algebra Λ is twisted (d + 2)-periodic  with respect to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' When σ = 1 or, equivalently, I ∼= Λ, the algebra Λ is said to be  (d+ 2)-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' For example, contraction algebras are known to be 4-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We  refer the reader to [ES08] and the references therein for information on (twisted)  periodic algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1  follows  from  the  injectivity  of  the  correspondence  in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 with d = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 outlined in this  note effectively goes through the proof of the latter in the special case of  2Z-derived contraction algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  We record the resulting characterisation of  contraction algebras for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Let R be an isolated cDV singularity with a crepant resolution  p: X → Spec(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Up to quasi-isomorphism, the 2Z-derived contraction algebra  Λ = Λcon(p) is the unique DG algebra with the following properties:  (1) Λ ∈ Dc(Λ) is a 2Z-cluster tilting object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (2) There is an isomorphism of algebras H0(Λ) ∼= Λ = Λcon(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (3) There is an isomorphism of Λ-bimodules H−2(Λ) ∼= Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  In other words, Λ is determined up to quasi-isomorphism by its image (Λ, Λ) under  the Triangulated Auslander–Iyama Correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' That the 2Z-derived contraction algebra satisfies the first two properties  follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 since, by definition, Λ is the derived endomorphism  algebra of a 2Z-cluster tilting object in Dsg(R) ≃ Dc(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Therefore Λ belongs to  the class of DG algebras in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1(1) with d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Given that Λ is 4-periodic,  the third property follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Moreover, Λ is determined up to quasi-isomorphism by  its image (Λ, Λ) under the Triangulated Auslander–Iyama Correspondence, which  is what we needed to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Isolated cDV singularities with non-smooth minimal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Con-  traction algebras are defined for an arbitrary cDV singularity R that is neither  isolated nor admits a crepant resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' However, contraction algebras are finite-  dimensional if and only if R defines an isolated singularity [DW19, Summary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='6],  and this finite-dimensionality is crucial to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' On the other hand, R  admits a crepant resolution if and only if the singularity category Dsg(R) admits a  2Z-cluster tilting object, see [BIKR08, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' If, on  the other hand, the minimal models9 of R are singular, then the contraction algebras  of R are the endomorphism algebras of maximal rigid objects in Dsg(R) [Wem18],  that is objects T ∈ Dsg(R) such that  add(T ) = {X ∈ Dsg(R)| Hom(T ⊕ X, (T ⊕ X)[1]) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  It is easy to verify that 2-cluster tilting objects are maximal rigid, but the converse is  false in general [BIKR08, BMV10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' moreover, if there exists a 2-cluster tilting object  then every maximal rigid object is also 2-cluster tilting [BIRS09, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='10 In  any case, one may still define the 2Z-derived contraction algebras of R as the derived  endomorphism algebras of maximal rigid objects in Dsg(R), computed in terms of  the canonical DG enhancement of the latter triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Note, however,  that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='1 does not cover the case of maximal rigid objects that are not  9In the context of the MMP in dimension three and higher, minimal models play the role of  minimal resolutions of surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' We do not recall the technical definition in this note, but only  mention that minimal models are permitted to have ‘mild’ singularities as long as they remain  ‘closer’ to the original space than a smooth resolution (which always exists by a famous theorem  of Hironaka [Hir64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=')  10The reader should compare this statement with the following geometric fact: If one minimal  model of Spec(R) is smooth, then all of its minimal models are also smooth [Kol89, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  THE DONOVAN–WEMYSS CONJECTURE  23  2Z-cluster tilting and hence it cannot be applied to prove a version of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4  for isolated cDV singularities that do not admit a crepant resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Finally, we  mention that the apparent variant of Conjecture A does not hold for isolated cDV  singularities whose minimal models are not smooth, see [Boo21, Ex 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='2] for an  explicit counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  References  [Ami07]  Claire Amiot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' On the structure of triangulated categories with finitely many indecom-  posables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' France, 135(3):435–474, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [AR91]  Maurice Auslander and Idun Reiten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' On a theorem of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Green on the dual of the trans-  pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In Representations of finite-dimensional algebras (Tsukuba, 1990), volume 11 of  CMS Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content=' Noncommutative deformations and flops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Duke  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content='  [DW19]  Will Donovan and Michael Wemyss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Contractions and deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=',  141(3):563–592, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [Dyc11]  Tobias Dyckerhoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Compact generators in categories of matrix factorizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Duke  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content='  [Eis80]  David Eisenbud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Homological algebra on a complete intersection, with an application  to group representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', 260(1):35–64, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [ES08]  Karin Erdmann and Andrzej Skowro´nski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Periodic algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In Trends in representa-  tion theory of algebras and related topics, EMS Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Congr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', pages 201–251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', Z¨urich, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  24  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' JASSO, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' KELLER, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' MURO  [GGRV92] Jorge Alberto Guccione, Juan Jose Guccione, Maria Julia Redondo, and Orlando Eu-  genio Villamayor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Hochschild and cyclic homology of hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Advances in Math-  ematics, 95(1):18–60, September 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [GKO13]  Christof Geiss, Bernhard Keller, and Steffen Oppermann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' n-angulated categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', 675:101–120, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [HI11]  Martin Herschend and Osamu Iyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Selfinjective quivers with potential and 2-  representation-finite algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content='  [Hir64]  Heisuke Hironaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Resolution of singularities of an algebraic variety over a field of  characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' I, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content=' ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' (2), 79:205–326,  1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [HK18]  Zheng Hua and Bernhard Keller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Cluster categories and rational curves, 2018,  1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content='AG].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [IY08]  Osamu Iyama and Yuji Yoshino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Mutation in triangulated categories and rigid Cohen-  Macaulay modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content='  [JM22]  Gustavo Jasso and Fernando Muro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The Triangulated Auslander–Iyama Correspon-  dence, with an appendix by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Keller, 2022, 2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='14413 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='RT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [Kad82]  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Kadeishvili.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The algebraic structure in the homology of an A(∞)-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Soob-  shch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Nauk Gruzin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' SSR, 108(2):249–252 (1983), 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [Kad88]  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Kadeishvili.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' The structure of the A(∞)-algebra, and the Hochschild and Harrison  cohomologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Trudy Tbiliss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Razmadze Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Nauk Gruzin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' SSR, 91:19–27,  1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content=' Genus zero Gopakumar-Vafa invariants of contractible curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' ´Ecole Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' (4), 27(1):63–  102, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content=' On the cyclic homology of exact categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content=' Algebra,  136(1):1–56, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content=' On differential graded categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' In International Congress of Math-  ematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' II, pages 151–190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', Z¨urich, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content=' Singular Hochschild cohomology via the singularity category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content='  Non-commutative  width  and  Gopakumar-Vafa  invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=', 211(2):435–521, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [Wem21]  Michael Wemyss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' A lockdown survey on cDV singularities, 2021, arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='16990  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='AG].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [Yos90]  Yuji Yoshino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Cohen-Macaulay modules over Cohen-Macaulay rings, volume 146 of  London Mathematical Society Lecture Note Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Cambridge University Press, Cam-  bridge, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  [Zim14]  Alexander Zimmermann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Representation theory, volume 19 of Algebra and Applica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Springer, Cham, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' A homological algebra point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Jasso) Lund University, Centre for Mathematical Sciences, S¨olvegatan 18A, 22100  Lund, Sweden  Email address: gustavo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='jasso@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='se  URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='se/staff/gustavo-jasso/  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Keller) Universit´e Paris Cit´e, UFR de Math´ematiques, Case 7012, Bˆatiment Sophie  Germain, 8 place Aur´elie Nemours, 75013 Paris Cedex 13, France  Email address: bernhard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='keller@imj-prg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='fr  URL: https://webusers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='imj-prg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='fr/~bernhard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='keller/  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content=' Muro) Universidad de Sevilla,  Facultad de Matem´aticas,  Departamento de  ´Algebra, Calle Tarfia s/n, 41012 Sevilla, Spain  Email address: fmuro@us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='es  URL: https://personal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
+page_content='es/fmuro/' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFJT4oBgHgl3EQfnyx6/content/2301.11593v1.pdf'}
diff --git a/99E4T4oBgHgl3EQfDgse/content/tmp_files/2301.04869v1.pdf.txt b/99E4T4oBgHgl3EQfDgse/content/tmp_files/2301.04869v1.pdf.txt
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+Parallel Interior-Point Solver for Block-Structured Nonlinear
+Programs on SIMD/GPU Architectures
+Fran¸cois Pacauda, Michel Schanenb, Sungho Shinb, Daniel Adrian Maldonadob,
+Mihai Anitescub
+a Centre Automatique et Syst`emes, Mines Paris - PSL, Paris, France; b Mathematics and
+Computer Science Department, Argonne National Laboratory, Lemont, USA
+ARTICLE HISTORY
+Compiled January 13, 2023
+ABSTRACT
+We investigate how to port the standard interior-point method to new exascale
+architectures for block-structured nonlinear programs with state equations. Compu-
+tationally, we decompose the interior-point algorithm into two successive operations:
+the evaluation of the derivatives and the solution of the associated Karush-Kuhn-
+Tucker (KKT) linear system. Our method accelerates both operations using two
+levels of parallelism. First, we distribute the computations on multiple processes
+using coarse parallelism. Second, each process uses a SIMD/GPU accelerator lo-
+cally to accelerate the operations using fine-grained parallelism. The KKT system
+is reduced by eliminating the inequalities and the state variables from the corre-
+sponding equations, to a dense matrix encoding the sensitivities of the problem’s
+degrees of freedom, drastically minimizing the memory exchange. We demonstrate
+the method’s capability on the supercomputer Polaris, a testbed for the future exas-
+cale Aurora system. Each node is equipped with four GPUs, a setup amenable to our
+two-level approach. Our experiments on the stochastic optimal power flow problem
+show that the method can achieve a 50x speed-up compared to the state-of-the-art
+method.
+1. Introduction
+Solving complex engineering problems often resorts to the solution of large-scale block-
+structured nonlinear programs. As such, there has been a long interest in designing
+efficient nonlinear optimization algorithms, particularly by using parallel computing.
+Parallelism can happen at two levels. At first, coarse parallelism splits the program
+into large computational chunks, usually dispatched to multiple processors using a
+message-passing interface in distributed memory. In this paradigm, the parallel al-
+gorithm is designed to minimize the communication between the different processes.
+In a complementary direction, fine-grained parallelism breaks down the program into
+small tasks, fast to compute in shared memory. This method requires a large num-
+ber of processors to be efficient, and it is usually better on SIMD architectures with
+low communication overhead, as provided by Graphical Processing Units (GPUs). In
+the mathematical optimization community, coarse parallelism has traditionally been
+used to solve large-scale block-structured optimization problems, as encountered in
+dynamic or stochastic nonlinear programs. On the contrary, fine-grained parallelism
+has gained attraction only recently, with the renewed interests for machine learning
+arXiv:2301.04869v1  [math.OC]  12 Jan 2023
+
+applications and stochastic gradient algorithms. In this work, we combine coarse and
+fine-grained parallelism to solve block-structured nonlinear problems on new exascale
+architectures, where the solution algorithm is streamlined on different GPUs using
+CUDA-aware MPI.
+1.1. Literature review
+In his pioneering work [39, 40], Robert Schnabel identified three practical approaches
+to run optimization algorithms in parallel: (i) parallelize the function evaluations; (ii)
+parallelize the linear algebra; and (iii) parallelize the optimization algorithm itself.
+The first attempt to parallelize the evaluations has been to streamline the com-
+putation of the derivatives using finite-differences [29]. Soon, it has been noted that
+parallelizing the forward pass in automatic differentiation (AD) is also straightforward,
+provided that we can propagate the tangents (encoding the first-order sensitivity) in
+parallel [20]. Unfortunately, doing the same in the reverse pass is not trivial, as ad-
+joining a mutable code leads to race conditions (e.g., every read becomes a write
+operation). This has led to extensive research on adapting automatic differentiation
+to parallel environments [4, 19, 27]. Now, most state-of-the-art differentiable tools em-
+ploy a Domain Specific Language (DSL) constraining the user to specific differentiable
+operations. In particular, this approach has been adopted mainly in machine learning,
+leading to the development of fast AD libraries efficiently generating the derivatives
+efficiently on hardware accelerators such as GPUs or TPUs [3, 32].
+The parallelization of linear algebra is usually more involved, as most large-scale op-
+timization methods fall back on the solution of sparse indefinite Karush-Kuhn-Tucker
+(KKT) systems [30]. In the 1980s, preliminary results were obtained by running itera-
+tive methods in parallel, using block-Krylov [36] or block-truncated Newton methods
+[28]. However, block iterative algorithms are quickly limited by the lack of generic
+preconditioners for KKT systems. The 1990s witnessed the emergence of the interior-
+point methods (IPM), together with the development of large-scale sparse direct linear
+solvers [12, 38]. In IPM, a significant portion of the time is spent solving a sequence
+of (indefinite) KKT systems, hence the method directly benefits from efficient sparse
+linear solvers able to run in parallel [1, 13]. In the 2000s, it was shown that, for block-
+structured optimization problems as we consider here, the layout of the optimization
+problem can be exploited further in a Schur complement approach to solve the Newton
+step in parallel [2, 9, 17, 22, 33, 44, 45]. These developments led to the development
+of mature decomposition-based parallel nonlinear solvers for scenario-based problems
+in the 2010s [8, 16, 34, 41, 46].
+Eventually, running an optimization algorithm fully in parallel generally requires a
+subtle combination of (i) and (ii), often devolving to a software engineering problem.
+The challenge is to evaluate the derivatives and solve the resulting KKT system each in
+parallel; all this while minimizing the communication between the different processes.
+This has led to the development of different prototypes for MPI-parallel modelers [10,
+21, 34, 43], most of them extending a specific AD backend [5, 14, 15]. Such approaches
+have been successfully applied to solve large-scale block-structured nonlinear problems,
+as encountered in stochastic programming and dynamic optimization.
+2
+
+1.2. Contributions
+In this article, we introduce a new parallel algorithm to solve block-structured non-
+linear programs involving state equations on exascale supercomputers. Our algorithm
+uses the parallel interior-point solver MadNLP [41], using two layers of parallelism to
+streamline both the evaluation of the derivatives and the solution of the KKT system.
+This framework targets new exascale supercomputers, where each node is assigned
+to multiple GPUs connected with a unified memory (designed to have fast memory
+exchange between the different GPUs).
+We demonstrate the capability of the algorithm on scenario-based power flow prob-
+lems (block-OPF), here formulated as two-stage stochastic nonlinear programs. The
+scenarios can be stochastic or represent contingencies (which can be interpreted as
+stochastic outcomes with uniform distribution), as is the case of the very widely used
+security-constrained AC optimal power flow (SC-ACOPF) problem [7]. SC-ACOPF is
+one of the core analyses undertaken in the planning, operational planning, and real-
+time operation of transmission systems [7]. SC-ACOPF is run several times a day by
+many operators in the US and the world. For brevity, we will refer to such problems
+as stochastic.
+The block structure of such problems is given by the different scenarios associ-
+ated with the stochastic problem, leading to potential parallelism in both the evalu-
+ation of the derivatives and the solution of the resulting block-angular KKT system.
+The parallel solution of the block-OPF problem with a Schur complement approach
+has been studied extensively both with PIPS-NLP [8, 37] (multiprocessing) and with
+Beltistos [23, 25] (multiprocessing + factorization of the dense Schur complement on
+the GPU). Compared to the state-of-the-art solver Beltistos, our approach carries out
+almost all computation on the GPUs including a global CUDA-aware MPI reduction,
+from the evaluation of the derivatives to the assembling of the Schur complement. We
+test our implementation on the pre-exascale supercomputer Polaris, where each node
+is equipped with 4 A100 GPUs, and we solve block-OPF problems with up to 9,251
+nodes.
+2. Problem statement
+In systems engineering, it is common to encounter optimization problems with rela-
+tively few degrees of freedom – ”controls”. Then, the goal is to appropriately fix the
+values for the degrees of freedom, e.g., by minimizing a given operational cost while
+satisfying the physical equations of the problem. In that context, the internal state of
+the system is described by a state variable x ∈ Rnx, whose values depend on the cur-
+rent controls u ∈ Rnu associated with the problem’s degrees of freedom. If the problem
+is well-posed, this translates to the state equation g(x, u) = 0, where the function g
+exhibits the physical structure of the problem (e.g., a differential equation encoding a
+dynamics, or a nonlinear network flow associated with static balance equations). When
+the system faces uncertainties, it is often appropriate to choose a control u feasible
+under a finite set of conditions (or scenarios). That is, the control u must satisfy N
+different state equations
+gi(xi, u) = 0
+for all
+i = 1, · · · , N,
+(1)
+3
+
+where the state xi now depend on the current scenario i. The variables xi can be assim-
+ilated into a recourse variable. The N functions g1, · · · , gN define the block structure
+of the problem.
+2.1. Block-structured nonlinear programs
+In addition to satisfying the N state equations (1), we aim at minimizing the average
+operating costs on the N different scenarios. The corresponding problem formulates
+as a two-stage nonlinear program, which, in our case, is a nonlinear program with
+partially separable structure [11]:
+min
+x1,··· ,xN,
+u
+N
+�
+i=1
+fi(xi, u)
+s.t.
+�
+�
+�
+�
+�
+xi ≥ 0 ,
+u ≥ 0
+gi(xi, u) = 0 ,
+hi(xi, u) ≤ 0 ,
+∀i = 1, · · · , N ,
+(2)
+with fi : Rnx × Rnu → R, gi : Rnx × Rnu → Rnx, hi : Rnx × Rnu → Rm smooth
+functions encoding the objective, the state equations, and the operational constraints,
+respectively. We note that the number of variables (N × nx + nu) and constraints
+(N × (m + nx)) are linearly proportional to the number of blocks N.
+In addition, if we introduce local control variables u1, · · · , uN with the additional
+coupling constraint u1 = · · · = uN = u, we get a problem with a separable structure,
+solvable using the primal decomposition method; at the expense of increasing the
+search space [11, 35].
+By introducing slack variables s1, · · · , sN, we rewrite (2) in standard form:
+min
+x1,··· ,xN,
+s1,··· ,sN,
+u
+N
+�
+i=1
+fi(xi, u)
+s.t.
+�
+�
+�
+�
+�
+u ≥ 0 ,
+xi ≥ 0 ,
+si ≥ 0
+gi(xi, u) = 0 ,
+hi(xi, u) + si = 0 ,
+∀i = 1, · · · , N .
+(3)
+We define yi ∈ Rnx the multipliers (or adjoints) associated to the equality con-
+straints gi(xi, u) = 0, zi ∈ Rm the multipliers associated to the operational constraints
+hi(xi, u) + si = 0, as well as λ, κi, νi the three multipliers associated to the respective
+bound constraints u ≥ 0, xi ≥ 0, si ≥ 0. The Lagrangian associated to (3) is:
+L(x, u, s; y, z, λ, µ, ν) :=
+N
+�
+i=1
+�
+fi(xi, u) + y⊤
+i gi(xi, u) + z⊤
+i
+�
+hi(xi, u) + si
+�
+− κixi − νisi
+�
+− λu ,
+(4)
+with x := (x1, · · · , xN), s := (s1, · · · , sN), y := (y1, · · · , yN), z := (z1, · · · , zN). To
+simplify the notations, we define the extended objective function and the extended
+constraints:
+f(x, u) :=
+�
+i=1
+fi(xi, u) ,
+g(x, u) :=
+�
+��
+g1(x1, u)
+...
+gN(xN, u)
+�
+�� ,
+h(x, u) :=
+�
+��
+h1(x1, u)
+...
+hN(xN, u)
+�
+�� .
+4
+
+We assume the functions f, g, h are twice differentiable. We denote
+H = ∂(x,u)h(x, u) ∈ RNm×(Nnx+nu)
+Jacobian of the inequality cons.
+G = ∂(x,u)g(x, u) ∈ RNnx×(Nnx+nu)
+Jacobian of the equality cons.
+W = ∇2
+(x,u)L(x, u, s; ·) ∈ R(Nnx+nu)×(Nnx+nu)
+Hessian of Lagrangian.
+2.2. Interior-point method
+The interior-point method (IPM) [30, Chapter 19] is a classical approach to solve (3).
+2.2.1. KKT system
+The Karush-Kuhn-Tucker (KKT) equations associated to (3) can be expressed as
+∇xfi + (Gi
+x)⊤yi + (Hi
+x)⊤zi − κi = 0,
+∀i = 1, · · · , N
+(5a)
+N
+�
+i=1
+�
+∇ufi + (Gi
+u)⊤yi + (Hi
+u)⊤zi
+�
+− λ = 0,
+(coupling)
+(5b)
+zi − νi = 0,
+∀i = 1, · · · , N
+(5c)
+gi(xi, u) = 0,
+∀i = 1, · · · , N
+(5d)
+hi(xi, u) + si = 0,
+∀i = 1, · · · , N
+(5e)
+Xiκi = 0, (xi, κi) ≥ 0,
+∀i = 1, · · · , N
+(5f)
+Siνi = 0, (si, νi) ≥ 0,
+∀i = 1, · · · , N
+(5g)
+Uλ = 0, (u, λ) ≥ 0,
+(5h)
+where U = diag(u), Xi = diag(xi), Si = diag(si).
+The interior-point method uses a homotopy parameter µ > 0 to replace the
+complementarity constraints (5f)-(5g)-(5h) by the smooth approximations: Xiκi =
+µenx, Siνi = µem, Uλ = µenu (en being the vector of all ones of dimension n). The
+resulting (smooth) system of nonlinear equations can be solved iteratively using New-
+ton method, where at each iteration, the descent direction is updated by solving the
+following augmented linear system:
+�
+���
+W + Σp
+0
+G⊤
+H⊤
+0
+Σs
+0
+I
+G
+0
+0
+0
+H
+I
+0
+0
+�
+���
+�
+���
+pd
+ps
+py
+pz
+�
+��� = −
+�
+���
+r1
+r2
+r3
+r4
+�
+���
+(6)
+with r1 =
+�
+∇xf + G⊤
+x y + H⊤
+x z − µX−1enx
+∇uf + G⊤
+u y + H⊤
+u z − µU−1enu
+�
+, r2 = z − µS−1em, r3 = g(x, u), r4 =
+h(x, u) + s. The primal descent direction pd decomposes as pd = (px1, · · · , pxN, pu).
+2.2.2. Block angular structure
+The linear system (6) is sparse and symmetric indefinite, and can be factorized using
+the Bunch-Kaufman algorithm. However, it is often beneficial to exploit its block-
+angular structure. Indeed, both the Hessian of the Lagrangian and the Jacobians have
+5
+
+a block-angular structure, given as
+W =
+�
+����
+Wx1x1
+Wx1u
+...
+...
+WxNxN
+WxNu
+Wux1
+. . .
+WuxN
+Wuu
+�
+���� ,
+G =
+�
+��
+G1
+x1
+G1
+u
+...
+...
+GN
+xN
+GN
+u
+�
+�� .
+By reordering the linear system (6), we can expose the block-angular structure of the
+KKT system as:
+�
+����
+A1
+B⊤
+1
+...
+...
+AN
+B⊤
+N
+B1
+. . .
+BN
+A0
+�
+����
+(7)
+with
+A0 = Wuu,
+Ai =
+�
+���
+Wxixi + Σxi
+0
+G⊤
+xi
+H⊤
+xi
+0
+Σsi
+0
+I
+Gxi
+0
+0
+0
+Hxi
+I
+0
+0
+�
+��� ,
+Bi =
+�
+�
+Wxiu
+(Gi
+u)⊤
+(Hi
+u)⊤
+�
+�
+⊤
+.
+The block-angular structure (7) can be exploited to solve the KKT linear system in
+parallel using a Schur complement approach. In that case, the submatrices Ai can be
+factorized independently to assemble the Schur complement in parallel [8].
+2.3. Condensation and reduction
+Instead of reordering the augmented KKT system (6) as a block angular matrix (7),
+we propose an alternative approach based on successive condensation and reduction
+of the KKT system, following the method introduced in [31]. If the structure is well-
+defined, we show that we can condense the KKT system (6) to a dense matrix with
+size nu × nu in two steps: first, by removing the inequality constraints in (6), then by
+exploiting the structure of the equality constraints to reduce the condensed system to
+a dense matrix. The condensation and reduction steps are illustrated in Figure 1.
+2.3.1. Condensation step
+The condensation step allows reducing the size of the KKT system drastically if the
+number of inequality constraints is large1.
+Proposition 2.1 (Condensed KKT system). The linear system (6) is equivalent to
+�
+K + Σp
+G⊤
+G
+0
+� �
+pd
+py
+�
+= −
+�
+r1 + H⊤(Σsr4 − r2)
+r3
+�
+,
+(8)
+where K ∈ R(Nnx+nu)×(Nnx+nu) is the condensed matrix K := W + H⊤ΣsH. The
+1It is equivalent to the normal equations in linear programming [30, Chapter 16, p.412]
+6
+
+Figure 1.. Successive reductions for a block-structured nonlinear problem with N = 3: Aug-
+mented system (6), Condensed system (8), Reduced system (11).
+descent directions ps and pz are recovered as
+�
+pz = Σs
+�
+Hpd + r4
+�
+− r2 ,
+ps = −Σ−1
+s
+�
+r2 + pz
+�
+.
+(9)
+Proof. See [31, Theorem 2.2].
+The condensed matrix K inherits the block-angular structure of the Hessian of the
+Lagrangian W.
+Proposition 2.2. The condensed matrix K = W +H⊤ΣsH has a block-angular struc-
+ture, given as
+K =
+�
+����
+Kx1x1
+Kx1u
+...
+...
+KxNxN
+KxNu
+Kux1
+. . .
+KuxN
+Kuu
+�
+����
+(10)
+where we have defined the condensed blocks Kxixi := Wxixi + (Hi
+xi)⊤ΣsiHi
+xi, Kuxi :=
+Wuxi + (Hi
+u)⊤ΣsiHi
+xi and Kuu := Wuu + �N
+i=1(Hi
+u)⊤ΣsiHi
+u.
+Proof. This is proved by induction.
+2.3.2. Reduction step
+In addition, we can exploit the structure of the equality constraints g1, · · · , gN to
+further reduce the size of the linear system (8) down to a dense matrix with size
+nu × nu. Equation (10) exhibits the structure w.r.t. the state x and the control u, we
+7
+
+3755 x 3755
+1193 x 1193
+107 x 107rewrite as such the condensed KKT system (8) as
+�
+�����������
+Kx1x1
+Kx1u
+(G1
+x1)⊤
+...
+...
+...
+KxNxN
+KxNu
+(GN
+xN)⊤
+Kux1
+. . .
+KuxN
+Kuu
+(G1
+u)⊤
+. . .
+(G1
+u)⊤
+G1
+x1
+G1
+u
+...
+...
+GN
+xN
+GN
+u
+�
+�����������
+�
+�����������
+px1...
+pxN
+pu
+p1
+y...
+pN
+y
+�
+�����������
+= −
+�
+����������
+ˆr1
+1...
+ˆrN
+1
+ˆr2
+ˆr1
+3...
+ˆrN
+3
+�
+����������
+,
+where we have renamed the right-hand-side in (8) as ˆr.
+Proposition 2.3 (Reduction). Assume that for all i = 1, · · · , N the Jacobian matri-
+ces Gi
+x ∈ Rnx×nx are invertible. Then the linear system (8) is equivalent to
+ˆKuu pu = −ˆr2 +
+N
+�
+i=1
+�
+(Gi
+u)⊤(Gi
+x)−⊤ˆri
+1 +
+�
+Kuxi − (Gi
+u)⊤(Gi
+x)−⊤Kxixi
+�
+(Gi
+x)−1ˆri
+3
+�
+(11)
+with ˆKuu := Z⊤KZ and Z ∈ R(nu+Nnx)×nu is the reduction operator defined as
+Z =
+�
+����
+−(G1
+x)−1G1
+u
+...
+−(GN
+x )−1GN
+u
+I
+�
+���� .
+(12)
+The descent directions px and py are recovered as
+�
+pi
+x = −(Gi
+x)−1�
+ˆri
+3 + Gi
+upu
+�
+pi
+y = −(Gi
+x)−⊤�
+ˆri
+1 + Kxixipi
+x + Kxiupu
+�
+.
+(13)
+Proof. See [31, Theorem 2.1].
+The reduction (11) is equivalent to a Schur complement approach applied to the
+condensed KKT system (8). In Proposition (2.1), we have shown that the condensed
+matrix K has a block-angular structure. The associated condensed KKT system (8)
+is also inheriting a block-angular structure in the form of (7), where the blocks are
+given by
+A0 = Kuu ,
+Ai =
+�
+Kxixi
+(Gi
+x)⊤
+Gi
+x
+0
+�
+,
+Bi =
+�
+Kxiu
+Gi
+u
+�⊤
+.
+(14)
+Proposition 2.4. Assume that for each i = 1, · · · , N the Jacobian Gi
+x is invertible.
+Let Suu = A0 − �N
+i=1 BiA−1
+i B⊤
+i
+be the Schur complement associated to the block-
+angular system (7) with the matrices (Ai, Bi) defined in (14). Then, the Schur com-
+plement Suu is equal to the reduced matrix �Kuu defined in (11): Suu = Z⊤KZ.
+8
+
+Proof. First, note that if the Jacobian Gi
+x is invertible, then the block matrix Ai
+defined in (14) is also invertible, with
+A−1
+i
+=
+�
+0
+(Gi
+x)−1
+(Gi
+x)−⊤
+−(Gi
+x)−⊤Kxixi(Gi
+x)−1
+�
+.
+(15)
+Using (14)-(15), we expand the expression of the terms in the sum constituting the
+Schur complement Suu:
+BiA−1
+i B⊤
+i =
+�
+Kuxi
+(Gi
+u)⊤� �
+0
+(Gi
+x)−1
+(Gi
+x)−⊤
+−(Gi
+x)−⊤Kxixi(Gi
+x)−1
+� �
+Kxiu
+Gi
+u
+�
+,
+= (Gi
+u)⊤(Gi
+x)−⊤Kxiu + Kuxi(Gi
+x)−1(Gi
+u) − (Gi
+u)⊤(Gi
+x)−⊤Kxixi(Gi
+x)−1Gi
+u .
+Hence, the Schur complement Suu = A0 − �N
+i=1 BiA−1
+i B⊤
+i expands as
+Suu = Kuu −
+N
+�
+i=1
+�
+(Gi
+u)⊤(Gi
+x)−⊤Kxiu + Kuxi(Gi
+x)−1(Gi
+u) − (Gi
+u)⊤(Gi
+x)−⊤Kxixi(Gi
+x)−1Gi
+u
+�
+= Z⊤KZ .
+We recover the expression of the reduced matrix �Kuu in Proposition 2.3.
+2.4. Discussion
+Hence, we can interpret the reduction step as a Schur complement approach. Forming
+the Schur complement has always been the bottleneck when solving distributed block
+angular problems in parallel [8, 26]. Its reduction operation involves large memory
+transfers between the processes, with the number of transfers being on the order
+of O(log(p)), where p is the number of processes. Due to the quasi-shared memory
+architecture on GPUs, the reduction can be implemented efficiently [31]. In the next
+section, we propose to extend [31] to assemble the reduced matrix �Kuu using two levels
+of parallelism, using both MPI and CUDA, thus reducing the reliance on distributed
+memory.
+3. Parallel implementation
+In the previous section, we have detailed the structure of block-angular nonlinear
+programs and presented the condensation and reduction steps for the KKT system.
+The loose coupling between the blocks is favorable for parallelizing the evaluation of
+the derivatives and the solution of the block-angular KKT system. Globally, we can
+distribute the computation on different processes using MPI (coarse parallelism). Lo-
+cally, we can further streamline the computation using GPU accelerators (fine-grained
+parallelism). This paradigm, with its two levels of parallelism, is directly in line with
+what is currently offered by the new exascale architectures, where each node has 4 to
+8 GPUs, all sharing a unified memory for fast communication. We present in §3.1 how
+we streamline the evaluation of the model using automatic differentiation, and in §3.2
+how we parallelize the solution of the KKT system.
+9
+
+3.1. Parallel automatic-differentiation
+First, we present how to evaluate the model in parallel using automatic differentiation
+[18]. We illustrate the procedure in Figure 2. The goal of the algorithm is to streamline
+the evaluation of the N scenarios on N/M GPUs, M being the number of scenarios
+evaluated locally on each GPU (we suppose here that N is a multiple of M).
+root
+g1, · · · , g4
+g13, · · · , g16
+g1
+g4
+· · ·
+g13
+g16
+· · ·
+· · ·
+· · ·
+Figure 2.. Parallel evaluation of the derivatives for g1, · · · gN on 4 GPUs: we have a total of
+N = 16 scenarios, each GPU evaluating M = 16/4 = 4 scenarios locally.
+3.1.1. Local parallelism
+The first level of parallelism streamlines the evaluation of the model on SIMD/GPU
+devices. We have designed our implementation to run entirely on the GPU device, to
+avoid any data transfer between the host and the device.
+3.1.1.1. Block evaluation. We suppose that the nonlinear functions (fi, gi, hi)
+share the same structure, its expressions yielding the same Abstract Syntax Tree (AST)
+for all i = 1, · · · , M. We illustrate the block evaluation on a simple abstract tree, but
+the reasoning extends to more complicated structures. We suppose that for all i, the
+functions fi, gi, hi depend linearly on a nonlinear basis matrix ψ : Rnx × Rnu → Rnb:
+that is, there exists three sparse matrices Lf, Lg, Lh such that
+fi(xi, u) = Lfψ(xi, u) ,
+gi(xi, u) = Lgψ(xi, u) ,
+hi(xi, u) = Lhψ(xi, u) .
+(16)
+Suppose we aim to evaluate the M functions g1, · · · , gM in batch for the states
+x1, · · · , xM. The structure (16) is directly amenable for SIMD evaluation. We de-
+note by XM = (x1, · · · , xM) ∈ Rnx×M the dense matrix obtained by concatenat-
+ing the M states together. By using a proper GPU kernel or a parallel modeler,
+we can evaluate the basis in a SIMD fashion and build the matrix Ψ(XM, u) :=
+�
+ψ(x1, u), · · · , ψ(xM, u)
+�
+∈ Rnb×M. Then, evaluating the functions g1, · · · , gM simul-
+taneously translates to the evaluation of one SpMM product:
+�
+g1(x1, u), · · · , gM(xM, u)
+�
+= LgΨ(XM, u) ∈ Rnx×M .
+(17)
+The total memory required in the two successive operations is O((nx + nb) × M), and
+depends linearly on the number of blocks M. We note the SpMM operations are generally
+implemented efficiently in the vendor library (cusparse for CUDA, rocSPARSE for
+AMDGPU).
+10
+
+3.1.1.2. First-order derivatives. Suppose that for a given i we have a differen-
+tiable implementation gbi : Rnd → Rnx associated to the function gi. We aim to
+evaluate the Jacobian-matrix products (∇gi)D for p tangents encoded in a matrix
+D ∈ Rnd×p using forward-mode AD and operator overloading. This operation trans-
+lates to propagating forward a vector of dual numbers. Denoting by d ∈ Dnd
+p
+the dual
+number encoding the p tangents stored in D, evaluating (∇gi)D simply amounts to
+call gbi(d) and extract the results in the dual numbers returned as a result. As Gi
+is sparse, we can apply the technique of Jacobian coloring [18] to compress the inde-
+pendent columns of the sparse matrix Gi and reduces the number of required seeding
+tangents p needed to evaluate the full Jacobian.
+Suppose now we want to evaluate the sparse Jacobians G1, · · · , GM in batch. As
+the functions gi are based on the same AST, their respective Jacobians G1, · · · , GM
+are sharing the same sparsity pattern. By seeding a matrix of dual numbers DM =
+(d1, · · · , dM) ∈ Dnd×M
+p
+, we can use the same operation as (17) to streamline the
+evaluation of the M Jacobian-vector products using the SIMD kernel Ψ(·) and SpMM
+operations:
+�
+gb1(d1), · · · , gbM(dM)
+�
+:= LgΨ(DM) ∈ Dnx×M
+p
+.
+(18)
+Once the results are evaluated, it remains to uncompress the dual outputs to build
+the M sparse Jacobians G1, · · · , GM. Hence, we can streamline the evaluation of the
+Jacobian along with the number of tangents p and the number of blocks M. This comes
+at the expense of increasing memory usage to O((nx + nb + nd) × M × p) (to store the
+dual matrices associated to the input, the intermediate basis Ψ and the output).
+3.1.1.3. Second-order derivatives. The evaluation of the second-order deriva-
+tives follows the same procedure, using forward-over-reverse AD. For each i, we sup-
+pose available an adjoint function adj gbi : Rnd × Rnx → Rnd which for any pri-
+mal x ∈ Rnd and adjoint y ∈ Rnx evaluates the Jacobian-transpose vector product
+(Gi(x))⊤y (reverse-mode). Using forward-mode AD on top of adj gbi, we can compute
+the second-order derivatives y⊤∇2gi(x)V for p directions V by calling adj gbi(x, y).
+Using Hessian coloring, we can compress the independent columns of the sparse matrix
+y⊤∇2g(x) and reduce the number of seeding tangents p required to evaluate the full
+Hessian. We note that in general obtaining an adjoint adj gbi running in parallel is
+nontrivial due to potential race conditions incurred by the control flow reversal of the
+original code.
+Computing the Hessian y⊤∇2gi(x) in parallel for i = 1, · · · M amounts to defining
+two matrices of dual numbers XM = (X1, · · · , XM) ∈ Dnd×M
+p
+, Y M = (y1, · · · , yM) ∈
+Dnx×M
+p
+and evaluate ∇Ψ(XM)⊤L⊤
+g Y M. The dual outputs are uncompressed to build
+the M sparse Hessians (as the sparsity pattern of the Hessians is different than those
+of the Jacobians, the matrix XM employed here is different than the one used in (18)).
+The total memory required to store the duals is O((2nx +nd +nb)×M ×p). For more
+details, we refer to the vector forward mode as described in [18].
+3.1.2. Global parallelism
+Now, if we have several GPUs at our disposal, we can push the parallelism further
+by distributing the evaluations using multiprocessing and a Message Passing Interface
+(MPI) library. Coming back at our original problem (2), we illustrate in Figure 2 how to
+dispatch the evaluation of the N nonlinear constraints g1, · · · , gN (the same reasoning
+11
+
+applies to the objectives f1, · · · , fN and the inequality constraints h1, · · · , hN). We use
+the streamlined implementation described in the previous subsection to evaluate the
+constraints in a batch of size M: the first GPU evaluates the constraints g1, · · · , gM,
+the second GPU evaluates gM+1, · · · , g2M, and so on. In total, the evaluation of the
+N constraints requires N/M GPUs (if M = 1, each GPU evaluate one constraint; if
+M = N, we use only one GPU evaluating all the constraints).
+The implementation has been designed to minimize the communication between
+the different processes: each batch g1, · · · , gM stores the data it needs locally, the only
+data exchange with the other processes being the vector of input and the vector of
+output. In addition, we will see in the next section we do not have to transfer the first-
+and second-order information if a parallel linear solver is being used.
+3.2. Parallel KKT solver
+By exploiting the block-angular structure of the KKT system, we can solve the New-
+ton step in parallel using a Schur complement approach. The challenge lies in the
+computation of the Schur complement matrix S = A0 − �N
+i=1 BiA−1
+i B⊤
+i . Each prod-
+uct BiA−1
+i B⊤
+i requires the factorization of the matrix Ai and the solution of a linear
+system with multiple (sparse) right-hand-side A−1
+i Bi. State-of-the-art methods are
+evaluating the Schur complement using an incomplete augmented factorization ap-
+plied on the auxiliary matrix
+�
+Ai
+B⊤
+i
+Bi
+0
+�
+, as currently implemented in the Pardiso
+linear solver [33]. Here, we use an alternative approach building on the reduced KKT
+system §2.3.2 (equivalent to the Schur complement approach). As the reduction can
+be streamlined on GPU accelerators [31], this approach can assemble the Schur com-
+plement in parallel using CUDA-aware MPI. We illustrate the parallel computation of
+the Schur complement in Figure 3.
+root
+(Assembling)
+K, G
+K, G
+K, G
+K, G
+AD
+AD
+AD
+AD
+ˆK1:4
+uu
+ˆK5:8
+uu
+ˆK9:12
+uu
+ˆK13:16
+uu
+(Reduction)
+Z⊤KZ
+Z⊤KZ
+Z⊤KZ
+Z⊤KZ
+MPI AllReduce
++
+(Schur compl.)
+ˆKuu
+Figure 3.. Parallel computation of the Schur complement.
+Assembling the sparse matrices. Using the procedure introduced in §3.1.1, we
+evaluate locally the Jacobians G1, · · · , GM the Jacobians H1, · · · , HM and the Hes-
+sians W 1, · · · , W M. Using dedicated kernels, we uncompress the results in the block-
+12
+
+angular sparse Jacobians
+G1:M
+x
+=
+�
+��
+G1
+x1
+...
+GM
+xM
+�
+�� , G1:M
+u
+=
+�
+��
+G1
+u...
+GM
+u
+�
+�� , H1:M =
+�
+��
+H1
+x1
+H1
+u
+...
+...
+HM
+xM
+HM
+u
+�
+�� ,
+and sparse Hessian
+W 1:M =
+�
+����
+Wx1x1
+Wx1u
+...
+...
+WxMxM
+WxMu
+Wux1
+. . .
+WuxM
+Wuu
+�
+���� .
+Once the sparse matrices are obtained, we recover the condensed matrix K1:M =
+W 1:M + (H1:M)⊤Σ(H1:M) (Proposition 2.1) using one SpGEMM operation and we fac-
+torize the matrix G1:M
+x
+using a sparse LU factorization (potentially running in batch
+as the matrices G1
+x1, · · · , GM
+xM are sharing the same sparsity pattern). Once the ma-
+trix G1:M
+x
+is factorized as PG1:M
+x
+Q = LU (P, Q being two permutation matrices),
+computing (G1:M
+x
+)−1b translates to two backsolves (SpSV) and two matrix-vector mul-
+tiplications (SpMV), as (G1:M
+x
+)−1b = QU−1L−1Pb.
+Local reduction. Once the sparse matrices are built, we evaluate locally the reduced
+matrix �K1:M
+uu
+on the GPU, using div(nu, nbatch) + 1 matrix-matrix product �K1:M
+uu V
+(with V ∈ Rnu×nbatch a dense matrix encoding nbatch vectors of the Cartesian basis of
+Rnu). The evaluation of one batched matrix-matrix product �K1:M
+uu V = (Z⊤K1:MZ)V
+proceeds in three steps
+(1) Solve Tx = −(G1:M
+x
+)−1(G1:M
+u
+V ).
+(2) Evaluate
+�
+Lx
+Lu
+�
+:=
+�
+K1:M
+xx
+K1:M
+xu
+K1:M
+ux
+K1:M
+uu
+� �
+Tx
+V
+�
+.
+(3) Set �K1:M
+uu V = Lu − G1:M
+u
+(G1:M
+x
+)−⊤Lx.
+In total, we need 2 SpSM and 3 SpMM operations in the first step, 1 SpMM in the second
+step, and 2 SpSM and 3 SpMM operations in the third step, giving a total of 4 SpSM
+and 7 SpMM operations. More than the computation, the reduction is limited by the
+memory, as we have to store the three buffers Lx, Tx, Tu with a total size of (2M ×
+nx + nu) × nbatch. If nx is too large, it is in our interest to reduce M (by using more
+GPUs) or to reduce nbatch (at the expense of computing more matrix-matrix product
+�K1:M
+uu V ).
+Global reduction. Once we obtain the locally reduced matrices �KnM+1:(n+1)M
+uu
+for n = 0, · · · , N/M − 1, we can assemble the global reduced matrix
+�Kuu =
+�N/M−1
+n=0
+�KnM+1:(n+1)M
+uu
+using one all reduce (MPI Allreduce) operation. The size of
+the reduced matrix �Kuu is nu × nu, hence limiting the memory transfer required in
+the algorithm.
+13
+
+3.3. Discussion
+We have presented a practical way to assemble the Schur complement on multi-GPU
+architectures. The parallelism occurs both at the local level (SIMD evaluations on the
+GPUs) and at the global level (distributed computation with MPI). The algorithm
+has the advantage of assembling the sparse Jacobians and Hessians only locally, as
+the reduction occurs before proceeding to the memory transfer with MPI Allreduce.
+The reduced matrix has a dimension nu × nu, which compresses the memory transfer
+significantly if the number of degrees of freedom nu is small. However, this comes at the
+expense of storing a vector of dual numbers (whose memory is linearly proportional
+to the number of blocks M evaluated locally and the number of tangents p being
+employed to evaluate the sparse derivatives) and additional buffers in the reduction
+algorithm. In the next section, we will test an implementation of the algorithm on
+CUDA GPUs, and show that the algorithm is practical.
+4. Numerical results
+We demonstrate the capabilities of the algorithm we introduced in Section §3 on the
+supercomputer Polaris, using CUDA-aware MPI to dispatch the solution on multiple
+GPUs. We present in §4.1 the stochastic optimal power flow problem, and give in §4.2
+detailed assessments of the algorithms we have introduced earlier in §3. Eventually,
+we present in §4.3 a benchmark comparing our parallel solution algorithm with a
+state-of-the-art solution method running on the CPU.
+4.1. Settings
+4.1.1. Case study: the block-structured optimal power flow
+The stochastic optimal power flow problem aims at finding an optimal dispatch for the
+generators u. The solution u should minimize the operational costs while satisfying the
+physical constraints (power flow equations g(x, u) = 0, here playing the role of the state
+equations) and operational constraints (line flow constraints h(x, u) ≤ 0) on a given
+set of scenarios. Each scenario is assigned given load parameters (energy demands) and
+potential contingencies (line tripping). The values of the state x depend on the local
+scenario we are in, the state x being the recourse variable in our case. As such, the
+problem has a partially separable structure as introduced in Problem (2), the control
+u being shared across all scenarios. We refer to [7] for the original presentation of the
+stochastic optimal power flow problem and to [8, 23, 24, 26] for practical algorithms
+solving the stochastic optimal power flow problem (some also focus on the multistage
+setting, which is not covered in this article). For our benchmark, we look at reference
+instances provided by MATPOWER [47], whose characteristics are detailed in Table 1.
+We recall that in our case, the size of the Schur complement matrix ˆKuu is given by
+the number of controls nu.
+4.1.2. Implementation
+The algorithm has been implemented entirely in Julia 1.8. The Schur complement
+approach has been developed as an extension of the nonlinear optimization solver
+MadNLP [41], using CUDA-aware MPI as provided in [6]. We have used the package
+14
+
+Name
+#bus
+#lines
+#gen
+nx
+nu
+case118
+118
+186
+54
+181
+107
+case1354pegase
+1,354
+1,991
+260
+2,447
+519
+case2869pegase
+2,869
+4,582
+510
+5,227
+1,019
+case9241pegase
+9,241
+16,049
+1,445
+17,036
+2,889
+Table 1.. MATPOWER instances used in the benchmark.
+ExaPF as a nonlinear modeler for the optimal power flow problem. All the results
+presented here have been generated on the supercomputer Polaris equipped with a
+total of 560 nodes, each node having with 1 CPU and 4 A100 GPUs.
+4.2. Assessment of the parallel implementation
+4.2.1. Assessing the performance of the parallel automatic differentation
+We first assess the performance of the parallel automatic differentiation we introduced
+in §3.1 in a multi-GPU setting. We compare the performance we obtain with a CPU
+implementation. We use case1354pegase as a representative instance, and display the
+time spent in the automatic differentiation as we increase the total number of scenarios
+N. The results are displayed in Figure 4.
+We observe that the computation time depends linearly on the number of scenarios,
+as expected. For N = 8, it is not worthwhile dispatching the evaluation on multiple
+GPUs as the problem is small enough to be evaluated on a single GPU. For N = 512,
+the evaluation time is 12.3s on the CPU, compared to 0.50, 0.41, 0.31, and 0.28s using
+1, 2, 4 and 8 GPUs, respectively. Hence, we get a 40x speed-up when evaluating the
+derivatives in a multi-GPU setting, and it is not worthwhile to use more than 4 GPUs
+(one node).
+8
+16
+32
+64
+128
+256
+512
+N scenarios
+10
+1
+100
+101
+Evaluation time [s]
+case1354pegase
+CPU
+1 GPUs
+2 GPUs
+4 GPUs
+8 GPUs
+Figure 4.. Time spent to evaluate the model and its derivatives with automatic differentiation.
+15
+
+4.2.2. Assessing the performance of the parallel KKT solver
+We proceed to the same performance analysis to assess the performance of the parallel
+KKT solver detailed in §3.2. We compare the time required to evaluate the full solution
+of the KKT system afresh (including reduction time, factorization time and backsolve
+time) on case1354pegase as we increase the number of scenarios N. As a reference,
+we give the time taken by the sparse linear solvers HSL MA27 (single-threaded) and
+HSL MA57 (multi-threaded). The results are displayed in Figure 5.
+On the left, we display the evolution of the time spent in the linear solver as we
+increase the number of scenarios. For N = 512, we observe that we get a linear speed-
+up as we increase the number of GPUs: using 8 GPUs, the parallel KKT solver is
+40x faster than using HSL MA27 on the CPU. Interestingly, we observe that HSL
+MA57 is not faster than HSL MA27, despite being multithreaded. This is consistent
+with the observation made in [42], and illustrates the difficulty of parallelizing ef-
+fectively the sparse LDL factorization (Bunch-Kaufman). On the right, we display a
+performance profile detailing the time spent in MA27 and the parallel KKT solver on
+case1354pegase with N = 512 scenarios. We observe that most of the time in HSL
+MA27 is spent on factorizing the sparse augmented KKT system (6). On the other
+side, the factorization of the dense reduced matrix ˆKuu is trivial using LAPACK on
+the GPU; the bottleneck in the parallel KKT solver is the reduction algorithm itself.
+Fortunately, the reduction algorithm can run in parallel: we get a linear speed-up as
+we increase the number of GPUs used in the reduction algorithm.
+8
+16
+32
+64
+128
+256
+512
+N scenarios
+10
+1
+100
+101
+Time [s]
+Linear solver time against N
+ma27
+ma57
+1 GPUs
+2 GPUs
+4 GPUs
+8 GPUs
+ma27
+1 GPU
+2 GPUs
+4 GPUs
+8 GPUs
+0
+2
+4
+6
+8
+Time [s]
+Performance profile for N = 512
+Factorization
+Backsolve
+Reduction
+Figure 5.. Time spent to solve the KKT system for case1354pegase.
+4.2.3. Assessing the memory consumption
+We have observed in §3.1 that the total memory required to store the duals is O((2nx+
+nd + nb) × M × p), with M being the number of scenarios stored locally (M = N on
+1 GPU, M = N/2 on 2 GPUs) and p the number of tangents. We display in Table 2
+the memory taken by the automatic differentiation backend and by the parallel KKT
+solver for case1354pegase as we increase the number of scenarios N. We note that
+storing the duals is expensive in terms of memory, with up to 10.9GB for N = 512 on
+one GPU (as a reference, each NVIDIA A100 GPU on Polaris has 40GB of memory
+available). By evaluating the model on different processes with MPI, we can split the
+memory consumption on the different GPUs we are using, leading to better use of the
+resource at our disposal.
+16
+
+1 GPU
+2 GPUs
+N
+AD
+KKT solver
+AD
+KKT solver
+8
+171.1
+92.3
+85.5
+48.1
+16
+342.2
+181.5
+171.1
+93.1
+32
+684.3
+360.0
+342.2
+183.2
+64
+1,368.7
+716.8
+684.3
+363.2
+128
+2,737.3
+1,430.5
+1,368.7
+723.4
+256
+5,474.7
+2,858.0
+2,737.3
+1,443.6
+512
+10,949.3
+5,712.8
+5,474.7
+2,884.1
+Table 2.. Memory consumption in MB
+4.3. Parallel solution of the block-structured OPF problem
+We analyze the parallel performance of our implementation on block-structured OPF
+problems.
+4.3.1. Assessing the parallel performance w.r.t. the number of scenarios
+First, we are interested in the scaling of the parallel algorithm in relation to the total
+number of scenarios N. We consider the case118 instance, and increase the number
+of scenarios N from 8 up to 2,048. For each N, we solve the block-structured OPF
+problem with MadNLP using our parallel KKT solver, and we compare with the
+performance we obtained with HSL MA27. The results are displayed in Figure 6. We
+observe that the solver HSL MA27 is initially faster than our parallel KKT solver, as
+the problem is too small to benefit from parallelism. However, as soon as N ≥ 16 the
+parallel KKT solver becomes competitive with HSL MA27. The relative performance is
+improving as we increase the number of scenarios N: for N = 512, we get a 68x speed-
+up when using 8 GPUs, compared to the reference HSL MA27 (10.4s versus 712s).
+Interestingly, using 2 nodes (=8 GPUs) does not lead to any speed-up compared to
+a single node (=4 GPUs) if N ≤ 256; this setting is attractive only when the size of
+the problem becomes sufficiently large (N ≥ 1024) to compensate for the additional
+memory exchange.
+4.3.2. Assessing the parallel performance w.r.t. the size of the problem
+Second, we increase the size of the problems. We set a fixed number of scenarios
+N = 8, and look at the time to solution for case1354pegase, case2869pegase and
+case9241pegase. We detail the respective dimension of each problem in Table 3. We
+display the results in Figure 7, and give the detailed benchmark in Table 4. On the
+left (a), we display the total time required to find the solution of the three instances as
+a function of the number of GPUs; on the right (b), we show the performance profile
+associated to case9241pegase. In (a), we observe that overall the parallel algorithm is
+faster than the CPU implementation. The parallel algorithm scales well as we increase
+the number of GPUs we are using, the parallel algorithm being 35x faster than the
+reference when using 8 GPUs to solve case9241pegase. In (b), we detail the time
+17
+
+8
+16
+32
+64
+128
+256
+512
+1,024 2,048
+N scenarios
+100
+101
+102
+103
+Time to solution [s]
+case118
+ma27 (CPU)
+polaris (1 GPUs)
+polaris (2 GPUs)
+polaris (4 GPUs)
+polaris (8 GPUs)
+Figure 6.. Time to solve the block-structured OPF problem case118 as a function of the
+number of scenarios N.
+spent in the different operations for case9241pegase: the time spent to factorize the
+Schur complement with Lapack (using cusolve) is constant as the size of the Schur
+complement remains the same as we increase the number of GPUs. We observe that
+the time spent in the AD decreases linearly with the number of GPUs exploited, but
+the relative time spent in AD is negligible (less than 5% of the total time). Most of
+the time is spent in the parallel reduction, as discussed earlier in §4.2.2.
+N
+nvar
+ncon
+ˆKuu (mb)
+1354pegase
+8
+20,095
+53,520
+2.1
+2869pegase
+8
+42,835
+119,216
+7.9
+9241pegase
+8
+139,177
+404,640
+63.7
+1354pegase
+512
+1,253,383
+4,425,280
+2.1
+Table 3.. Dimension of the instances we have used in our benchmark.
+4.3.3. Assessing the parallel performance on a very large-scale instance
+We finish our numerical experiments by solving a very large-scale instance:
+case1354pegase with N = 512 scenarios. The dimension of the resulting optimization
+problem is displayed in Table 3: the problem has more than 1 million variables, and 4
+millions constraints. We solve this instance on resp. 1 node, 2, 4 and 8 nodes (resp. 4,
+8, 16 and 32 GPUs). The results are displayed in Figure 8. We observe that the scaling
+is almost perfect when we use 2 nodes (8 GPUs) instead of a single node (4 GPUs)
+but we do not observe the same behavior when we increase the number of nodes to 4
+and 8. On that instance, the gain we get when using 8 nodes (32 GPUs) is marginal
+18
+
+CPU
+1 GPU
+2 GPU
+4 GPU
+8 GPU
+100
+101
+102
+103
+Time to solution [s]
+Benchmark
+1354pegase
+2849pegase
+9241pegase
+1 GPU
+2 GPUs
+4 GPUs
+8 GPUs
+10
+1
+100
+101
+102
+Time [s]
+Performance profile, 9241pegase
+Total time
+linear scaling
+AD
+Factorization
+Figure 7.. For a fixed number of scenarios N = 8, (a) total time spent solving the block-
+OPF case1354pegase, case2869pegase and case9241pegase with MadNLP (b) performance
+profile for case9241pegase with varying number of GPUs.
+1354pegase
+2869pegase
+9241pegase
+#it
+AD
+KKT
+Tot.
+#it
+AD
+KKT
+Tot.
+#it
+AD
+KKT
+Tot.
+CPU
+44
+2.6
+4.2
+7.0
+77
+11.9
+27.4
+40.3
+136
+205.6
+771.8
+984.1
+1 GPU
+44
+0.3
+1.8
+2.1
+93
+1.1
+11.7
+12.8
+98
+5.5
+112.3
+117.8
+2 GPUs
+44
+0.3
+1.1
+1.4
+93
+0.8
+7.4
+8.2
+98
+3.4
+56.8
+60.2
+4 GPUs
+44
+0.3
+1.0
+1.3
+93
+0.8
+5.7
+6.5
+98
+2.3
+35.8
+38.1
+8 GPUs
+44
+0.2
+1.0
+1.2
+93
+0.6
+5.1
+5.7
+98
+1.4
+26.4
+27.7
+Table 4.. Detailed results
+compared to when using 4 nodes (16 GPUs): the solving time only decreases from
+67s to 58s. This corroborate our observations: it is better to pack all the computa-
+tion on a single node to use four A100 GPUs connected together via unified memory
+(NVLINK has a transfer rate of 600GB/s). When we have to use more than 2 nodes,
+the memory transfers are more involved as they have to pass through the network of
+the supercomputer.
+5. Conclusion
+We show promising results for leveraging massively parallel SIMD architectures like
+GPUs for block-structured nonlinear programs. The parallelism is applied to both the
+derivative evaluation and the solution of the KKT linear system. The main operation
+in the KKT algorithm is the assembling of the Schur complement, the factorization of
+the dense Schur complement being fast to carry on the GPU.
+At all levels, the method benefits significantly from the massive parallelism, achiev-
+ing a speedup of around 40 for the derivatives compared to a sequential CPU im-
+plementation. The speedup is very application dependent, not least on the Hessian
+coloring and the problem’s structure. The assembling of the Schur complement is bot-
+tlenecked by a distributed reduction operation bound by the interconnect’s latency
+and throughput between GPUs. Current, so-called super nodes with multiple GPUs
+19
+
+4
+8
+16
+32
+#GPUs
+102
+3 × 101
+4 × 101
+6 × 101
+2 × 102
+Time to solution [s]
+Benchmark 1354pegase (N=512)
+Measured
+linear scaling
+Figure 8.. Solving case1354pegase with N = 512
+connected via fast networks like NVLINK greatly accelerate this operation. Lastly, our
+method is limited by the memory capacity of the GPU accelerators as it grows linearly
+with the number of problem blocks. In the context of ACOPF we are confident that
+upcoming GPUs will provide enough memory to solve a large number of scenarios in
+parallel, even for the largest grid instances (e.g., Eastern Interconnection with 70,000
+nodes).
+With the upcoming release of the Aurora supercomputer, these SIMD architectures
+will allow new science in regimes that were impossible with previous CPU architec-
+tures.
+Acknowledgment
+This material was based upon work supported by the U.S. Department of Energy, Of-
+fice of Science, Office of Advanced Scientific Computing Research (ASCR) under Con-
+tract DE-AC02-06CH11347 and by NSF through award CNS-1545046. The authors
+gratefully acknowledge the funding support from the Applied Mathematics Program
+within the U.S. Department of Energy’s (DOE) Office of Advanced Scientific Com-
+puting Research (ASCR) as part of the project ExaSGD. This research used resources
+of the Argonne Leadership Computing Facility, which is a DOE Office of Science User
+Facility supported under Contract DE-AC02-06CH11357.
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+Government License: The submitted manuscript
+has been created by UChicago Argonne, LLC,
+Operator of Argonne National Laboratory (“Ar-
+gonne”). Argonne, a U.S. Department of Energy
+Office of Science laboratory, is operated under
+Contract No. DE-AC02-06CH11357. The U.S.
+Government retains for itself, and others acting
+on its behalf, a paid-up nonexclusive, irrevoca-
+ble worldwide license in said article to repro-
+duce, prepare derivative works, distribute copies
+to the public, and perform publicly and display
+publicly, by or on behalf of the Government.
+The Department of Energy will provide pub-
+lic access to these results of federally sponsored
+research in accordance with the DOE Public
+Access Plan. http://energy.gov/downloads/doe-
+public-access-plan.
+23
+
diff --git a/99E4T4oBgHgl3EQfDgse/content/tmp_files/load_file.txt b/99E4T4oBgHgl3EQfDgse/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..260a7cbd729afe1812842da6610982bc4ebf0be9
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@@ -0,0 +1,1007 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf,len=1006
+page_content='Parallel Interior-Point Solver for Block-Structured Nonlinear  Programs on SIMD/GPU Architectures  Fran¸cois Pacauda, Michel Schanenb, Sungho Shinb, Daniel Adrian Maldonadob,  Mihai Anitescub  a Centre Automatique et Syst`emes, Mines Paris - PSL, Paris, France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' b Mathematics and  Computer Science Department, Argonne National Laboratory, Lemont, USA  ARTICLE HISTORY  Compiled January 13, 2023  ABSTRACT  We investigate how to port the standard interior-point method to new exascale  architectures for block-structured nonlinear programs with state equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Compu-  tationally, we decompose the interior-point algorithm into two successive operations:  the evaluation of the derivatives and the solution of the associated Karush-Kuhn-  Tucker (KKT) linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Our method accelerates both operations using two  levels of parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' First, we distribute the computations on multiple processes  using coarse parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Second, each process uses a SIMD/GPU accelerator lo-  cally to accelerate the operations using fine-grained parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The KKT system  is reduced by eliminating the inequalities and the state variables from the corre-  sponding equations, to a dense matrix encoding the sensitivities of the problem’s  degrees of freedom, drastically minimizing the memory exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We demonstrate  the method’s capability on the supercomputer Polaris, a testbed for the future exas-  cale Aurora system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Each node is equipped with four GPUs, a setup amenable to our  two-level approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Our experiments on the stochastic optimal power flow problem  show that the method can achieve a 50x speed-up compared to the state-of-the-art  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Introduction  Solving complex engineering problems often resorts to the solution of large-scale block-  structured nonlinear programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' As such, there has been a long interest in designing  efficient nonlinear optimization algorithms, particularly by using parallel computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Parallelism can happen at two levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' At first, coarse parallelism splits the program  into large computational chunks, usually dispatched to multiple processors using a  message-passing interface in distributed memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In this paradigm, the parallel al-  gorithm is designed to minimize the communication between the different processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  In a complementary direction, fine-grained parallelism breaks down the program into  small tasks, fast to compute in shared memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' This method requires a large num-  ber of processors to be efficient, and it is usually better on SIMD architectures with  low communication overhead, as provided by Graphical Processing Units (GPUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In  the mathematical optimization community, coarse parallelism has traditionally been  used to solve large-scale block-structured optimization problems, as encountered in  dynamic or stochastic nonlinear programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' On the contrary, fine-grained parallelism  has gained attraction only recently, with the renewed interests for machine learning  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='04869v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='OC]  12 Jan 2023  applications and stochastic gradient algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In this work, we combine coarse and  fine-grained parallelism to solve block-structured nonlinear problems on new exascale  architectures, where the solution algorithm is streamlined on different GPUs using  CUDA-aware MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Literature review  In his pioneering work [39, 40], Robert Schnabel identified three practical approaches  to run optimization algorithms in parallel: (i) parallelize the function evaluations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' (ii)  parallelize the linear algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' and (iii) parallelize the optimization algorithm itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The first attempt to parallelize the evaluations has been to streamline the com-  putation of the derivatives using finite-differences [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Soon, it has been noted that  parallelizing the forward pass in automatic differentiation (AD) is also straightforward,  provided that we can propagate the tangents (encoding the first-order sensitivity) in  parallel [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Unfortunately, doing the same in the reverse pass is not trivial, as ad-  joining a mutable code leads to race conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=', every read becomes a write  operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' This has led to extensive research on adapting automatic differentiation  to parallel environments [4, 19, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Now, most state-of-the-art differentiable tools em-  ploy a Domain Specific Language (DSL) constraining the user to specific differentiable  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In particular, this approach has been adopted mainly in machine learning,  leading to the development of fast AD libraries efficiently generating the derivatives  efficiently on hardware accelerators such as GPUs or TPUs [3, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The parallelization of linear algebra is usually more involved, as most large-scale op-  timization methods fall back on the solution of sparse indefinite Karush-Kuhn-Tucker  (KKT) systems [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In the 1980s, preliminary results were obtained by running itera-  tive methods in parallel, using block-Krylov [36] or block-truncated Newton methods  [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' However, block iterative algorithms are quickly limited by the lack of generic  preconditioners for KKT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The 1990s witnessed the emergence of the interior-  point methods (IPM), together with the development of large-scale sparse direct linear  solvers [12, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In IPM, a significant portion of the time is spent solving a sequence  of (indefinite) KKT systems, hence the method directly benefits from efficient sparse  linear solvers able to run in parallel [1, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In the 2000s, it was shown that, for block-  structured optimization problems as we consider here, the layout of the optimization  problem can be exploited further in a Schur complement approach to solve the Newton  step in parallel [2, 9, 17, 22, 33, 44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' These developments led to the development  of mature decomposition-based parallel nonlinear solvers for scenario-based problems  in the 2010s [8, 16, 34, 41, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Eventually, running an optimization algorithm fully in parallel generally requires a  subtle combination of (i) and (ii), often devolving to a software engineering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The challenge is to evaluate the derivatives and solve the resulting KKT system each in  parallel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' all this while minimizing the communication between the different processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  This has led to the development of different prototypes for MPI-parallel modelers [10,  21, 34, 43], most of them extending a specific AD backend [5, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Such approaches  have been successfully applied to solve large-scale block-structured nonlinear problems,  as encountered in stochastic programming and dynamic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Contributions  In this article, we introduce a new parallel algorithm to solve block-structured non-  linear programs involving state equations on exascale supercomputers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Our algorithm  uses the parallel interior-point solver MadNLP [41], using two layers of parallelism to  streamline both the evaluation of the derivatives and the solution of the KKT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  This framework targets new exascale supercomputers, where each node is assigned  to multiple GPUs connected with a unified memory (designed to have fast memory  exchange between the different GPUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  We demonstrate the capability of the algorithm on scenario-based power flow prob-  lems (block-OPF), here formulated as two-stage stochastic nonlinear programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The  scenarios can be stochastic or represent contingencies (which can be interpreted as  stochastic outcomes with uniform distribution), as is the case of the very widely used  security-constrained AC optimal power flow (SC-ACOPF) problem [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' SC-ACOPF is  one of the core analyses undertaken in the planning, operational planning, and real-  time operation of transmission systems [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' SC-ACOPF is run several times a day by  many operators in the US and the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' For brevity, we will refer to such problems  as stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The block structure of such problems is given by the different scenarios associ-  ated with the stochastic problem, leading to potential parallelism in both the evalu-  ation of the derivatives and the solution of the resulting block-angular KKT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The parallel solution of the block-OPF problem with a Schur complement approach  has been studied extensively both with PIPS-NLP [8, 37] (multiprocessing) and with  Beltistos [23, 25] (multiprocessing + factorization of the dense Schur complement on  the GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Compared to the state-of-the-art solver Beltistos, our approach carries out  almost all computation on the GPUs including a global CUDA-aware MPI reduction,  from the evaluation of the derivatives to the assembling of the Schur complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We  test our implementation on the pre-exascale supercomputer Polaris, where each node  is equipped with 4 A100 GPUs, and we solve block-OPF problems with up to 9,251  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Problem statement  In systems engineering, it is common to encounter optimization problems with rela-  tively few degrees of freedom – ”controls”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Then, the goal is to appropriately fix the  values for the degrees of freedom, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=', by minimizing a given operational cost while  satisfying the physical equations of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In that context, the internal state of  the system is described by a state variable x ∈ Rnx, whose values depend on the cur-  rent controls u ∈ Rnu associated with the problem’s degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' If the problem  is well-posed, this translates to the state equation g(x, u) = 0, where the function g  exhibits the physical structure of the problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=', a differential equation encoding a  dynamics, or a nonlinear network flow associated with static balance equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' When  the system faces uncertainties, it is often appropriate to choose a control u feasible  under a finite set of conditions (or scenarios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' That is, the control u must satisfy N  different state equations  gi(xi, u) = 0  for all  i = 1, · · · , N,  (1)  3  where the state xi now depend on the current scenario i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The variables xi can be assim-  ilated into a recourse variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The N functions g1, · · · , gN define the block structure  of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Block-structured nonlinear programs  In addition to satisfying the N state equations (1), we aim at minimizing the average  operating costs on the N different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The corresponding problem formulates  as a two-stage nonlinear program, which, in our case, is a nonlinear program with  partially separable structure [11]:  min  x1,··· ,xN,  u  N  �  i=1  fi(xi, u)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  �  �  �  �  �  xi ≥ 0 ,  u ≥ 0  gi(xi, u) = 0 ,  hi(xi, u) ≤ 0 ,  ∀i = 1, · · · , N ,  (2)  with fi : Rnx × Rnu → R, gi : Rnx × Rnu → Rnx, hi : Rnx × Rnu → Rm smooth  functions encoding the objective, the state equations, and the operational constraints,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We note that the number of variables (N × nx + nu) and constraints  (N × (m + nx)) are linearly proportional to the number of blocks N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  In addition, if we introduce local control variables u1, · · · , uN with the additional  coupling constraint u1 = · · · = uN = u, we get a problem with a separable structure,  solvable using the primal decomposition method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' at the expense of increasing the  search space [11, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  By introducing slack variables s1, · · · , sN, we rewrite (2) in standard form:  min  x1,··· ,xN,  s1,··· ,sN,  u  N  �  i=1  fi(xi, u)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  �  �  �  �  �  u ≥ 0 ,  xi ≥ 0 ,  si ≥ 0  gi(xi, u) = 0 ,  hi(xi, u) + si = 0 ,  ∀i = 1, · · · , N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (3)  We define yi ∈ Rnx the multipliers (or adjoints) associated to the equality con-  straints gi(xi, u) = 0, zi ∈ Rm the multipliers associated to the operational constraints  hi(xi, u) + si = 0, as well as λ, κi, νi the three multipliers associated to the respective  bound constraints u ≥ 0, xi ≥ 0, si ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The Lagrangian associated to (3) is:  L(x, u, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' y, z, λ, µ, ν) :=  N  �  i=1  �  fi(xi, u) + y⊤  i gi(xi, u) + z⊤  i  �  hi(xi, u) + si  �  − κixi − νisi  �  − λu ,  (4)  with x := (x1, · · · , xN), s := (s1, · · · , sN), y := (y1, · · · , yN), z := (z1, · · · , zN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' To  simplify the notations, we define the extended objective function and the extended  constraints:  f(x, u) :=  �  i=1  fi(xi, u) ,  g(x, u) :=  �  ��  g1(x1, u)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  gN(xN, u)  �  �� ,  h(x, u) :=  �  ��  h1(x1, u)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  hN(xN, u)  �  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  4  We assume the functions f, g, h are twice differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We denote  H = ∂(x,u)h(x, u) ∈ RNm×(Nnx+nu)  Jacobian of the inequality cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  G = ∂(x,u)g(x, u) ∈ RNnx×(Nnx+nu)  Jacobian of the equality cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  W = ∇2  (x,u)L(x, u, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' ·) ∈ R(Nnx+nu)×(Nnx+nu)  Hessian of Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Interior-point method  The interior-point method (IPM) [30, Chapter 19] is a classical approach to solve (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' KKT system  The Karush-Kuhn-Tucker (KKT) equations associated to (3) can be expressed as  ∇xfi + (Gi  x)⊤yi + (Hi  x)⊤zi − κi = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  ∀i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' N  (5a)  N  �  i=1  �  ∇ufi + (Gi  u)⊤yi + (Hi  u)⊤zi  �  − λ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (coupling)  (5b)  zi − νi = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  ∀i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' N  (5c)  gi(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' u) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  ∀i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' N  (5d)  hi(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' u) + si = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  ∀i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' N  (5e)  Xiκi = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' κi) ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  ∀i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' N  (5f)  Siνi = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' (si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' νi) ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  ∀i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' N  (5g)  Uλ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' λ) ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (5h)  where U = diag(u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Xi = diag(xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Si = diag(si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The interior-point method uses a homotopy parameter µ > 0 to replace the  complementarity constraints (5f)-(5g)-(5h) by the smooth approximations: Xiκi =  µenx, Siνi = µem, Uλ = µenu (en being the vector of all ones of dimension n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The  resulting (smooth) system of nonlinear equations can be solved iteratively using New-  ton method, where at each iteration, the descent direction is updated by solving the  following augmented linear system:  �  ���  W + Σp  0  G⊤  H⊤  0  Σs  0  I  G  0  0  0  H  I  0  0  �  ���  �  ���  pd  ps  py  pz  �  ��� = −  �  ���  r1  r2  r3  r4  �  ���  (6)  with r1 =  �  ∇xf + G⊤  x y + H⊤  x z − µX−1enx  ∇uf + G⊤  u y + H⊤  u z − µU−1enu  �  , r2 = z − µS−1em, r3 = g(x, u), r4 =  h(x, u) + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The primal descent direction pd decomposes as pd = (px1, · · · , pxN, pu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Block angular structure  The linear system (6) is sparse and symmetric indefinite, and can be factorized using  the Bunch-Kaufman algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' However, it is often beneficial to exploit its block-  angular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Indeed, both the Hessian of the Lagrangian and the Jacobians have  5  a block-angular structure, given as  W =  �  ����  Wx1x1  Wx1u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  WxNxN  WxNu  Wux1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  WuxN  Wuu  �  ���� ,  G =  �  ��  G1  x1  G1  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  GN  xN  GN  u  �  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  By reordering the linear system (6), we can expose the block-angular structure of the  KKT system as:  �  ����  A1  B⊤  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  AN  B⊤  N  B1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  BN  A0  �  ����  (7)  with  A0 = Wuu,  Ai =  �  ���  Wxixi + Σxi  0  G⊤  xi  H⊤  xi  0  Σsi  0  I  Gxi  0  0  0  Hxi  I  0  0  �  ��� ,  Bi =  �  �  Wxiu  (Gi  u)⊤  (Hi  u)⊤  �  �  ⊤  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The block-angular structure (7) can be exploited to solve the KKT linear system in  parallel using a Schur complement approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In that case, the submatrices Ai can be  factorized independently to assemble the Schur complement in parallel [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Condensation and reduction  Instead of reordering the augmented KKT system (6) as a block angular matrix (7),  we propose an alternative approach based on successive condensation and reduction  of the KKT system, following the method introduced in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' If the structure is well-  defined, we show that we can condense the KKT system (6) to a dense matrix with  size nu × nu in two steps: first, by removing the inequality constraints in (6), then by  exploiting the structure of the equality constraints to reduce the condensed system to  a dense matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The condensation and reduction steps are illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Condensation step  The condensation step allows reducing the size of the KKT system drastically if the  number of inequality constraints is large1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1 (Condensed KKT system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The linear system (6) is equivalent to  �  K + Σp  G⊤  G  0  � �  pd  py  �  = −  �  r1 + H⊤(Σsr4 − r2)  r3  �  ,  (8)  where K ∈ R(Nnx+nu)×(Nnx+nu) is the condensed matrix K := W + H⊤ΣsH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The  1It is equivalent to the normal equations in linear programming [30, Chapter 16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='412]  6  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='. Successive reductions for a block-structured nonlinear problem with N = 3: Aug-  mented system (6), Condensed system (8), Reduced system (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  descent directions ps and pz are recovered as  �  pz = Σs  �  Hpd + r4  �  − r2 ,  ps = −Σ−1  s  �  r2 + pz  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (9)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' See [31, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The condensed matrix K inherits the block-angular structure of the Hessian of the  Lagrangian W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The condensed matrix K = W +H⊤ΣsH has a block-angular struc-  ture, given as  K =  �  ����  Kx1x1  Kx1u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  KxNxN  KxNu  Kux1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  KuxN  Kuu  �  ����  (10)  where we have defined the condensed blocks Kxixi := Wxixi + (Hi  xi)⊤ΣsiHi  xi, Kuxi :=  Wuxi + (Hi  u)⊤ΣsiHi  xi and Kuu := Wuu + �N  i=1(Hi  u)⊤ΣsiHi  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' This is proved by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Reduction step  In addition, we can exploit the structure of the equality constraints g1, · · · , gN to  further reduce the size of the linear system (8) down to a dense matrix with size  nu × nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Equation (10) exhibits the structure w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' the state x and the control u, we  7  3755 x 3755  1193 x 1193  107 x 107rewrite as such the condensed KKT system (8) as  �  �����������  Kx1x1  Kx1u  (G1  x1)⊤  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  KxNxN  KxNu  (GN  xN)⊤  Kux1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  KuxN  Kuu  (G1  u)⊤  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (G1  u)⊤  G1  x1  G1  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  GN  xN  GN  u  �  �����������  �  �����������  px1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  pxN  pu  p1  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  pN  y  �  �����������  = −  �  ����������  ˆr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  ˆrN  1  ˆr2  ˆr1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  ˆrN  3  �  ����������  ,  where we have renamed the right-hand-side in (8) as ˆr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3 (Reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Assume that for all i = 1, · · · , N the Jacobian matri-  ces Gi  x ∈ Rnx×nx are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Then the linear system (8) is equivalent to  ˆKuu pu = −ˆr2 +  N  �  i=1  �  (Gi  u)⊤(Gi  x)−⊤ˆri  1 +  �  Kuxi − (Gi  u)⊤(Gi  x)−⊤Kxixi  �  (Gi  x)−1ˆri  3  �  (11)  with ˆKuu := Z⊤KZ and Z ∈ R(nu+Nnx)×nu is the reduction operator defined as  Z =  �  ����  −(G1  x)−1G1  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  −(GN  x )−1GN  u  I  �  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (12)  The descent directions px and py are recovered as  �  pi  x = −(Gi  x)−1�  ˆri  3 + Gi  upu  �  pi  y = −(Gi  x)−⊤�  ˆri  1 + Kxixipi  x + Kxiupu  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (13)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' See [31, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The reduction (11) is equivalent to a Schur complement approach applied to the  condensed KKT system (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1), we have shown that the condensed  matrix K has a block-angular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The associated condensed KKT system (8)  is also inheriting a block-angular structure in the form of (7), where the blocks are  given by  A0 = Kuu ,  Ai =  �  Kxixi  (Gi  x)⊤  Gi  x  0  �  ,  Bi =  �  Kxiu  Gi  u  �⊤  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (14)  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Assume that for each i = 1, · · · , N the Jacobian Gi  x is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Let Suu = A0 − �N  i=1 BiA−1  i B⊤  i  be the Schur complement associated to the block-  angular system (7) with the matrices (Ai, Bi) defined in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Then, the Schur com-  plement Suu is equal to the reduced matrix �Kuu defined in (11): Suu = Z⊤KZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  8  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' First, note that if the Jacobian Gi  x is invertible, then the block matrix Ai  defined in (14) is also invertible, with  A−1  i  =  �  0  (Gi  x)−1  (Gi  x)−⊤  −(Gi  x)−⊤Kxixi(Gi  x)−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (15)  Using (14)-(15), we expand the expression of the terms in the sum constituting the  Schur complement Suu:  BiA−1  i B⊤  i =  �  Kuxi  (Gi  u)⊤� �  0  (Gi  x)−1  (Gi  x)−⊤  −(Gi  x)−⊤Kxixi(Gi  x)−1  � �  Kxiu  Gi  u  �  ,  = (Gi  u)⊤(Gi  x)−⊤Kxiu + Kuxi(Gi  x)−1(Gi  u) − (Gi  u)⊤(Gi  x)−⊤Kxixi(Gi  x)−1Gi  u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Hence, the Schur complement Suu = A0 − �N  i=1 BiA−1  i B⊤  i expands as  Suu = Kuu −  N  �  i=1  �  (Gi  u)⊤(Gi  x)−⊤Kxiu + Kuxi(Gi  x)−1(Gi  u) − (Gi  u)⊤(Gi  x)−⊤Kxixi(Gi  x)−1Gi  u  �  = Z⊤KZ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  We recover the expression of the reduced matrix �Kuu in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Discussion  Hence, we can interpret the reduction step as a Schur complement approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Forming  the Schur complement has always been the bottleneck when solving distributed block  angular problems in parallel [8, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Its reduction operation involves large memory  transfers between the processes, with the number of transfers being on the order  of O(log(p)), where p is the number of processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Due to the quasi-shared memory  architecture on GPUs, the reduction can be implemented efficiently [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In the next  section, we propose to extend [31] to assemble the reduced matrix �Kuu using two levels  of parallelism, using both MPI and CUDA, thus reducing the reliance on distributed  memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Parallel implementation  In the previous section, we have detailed the structure of block-angular nonlinear  programs and presented the condensation and reduction steps for the KKT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The loose coupling between the blocks is favorable for parallelizing the evaluation of  the derivatives and the solution of the block-angular KKT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Globally, we can  distribute the computation on different processes using MPI (coarse parallelism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Lo-  cally, we can further streamline the computation using GPU accelerators (fine-grained  parallelism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' This paradigm, with its two levels of parallelism, is directly in line with  what is currently offered by the new exascale architectures, where each node has 4 to  8 GPUs, all sharing a unified memory for fast communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We present in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1 how  we streamline the evaluation of the model using automatic differentiation, and in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2  how we parallelize the solution of the KKT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Parallel automatic-differentiation  First, we present how to evaluate the model in parallel using automatic differentiation  [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We illustrate the procedure in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The goal of the algorithm is to streamline  the evaluation of the N scenarios on N/M GPUs, M being the number of scenarios  evaluated locally on each GPU (we suppose here that N is a multiple of M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  root  g1, · · · , g4  g13, · · · , g16  g1  g4  · ·  g13  g16  · ·  · ·  · ·  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='. Parallel evaluation of the derivatives for g1, · · · gN on 4 GPUs: we have a total of  N = 16 scenarios, each GPU evaluating M = 16/4 = 4 scenarios locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Local parallelism  The first level of parallelism streamlines the evaluation of the model on SIMD/GPU  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We have designed our implementation to run entirely on the GPU device, to  avoid any data transfer between the host and the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Block evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We suppose that the nonlinear functions (fi, gi, hi)  share the same structure, its expressions yielding the same Abstract Syntax Tree (AST)  for all i = 1, · · · , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We illustrate the block evaluation on a simple abstract tree, but  the reasoning extends to more complicated structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We suppose that for all i, the  functions fi, gi, hi depend linearly on a nonlinear basis matrix ψ : Rnx × Rnu → Rnb:  that is, there exists three sparse matrices Lf, Lg, Lh such that  fi(xi, u) = Lfψ(xi, u) ,  gi(xi, u) = Lgψ(xi, u) ,  hi(xi, u) = Lhψ(xi, u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (16)  Suppose we aim to evaluate the M functions g1, · · · , gM in batch for the states  x1, · · · , xM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The structure (16) is directly amenable for SIMD evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We de-  note by XM = (x1, · · · , xM) ∈ Rnx×M the dense matrix obtained by concatenat-  ing the M states together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' By using a proper GPU kernel or a parallel modeler,  we can evaluate the basis in a SIMD fashion and build the matrix Ψ(XM, u) :=  �  ψ(x1, u), · · · , ψ(xM, u)  �  ∈ Rnb×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Then, evaluating the functions g1, · · · , gM simul-  taneously translates to the evaluation of one SpMM product:  �  g1(x1, u), · · · , gM(xM, u)  �  = LgΨ(XM, u) ∈ Rnx×M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (17)  The total memory required in the two successive operations is O((nx + nb) × M), and  depends linearly on the number of blocks M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We note the SpMM operations are generally  implemented efficiently in the vendor library (cusparse for CUDA, rocSPARSE for  AMDGPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' First-order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Suppose that for a given i we have a differen-  tiable implementation gbi : Rnd → Rnx associated to the function gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We aim to  evaluate the Jacobian-matrix products (∇gi)D for p tangents encoded in a matrix  D ∈ Rnd×p using forward-mode AD and operator overloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' This operation trans-  lates to propagating forward a vector of dual numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Denoting by d ∈ Dnd  p  the dual  number encoding the p tangents stored in D, evaluating (∇gi)D simply amounts to  call gbi(d) and extract the results in the dual numbers returned as a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' As Gi  is sparse, we can apply the technique of Jacobian coloring [18] to compress the inde-  pendent columns of the sparse matrix Gi and reduces the number of required seeding  tangents p needed to evaluate the full Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Suppose now we want to evaluate the sparse Jacobians G1, · · · , GM in batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' As  the functions gi are based on the same AST, their respective Jacobians G1, · · · , GM  are sharing the same sparsity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' By seeding a matrix of dual numbers DM =  (d1, · · · , dM) ∈ Dnd×M  p  , we can use the same operation as (17) to streamline the  evaluation of the M Jacobian-vector products using the SIMD kernel Ψ(·) and SpMM  operations:  �  gb1(d1), · · · , gbM(dM)  �  := LgΨ(DM) ∈ Dnx×M  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (18)  Once the results are evaluated, it remains to uncompress the dual outputs to build  the M sparse Jacobians G1, · · · , GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Hence, we can streamline the evaluation of the  Jacobian along with the number of tangents p and the number of blocks M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' This comes  at the expense of increasing memory usage to O((nx + nb + nd) × M × p) (to store the  dual matrices associated to the input, the intermediate basis Ψ and the output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Second-order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The evaluation of the second-order deriva-  tives follows the same procedure, using forward-over-reverse AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' For each i, we sup-  pose available an adjoint function adj gbi : Rnd × Rnx → Rnd which for any pri-  mal x ∈ Rnd and adjoint y ∈ Rnx evaluates the Jacobian-transpose vector product  (Gi(x))⊤y (reverse-mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Using forward-mode AD on top of adj gbi, we can compute  the second-order derivatives y⊤∇2gi(x)V for p directions V by calling adj gbi(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Using Hessian coloring, we can compress the independent columns of the sparse matrix  y⊤∇2g(x) and reduce the number of seeding tangents p required to evaluate the full  Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We note that in general obtaining an adjoint adj gbi running in parallel is  nontrivial due to potential race conditions incurred by the control flow reversal of the  original code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Computing the Hessian y⊤∇2gi(x) in parallel for i = 1, · · · M amounts to defining  two matrices of dual numbers XM = (X1, · · · , XM) ∈ Dnd×M  p  , Y M = (y1, · · · , yM) ∈  Dnx×M  p  and evaluate ∇Ψ(XM)⊤L⊤  g Y M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The dual outputs are uncompressed to build  the M sparse Hessians (as the sparsity pattern of the Hessians is different than those  of the Jacobians, the matrix XM employed here is different than the one used in (18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The total memory required to store the duals is O((2nx +nd +nb)×M ×p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' For more  details, we refer to the vector forward mode as described in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Global parallelism  Now, if we have several GPUs at our disposal, we can push the parallelism further  by distributing the evaluations using multiprocessing and a Message Passing Interface  (MPI) library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Coming back at our original problem (2), we illustrate in Figure 2 how to  dispatch the evaluation of the N nonlinear constraints g1, · · · , gN (the same reasoning  11  applies to the objectives f1, · · · , fN and the inequality constraints h1, · · · , hN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We use  the streamlined implementation described in the previous subsection to evaluate the  constraints in a batch of size M: the first GPU evaluates the constraints g1, · · · , gM,  the second GPU evaluates gM+1, · · · , g2M, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In total, the evaluation of the  N constraints requires N/M GPUs (if M = 1, each GPU evaluate one constraint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' if  M = N, we use only one GPU evaluating all the constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The implementation has been designed to minimize the communication between  the different processes: each batch g1, · · · , gM stores the data it needs locally, the only  data exchange with the other processes being the vector of input and the vector of  output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In addition, we will see in the next section we do not have to transfer the first-  and second-order information if a parallel linear solver is being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Parallel KKT solver  By exploiting the block-angular structure of the KKT system, we can solve the New-  ton step in parallel using a Schur complement approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The challenge lies in the  computation of the Schur complement matrix S = A0 − �N  i=1 BiA−1  i B⊤  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Each prod-  uct BiA−1  i B⊤  i requires the factorization of the matrix Ai and the solution of a linear  system with multiple (sparse) right-hand-side A−1  i Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' State-of-the-art methods are  evaluating the Schur complement using an incomplete augmented factorization ap-  plied on the auxiliary matrix  �  Ai  B⊤  i  Bi  0  �  , as currently implemented in the Pardiso  linear solver [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Here, we use an alternative approach building on the reduced KKT  system §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2 (equivalent to the Schur complement approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' As the reduction can  be streamlined on GPU accelerators [31], this approach can assemble the Schur com-  plement in parallel using CUDA-aware MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We illustrate the parallel computation of  the Schur complement in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  root  (Assembling)  K, G  K, G  K, G  K, G  AD  AD  AD  AD  ˆK1:4  uu  ˆK5:8  uu  ˆK9:12  uu  ˆK13:16  uu  (Reduction)  Z⊤KZ  Z⊤KZ  Z⊤KZ  Z⊤KZ  MPI AllReduce  +  (Schur compl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=')  ˆKuu  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='. Parallel computation of the Schur complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Assembling the sparse matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Using the procedure introduced in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1, we  evaluate locally the Jacobians G1, · · · , GM the Jacobians H1, · · · , HM and the Hes-  sians W 1, · · · , W M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Using dedicated kernels, we uncompress the results in the block-  12  angular sparse Jacobians  G1:M  x  =  �  ��  G1  x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  GM  xM  �  �� , G1:M  u  =  �  ��  G1  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  GM  u  �  �� , H1:M =  �  ��  H1  x1  H1  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  HM  xM  HM  u  �  �� ,  and sparse Hessian  W 1:M =  �  ����  Wx1x1  Wx1u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  WxMxM  WxMu  Wux1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  WuxM  Wuu  �  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Once the sparse matrices are obtained, we recover the condensed matrix K1:M =  W 1:M + (H1:M)⊤Σ(H1:M) (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1) using one SpGEMM operation and we fac-  torize the matrix G1:M  x  using a sparse LU factorization (potentially running in batch  as the matrices G1  x1, · · · , GM  xM are sharing the same sparsity pattern).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Once the ma-  trix G1:M  x  is factorized as PG1:M  x  Q = LU (P, Q being two permutation matrices),  computing (G1:M  x  )−1b translates to two backsolves (SpSV) and two matrix-vector mul-  tiplications (SpMV), as (G1:M  x  )−1b = QU−1L−1Pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Local reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Once the sparse matrices are built, we evaluate locally the reduced  matrix �K1:M  uu  on the GPU, using div(nu, nbatch) + 1 matrix-matrix product �K1:M  uu V  (with V ∈ Rnu×nbatch a dense matrix encoding nbatch vectors of the Cartesian basis of  Rnu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The evaluation of one batched matrix-matrix product �K1:M  uu V = (Z⊤K1:MZ)V  proceeds in three steps  (1) Solve Tx = −(G1:M  x  )−1(G1:M  u  V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (2) Evaluate  �  Lx  Lu  �  :=  �  K1:M  xx  K1:M  xu  K1:M  ux  K1:M  uu  � �  Tx  V  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  (3) Set �K1:M  uu V = Lu − G1:M  u  (G1:M  x  )−⊤Lx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  In total, we need 2 SpSM and 3 SpMM operations in the first step, 1 SpMM in the second  step, and 2 SpSM and 3 SpMM operations in the third step, giving a total of 4 SpSM  and 7 SpMM operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' More than the computation, the reduction is limited by the  memory, as we have to store the three buffers Lx, Tx, Tu with a total size of (2M ×  nx + nu) × nbatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' If nx is too large, it is in our interest to reduce M (by using more  GPUs) or to reduce nbatch (at the expense of computing more matrix-matrix product  �K1:M  uu V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Global reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Once we obtain the locally reduced matrices �KnM+1:(n+1)M  uu  for n = 0, · · · , N/M − 1, we can assemble the global reduced matrix  �Kuu =  �N/M−1  n=0  �KnM+1:(n+1)M  uu  using one all reduce (MPI Allreduce) operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The size of  the reduced matrix �Kuu is nu × nu, hence limiting the memory transfer required in  the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Discussion  We have presented a practical way to assemble the Schur complement on multi-GPU  architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The parallelism occurs both at the local level (SIMD evaluations on the  GPUs) and at the global level (distributed computation with MPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The algorithm  has the advantage of assembling the sparse Jacobians and Hessians only locally, as  the reduction occurs before proceeding to the memory transfer with MPI Allreduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The reduced matrix has a dimension nu × nu, which compresses the memory transfer  significantly if the number of degrees of freedom nu is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' However, this comes at the  expense of storing a vector of dual numbers (whose memory is linearly proportional  to the number of blocks M evaluated locally and the number of tangents p being  employed to evaluate the sparse derivatives) and additional buffers in the reduction  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In the next section, we will test an implementation of the algorithm on  CUDA GPUs, and show that the algorithm is practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Numerical results  We demonstrate the capabilities of the algorithm we introduced in Section §3 on the  supercomputer Polaris, using CUDA-aware MPI to dispatch the solution on multiple  GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We present in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1 the stochastic optimal power flow problem, and give in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2  detailed assessments of the algorithms we have introduced earlier in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Eventually,  we present in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3 a benchmark comparing our parallel solution algorithm with a  state-of-the-art solution method running on the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Settings  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Case study: the block-structured optimal power flow  The stochastic optimal power flow problem aims at finding an optimal dispatch for the  generators u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The solution u should minimize the operational costs while satisfying the  physical constraints (power flow equations g(x, u) = 0, here playing the role of the state  equations) and operational constraints (line flow constraints h(x, u) ≤ 0) on a given  set of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Each scenario is assigned given load parameters (energy demands) and  potential contingencies (line tripping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The values of the state x depend on the local  scenario we are in, the state x being the recourse variable in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' As such, the  problem has a partially separable structure as introduced in Problem (2), the control  u being shared across all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We refer to [7] for the original presentation of the  stochastic optimal power flow problem and to [8, 23, 24, 26] for practical algorithms  solving the stochastic optimal power flow problem (some also focus on the multistage  setting, which is not covered in this article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' For our benchmark, we look at reference  instances provided by MATPOWER [47], whose characteristics are detailed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  We recall that in our case, the size of the Schur complement matrix ˆKuu is given by  the number of controls nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Implementation  The algorithm has been implemented entirely in Julia 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The Schur complement  approach has been developed as an extension of the nonlinear optimization solver  MadNLP [41], using CUDA-aware MPI as provided in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We have used the package  14  Name  #bus  #lines  #gen  nx  nu  case118  118  186  54  181  107  case1354pegase  1,354  1,991  260  2,447  519  case2869pegase  2,869  4,582  510  5,227  1,019  case9241pegase  9,241  16,049  1,445  17,036  2,889  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='. MATPOWER instances used in the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  ExaPF as a nonlinear modeler for the optimal power flow problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' All the results  presented here have been generated on the supercomputer Polaris equipped with a  total of 560 nodes, each node having with 1 CPU and 4 A100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Assessment of the parallel implementation  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Assessing the performance of the parallel automatic differentation  We first assess the performance of the parallel automatic differentiation we introduced  in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1 in a multi-GPU setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We compare the performance we obtain with a CPU  implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We use case1354pegase as a representative instance, and display the  time spent in the automatic differentiation as we increase the total number of scenarios  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The results are displayed in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  We observe that the computation time depends linearly on the number of scenarios,  as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' For N = 8, it is not worthwhile dispatching the evaluation on multiple  GPUs as the problem is small enough to be evaluated on a single GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' For N = 512,  the evaluation time is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3s on the CPU, compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='41, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='31, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='28s using  1, 2, 4 and 8 GPUs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Hence, we get a 40x speed-up when evaluating the  derivatives in a multi-GPU setting, and it is not worthwhile to use more than 4 GPUs  (one node).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  8  16  32  64  128  256  512  N scenarios  10  1  100  101  Evaluation time [s]  case1354pegase  CPU  1 GPUs  2 GPUs  4 GPUs  8 GPUs  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='. Time spent to evaluate the model and its derivatives with automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Assessing the performance of the parallel KKT solver  We proceed to the same performance analysis to assess the performance of the parallel  KKT solver detailed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We compare the time required to evaluate the full solution  of the KKT system afresh (including reduction time, factorization time and backsolve  time) on case1354pegase as we increase the number of scenarios N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' As a reference,  we give the time taken by the sparse linear solvers HSL MA27 (single-threaded) and  HSL MA57 (multi-threaded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The results are displayed in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  On the left, we display the evolution of the time spent in the linear solver as we  increase the number of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' For N = 512, we observe that we get a linear speed-  up as we increase the number of GPUs: using 8 GPUs, the parallel KKT solver is  40x faster than using HSL MA27 on the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Interestingly, we observe that HSL  MA57 is not faster than HSL MA27, despite being multithreaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' This is consistent  with the observation made in [42], and illustrates the difficulty of parallelizing ef-  fectively the sparse LDL factorization (Bunch-Kaufman).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' On the right, we display a  performance profile detailing the time spent in MA27 and the parallel KKT solver on  case1354pegase with N = 512 scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We observe that most of the time in HSL  MA27 is spent on factorizing the sparse augmented KKT system (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' On the other  side, the factorization of the dense reduced matrix ˆKuu is trivial using LAPACK on  the GPU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' the bottleneck in the parallel KKT solver is the reduction algorithm itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Fortunately, the reduction algorithm can run in parallel: we get a linear speed-up as  we increase the number of GPUs used in the reduction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  8  16  32  64  128  256  512  N scenarios  10  1  100  101  Time [s]  Linear solver time against N  ma27  ma57  1 GPUs  2 GPUs  4 GPUs  8 GPUs  ma27  1 GPU  2 GPUs  4 GPUs  8 GPUs  0  2  4  6  8  Time [s]  Performance profile for N = 512  Factorization  Backsolve  Reduction  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='. Time spent to solve the KKT system for case1354pegase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Assessing the memory consumption  We have observed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1 that the total memory required to store the duals is O((2nx+  nd + nb) × M × p), with M being the number of scenarios stored locally (M = N on  1 GPU, M = N/2 on 2 GPUs) and p the number of tangents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We display in Table 2  the memory taken by the automatic differentiation backend and by the parallel KKT  solver for case1354pegase as we increase the number of scenarios N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We note that  storing the duals is expensive in terms of memory, with up to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='9GB for N = 512 on  one GPU (as a reference, each NVIDIA A100 GPU on Polaris has 40GB of memory  available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' By evaluating the model on different processes with MPI, we can split the  memory consumption on the different GPUs we are using, leading to better use of the  resource at our disposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  16  1 GPU  2 GPUs  N  AD  KKT solver  AD  KKT solver  8  171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='5  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1  16  342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2  181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='5  171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1  32  684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3  360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='0  342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2  183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2  64  1,368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='7  716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='8  684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3  363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2  128  2,737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3  1,430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='5  1,368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='7  723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='4  256  5,474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='7  2,858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='0  2,737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3  1,443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='6  512  10,949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3  5,712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='8  5,474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='7  2,884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='. Memory consumption in MB  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Parallel solution of the block-structured OPF problem  We analyze the parallel performance of our implementation on block-structured OPF  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Assessing the parallel performance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' the number of scenarios  First, we are interested in the scaling of the parallel algorithm in relation to the total  number of scenarios N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We consider the case118 instance, and increase the number  of scenarios N from 8 up to 2,048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' For each N, we solve the block-structured OPF  problem with MadNLP using our parallel KKT solver, and we compare with the  performance we obtained with HSL MA27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The results are displayed in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We  observe that the solver HSL MA27 is initially faster than our parallel KKT solver, as  the problem is too small to benefit from parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' However, as soon as N ≥ 16 the  parallel KKT solver becomes competitive with HSL MA27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The relative performance is  improving as we increase the number of scenarios N: for N = 512, we get a 68x speed-  up when using 8 GPUs, compared to the reference HSL MA27 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='4s versus 712s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Interestingly, using 2 nodes (=8 GPUs) does not lead to any speed-up compared to  a single node (=4 GPUs) if N ≤ 256;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' this setting is attractive only when the size of  the problem becomes sufficiently large (N ≥ 1024) to compensate for the additional  memory exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Assessing the parallel performance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' the size of the problem  Second, we increase the size of the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We set a fixed number of scenarios  N = 8, and look at the time to solution for case1354pegase, case2869pegase and  case9241pegase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We detail the respective dimension of each problem in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We  display the results in Figure 7, and give the detailed benchmark in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' On the  left (a), we display the total time required to find the solution of the three instances as  a function of the number of GPUs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' on the right (b), we show the performance profile  associated to case9241pegase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In (a), we observe that overall the parallel algorithm is  faster than the CPU implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The parallel algorithm scales well as we increase  the number of GPUs we are using, the parallel algorithm being 35x faster than the  reference when using 8 GPUs to solve case9241pegase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In (b), we detail the time  17  8  16  32  64  128  256  512  1,024 2,048  N scenarios  100  101  102  103  Time to solution [s]  case118  ma27 (CPU)  polaris (1 GPUs)  polaris (2 GPUs)  polaris (4 GPUs)  polaris (8 GPUs)  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='. Time to solve the block-structured OPF problem case118 as a function of the  number of scenarios N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  spent in the different operations for case9241pegase: the time spent to factorize the  Schur complement with Lapack (using cusolve) is constant as the size of the Schur  complement remains the same as we increase the number of GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We observe that  the time spent in the AD decreases linearly with the number of GPUs exploited, but  the relative time spent in AD is negligible (less than 5% of the total time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Most of  the time is spent in the parallel reduction, as discussed earlier in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  N  nvar  ncon  ˆKuu (mb)  1354pegase  8  20,095  53,520  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1  2869pegase  8  42,835  119,216  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='9  9241pegase  8  139,177  404,640  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='7  1354pegase  512  1,253,383  4,425,280  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='. Dimension of the instances we have used in our benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Assessing the parallel performance on a very large-scale instance  We finish our numerical experiments by solving a very large-scale instance:  case1354pegase with N = 512 scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The dimension of the resulting optimization  problem is displayed in Table 3: the problem has more than 1 million variables, and 4  millions constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We solve this instance on resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' 1 node, 2, 4 and 8 nodes (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' 4,  8, 16 and 32 GPUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The results are displayed in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' We observe that the scaling  is almost perfect when we use 2 nodes (8 GPUs) instead of a single node (4 GPUs)  but we do not observe the same behavior when we increase the number of nodes to 4  and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' On that instance, the gain we get when using 8 nodes (32 GPUs) is marginal  18  CPU  1 GPU  2 GPU  4 GPU  8 GPU  100  101  102  103  Time to solution [s]  Benchmark  1354pegase  2849pegase  9241pegase  1 GPU  2 GPUs  4 GPUs  8 GPUs  10  1  100  101  102  Time [s]  Performance profile, 9241pegase  Total time  linear scaling  AD  Factorization  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='. For a fixed number of scenarios N = 8, (a) total time spent solving the block-  OPF case1354pegase, case2869pegase and case9241pegase with MadNLP (b) performance  profile for case9241pegase with varying number of GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  1354pegase  2869pegase  9241pegase  #it  AD  KKT  Tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  #it  AD  KKT  Tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  #it  AD  KKT  Tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  CPU  44  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='2  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='0  77  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='9  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='4  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3  136  205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='6  771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='8  984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1  1 GPU  44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1  93  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='7  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='8  98  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='5  112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3  117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='8  2 GPUs  44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='4  93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
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+page_content='8  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='1  8 GPUs  44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
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+page_content='1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='7  98  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='4  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='4  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='7  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='. Detailed results  compared to when using 4 nodes (16 GPUs): the solving time only decreases from  67s to 58s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' This corroborate our observations: it is better to pack all the computa-  tion on a single node to use four A100 GPUs connected together via unified memory  (NVLINK has a transfer rate of 600GB/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' When we have to use more than 2 nodes,  the memory transfers are more involved as they have to pass through the network of  the supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Conclusion  We show promising results for leveraging massively parallel SIMD architectures like  GPUs for block-structured nonlinear programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The parallelism is applied to both the  derivative evaluation and the solution of the KKT linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The main operation  in the KKT algorithm is the assembling of the Schur complement, the factorization of  the dense Schur complement being fast to carry on the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  At all levels, the method benefits significantly from the massive parallelism, achiev-  ing a speedup of around 40 for the derivatives compared to a sequential CPU im-  plementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The speedup is very application dependent, not least on the Hessian  coloring and the problem’s structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The assembling of the Schur complement is bot-  tlenecked by a distributed reduction operation bound by the interconnect’s latency  and throughput between GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Current, so-called super nodes with multiple GPUs  19  4  8  16  32  #GPUs  102  3 × 101  4 × 101  6 × 101  2 × 102  Time to solution [s]  Benchmark 1354pegase (N=512)  Measured  linear scaling  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='. Solving case1354pegase with N = 512  connected via fast networks like NVLINK greatly accelerate this operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Lastly, our  method is limited by the memory capacity of the GPU accelerators as it grows linearly  with the number of problem blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' In the context of ACOPF we are confident that  upcoming GPUs will provide enough memory to solve a large number of scenarios in  parallel, even for the largest grid instances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=', Eastern Interconnection with 70,000  nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  With the upcoming release of the Aurora supercomputer, these SIMD architectures  will allow new science in regimes that were impossible with previous CPU architec-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Acknowledgment  This material was based upon work supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Department of Energy, Of-  fice of Science, Office of Advanced Scientific Computing Research (ASCR) under Con-  tract DE-AC02-06CH11347 and by NSF through award CNS-1545046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The authors  gratefully acknowledge the funding support from the Applied Mathematics Program  within the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Department of Energy’s (DOE) Office of Advanced Scientific Com-  puting Research (ASCR) as part of the project ExaSGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' This research used resources  of the Argonne Leadership Computing Facility, which is a DOE Office of Science User  Facility supported under Contract DE-AC02-06CH11357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
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+page_content='  Government License: The submitted manuscript  has been created by UChicago Argonne, LLC,  Operator of Argonne National Laboratory (“Ar-  gonne”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Argonne, a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' Department of Energy  Office of Science laboratory, is operated under  Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' DE-AC02-06CH11357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' The U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  Government retains for itself, and others acting  on its behalf, a paid-up nonexclusive, irrevoca-  ble worldwide license in said article to repro-  duce, prepare derivative works, distribute copies  to the public, and perform publicly and display  publicly, by or on behalf of the Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  The Department of Energy will provide pub-  lic access to these results of federally sponsored  research in accordance with the DOE Public  Access Plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content=' http://energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='gov/downloads/doe-  public-access-plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
+page_content='  23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E4T4oBgHgl3EQfDgse/content/2301.04869v1.pdf'}
diff --git a/9tFLT4oBgHgl3EQfCC72/content/tmp_files/2301.11974v1.pdf.txt b/9tFLT4oBgHgl3EQfCC72/content/tmp_files/2301.11974v1.pdf.txt
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+arXiv:2301.11974v1  [math.OC]  27 Jan 2023
+Augmenting Bi-objective Branch and Bound
+by Scalarization-Based Information
+Julius Bauß∗ and Michael Stiglmayr
+University of Wuppertal, School of Mathematics and Natural Sciences,
+IMACM, Gaußstr. 20, 42119 Wuppertal, Germany
+{bauss,stiglmayr}@math.uni-wuppertal.de
+While Branch and Bound based algorithms are a standard approach to solve single-
+objective (mixed-)integer optimization problems, multi-objective Branch and Bound meth-
+ods are only rarely applied compared to the predominant objective space methods. In this
+paper we propose modifications to increase the performance of multi-objective Branch
+and Bound algorithms by utilizing scalarization-based information. We use the hypervol-
+ume indicator as a measure for the gap between lower and upper bound set to implement
+a multi-objective best-first strategy. By adaptively solving scalarizations in the root node
+to integer optimality we improve both, upper and lower bound set. The obtained lower
+bound can then be integrated into the lower bounds of all active nodes, while the deter-
+mined solution is added to the upper bound set. Numerical experiments show that the
+number of investigated nodes can be significantly reduced by up to 83% and the total
+computation time can be reduced by up to 80%.
+Keywords: multi-objective optimization, mult-iobjective branch and bound, integer pro-
+gramming, hypervolume indicator
+1 Introduction
+Many optimization problems occurring in real-word applications include a conflict of interests and
+goals, or secondary objectives, in a word, they are multi-objective. Thus, there is (in general) not
+one solution that optimizes all objectives at once. Following the a posteriori paradigm of decision
+making, we aim at determining the set of so-called efficient solutions or the images, the so-called
+non-dominated points, which cannot be improved in one objective without deterioration in at least
+one other objective. Thus, efficient solutions are reasonable choices for decision makers.
+As we are considering specifically bi-objective integer linear programs and their solution with
+multi-objective Branch and Bound methods, the following literature survey will also focus on this
+and closely related topics. A comprehensive introduction to multi-objective optimization in general
+is given, e.g., in Steuer (1986); Ehrgott (2005).
+Solution approaches for multi-objective optimization problems are often categorized in: objective
+space and decision space methods. Objective space methods scalarize the underlying problem, i. e.,
+it is replaced by a series of single-objective problems to determine successively the set of efficient
+∗Corresponding author
+1
+
+solutions.
+In the case of multi-objective integer programming, these scalarized problems can be
+solved with commercial integer programming solvers like CPLEX or Gurobi. The utilization of these
+optimized, single-criteria solvers are a major advantage and one of the reasons why those methods
+are predominant in multi-objective optimization.
+There are numerous objective space methods and a popular one is the ε-constraint method that
+was introduced for two objectives by Haimes et al. (1971). In every iteration the first objective is
+optimized with an updated constraint to ensure an improvement regarding the second objective. In
+Laumanns et al. (2006) an extension to three and more objectives is presented. Many approaches
+based on the ε-constraint method have been published in the last decades, for example Boland et al.
+(2017) and Kirlik and Sayın (2014) combine the method with reduction of dimension in the tri-
+respectively multi-dimensional case.
+The weighted sum scalarization is an objective space method based on the optimization of a
+weighted sum of the objective functions using non-negative weights.
+Note that not all efficient
+solutions can be determined as optimal solutions of the weighted sum scalarization using suit-
+able weights (see, e. g. Aneja and Nair, 1979).
+Efficient solutions which can be obtained by us-
+ing weighted sum scalarization are denoted as supported efficicent and their corresponding non-
+dominated points are located on the boundary of the convex hull of feasible image points. Extensions
+of the weighted sum method to the multi-objective case are proposed in Przybylski et al. (2010a),
+Özpeynirci and Köksalan (2010), Bö¸kler and Mutzel (2015), and Przybylski et al. (2019).
+Ulungu and Teghem (1995) introduced the so-called two-phase method for bi-objective problems.
+In the first phase the extreme supported non-dominated points are generated with an algorithm sim-
+ilar to the initial weighted sum approach. In the second phase the remaining non-dominated points
+are generated by searching in triangles defined by two consecutive extreme supported non-dominated
+points. In Przybylski et al. (2008) and Tuyttens et al. (2000) problem specific algorithms are sug-
+gested for the second phase, while in Przybylski et al. (2010b) a two-phase method for problems
+with more than two objectives is proposed.
+The augmented weighted Tchebycheff method, first presented in Steuer and Choo (1983), mini-
+mizes the augmented weighted Tchebycheff distance between a predefined reference point and the
+set of feasible image points. Dächert et al. (2012) suggested an adaptive choice of the augmentation
+term for the bi-objective case.
+In Boland et al. (2015a), Boland et al. (2015b) (for the bi-objective case), Dächert and Klamroth
+(2014), and Klamroth et al. (2015) (for the tri- respectively multi-objective case) search region split-
+ting methods are proposed. In this class of objective space methods, the search region (based on the
+already determined non-dominated points) is splitted into so-called search zones on which scalariza-
+tions are solved indpendently.
+Besides their advantages, objective space methods share a shortcoming: In each iteration a scalar-
+ized integer program is solved from scratch. Even though in some objective space methods starting
+solutions can be transfered from previous iterations, a large number of very similar problems has
+to be solved. In order to avoid this effort, decision space methods, mainly the Branch and Bound
+method, have been increasingly investigated in the recent years.
+Klein and Hannan (1982) developed one of the first Branch and Bound algorithms for multi-
+objective integeger programs with a typical one tree structure. In Kiziltan and Yucaoğlu (1983) a
+general Branch and Bound framework for multi-objective integer programs with binary variables
+is presented. Ulungu and Teghem (1997) and Visée et al. (1998) proposed problem specific Branch
+and Bound approaches for bi-objective Knapsack problems, where the latter approach is integrated
+in a two-phase method.
+Mavrotas and Diakoulaki (1998) extend the Branch and Bound approach to multi-objective mixed
+integer programs. Parts of the algorithm are refined in Mavrotas and Diakoulaki (2005). In Vincent et al.
+(2013) this algorithm is improved and it is shown that the original algorithm is not correct because
+the final dominance test is incomplete.
+In Belotti et al. (2012) a Branch and Bound method is
+presented that can handle bi-objective mixed integer programs with continious variables in both
+objective functions.
+2
+
+The Branch and Bound method proposed in Sourd and Spanjaard (2008) uses a set of points as
+lower bound instead of just using a single point. Furthermore hyperplanes are used to fathom nodes
+by dominance. In Stidsen et al. (2014) this idea is continued. They use hyperplanes as a lower
+bound set that are generated by solving weighted sum scalarizations. Additionally they present
+the so-called Pareto branching and the slicing technique. With Pareto branching it is possible to
+divide the objective space to possibly ignore parts of it in specific nodes.
+Slicing partitions the
+objective space in equally large parts and a respective slice can be fathomed if it is dominated by
+an already found integer point.
+In Stidsen and Andersen (2018) this algorithm is improved and
+an approach to parallelize the algorithm is presented. Based on this, the idea Pareto branching is
+further investigated in Parragh and Tricoire (2019) and Gadegaard et al. (2019) for the bi-objective
+case and Forget et al. (2022) for the tri-objective case. A self-contained survey of multi-objective
+Branch and Bound approaches is given in Przybylski and Gandibleux (2017).
+In this paper we present a bi-objective Branch and Bound algorithm that is augmented by
+scalarization-based information. We make use of optimized single-objective solvers for scalar in-
+teger programs and integrate the resulting information into the bi-objective Branch and Bound by
+improving lower and upper bounds. Furthermore, we propose a new adaptive node selection strategy,
+which relies on objective space information. In our numerical analysis we show the effectiveness of
+these improvements by comparing them with a generic multi-objective Branch and Bound algorithm,
+which we use as our baseline algorithm.
+The remainder of the article is organized as follows: In Section 2, we introduce notations and def-
+initions for multi-objective optimization. In Section 3, we present a general multi-objective Branch
+and Bound framework and its key components. Furthermore, we describe a specific (however stan-
+dard) multi-objective Branch and Bound algorithm, which will be used as baseline implementation
+in our numerical tests. In Section 4, we present augmentations of the multi-objective Branch and
+Bound, that utilize objective space information to improve the node selection as well as the compu-
+tation of upper and lower bounds. We provide numerical results in Section 5 and in Section 6, we
+outline conclusions and outlooks for further research.
+2 Preliminaries
+We introduce a general multi-objective integer linear program which can be written in the form:
+min
+�
+z1(x), . . . , zp(x)
+�⊤
+s.t.
+A x ≤ b
+x ≥ 0
+x ∈ Zn.
+(MOILP)
+Thereby, z(x) := (z1(x), . . . , zp(x))⊤ = C ·x ∈ Rp (with p ≥ 2) denotes the objective function vector,
+with C ∈ Rp×n the matrix of objective coefficients. The set of feasible solutions X := {x ∈ Zn : A ≤
+b, x ≥ 0} is a subset of the decision space Rn, while its image Y := {C x: x ∈ X} is a subset of the
+objective space Rp.
+We use the Pareto concept of optimality which relies on the componentwise order. Let y1, y2 ∈ Rp,
+then we define the corresponding dominance relations as follows:
+• y1 ≦ y2, i.e., y1 weakly dominates y2 if y1
+k ≤ y2
+k for k = 1, ..., p,
+• y1 < y2, i.e., y1 strictly dominates y2 if y1
+k < y2
+k for k = 1, ..., p,
+• y1 ≤ y2, i.e., y1 dominates y2 if y1 ≤ y2 and y1 ̸= y2.
+A feasible solution x ∈ X is called efficient if there is no other solution ˆx ∈ X dominating it, i.e.,
+z(ˆx) ≤ z(x). A feasible solution x ∈ X is called weakly efficient if there is no ˆx ∈ X such that
+z(ˆx) < z(x). The set of efficient solutions is denoted by XE. By YN = {z(x) ∈ Y : x ∈ XE} we
+3
+
+denote the set of the non-dominated points in the objective space. Moreover, for any set Q ⊆ Rp
+we denote by QN the set of its non-dominated points (i.e., q ∈ QN
+⇐⇒ ∄q′ ∈ Q: q′ ≤ q). For a
+comprehensive introduction to multi-objective optimization see, e. g., Ehrgott (2005).
+In this article we consider a minimal complete set as solution of a multi-objective optimization
+problem. A minimal complete set denotes the set of all non-dominated points YN and one efficient
+solution for each non-dominated point. See Serafini (1987) for a comparison of solution concepts in
+multi-objective optimization.
+A standard solution approach in multi-objective optimization is the weighted sum scalarization
+given in (WSλ).
+min WSλ(x) := λ⊤z(x) =
+p
+�
+i=1
+λi zi(x)
+s.t. x ∈ X
+(WSλ)
+Obviously, every optimal solution of the weighted sum scalarization for λ ∈ Rp
+> := {λ ∈ Rp : λ > 0}
+is efficient for (MOILP). However, in general not all efficient solutions are optimal solutions of a
+corresponding weighted sum problem. An efficient solution x′ ∈ XE is called supported if there is a
+weighting vector λ′ ∈ Rp
+> such that x′ is optimal for (WSλ) for λ = λ′, otherwise x′ is unsupported.
+Note that the non-dominated points corresponding to supported efficient solutions are located on
+the boundary of the convex hull of Y , while the unsupported non-dominated points are located in
+its (relative) interior.
+As already mentioned in the introduction the computation of upper and lower bounds on the
+non-dominated set is a crucial component of any multi-objective Branch and Bound algorithm. The
+tightest componentwise upper and lower bounds of YN are the ideal point yI and the Nadir point
+yN given by:
+yI
+k = min
+y∈Y yk
+and
+yN
+k = max
+y∈YN yk
+for k = 1, . . . p.
+Obviously, yI ≦ y ≦ yN holds for every y ∈ YN, i.e.; YN is contained in the hyperbox spanned by
+the corner points yI and yN. However, these single point bounds are in general very weak except for
+the degenerate case of yI = yN. This motivates to consider bound sets instead of bounds consisting
+of a single point. We will rely on the definition of bound sets proposed in Ehrgott and Gandibleux
+(2007). Let Rp
+≧ := {y ∈ Rp : y ≧ 0}, then
+• A lower bound set L ⊂ Rp for YN is a
+– Rp
+≧-closed (i.e., the set L + Rp
+≧ is closed),
+– Rp
+≧-bounded (i.e., there exists a y ∈ Rp such that L ⊂ y + Rp
+≧)
+– stable set (i.e., L ⊂ (L + Rp
+≧)N),
+such that YN ⊂ (L + Rp
+≧).
+• An upper bound set U ⊂ Rp for YN is a
+– Rp
+≧-closed,
+– Rp
+≧-bounded,
+– stable sets,
+such that YN ⊂ cl
+�
+(U + Rp
+≧)∁�
+.
+The upper bound and lower bound that we will define for our branch and bound framework in
+Section 3 will suit these definitions. We say a lower bound L is weakly dominated by an upper
+bound U if for all l ∈ L there exists an u ∈ U such that u ≦ l.
+4
+
+In the following we restrict ourselves to bi-objective binary linear optimization problems, i. e.,
+problems with two linear objective functions and variables x ∈ {0, 1}n:
+min
+z(x) =
+�
+z1(x), z2(x)
+�⊤
+s.t.
+A x ≤ b
+x ∈ {0, 1}n.
+(BO01LP)
+3 A Generic Multi-objective Branch and Bound Framework
+In this section we present a generic multi-objective Branch and Bound framework, which we specify
+and augment by using scalarization based information in the then following sections.
+Branch and Bound methods follow a “divide and conquer” paradigm. A problem that is too hard
+to be solved directly, is divided into smaller and thus easier subproblems. Thereby, subproblems are
+associated with nodes in a tree data structure according to their descent, i.e., node i is a descendant
+node of node j iff the feasible set of the subprobem associated with node i is a subset of the feasible
+set of the subproblem associated with node j. The corresponding subproblems of the child nodes are
+created by subdividing the feasible set of the corresponding (sub)problem of the parent node. Starting
+with the root node, to which the original optimization problem is associated, the algorithm selects in
+each iteration one active node and updates its lower bound and upper bound. Then the active node
+can be fathomed if the corresponding subproblem is either solved or irrelevant for the determination
+of a minimal complete set. If we cannot prune we subdivide the corresponding problem into new
+subproblems and create corresponding child nodes (branching). For a more detailed introduction
+and survey of multi-objective Branch and Bound algorithms see Przybylski and Gandibleux (2017).
+A recent survey of single-objective Branch and Bound frameworks is given e.g. in Morrison et al.
+(2016). In the following we specify the lower bound, upper bound, branching rule and node selection
+we use in our framework.
+Lower bound:
+Lower bound sets are often determined by solving relaxations of the respective
+subproblem. Like in the single-objective case, the most frequently used relaxations are linear and
+convex relaxations. In order to solve the linear relaxation we are using in our framework, we apply
+Benson’s outer approximation algorithm (Benson, 1998). The algorithm is initiated with a lower
+bound, which is improved in every iteration by generating cuts. Due to the outer approximation
+structure the algorithm can be aborted at any time returning a valid lower bound.
+Alternatively, linear (or convex) relaxations can be obtained using a dichotomic scheme (see, for
+example, Aneja and Nair, 1979; Özpeynirci and Köksalan, 2010; Przybylski et al., 2010a).
+Upper bound:
+The upper bound set, in the following denoted by U, is stored in the form of a so-
+called incumbent list. Throughout the run of the algorithm, it contains all integer feasible solutions
+and their corresponding outcome vectors that are not dominated by another feasible solution found
+so far. In every iteration the extreme supported solutions of the computed lower bound sets are
+checked for integer feasibility. An integer feasible solution ¯x ∈ X is then appended to the incumbent
+list, if there is no x ∈ U dominating ¯x, i. e., C(x) ≤ C(¯x). If a new solution ¯x is added to the
+incumbent list U all solutions x ∈ U which are dominated by ¯x (C(¯x) ≤ C(x)) are removed from
+it. Note that an update of the incubent list requires a subsequent update of the list of local upper
+bounds. A detailed description of local upper bounds, their computation and update in an arbitrary
+number of criteria is given in Klamroth et al. (2015). In this framework we start with an empty
+upper bound set. However, it is also possible to initialize the incumbent list by heuristic methods, or
+by solving scalarizations like, e.g., in the two-phase method (Ulungu and Teghem, 1995; Visée et al.,
+1998).
+U ⊎ {¯x} :=
+�
+U
+if ∃x ∈ U : C(x) ≤ C(¯x)
+{¯x} ∪ {x ∈ U : C(¯x) ≰ C(x)}
+otherwise
+5
+
+Node selection:
+In every iteration of the algorithm an unexplored node is selected from the tree of
+subproblems. This node is called active node. The order in which the nodes of the tree are considered
+has a significant impact on the number of created nodes that have to be explored and thus on the
+computation time.
+Two types of strategies need to be distinguished: static strategies and dynamic strategies. The two
+most common examples of static strategies are the depth-first strategy and the breadth-first strategy.
+Most multi-objective Branch and Bound algorithms in literature follow a depth-first strategy. Thus,
+we use this strategy for our baseline implementation as well.
+In contrast to the single-objective case, dynamic node selection strategies are rarely applied in the
+multi-objective case. The usage of dynamic strategies for the choice of the active node can be seen
+in (Stidsen et al., 2014), (Belotti et al., 2012) and (Jesus et al., 2021), for example.
+Fathoming:
+In order to avoid the total enumeration of all feasible solutions, nodes are fathomed
+if the respective subproblem is either solved to optimality or does not contain solutions which are
+necessary to determine a minimal complete set. In particular, there are three different situations in
+which a node can be fathomed:
+i) Fathoming by infeasibility: If the LP-relaxation of a subproblem is infeasible then the corre-
+sponding subproblem is infeasible as well, since the feasible set of the subproblem is a subset
+of the feasible set of its relaxation.
+ii) Fathoming by optimality: Similar to the single-objective case we can fathom a node by opti-
+mality if the lower bound L is equal to the upper bound U. This implies the subproblem is
+solved to optimality and the associated node must not be subdiveded further. However, this
+can happen in the multi-objective case only if the lower and upper bound consist of the same
+single point, namely the ideal point.
+iii) Fathoming by dominance: A node can be fathomed by dominance if all feasible solutions of this
+subproblem are dominated by points in the incumbent list. In order to check dominance for all
+feasible outcome vectors of a subproblem we compare the lower bound L of the corresponding
+node to the current upper bound U. If for all l ∈ L there is a point in the incumbent list u ∈ U
+with u ≦ l then all feasible points in the subtree are dominated by the current incumbent list.
+In other words, if there is no local upper bound defined by U above the computed lower bound
+the node can be fathomed by dominance.
+Branching:
+As mentioned in the beginning of this section, one of the key aspects of Branch and
+Bound is iterative subdivision into smaller subproblems. Thereby subproblems are associated with
+nodes in a tree, such that the subproblem associated to a child node is obtained by one branching
+step. Since we consider binary optimization problems (BO01LP), we can divide a (sub)problem into
+two new subproblems by fixing a specific variable to 0 and respectively to 1 in the other subproblem.
+This results in a binary Branch and Bound tree.
+The branching rule determines which variable is selected as branching variable in each iteration.
+Thereby, one distinguishes between static and dynamic strategies. Static strategies determine an
+order of the variables in advance. In each iteration of the algorithm the next variable in this list is
+used as branching variable. With dynamic strategies the branching variable is selected by consid-
+ering information obtained from previous iterations, i.e., from the solution of (linear) relaxations of
+(sub)problems.
+The basic idea of static strategies for single-objective problems is to sort the variables, beginning
+with the most promising according to the objective function values (see, e. g., (Kellerer et al., 2004)).
+However, this cannot be easily extended to the multi-objective case due to conflicting objective
+functions. Nevertheless there are some approaches to extend static strategies to the multi-objective
+case (see for example (Ulungu and Teghem, 1997) and (Bazgan et al., 2009)).
+6
+
+In contrast to most of the published papers which apply static strategies we use a dynamic strategy
+as proposed in Belotti et al. (2012). By solving the linear relaxation of a (sub)problem we obtain
+the lower bound set L. For all extreme points of L we check how often a variable is fractional in the
+corresponding solutions. As branching variable we choose the one which is most often fractional.
+4 Using Objective Space Information in Multi-objective Branch
+and Bound
+In this section, we propose modifications which improve the computational efficiency of bi-objective
+Branch and Bound algorithms in two critical aspects.
+One of the weaknesses of multi-objective
+Branch and Bound as compared to its single-objective counterpart is the bounding procedure. While
+any feasible solution ¯x ∈ X dominates w.r.t. one (linear) objective a half-space in decision space (i.e.,
+{x ∈ Rn : c⊤x ≥ c⊤¯x}), the set of feasible solutions which are dominated by a solution ¯x in p ≥ 2
+objective functions (C ∈ Rp×n) forms a cone {x ∈ Rn : C x ≧ C ¯x}. The cone of dominated solutions
+is smaller the more the objective functions are in conflict, leading also to a larger number of efficient
+solutions. This implies that a significant part of the Branch and Bound tree has to be enumerated
+and only a small number of branches can be pruned by dominance. Despite of this general problem
+in multi-objective optimization, this asks for good bounding procedures to avoid the unnecessary
+evaluation of dominated branches. This however, requires good solutions in the incumbent list as
+well as tight lower bounds.
+In order to achieve this, we suggest a new branching strategy and the hybridization of Branch and
+Bound with objective space methods. We determine scalarized subproblems adapted to the state of
+the Branch and Bound and solve these to integer optimality.
+4.1 Branching Strategy
+The branching strategy comprises two subsequent decisions: the choice of the active node and its
+branching into subproblems, i. e. the decision on which variable the subproblem is branched. This
+second step is denoted as branching rule. We discuss these two steps together since the order in
+which the nodes are considered has a significant impact on the branched variable. Instead of the
+static depth- or breadth-first we use a dynamic node selection strategy, while we rely on the most
+fractional rule as branching rule.
+The basic idea of our strategy is quite simple and a natural extension of choosing the largest gap
+in the single-objective case (see, for example, Dechter and Pearl, 1985). For every created node we
+compute the approximate hypervolume gap between lower and upper bound. We use the definition of
+hypervolume proposed in Zitzler and Thiele (1999). In every iteration we choose the node with the
+largest hypervolume gap as active node (cf. Jesus et al. (2021)). Note that when a node is created
+during the branching process, the approximated hypervolume gap of the parent node is assigned to
+it. We distinguish two variants of the hypervolume gap: the total hypervolume gap and the local
+hypervolume gap. While the total hypervolume gap measures the volume of the search region, i. e.
+the volume between lower and upper bound set, the local hypervolume gap approach considers only
+the volume of the largest search zone, i. e. the gap between a local upper bound and the lower bound
+set. For a more detailed definition of search regions and search zones we refer to Klamroth et al.
+(2015).
+Figure 1 illustrates the two different approaches. Here, z1, . . . , z4 ∈ K ⊂ U are points of the
+incumbent list and lu1, . . . , lu3 are their corresponding local upper bounds, where K is a subset
+of the incumbent list containing just the points above the lower bound of node ¯n. The green line
+represents the lower bound. Figure 1a shows how to measure the total hypervolume gap of a node ¯n,
+in the following denoted by thg(¯n). For this approach we consider the approximated search region
+of the corresponding node. Since there is a natural order in the bi-objective case, it is possible to
+consider the approximated search zone of the first local upper bound, i.e. the local upper bound
+7
+
+z1
+z2
+lu1
+lu2
+lu3
+z1
+z2
+z3
+z4
+(a)
+z1
+z2
+A
+lu1
+lu2
+lu3
+z1
+z2
+z3
+z4
+(b)
+z1
+z2
+B
+lu1
+lu2
+lu3
+z1
+z2
+z3
+z4
+(c)
+z1
+z2
+C
+lu1
+lu2
+lu3
+z1
+z2
+z3
+z4
+(d)
+Figure 1: Example of computation of the two different approximated hypervolume gap approaches.
+8
+
+with the smallest z1-value. Therefore we define the two spanning points, which, together with the
+corresponding local upper bound, define a triangle. The spanning points of a local upper bound lu
+are defined by spi(lu) := {l ∈ L: l3−i = lu3−i}, i = 1, 2. So, the approximate hypervolume gap of lu
+is given by
+hg(lu) := 1
+2
+��sp1(lu)1 − lu1
+�� ·
+��sp2(lu)2 − lu2
+��.
+For the remaining local upper bounds we compute the hypervolume of slices as shown in the illus-
+tration. The hypervolume of the slice of lui, i = 1, . . . , |K| − 1 is defined as
+sl(lui) :=
+��zi
+2 − sp2(lui−1)2
+�� +
+��lui
+2 − sp2(lui)2
+��
+2
+·
+��zi
+1 − lui
+1
+��.
+So, the total (approximated) hypervolume gap, which is assigned to node ¯n, is given by
+thg(¯n) := hg(lu1) + sl(lu2) + . . . + sl(lu|K|−1).
+The Figures 1b, 1c and 1d show the computation of the local hypervolume gap. The local hyper-
+volume gap of a node ¯n is considered as the largest approximated hypervolume gap of a local upper
+bound corresponding to points in K. Therefore, the the local hypervolume gap, which is assigned
+to node ¯n, is defined by
+lhg(¯n) :=
+max
+i=1,...,|K|−1hg(lui).
+In the given example, B is the largest approximated hypervolume and therefore is assigned to node
+¯n.
+Note that in our presented algorithms in Section 4.2 the local upper bound is initialized with the
+point (∞, ∞)⊤. Therefore, it is possible to apply the new branching strategies immediately at the
+beginning of the algorithm. Obviously, this approximation may neglect significantly large parts of
+the search regions and search zones. However, the idea of the approximated hypervolume gap eases
+computation and saves time. The efficiency of these new dynamic branching strategies is shown in
+Section 5.
+4.2 Augmenting Branch and Bound with IP Scalarizations
+In this subsection, we introduce a method to incorporate scalarizations into Branch and Bound. We
+build a hybrid Branch and Bound algorithm combining the partial enumeration of decision space
+with objective space information by solving scalarizations to integer optimality.
+An integer optimal solution ¯x of a scalarization can be used to update upper and lower bound.
+Obviously, the corresponding image point z(¯x) can be added to the incumbent list. Moreover, a
+scalarizing function and its optimal solution ¯x define a level set, which can be included in the lower
+bound set for all descendant nodes. In order to utilize these improved lower bounds in all nodes we
+solve the IP scalarizations in the root node.
+4.2.1 Using Weighted Sum Scalarization
+During the run of the Branch and Bound algorithm, a strategy triggers the IP solution of weighted
+sum scalarizations in the root node. Thus, we solve problem (WSλ) for for adaptively chosen values
+of λ ∈ R2
+>. Although we solve the IP scalarization in the root node the parameter λ is gained from
+the currently active node. Thereby, λ is determined by the largest approximated local hypervolume
+gap in the active node. This gap is spanned by two points in the incumbent list together with their
+local upper bound. Note that these points spanning the largest gap are already determined if the
+local hypervolume gap branching strategy is applied. The corresponding value of λ is determined by
+computing the normal to the hyperplane that is defined by those two points. Once λ is obtained,
+we can solve problem (WSλ) with a single-objective integer linear programming solver. Let ¯xλ be
+9
+
+the optimal solution of the weighted sum scalarization with weighting vector λ, then z(¯xλ) is a
+supported non-dominated point of (BO01LP). Thus, we can add this point to the incumbent list (if
+it was not found in previous iterations) and filter the resulting list for non-dominance. Moreover,
+the solution of integer scalarizations can also be used to tighten the lower bound set, since the level
+set {z ∈ R2 : λ⊤z = WSλ(¯xλ)} provides the valid inequality λ⊤z(x) ≥ WSλ(¯xλ) for all x ∈ X.
+z1
+z2
+lu1
+lu2
+lu3
+z1
+z2
+z3
+z4
+z(¯xλ)
+(a)
+z1
+z2
+lu4
+lu5
+z1
+z5
+z4
+(b)
+z1
+z2
+lu4
+lu5
+z1
+z5
+z4
+(c)
+z1
+z2
+lu4
+lu5
+z1
+z5
+z4
+(d)
+Figure 2: Example of updating the lower and upper bound with the usage of the weighted sum
+scalarization.
+Figure 2 illustrates the update of the lower and upper bound set. In Figure 2a, z1, . . . , z4 indicate
+points that are currently in the incumbent list U¸ and lu1, . . . , lu3 are the corresponding local upper
+bounds.
+The point z(¯xλ) is obtained by solving a weighted sum scalarization (WSλ) to integer
+optimality. Since the new point is not contained in the incumbent list so far, we can update the
+upper bound as it is shown in Figure 2b. The new incumbent list then reads as U := {z(¯xλ)} ∪ {z ∈
+U : z(¯xλ) � z}. Moreover, the lower bound set L can be updated by integrating the blue hyperplane
+into the lower bound set, i. e. L := {z ∈ L + R2
+≧ : λ⊤z ≥ WS(¯xλ)}N as it is shown in Figure 2c and
+2d. In this situation, both —the lower and upper bound— are updated, which is not the case in
+10
+
+general.
+The example illustrates the benefits of hybridizing multi-objective Branch and Bound with IP
+scalarizations. Due to weak bounding, nodes may not be fathomed by dominance even if they do
+not contain additional non-dominated points. The tighter upper bound increases the probability of
+fathoming a node by dominance in later iterations of the algorithm. Also, the lower bound might be
+improved. Since we are solving an IP scalarization in the root node, the obtained optimal level set is
+a valid inequality for all subproblems. We combine our new branching strategy and the augmentation
+with IP scalarizations to our first hybrid Branch and Bound approach.
+Hybrid Branch and Bound Algorithm using Weighted Sum Scalarization
+• Lower bound: linear relaxation
+• Upper bound: incumbent list
+• Node selection: node with the largest total/local hypervolume gap
+• Branching rule: most fractional
+• Adaptively solve weighted sum scalarizations in the root node to integer optimality to improve
+lower and upper bounds by objective space information
+Instead of using a static depth-first strategy (as in the general Branch and Bound framework in
+Section 3) we apply the dynamic strategy based on the hypervolume gap (c.f. Section 4.1). Even
+though the extreme points of the lower bound sets might be updated by the weighted sum scalar-
+ization, the branching variable is selected based on the original lower bounds. This is due to the
+fact that the preimages of such intersection points of IP scalarizations and the lower bound set
+are in general not available. Note that the weighted sum IP scalarizations are included adaptively
+into the Branch and Bound. The description of their algorithmic control, however, is postponed to
+Section 4.3.
+In order to conclude the description of the proposed hybrid Branch and Bound algorithm using
+weighted sum scalarizations, we want to briefly discuss its advantages and shortcomings. Firstly, it
+is easy to determine the scalarization parameter λ and to integrate the hyperplane into the lower
+bound set. Its advantage, however, is that the lower bound remains convex. Therefore, the check
+for fathoming by dominance remains intuitive. Unfortunately, the weighted sum scalarization can
+only find supported efficient solutions and the lower bound cannot be improved beyond the convex
+hull of YN. This motivates us to consider in the following the augmented weighted Tchebycheff
+scalarization, a scalarization approach which can determine also unsupported efficient solutions.
+4.2.2 Using Augmented Weighted Tchebycheff Scalarization
+We start by defining the weighted Tchebycheff norm: Let wi > 0, i = 1, . . . , p be positive weights
+with �p
+i=1 wi = 1. Then the weighted Tchebycheff norm of a vector z ∈ Rp is defined by
+∥z∥w
+∞ := max
+i=1,...,p
+�
+wi |zi|
+�
+.
+(1)
+So, the weighted Tchebycheff scalarization of a multi-objective optimization problem (MOILP) with
+respect to a given reference point s ∈ Rp can be written as:
+min
+�
+∥z(x) − s∥w
+∞ : x ∈ X
+�
+.
+(2)
+If the reference point is chosen such that s < z(x) for all x ∈ X, every efficient solution can be deter-
+mined as optimal solution of the weighted Tchebycheff scalarization (2) by variation of w ∈ Rp
++ (see,
+e.g., Miettinen, 1998). Nevertheless, optimal solutions of the weighted Tchebycheff scalarization cor-
+respond in general only to weakly efficient solutions of the multi-objective problem (Steuer and Choo,
+1983; Miettinen, 1998). This shortcoming is compensated by an additive augmentation term in the
+augmented weighted Tchebycheff norm
+∥z∥w
+τ := ∥z∥w
+∞ + τ ∥z∥1 ,
+(3)
+11
+
+where ∥z∥1 = |z1| + . . . + |zp| denotes the L1-norm, wi ≥ 0, i = 1, . . . , p, �p
+i=1 wi = 1 and τ >
+0.
+Steuer and Choo (1983) proposed the augmented weighted Tchebycheff scalarization given in
+(AWT w
+τ ).
+min AWT w
+τ (x) := ∥z(x) − s∥w
+τ
+s.t. x ∈ X
+(AWT w
+τ )
+Thereby, the augmentation term makes the augmented weighted Tchebycheff norm a strongly mono-
+tone norm and thus the objective function of (AWT w
+τ ) a strongly increasing achievement scalar-
+izing function (Miettinen, 1998). Consequently, every optimal solution of (AWT w
+τ ) is efficient for
+(MOILP).
+Note that an appropriate choice of the parameter τ is difficult in general. On the one hand, too
+small values of τ may lead to numerical difficulties. On the other hand, non-supported efficient
+solutions might be suboptimal for (AWT w
+τ ) if the value of τ is too large. However, for bi-objective
+integer programming Dächert et al. (2012) propose an adaptive method to determine an optimal
+value of τ. We use this proposed parameters w1, w2 and τ for our method.
+As a reference point s we use the local ideal point of two adjacent non-dominated points. Since the
+augmented weighted Tchebycheff scalarization can only determine non-dominated points (and the
+corresponding efficient solutions) which are (strictly) dominated by the reference point, we obtain a
+non-dominated point in this box.
+The goal to improve the lower bound set beyond the convex hull of non-dominated points is the
+motivation to solve augmented weighted Tchebycheff scalarizations to integer optimality. Figure
+3 shows an example how such an update of the bounds could look like. Here, z1 and z2 are two
+known non-dominated points (obtained with the weighted sum IP scalarization).
+Point z3 is a
+non-supported non-dominated point that has not been found yet in Figure 3a. By using the local
+ideal point of z1 and z2 as the reference point s, Figure 3b illustrates how the non-dominated point
+z3 is found by applying the augmented weighted Tchebycheff scalarization. In Figure 3c and 3d
+the resulting improvements of the lower and upper bound are shown. Obviously the lower bound
+is improved beyond the convex hull of YN. We now define our second hybrid Branch and Bound
+approach:
+Hybrid Branch and Bound Algorithm using augmented weighted Tchebycheff Scalarization
+• Lower bound: linear relaxation
+• Upper bound: incumbent list
+• Node selection: node with the biggest total/local hypervolume gap
+• Branching rule: most fractional
+• Adaptively solve weighted sum and augmented weighted Tchebycheff scalarizations in the root
+node to integer optimality to improve lower and upper bounds by objective space information
+Additionally to the weighted sum scalarization, we use the augmented weighted Tchebycheff scalar-
+ization. Since two adjacent non-dominated points are required as input of the augmented weighted
+Tchebycheff scalarization, we cannot rely on points in the incumbent list, which are only non-
+dominated so far. In fact, we apply augmented weighted Tchebycheff IP scalarizations only to boxes
+spanned by points obtained as optimal solutions of the weighted sum scalarization. Thus, we do not
+rely on parameters from the currently active node, but solve the augmented weighted Tchebycheff
+scalarization in the largest area defined by two adjacent known non-dominated points.
+When using augmented weighted Tchebycheff IP scalarizations, the lower bound can become
+tighter than the convex hull of the set of non-dominated points, which reduces the area where
+new non-dominated points can be found. Additionally, we can find non-supported non-dominated
+points in early stages of the algorithm. This improves the upper bound in the beginning resulting
+in a higher chance of fathoming a node by dominance. However, this also implies that the lower
+12
+
+z1
+z2
+z1
+z2
+z3
+lu1
+(a)
+z1
+z2
+z1
+z2
+z3
+lu1
+s
+(b)
+z1
+z2
+z1
+z2
+z3
+lu2
+lu3
+s
+(c)
+z1
+z2
+z1
+z2
+z3
+lu2
+lu3
+(d)
+Figure 3: Example of updating the lower and upper bound with the usage of the augmented weighted
+Tchebycheff scalarization.
+bound gets non-convex in general, which makes the fathoming tests significantly harder, and the
+lower bound improves only locally.
+4.3 Algorithmic Control of IP Scalarizations
+In the previous subsections we did not specify when to solve IP scalarizations, which implies a sig-
+nificant computational cost itself. However, this might be the most crucial part within the presented
+methods. Obviously, we aim at gaining as much information as possible by solving IP scalarizations.
+More objective space information will lead to tighter bounds that reduce the number of created
+nodes, due to a higher probability of fathoming by dominance and smaller search zones. Moreover, a
+reduced number of created nodes will reduce the total computation time. At the same time, solving
+overly many IP scalarizations will have a negative impact on the computation time. Furthermore,
+at a certain point the lower and upper bound will not improve anymore when solving additional IP
+13
+
+scalarizations.
+So, there exists a trade-off between the reduction of the number of created subproblems and the
+decrease of the computation time. The difficulty is to find an appropriate condition to trigger an IP
+scalarization. Obviously, solving IP scalarizations more frequently in the beginning of the Branch
+and Bound algorithm is very promising.
+The earlier the lower and upper bounds are improved
+the more nodes might be fathomed. Moreover, solving the IP scalarization when the active node
+has weak bounds will lead to stronger improvements than in later stages of the algorithm. This
+is complemented by our adaptive branching strategy, which tends to select subproblems with weak
+lower bounds first.
+The hybrid Branch and Bound algorithm using augmented weighted Tchebycheff scalarization
+entails also another problem. The augmented weighted Tchebycheff scalarization improves the lower
+bound just locally. If we use this scalarization at the beginning of the algorithm instead of the
+weighted sum scalarization, this could lead to an increase of created nodes. Once again, the intuitive
+idea is to start with the weighted sum IP scalarization more frequently in the beginning of the
+algorithm. This ensures that the lower bound improves globally at early stages of the Branch and
+Bound. The augmented weighted Tchebycheff scalarization should be used in later stages of the
+algorithm to find non-supported non-dominated points and to improve the lower bound locally. The
+efficiency of this idea and other approaches will be shown in the next section where we present
+numerical test results.
+5 Numerical Results
+All algorithms were implemented in Julia 1.7.1 and the linear relaxations were solved with Bensolve
+2.1. The numerical tests were executed on a single core of a 3.20 GHz Intel® Core™ i7-8700 CPU
+processor in a computer with 32 GB RAM, running under openSUSE linux Leap 15.3.
+We present numerical results of our new approaches and compare them to the general Branch
+and Bound framework presented in Section 3 which we use as baseline implementation. We consider
+three different types of problems: knapsack problems, assignment problems and discrete facility
+location problems. Multiple combinations of parameter settings are used to solve these test problems.
+Thereby, we compare the average number of explored nodes, the average number of solved IPs and
+the average computation time for 20 instances per problem size. The different evaluated approaches
+are
+• the generic bi-objective Branch and Bouch (BB),
+• bi-objective Branch and Bound using the local (BS1) respectively global (BS2) hypervolume
+gap as node selection criterion,
+• hybrid Branch and Bound including weighted sum IP scalarizations (WS), and
+• different combinations of the hybrid Branch and Bound algorithm using weighted sum IP scalar-
+ization (M1.α.β) and hybrid Branch and Bound algorithm using weighted sum and augmented
+weighted Tchebycheff IP scalarization (M2.α.β.γ).
+The parameter α ∈ {1, 2, 3} controls how often IP scalarizations are applied. Since the algorithmic
+control of IP scalarization is selected for each value of α problem dependent, it is described in
+detail in the following subsections. However, the larger the parameter α is chosen, the fewer IP
+scalarizations are applied. With β we distinguish between the local (β = 1) and the global (β = 2)
+hypervolume gap strategy. In the hybrid Branch and Bound algorithm using augmented weighted
+Tchebycheff scalarization we also distinguish between integrating the objective space information
+of the augmented weighted Tchebycheff into the lower bound (γ = 1) or not (γ = 2). Note that
+the tested parameter values are not optimized but have shown to provide good results on our test
+instances.
+14
+
+5.1 Bi-objective Multidimensional Knapsack Problems
+We consider bi-objective, multidimensional knapsack problems with one, two and three linear re-
+strictions (i. e. m = 1, 2, 3). For every problem size we randomly generate 20 instances of the form
+max
+n
+�
+i=1
+ck
+i xi
+k = 1, 2
+s.t.
+n
+�
+i=1
+wi xi ≤ b
+n
+�
+i=1
+vij xi ≤ dj
+j = 1, ..., m − 1
+x ∈ {0, 1}n
+with ck
+i ∈ [50, 100], wi ∈ [5, 15], b = 5 n, vij ∈ [5, 15] and dj =
+� r n
+2
+�
+with r ∈ [5, 15]. Depending on
+the parameter α we specify when and how often IP scalarizations are solved. In M1.1.β and WS
+we apply the weighted sum scalarization every 10-th iterations. In M1.2.β we apply it every 10-th
+iteration but only within the first n2 iterations. In M1.3.β we apply the weighted sum scalarization
+every 10-th iteration within the first n2/3 iterations, every n-th iteration within the next n2/3
+iterations and every 2n-th iteration within the third n2/3 iterations. In M2.1.β.γ we apply the
+weighted sum scalarization every 10-th iteration and every 50-th iteration the augmented weighted
+Tchebycheff scalarization is used instead. In M2.2.β.γ we operate like in M1.2.β but after the first
+n2 iterations we apply the augmented weighted Tchebycheff scalarization every 50-th iteration. In
+M2.3.β.γ we operate like in M1.3.β but after the first n2 iterations we apply the augmented weighted
+Tchebycheff scalarization every 50-th iteration. If a scalarization cannot be applied or the same IP
+scalarization has already been solved before, no IP scalarization is solved in that iteration.
+First of all, we notice that our branching strategies have a huge impact on the number of explored
+nodes and the computation time in knapsack problems. We observe that in general the local hyper-
+volume gap strategy works better than the global hypervolume gap strategy. With the local strategy
+we can reduce the number of explored nodes by up to 76% (Table 1c and 2b) and the computation
+time by up to 73% (Table 1c). Although the local strategy works better the global hypervolume
+gap strategy has also a significant impact. The number of explored nodes can be reduced by up to
+58% (Table 2c) and the computation time by up to 52% (Table 2c). The number of nodes and the
+computation time is reduced in all our approaches and we can notice that combinations with the
+local hypervolume strategy work better.
+By limiting the number of solved weighted sum IPs (i. e. in M1.2.β, M1.3.β, M2.2.β.γ and
+M2.3.β.γ) we notice two consequences. The number of nodes increases while the number of solved
+IPs decreases. Although the number of nodes (and thus the number of considered subproblems) is
+increasing, the total computation time decreases. This implies that the reduced computation time
+to solve IP scalarizations compensates the increase of nodes, which results in a trade-off between the
+number of explored nodes and the computation time. Another interesting aspect can be observed in
+M2.α.β.1 and M2.α.β.2. The computation time can be reduced if we do not integrate the augmented
+weighted Tchebycheff objective level set into the lower bound. This can be explained by the fact that
+the lower bound improvements of augmented weighted Tchebycheff are only local and do not com-
+pensate the computation time needed to integrate the information. The intuitive assumption that
+the number of explored nodes will then rise significantly is false. So, both our branching strategies
+work better, if we do not consider the local updates of the lower bound.
+We can reach a reduction of the explored nodes by up to 83% (Table 2b) and a reduction of the
+computation time by up to 80% (Table 2b) in the best case. The strategies M2.1.1.1 and M2.1.1.2
+seem to work best for knapsack problems. In most cases these two strategies have the largest impact
+on the number of explored nodes. Nevertheless, M2.1.1.2 achieves for all instance sizes the best
+computation times, since computation time is saved by not integrating the augmented weighted
+15
+
+knapsack problem, m = 1, n = 50
+version
+nodes
+time (s)
+solved
+IPs
+BB
+27916.3
+18.153
+0.0
+BS1
+11788.1
+8.339
+0.0
+WS
+14270.7
+10.507
+33.75
+M1.1.1
+10789.7
+8.452
+26.4
+M1.2.1
+10793.5
+8.188
+21.2
+M1.3.1
+10795.7
+8.116
+17.95
+M2.1.1.1
+9888.5
+10.873
+48.7
+M2.2.1.1
+10140.3
+9.437
+32.65
+M2.3.1.1
+10521.0
+8.774
+25.65
+M2.1.1.2
+9840.1
+8.396
+45.55
+M2.2.1.2
+10130.6
+8.422
+32.35
+M2.3.1.2
+10401.8
+8.288
+26.25
+BS2
+16739.8
+11.397
+0.0
+M1.1.2
+11026.3
+8.861
+26.1
+M1.2.2
+11024.5
+8.860
+19.85
+M1.3.2
+11047.4
+8.645
+16.05
+M2.1.2.1
+10071.8
+10.907
+45.85
+M2.2.2.1
+10421.4
+9.587
+31.8
+M2.3.2.1
+10583.2
+9.448
+24.45
+M2.1.2.2
+9994.1
+8.940
+46.65
+M2.2.2.2
+10413.4
+8.820
+32.55
+M2.3.2.2
+10568.9
+8.727
+25.15
+(a) Knapsack problem with m = 1 constraint
+and n = 50 variables
+knapsack problem, m = 1, n = 80
+version
+nodes
+time (s)
+solved
+IPs
+BB
+153938.9
+186.330
+0.0
+BS1
+36392.0
+50.952
+0.0
+WS
+58825.7
+79.545
+54.0
+M1.1.1
+34337.7
+50.431
+41.65
+M1.2.1
+34333.9
+50.312
+33.1
+M1.3.1
+34307.1
+50.505
+26.35
+M2.1.1.1
+31643.7
+81.625
+100.2
+M2.2.1.1
+32708.9
+68.939
+76.2
+M2.3.1.1
+32986.3
+69.848
+63.6
+M2.1.1.2
+31274.5
+46.721
+102.85
+M2.2.1.2
+32795.7
+48.576
+76.3
+M2.3.1.2
+33025.8
+48.358
+63.4
+BS2
+90976.0
+116.847
+0.0
+M1.1.2
+39745.1
+59.321
+45.25
+M1.2.2
+40083.2
+59.350
+31.2
+M1.3.2
+39918.1
+58.999
+24.5
+M2.1.2.1
+31905.8
+80.505
+99.7
+M2.2.2.1
+34496.9
+79.444
+84.0
+M2.3.2.1
+34571.7
+72.955
+65.15
+M2.1.2.2
+32074.9
+48.510
+104.85
+M2.2.2.2
+34169.8
+51.464
+87.15
+M2.3.2.2
+34943.3
+51.887
+63.1
+(b) Knapsack problem with m = 1 constraint
+and n = 80 variables
+knapsack problem, m = 1, n = 100
+version
+nodes
+time (s)
+solved
+IPs
+BB
+297345.3
+484.676
+0.0
+BS1
+68920.5
+128.967
+0.0
+WS
+128080.8
+224.587
+66.95
+M1.1.1
+67369.1
+128.665
+54.1
+M1.2.1
+67370.1
+128.924
+39.95
+M1.3.1
+67353.3
+128.993
+32.9
+M2.1.1.1
+58214.2
+198.683
+156.85
+M2.2.1.1
+61533.1
+179.516
+123.0
+M2.3.1.1
+62127.3
+177.621
+104.55
+M2.1.1.2
+58151.3
+112.575
+158.1
+M2.2.1.2
+61490.6
+118.600
+120.1
+M2.3.1.2
+61762.6
+118.306
+108.65
+BS2
+187306.9
+318.524
+0.0
+M1.1.2
+73766.2
+144.684
+54.75
+M1.2.2
+74065.4
+144.677
+37.9
+M1.3.2
+73865.0
+144.306
+31.0
+M2.1.2.1
+59512.7
+200.754
+158.05
+M2.2.2.1
+64489.2
+192.211
+127.0
+M2.3.2.1
+64330.5
+187.803
+114.65
+M2.1.2.2
+60470.8
+118.479
+157.75
+M2.2.2.2
+64943.8
+127.428
+123.5
+M2.3.2.2
+64711.1
+126.525
+113.5
+(c) Knapsack problem with m = 1 constraint
+and n = 100 variables
+knapsack problem, m = 2, n = 50
+version
+nodes
+time (s)
+solved
+IPs
+BB
+32655.6
+25.8684
+0.0
+BS1
+10982.3
+9.6578
+0.0
+WS
+14180.9
+13.1749
+33.25
+M1.1.1
+9784.7
+9.5159
+26.6
+M1.2.1
+9782.5
+9.2684
+19.65
+M1.3.1
+9791.1
+9.1580
+14.85
+M2.1.1.1
+8900.7
+12.6639
+47.75
+M2.2.1.1
+9407.5
+11.5112
+34.5
+M2.3.1.1
+9507.5
+11.0256
+24.8
+M2.1.1.2
+8892.9
+9.2702
+47.0
+M2.2.1.2
+9370.2
+9.2053
+33.05
+M2.3.1.2
+9484.3
+9.1161
+24.75
+BS2
+15639.2
+13.1246
+0.0
+M1.1.2
+10665.3
+10.8423
+28.5
+M1.2.2
+10671.3
+10.5141
+19.4
+M1.3.2
+10854.2
+10.5765
+15.7
+M2.1.2.1
+9045.3
+12.8916
+48.9
+M2.2.2.1
+9629.7
+12.1913
+34.9
+M2.3.2.1
+9814.7
+11.6068
+28.1
+M2.1.2.2
+9030.0
+9.5854
+48.2
+M2.2.2.2
+9608.5
+9.7621
+36.05
+M2.3.2.2
+9787.6
+9.6722
+28.35
+(d) Knapsack problem with m = 2 constraints
+and n = 50 variables
+Table 1: Numerical results of the bi-objective, multidimensional knapsack problems
+16
+
+knapsack problem, m = 2, n = 80
+version
+nodes
+time (s)
+solved
+IPs
+BB
+159911.4
+287.925
+0.0
+BS1
+41092.0
+88.121
+0.0
+WS
+63215.0
+130.338
+55.0
+M1.1.1
+37799.1
+82.654
+43.9
+M1.2.1
+37835.8
+82.544
+30.55
+M1.3.1
+37811.3
+82.369
+24.75
+M2.1.1.1
+31615.0
+115.164
+102.55
+M2.2.1.1
+34772.5
+102.965
+72.5
+M2.3.1.1
+35127.1
+100.706
+60.6
+M2.1.1.2
+31590.2
+69.290
+104.7
+M2.2.1.2
+34977.7
+77.063
+72.45
+M2.3.1.2
+35170.3
+77.279
+61.3
+BS2
+115223.3
+224.926
+0.0
+M1.1.2
+43581.8
+97.039
+47.8
+M1.2.2
+43744.8
+97.874
+29.1
+M1.3.2
+45481.1
+102.689
+23.45
+M2.1.2.1
+32388.8
+116.173
+106.25
+M2.2.2.1
+36453.2
+120.161
+78.3
+M2.3.2.1
+35942.7
+116.972
+69.05
+M2.1.2.2
+33264.8
+74.207
+104.75
+M2.2.2.2
+36578.1
+81.915
+77.2
+M2.3.2.2
+35505.1
+77.971
+69.55
+(a) Knapsack problem with m = 2 constraints
+and n = 80 variables
+knapsack problem, m = 2, n = 100
+version
+nodes
+time (s)
+solved
+IPs
+BB
+428526.3
+1074.21
+0.0
+BS1
+100962.6
+326.98
+0.0
+WS
+166108.1
+464.71
+67.25
+M1.1.1
+98831.5
+323.54
+54.8
+M1.2.1
+99313.5
+325.32
+38.95
+M1.3.1
+98770.6
+322.65
+32.6
+M2.1.1.1
+69951.9
+402.48
+149.35
+M2.2.1.1
+73433.8
+379.33
+119.6
+M2.3.1.1
+73424.7
+371.63
+102.95
+M2.1.1.2
+70172.3
+212.40
+153.15
+M2.2.1.2
+72824.8
+219.73
+121.55
+M2.3.1.2
+73651.5
+221.96
+107.5
+BS2
+271110.8
+720.46
+0.0
+M1.1.2
+113605.9
+381.46
+57.45
+M1.2.2
+117188.3
+394.57
+36.45
+M1.3.2
+113665.3
+378.63
+29.85
+M2.1.2.1
+70603.0
+404.28
+150.8
+M2.2.2.1
+77836.0
+400.18
+121.4
+M2.3.2.1
+76818.2
+399.89
+110.8
+M2.1.2.2
+72316.9
+219.91
+148.4
+M2.2.2.2
+78135.2
+240.57
+121.3
+M2.3.2.2
+77073.0
+235.49
+112.45
+(b) Knapsack problem with m = 2 constraints
+and n = 100 variables
+knapsack problem, m = 3, n = 50
+version
+nodes
+time (s)
+solved
+IPs
+BB
+54430.4
+51.5026
+0.0
+BS1
+15260.1
+17.9276
+0.0
+WS
+18112.9
+20.5208
+36.2
+M1.1.1
+13522.4
+16.8431
+28.8
+M1.2.1
+13530.7
+16.4735
+19.0
+M1.3.1
+13576.5
+16.4442
+14.95
+M2.1.1.1
+12345.3
+22.1289
+51.15
+M2.2.1.1
+12973.5
+19.7592
+34.5
+M2.3.1.1
+13014.0
+19.6348
+28.8
+M2.1.1.2
+12241.1
+16.1190
+53.55
+M2.2.1.2
+12934.3
+16.2933
+35.15
+M2.3.1.2
+12908.3
+16.1009
+30.35
+BS2
+22597.3
+24.5736
+0.0
+M1.1.2
+14645.7
+18.6425
+29.55
+M1.2.2
+14573.2
+18.0521
+16.75
+M1.3.2
+14597.0
+17.9793
+14.5
+M2.1.2.1
+12617.9
+22.3518
+54.65
+M2.2.2.1
+13324.9
+20.7655
+33.4
+M2.3.2.1
+13252.3
+20.4366
+30.55
+M2.1.2.2
+12682.4
+16.8670
+56.65
+M2.2.2.2
+13180.2
+16.7601
+33.6
+M2.3.2.2
+13274.4
+16.8497
+32.15
+(c) Knapsack problem with m = 3 constraints
+and n = 50 variables
+knapsack problem, m = 3, n = 80
+version
+nodes
+time (s)
+solved
+IPs
+BB
+263971.6
+724.999
+0.0
+BS1
+81609.9
+287.899
+0.0
+WS
+121360.8
+376.247
+56.4
+M1.1.1
+80406.5
+282.897
+47.35
+M1.2.1
+79971.5
+279.885
+32.2
+M1.3.1
+80089.6
+279.686
+26.55
+M2.1.1.1
+54187.1
+340.389
+115.7
+M2.2.1.1
+55915.9
+328.411
+85.45
+M2.3.1.1
+58486.8
+330.347
+66.9
+M2.1.1.2
+53396.0
+164.755
+116.0
+M2.2.1.2
+56572.6
+174.131
+86.25
+M2.3.1.2
+57452.9
+176.657
+67.85
+BS2
+140681.4
+390.578
+0.0
+M1.1.2
+92175.8
+334.414
+50.45
+M1.2.2
+96339.3
+350.148
+29.05
+M1.3.2
+96099.5
+348.915
+24.0
+M2.1.2.1
+54119.5
+349.261
+112.5
+M2.2.2.1
+59379.8
+344.899
+89.1
+M2.3.2.1
+60326.6
+349.874
+74.15
+M2.1.2.2
+54595.6
+176.090
+119.65
+M2.2.2.2
+62851.9
+211.373
+88.9
+M2.3.2.2
+61200.8
+205.413
+76.6
+(d) Knapsack problem with m = 3 constraints
+and n = 80 variables
+Table 2: Numerical results of the bi-objective, multidimensional knapsack problems
+17
+
+Tchebycheff objective space information into the lower bound. Note that with rising numbers of
+variables and constraints the hybridization techniques have larger impact on the performance of the
+Branch and Bound algorithm.
+5.2 Bi-objective Assignment Problems
+We consider bi-objective assignment problems having n = ℓ2 variables,
+max
+ℓ
+�
+i=1
+ℓ
+�
+j=1
+ck
+ij xij
+k = 1, 2
+s.t.
+ℓ
+�
+i=1
+xij = 1
+j = 1, ..., ℓ
+ℓ
+�
+j=1
+xij = 1
+i = 1, ..., ℓ
+x ∈ {0, 1}ℓ×ℓ
+where the cost coefficients ck
+ij ∈ [50, 100]. The algorithmic strategy for the solution of IP scalariza-
+tions depending on the value of the parameter α is chosen similarly to the previous case of knapsack
+problems. However, we adapt the boundaries due to the different number of nodes to explore in as-
+signment problems. In M1.1.β weighted sum scalarizations are solved every 10-th iteration to integer
+optimality. In M1.2.β we apply the weighted sum every 10-th iteration within the first n·ℓ iterations.
+In M1.3.β we apply the weighted sum every 10-th iteration within the first n · ℓ/3 iterations, every
+ℓ-th iteration in the next n · ℓ/3 iterations and every n-th iteration in the third n · ℓ/3 iterations.
+For M.2.α.β.γ we use the same algorithmic strategy as in hybrid Branch and Bound for knapsack
+problems. If a scalarization cannot be applied or an IP with identical objective function has been
+solved prior, no IP is solved in that iteration.
+Due to the total unimodularity of the assignment problem, the weighted sum scalarizations do in
+general not improve the lower bound sets of subproblems. However, in situations where the weighted
+sum IP scalarization generates a supported efficient solution, whose corresponding non-dominated
+point is not an extreme point of the lower bound set, the local upper bounds move closer to the lower
+bound set. This reduces the gap between upper and lower bound and may lead to a decrease of the
+explored subproblems. Note that this update of the upper bound set may also have the contrary
+effect (the number of considered subproblems increases), since it can change the order in which the
+subproblems are considered. Nevertheless, the weighted sum IP scalarizations are necessary to find
+non-dominated points on which the augmented weighted Tchebycheff scalarization can be applied.
+Our branching strategies have a significant impact on the number of explored nodes and the
+computation time. Again, the local hypervolume gap performs better than the global hypervolume
+gap strategy. With the local strategy we can reduce the number of explored nodes by up to 39%
+(Table 3c) and the computation time by up to 33% (Table 3c). Using the global hypervolume gap
+strategy we can reduce the number of explored nodes by up to 12% (Table 3b) and the computation
+time by up to 12% (Table 3b). We reach a reduction of the explored nodes by up to 46% (Table 3d)
+and a reduction of the computation time by up to 42% (Table 3d), in the best case. Again, the
+strategies M2.1.1.1 and M2.1.1.2 seem to work the best for assignment problems in terms of explored
+nodes. Nevertheless, M2.1.1.2 leads to a better computation time which can be explained by the
+same argument as before. Furthermore, strategy BS1 works very well and is able to compete with
+the previously mentioned strategies with respect to number of nodes and computation time without
+solving a single IP scalarization.
+18
+
+assignment problem, n = 100
+version
+nodes
+time (s)
+solved
+IPs
+BB
+3117.0
+5.1507
+0.0
+BS1
+2422.2
+4.1182
+0.0
+WS
+3117.0
+5.2631
+19.2
+M1.1.1
+2425.1
+4.1937
+13.95
+M1.2.1
+2425.1
+4.1564
+11.95
+M1.3.1
+2425.1
+4.1996
+8.4
+M2.1.1.1
+2418.2
+4.4063
+19.8
+M2.2.1.1
+2420.5
+4.2726
+14.4
+M2.3.1.1
+2419.5
+4.2242
+9.65
+M2.1.1.2
+2420.5
+4.3474
+19.9
+M2.2.1.2
+2420.5
+4.2014
+14.55
+M2.3.1.2
+2423.0
+4.2055
+9.8
+BS2
+2948.9
+4.9644
+0.0
+M1.1.2
+2519.8
+4.4349
+14.1
+M1.2.2
+2518.7
+4.4211
+11.05
+M1.3.2
+2516.2
+4.4203
+7.65
+M2.1.2.1
+2499.7
+4.6173
+20.4
+M2.2.2.1
+2518.7
+4.4661
+12.2
+M2.3.2.1
+2516.2
+4.4214
+8.35
+M2.1.2.2
+2498.1
+4.5400
+19.55
+M2.2.2.2
+2518.7
+4.4451
+12.25
+M2.3.2.2
+2516.2
+4.3972
+8.4
+(a) Assignment problem with n = 100 variables
+assignment problem, n = 144
+version
+nodes
+time (s)
+solved
+IPs
+BB
+9661.9
+28.2667
+0.0
+BS1
+6255.4
+18.9362
+0.0
+WS
+9610.8
+28.1363
+33.35
+M1.1.1
+6274.5
+18.9323
+22.3
+M1.2.1
+6274.5
+18.9869
+14.3
+M1.3.1
+6274.5
+18.9318
+9.85
+M2.1.1.1
+6210.9
+20.0216
+46.0
+M2.2.1.1
+6279.9
+19.4670
+23.55
+M2.3.1.1
+6271.2
+19.1542
+12.65
+M2.1.1.2
+6171.0
+19.4994
+43.6
+M2.2.1.2
+6261.5
+19.2851
+24.65
+M2.3.1.2
+6270.5
+18.8676
+12.45
+BS2
+8448.0
+24.7742
+0.0
+M1.1.2
+6781.8
+20.6825
+24.05
+M1.2.2
+6779.4
+20.5940
+13.25
+M1.3.2
+6734.2
+20.4270
+9.3
+M2.1.2.1
+6472.8
+20.8732
+44.1
+M2.2.2.1
+6724.2
+20.6838
+16.8
+M2.3.2.1
+6682.3
+20.4212
+12.6
+M2.1.2.2
+6500.3
+20.2536
+42.55
+M2.2.2.2
+6737.7
+20.5214
+16.75
+M2.3.2.2
+6689.0
+20.3987
+12.95
+(b) Assignment problem with n = 144 variables
+assignment problem, n = 225
+version
+nodes
+time (s)
+solved
+IPs
+BB
+26810.5
+160.712
+0.0
+BS1
+16142.2
+107.909
+0.0
+WS
+26810.5
+163.666
+46.85
+M1.1.1
+16150.0
+107.842
+32.8
+M1.2.1
+16151.2
+109.687
+19.7
+M1.3.1
+16164.7
+109.073
+12.05
+M2.1.1.1
+15424.1
+111.181
+87.4
+M2.2.1.1
+15964.3
+110.004
+43.05
+M2.3.1.1
+16112.2
+110.372
+19.5
+M2.1.1.2
+15554.1
+106.748
+89.5
+M2.2.1.2
+16008.4
+107.641
+41.35
+M2.3.1.2
+16106.5
+109.735
+19.25
+BS2
+24566.2
+152.341
+0.0
+M1.1.2
+17499.1
+117.481
+34.15
+M1.2.2
+17591.4
+118.804
+16.75
+M1.3.2
+17287.8
+116.681
+11.3
+M2.1.2.1
+16095.7
+115.012
+85.8
+M2.2.2.1
+17048.8
+116.275
+31.55
+M2.3.2.1
+17093.3
+117.976
+17.75
+M2.1.2.2
+16183.0
+110.920
+79.55
+M2.2.2.2
+17104.7
+117.885
+32.2
+M2.3.2.2
+17070.0
+117.557
+16.65
+(c) Assignment problem with n = 225 variables
+assignment problem, n = 324
+version
+nodes
+time (s)
+solved
+IPs
+BB
+76643.0
+798.644
+0.0
+BS1
+47311.6
+527.471
+0.0
+WS
+75179.2
+786.036
+61.75
+M1.1.1
+47327.0
+530.055
+47.75
+M1.2.1
+47332.8
+523.562
+21.9
+M1.3.1
+47375.6
+524.610
+15.25
+M2.1.1.1
+40978.0
+477.927
+157.5
+M2.2.1.1
+43760.5
+497.812
+82.0
+M2.3.1.1
+44343.9
+499.678
+52.35
+M2.1.1.2
+41294.4
+457.120
+159.05
+M2.2.1.2
+43864.6
+487.051
+81.25
+M2.3.1.2
+44499.7
+489.744
+52.05
+BS2
+69042.4
+723.853
+0.0
+M1.1.2
+49367.5
+565.719
+48.45
+M1.2.2
+49053.0
+553.130
+22.95
+M1.3.2
+49221.4
+554.594
+15.2
+M2.1.2.1
+41840.5
+489.647
+150.2
+M2.2.2.1
+45621.0
+520.469
+67.15
+M2.3.2.1
+46915.9
+533.692
+39.35
+M2.1.2.2
+42726.6
+474.220
+156.15
+M2.2.2.2
+45901.6
+513.111
+67.05
+M2.3.2.2
+46902.8
+526.440
+43.45
+(d) Assignment problem with n = 324 variables
+Table 3: Numerical results of the bi-objective assignment problems
+19
+
+5.3 Bi-objective Discrete Facility Location Problems
+We consider discrete facility location problems of the form
+max
+ℓ
+�
+i=1
+q
+�
+j=1
+ck
+ij xij +
+q
+�
+j=1
+f k
+j yj
+k = 1, 2
+s.t.
+q
+�
+j=1
+xij ≤ 1
+i = 1, ..., ℓ
+xij ≤ yj
+i = 1, ..., ℓ, j = 1, ..., q
+x ∈ {0, 1}ℓ×q
+y ∈ {0, 1}q
+where ℓ is the number of customers and q the number of facilities. We randomly generate coordinates
+of ℓ customers and q facilities in a square with length 200. The costs of the first objective function
+correspond to the l1-distances between the customers and facilities, while the costs of the second
+objective function are randomly generated (i. e. c2
+ij ∈ [1, 200]) and f k
+j ∈ [200, 400]. The number
+of variables is n = (ℓ + 1) q.
+We restrict the numerical tests to problems where the number of
+facilities is 20% of the number of customers. Again, we need to specify when and how often integer
+scalarizations are applied: We use the same methods as before but adapt the boundaries due to the
+different number of nodes to explore in discrete facility location problems. In M1.1.β we apply the
+weighted sum IP scalarization every 10-th iteration. In M1.2.β we apply the weighted sum every
+10-th iteration within the first n2/4 iterations. In M1.3.β we apply the weighted sum scalarization
+every 10-th iteration in the first n2/4 iterations, every n/2-th iteration in the next n2/4 iterations
+and every n-th iteration in the third n2/4 iterations. In M.2.α.β.γ we operate analogous to the
+methods used for knapsack and assignment problems. If a scalarization cannot be applied or an IP
+with identical objective function has been solved prior, no IP is solved in that iteration.
+Again, both new branching strategies have an impact on the number of explored nodes and the
+computation time. The local strategy, once more, leads to better results, namely reduction of the
+explored nodes by up to 52% (Table 4d) and reduction of the computation time by up to 45%
+(Table 4d). With the global hypervolume gap strategy we can reach a reduction of the explored
+nodes by up to 24% (Table 4d) and a reduction of the computation time by up to 21% (Table 4d).
+In the best case we can reach a reduction of the explored nodes by up to 57% (Table 4d) and of
+the computation time by up to 50% (Table 4d). Once again, M2.1.1.2 seems to be the best choice
+with respect to the number of explored nodes and with a rising number of variables it is also the
+best choice regarding the computation time. With a smaller number of variables, BS1 leads to good
+results with respect to both aspects without solving a single IP.
+5.4 Summary
+In all of the three tested problem classes (knapsack, assignment, discrete facility location) a signif-
+icant reduction of the number of explored nodes and the computation time can be realized with
+all presented combinations of the hybrid Branch and Bound approach. With increasing problem
+size (number of variables) the impact of the presented augmentations increases. Furthermore, the
+approaches perform better on problems where the gap between YN and the solution of the linear
+relaxation is larger compared to totally unimodular problems.
+The local hypervolume gap strategy for the node selection outperforms the global hypervolume gap
+strategy in our numerical tests. A reason for this is that in the global hypervolume gap strategy many
+small search zones can add up to a large gap although the lower bound might be quite close to the non-
+dominated points. The local hypervolume gap strategy chooses the node with the largest search zone,
+which has the biggest potential to reduce this gap. Moreover, the local hypervolume gap strategy
+aims at an uniform distribution of points in the incumbent list. In our numerical test, M2.1.1.2 turn
+20
+
+facility location problem, n = 48
+version
+nodes
+time (s)
+solved
+IPs
+BB
+1431.1
+1.0851
+0.0
+BS1
+999.8
+0.7640
+0.0
+WS
+1431.1
+1.4550
+15.35
+M1.1.1
+1000.2
+0.8354
+10.95
+M1.2.1
+1000.2
+0.8219
+8.8
+M1.3.1
+1000.2
+0.8243
+6.95
+M2.1.1.1
+999.7
+0.9361
+13.2
+M2.2.1.1
+1000.2
+0.8361
+9.85
+M2.3.1.1
+1000.2
+0.8079
+7.85
+M2.1.1.2
+999.3
+0.8536
+12.8
+M2.2.1.2
+1000.2
+0.8182
+9.85
+M2.3.1.2
+1000.2
+0.8037
+7.85
+BS2
+1268.6
+0.9714
+0.0
+M1.1.2
+1041.8
+0.8654
+12.3
+M1.2.2
+1041.8
+0.8527
+9.35
+M1.3.2
+1041.8
+0.8592
+7.35
+M2.1.2.1
+1036.5
+0.9722
+15.65
+M2.2.2.1
+1041.8
+0.8761
+10.25
+M2.3.2.1
+1041.8
+0.8735
+7.6
+M2.1.2.2
+1034.6
+0.9678
+15.1
+M2.2.2.2
+1041.8
+0.8843
+10.4
+M2.3.2.2
+1041.8
+0.8654
+7.6
+(a) Facility location problem with 15 customers
+and 3 facilities
+facility location problem, n = 84
+version
+nodes
+time (s)
+solved
+IPs
+BB
+7949.1
+12.6393
+0.0
+BS1
+4626.7
+7.9303
+0.0
+WS
+7490.2
+12.2058
+35.0
+M1.1.1
+4627.2
+8.1249
+26.45
+M1.2.1
+4627.2
+8.1149
+18.2
+M1.3.1
+4626.4
+8.0610
+13.85
+M2.1.1.1
+4527.8
+9.2394
+49.0
+M2.2.1.1
+4601.6
+8.4311
+26.45
+M2.3.1.1
+4610.4
+8.2569
+18.9
+M2.1.1.2
+4526.1
+8.4415
+45.3
+M2.2.1.2
+4591.1
+8.1289
+24.25
+M2.3.1.2
+4605.2
+8.1312
+19.8
+BS2
+7084.4
+11.5822
+0.0
+M1.1.2
+4873.8
+8.7719
+26.35
+M1.2.2
+4874.8
+8.7534
+17.45
+M1.3.2
+4894.1
+8.7031
+12.6
+M2.1.2.1
+4673.6
+9.5755
+49.65
+M2.2.2.1
+4832.0
+8.9275
+23.6
+M2.3.2.1
+4812.8
+8.8050
+17.4
+M2.1.2.2
+4628.9
+8.7019
+46.15
+M2.2.2.2
+4825.9
+8.7491
+24.4
+M2.3.2.2
+4807.5
+8.5984
+17.5
+(b) Facility location problem with 20 customers
+and 4 facilities
+facility location problem, n = 130
+version
+nodes
+time (s)
+solved
+IPs
+BB
+17461.9
+51.6307
+0.0
+BS1
+10684.0
+34.5157
+0.0
+WS
+16795.9
+50.9317
+51.35
+M1.1.1
+10753.9
+35.4356
+37.1
+M1.2.1
+10753.9
+35.2764
+25.5
+M1.3.1
+10753.9
+35.1007
+19.15
+M2.1.1.1
+10104.4
+38.3909
+89.65
+M2.2.1.1
+10678.1
+35.7434
+39.35
+M2.3.1.1
+10722.3
+35.6262
+27.7
+M2.1.1.2
+10103.0
+34.5003
+85.95
+M2.2.1.2
+10691.2
+35.3730
+39.05
+M2.3.1.2
+10718.6
+35.1690
+27.45
+BS2
+15474.7
+46.5130
+0.0
+M1.1.2
+11548.8
+38.4891
+39.3
+M1.2.2
+11601.4
+38.5848
+24.6
+M1.3.2
+11535.1
+38.2917
+18.0
+M2.1.2.1
+10684.3
+39.8639
+81.5
+M2.2.2.1
+11381.8
+39.1988
+37.15
+M2.3.2.1
+11336.0
+38.7754
+28.45
+M2.1.2.2
+10695.5
+36.9098
+78.25
+M2.2.2.2
+11341.6
+38.2646
+39.55
+M2.3.2.2
+11352.1
+38.0063
+25.9
+(c) Facility location problem with 25 customers
+and 5 facilites
+facility location problem, n = 186
+version
+nodes
+time (s)
+solved
+IPs
+BB
+67369.3
+373.238
+0.0
+BS1
+31844.1
+203.145
+0.0
+WS
+62192.8
+349.741
+69.95
+M1.1.1
+32106.6
+206.244
+53.05
+M1.2.1
+32097.9
+206.851
+33.4
+M1.3.1
+32148.5
+207.199
+23.75
+M2.1.1.1
+28384.2
+224.486
+186.8
+M2.2.1.1
+31074.5
+218.314
+101.15
+M2.3.1.1
+32004.8
+209.608
+50.1
+M2.1.1.2
+28558.8
+186.010
+172.7
+M2.2.1.2
+30946.4
+202.329
+97.75
+M2.3.1.2
+32011.8
+207.715
+47.5
+BS2
+50759.0
+292.344
+0.0
+M1.1.2
+35150.1
+230.687
+55.5
+M1.2.2
+35789.8
+233.967
+30.8
+M1.3.2
+35255.0
+233.163
+21.95
+M2.1.2.1
+29704.3
+230.016
+172.65
+M2.2.2.1
+33959.7
+236.351
+81.75
+M2.3.2.1
+34228.0
+233.553
+50.2
+M2.1.2.2
+30412.6
+202.719
+167.4
+M2.2.2.2
+34511.8
+229.970
+69.95
+M2.3.2.2
+34405.0
+229.021
+47.1
+(d) Facility location problem with 30 customers
+and 6 facilities
+Table 4: Numerical results of the bi-objective integer facility location problem
+21
+
+out to be the best choice in most cases with respect to the number of explored nodes and computation
+time. In this version, we use the local hypervolume gap strategy for the choice of the active node,
+every 10-th iteration the weighted sum IP scalarization is applied and every 50-th iteration we
+apply the augmented weighted Tchebycheff scalarization instead. Futhermore, the objective space
+information gained by the augmented weighted Tchebycheff scalarization is not used to update the
+lower bound set, since its local improvements do not compensate the increased computation time.
+Although we need to solve more IPs than in most other approaches, the computation time is the
+lowest compared to the others. So, using the augmented weighted Tchebycheff scalarization in the
+beginning of the Branch and Bound works best. Due to the likelihood of finding non-supported non-
+dominated points in the early stages of the algorithm, the upper bound can be further improved.
+This results to a higher probability of fathoming a node by dominance. Nevertheless, with version
+BS1 we also achive a remarkable reduction in terms of the number of explored nodes and computation
+time by using the local hypervolume gap strategy for node selection.
+6 Conclusion and Outlook
+In this paper, we propose two approaches to incorporate objective space information in bi-objective
+Branch and Bound. By using the local or global (approximated) hypervolume gap as a node selection
+criterion, we adapt the run of the Branch and Bound algorithm to the problem instance. Additionally,
+we adaptively solve scalarizations to integer optimality to improve the lower and the upper bound
+set by the obtained objective space information. Our numerical results show the effectiveness of
+both approaches and in particular of their combination. The dynamic branching rule based on the
+local (approximated) hypervolume gap has large impact on the number of explored subproblems,
+is compuationally efficient and can be easily integrated in other multi-objective Branch and Bound
+algorithms.
+While we tested in this paper the individual contributions of our augmentations on a generic
+bi-objective Branch and Bound, we will continue to extend our ideas to multiple dimensions and
+integrate them into a competetive multi-objective Branch and Bound framework. Particularly in
+higher dimensions, it may be promising to combine our approaches with objective space branching.
+Acknowledgment
+The authors thankfully acknowledge financial support by Deutsche Forschungsgemeinschaft, project
+number KL 1076/11-1.
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diff --git a/9tFLT4oBgHgl3EQfCC72/content/tmp_files/load_file.txt b/9tFLT4oBgHgl3EQfCC72/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf,len=2974
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='11974v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='OC]  27 Jan 2023  Augmenting Bi-objective Branch and Bound  by Scalarization-Based Information  Julius Bauß∗ and Michael Stiglmayr  University of Wuppertal, School of Mathematics and Natural Sciences,  IMACM, Gaußstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' 20, 42119 Wuppertal, Germany  {bauss,stiglmayr}@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='uni-wuppertal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='de  While Branch and Bound based algorithms are a standard approach to solve single-  objective (mixed-)integer optimization problems, multi-objective Branch and Bound meth-  ods are only rarely applied compared to the predominant objective space methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In this  paper we propose modifications to increase the performance of multi-objective Branch  and Bound algorithms by utilizing scalarization-based information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We use the hypervol-  ume indicator as a measure for the gap between lower and upper bound set to implement  a multi-objective best-first strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' By adaptively solving scalarizations in the root node  to integer optimality we improve both, upper and lower bound set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The obtained lower  bound can then be integrated into the lower bounds of all active nodes, while the deter-  mined solution is added to the upper bound set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Numerical experiments show that the  number of investigated nodes can be significantly reduced by up to 83% and the total  computation time can be reduced by up to 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Keywords: multi-objective optimization, mult-iobjective branch and bound, integer pro-  gramming, hypervolume indicator  1 Introduction  Many optimization problems occurring in real-word applications include a conflict of interests and  goals, or secondary objectives, in a word, they are multi-objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Thus, there is (in general) not  one solution that optimizes all objectives at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Following the a posteriori paradigm of decision  making, we aim at determining the set of so-called efficient solutions or the images, the so-called  non-dominated points, which cannot be improved in one objective without deterioration in at least  one other objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Thus, efficient solutions are reasonable choices for decision makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  As we are considering specifically bi-objective integer linear programs and their solution with  multi-objective Branch and Bound methods, the following literature survey will also focus on this  and closely related topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' A comprehensive introduction to multi-objective optimization in general  is given, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', in Steuer (1986);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Ehrgott (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Solution approaches for multi-objective optimization problems are often categorized in: objective  space and decision space methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Objective space methods scalarize the underlying problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=',  it is replaced by a series of single-objective problems to determine successively the set of efficient  ∗Corresponding author  1  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In the case of multi-objective integer programming, these scalarized problems can be  solved with commercial integer programming solvers like CPLEX or Gurobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The utilization of these  optimized, single-criteria solvers are a major advantage and one of the reasons why those methods  are predominant in multi-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  There are numerous objective space methods and a popular one is the ε-constraint method that  was introduced for two objectives by Haimes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In every iteration the first objective is  optimized with an updated constraint to ensure an improvement regarding the second objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In  Laumanns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2006) an extension to three and more objectives is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Many approaches  based on the ε-constraint method have been published in the last decades, for example Boland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  (2017) and Kirlik and Sayın (2014) combine the method with reduction of dimension in the tri-  respectively multi-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The weighted sum scalarization is an objective space method based on the optimization of a  weighted sum of the objective functions using non-negative weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Note that not all efficient  solutions can be determined as optimal solutions of the weighted sum scalarization using suit-  able weights (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Aneja and Nair, 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Efficient solutions which can be obtained by us-  ing weighted sum scalarization are denoted as supported efficicent and their corresponding non-  dominated points are located on the boundary of the convex hull of feasible image points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Extensions  of the weighted sum method to the multi-objective case are proposed in Przybylski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2010a),  Özpeynirci and Köksalan (2010), Bö¸kler and Mutzel (2015), and Przybylski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Ulungu and Teghem (1995) introduced the so-called two-phase method for bi-objective problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In the first phase the extreme supported non-dominated points are generated with an algorithm sim-  ilar to the initial weighted sum approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In the second phase the remaining non-dominated points  are generated by searching in triangles defined by two consecutive extreme supported non-dominated  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In Przybylski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2008) and Tuyttens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2000) problem specific algorithms are sug-  gested for the second phase, while in Przybylski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2010b) a two-phase method for problems  with more than two objectives is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The augmented weighted Tchebycheff method, first presented in Steuer and Choo (1983), mini-  mizes the augmented weighted Tchebycheff distance between a predefined reference point and the  set of feasible image points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Dächert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2012) suggested an adaptive choice of the augmentation  term for the bi-objective case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In Boland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2015a), Boland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2015b) (for the bi-objective case), Dächert and Klamroth  (2014), and Klamroth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2015) (for the tri- respectively multi-objective case) search region split-  ting methods are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In this class of objective space methods, the search region (based on the  already determined non-dominated points) is splitted into so-called search zones on which scalariza-  tions are solved indpendently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Besides their advantages, objective space methods share a shortcoming: In each iteration a scalar-  ized integer program is solved from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Even though in some objective space methods starting  solutions can be transfered from previous iterations, a large number of very similar problems has  to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In order to avoid this effort, decision space methods, mainly the Branch and Bound  method, have been increasingly investigated in the recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Klein and Hannan (1982) developed one of the first Branch and Bound algorithms for multi-  objective integeger programs with a typical one tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In Kiziltan and Yucaoğlu (1983) a  general Branch and Bound framework for multi-objective integer programs with binary variables  is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Ulungu and Teghem (1997) and Visée et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (1998) proposed problem specific Branch  and Bound approaches for bi-objective Knapsack problems, where the latter approach is integrated  in a two-phase method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Mavrotas and Diakoulaki (1998) extend the Branch and Bound approach to multi-objective mixed  integer programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Parts of the algorithm are refined in Mavrotas and Diakoulaki (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  (2013) this algorithm is improved and it is shown that the original algorithm is not correct because  the final dominance test is incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In Belotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2012) a Branch and Bound method is  presented that can handle bi-objective mixed integer programs with continious variables in both  objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  2  The Branch and Bound method proposed in Sourd and Spanjaard (2008) uses a set of points as  lower bound instead of just using a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Furthermore hyperplanes are used to fathom nodes  by dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In Stidsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2014) this idea is continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' They use hyperplanes as a lower  bound set that are generated by solving weighted sum scalarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Additionally they present  the so-called Pareto branching and the slicing technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' With Pareto branching it is possible to  divide the objective space to possibly ignore parts of it in specific nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Slicing partitions the  objective space in equally large parts and a respective slice can be fathomed if it is dominated by  an already found integer point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In Stidsen and Andersen (2018) this algorithm is improved and  an approach to parallelize the algorithm is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Based on this, the idea Pareto branching is  further investigated in Parragh and Tricoire (2019) and Gadegaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2019) for the bi-objective  case and Forget et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2022) for the tri-objective case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' A self-contained survey of multi-objective  Branch and Bound approaches is given in Przybylski and Gandibleux (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In this paper we present a bi-objective Branch and Bound algorithm that is augmented by  scalarization-based information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We make use of optimized single-objective solvers for scalar in-  teger programs and integrate the resulting information into the bi-objective Branch and Bound by  improving lower and upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Furthermore, we propose a new adaptive node selection strategy,  which relies on objective space information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In our numerical analysis we show the effectiveness of  these improvements by comparing them with a generic multi-objective Branch and Bound algorithm,  which we use as our baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The remainder of the article is organized as follows: In Section 2, we introduce notations and def-  initions for multi-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In Section 3, we present a general multi-objective Branch  and Bound framework and its key components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Furthermore, we describe a specific (however stan-  dard) multi-objective Branch and Bound algorithm, which will be used as baseline implementation  in our numerical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In Section 4, we present augmentations of the multi-objective Branch and  Bound, that utilize objective space information to improve the node selection as well as the compu-  tation of upper and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We provide numerical results in Section 5 and in Section 6, we  outline conclusions and outlooks for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  2 Preliminaries  We introduce a general multi-objective integer linear program which can be written in the form:  min  �  z1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' , zp(x)  �⊤  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  A x ≤ b  x ≥ 0  x ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  (MOILP)  Thereby, z(x) := (z1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' , zp(x))⊤ = C ·x ∈ Rp (with p ≥ 2) denotes the objective function vector,  with C ∈ Rp×n the matrix of objective coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The set of feasible solutions X := {x ∈ Zn : A ≤  b, x ≥ 0} is a subset of the decision space Rn, while its image Y := {C x: x ∈ X} is a subset of the  objective space Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  We use the Pareto concept of optimality which relies on the componentwise order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Let y1, y2 ∈ Rp,  then we define the corresponding dominance relations as follows:  y1 ≦ y2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', y1 weakly dominates y2 if y1  k ≤ y2  k for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', p,  y1 < y2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', y1 strictly dominates y2 if y1  k < y2  k for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', p,  y1 ≤ y2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', y1 dominates y2 if y1 ≤ y2 and y1 ̸= y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  A feasible solution x ∈ X is called efficient if there is no other solution ˆx ∈ X dominating it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=',  z(ˆx) ≤ z(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' A feasible solution x ∈ X is called weakly efficient if there is no ˆx ∈ X such that  z(ˆx) < z(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The set of efficient solutions is denoted by XE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' By YN = {z(x) ∈ Y : x ∈ XE} we  3  denote the set of the non-dominated points in the objective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Moreover, for any set Q ⊆ Rp  we denote by QN the set of its non-dominated points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', q ∈ QN  ⇐⇒ ∄q′ ∈ Q: q′ ≤ q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' For a  comprehensive introduction to multi-objective optimization see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', Ehrgott (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In this article we consider a minimal complete set as solution of a multi-objective optimization  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' A minimal complete set denotes the set of all non-dominated points YN and one efficient  solution for each non-dominated point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' See Serafini (1987) for a comparison of solution concepts in  multi-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  A standard solution approach in multi-objective optimization is the weighted sum scalarization  given in (WSλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  min WSλ(x) := λ⊤z(x) =  p  �  i=1  λi zi(x)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' x ∈ X  (WSλ)  Obviously, every optimal solution of the weighted sum scalarization for λ ∈ Rp  > := {λ ∈ Rp : λ > 0}  is efficient for (MOILP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' However, in general not all efficient solutions are optimal solutions of a  corresponding weighted sum problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' An efficient solution x′ ∈ XE is called supported if there is a  weighting vector λ′ ∈ Rp  > such that x′ is optimal for (WSλ) for λ = λ′, otherwise x′ is unsupported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Note that the non-dominated points corresponding to supported efficient solutions are located on  the boundary of the convex hull of Y , while the unsupported non-dominated points are located in  its (relative) interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  As already mentioned in the introduction the computation of upper and lower bounds on the  non-dominated set is a crucial component of any multi-objective Branch and Bound algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The  tightest componentwise upper and lower bounds of YN are the ideal point yI and the Nadir point  yN given by:  yI  k = min  y∈Y yk  and  yN  k = max  y∈YN yk  for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Obviously, yI ≦ y ≦ yN holds for every y ∈ YN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' YN is contained in the hyperbox spanned by  the corner points yI and yN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' However, these single point bounds are in general very weak except for  the degenerate case of yI = yN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This motivates to consider bound sets instead of bounds consisting  of a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We will rely on the definition of bound sets proposed in Ehrgott and Gandibleux  (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Let Rp  ≧ := {y ∈ Rp : y ≧ 0}, then  A lower bound set L ⊂ Rp for YN is a  – Rp  ≧-closed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', the set L + Rp  ≧ is closed),  – Rp  ≧-bounded (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', there exists a y ∈ Rp such that L ⊂ y + Rp  ≧)  – stable set (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', L ⊂ (L + Rp  ≧)N),  such that YN ⊂ (L + Rp  ≧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  An upper bound set U ⊂ Rp for YN is a  – Rp  ≧-closed,  – Rp  ≧-bounded,  – stable sets,  such that YN ⊂ cl  �  (U + Rp  ≧)∁�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The upper bound and lower bound that we will define for our branch and bound framework in  Section 3 will suit these definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We say a lower bound L is weakly dominated by an upper  bound U if for all l ∈ L there exists an u ∈ U such that u ≦ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  4  In the following we restrict ourselves to bi-objective binary linear optimization problems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=',  problems with two linear objective functions and variables x ∈ {0, 1}n:  min  z(x) =  �  z1(x), z2(x)  �⊤  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  A x ≤ b  x ∈ {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  (BO01LP)  3 A Generic Multi-objective Branch and Bound Framework  In this section we present a generic multi-objective Branch and Bound framework, which we specify  and augment by using scalarization based information in the then following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Branch and Bound methods follow a “divide and conquer” paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' A problem that is too hard  to be solved directly, is divided into smaller and thus easier subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Thereby, subproblems are  associated with nodes in a tree data structure according to their descent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', node i is a descendant  node of node j iff the feasible set of the subprobem associated with node i is a subset of the feasible  set of the subproblem associated with node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The corresponding subproblems of the child nodes are  created by subdividing the feasible set of the corresponding (sub)problem of the parent node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Starting  with the root node, to which the original optimization problem is associated, the algorithm selects in  each iteration one active node and updates its lower bound and upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Then the active node  can be fathomed if the corresponding subproblem is either solved or irrelevant for the determination  of a minimal complete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' If we cannot prune we subdivide the corresponding problem into new  subproblems and create corresponding child nodes (branching).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' For a more detailed introduction  and survey of multi-objective Branch and Bound algorithms see Przybylski and Gandibleux (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  A recent survey of single-objective Branch and Bound frameworks is given e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' in Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In the following we specify the lower bound, upper bound, branching rule and node selection  we use in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Lower bound:  Lower bound sets are often determined by solving relaxations of the respective  subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Like in the single-objective case, the most frequently used relaxations are linear and  convex relaxations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In order to solve the linear relaxation we are using in our framework, we apply  Benson’s outer approximation algorithm (Benson, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The algorithm is initiated with a lower  bound, which is improved in every iteration by generating cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Due to the outer approximation  structure the algorithm can be aborted at any time returning a valid lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Alternatively, linear (or convex) relaxations can be obtained using a dichotomic scheme (see, for  example, Aneja and Nair, 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Özpeynirci and Köksalan, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Przybylski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', 2010a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Upper bound:  The upper bound set, in the following denoted by U, is stored in the form of a so-  called incumbent list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Throughout the run of the algorithm, it contains all integer feasible solutions  and their corresponding outcome vectors that are not dominated by another feasible solution found  so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In every iteration the extreme supported solutions of the computed lower bound sets are  checked for integer feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' An integer feasible solution ¯x ∈ X is then appended to the incumbent  list, if there is no x ∈ U dominating ¯x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', C(x) ≤ C(¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' If a new solution ¯x is added to the  incumbent list U all solutions x ∈ U which are dominated by ¯x (C(¯x) ≤ C(x)) are removed from  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Note that an update of the incubent list requires a subsequent update of the list of local upper  bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' A detailed description of local upper bounds, their computation and update in an arbitrary  number of criteria is given in Klamroth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In this framework we start with an empty  upper bound set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' However, it is also possible to initialize the incumbent list by heuristic methods, or  by solving scalarizations like, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', in the two-phase method (Ulungu and Teghem, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Visée et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=',  1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  U ⊎ {¯x} :=  �  U  if ∃x ∈ U : C(x) ≤ C(¯x)  {¯x} ∪ {x ∈ U : C(¯x) ≰ C(x)}  otherwise  5  Node selection:  In every iteration of the algorithm an unexplored node is selected from the tree of  subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This node is called active node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The order in which the nodes of the tree are considered  has a significant impact on the number of created nodes that have to be explored and thus on the  computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Two types of strategies need to be distinguished: static strategies and dynamic strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The two  most common examples of static strategies are the depth-first strategy and the breadth-first strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Most multi-objective Branch and Bound algorithms in literature follow a depth-first strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Thus,  we use this strategy for our baseline implementation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In contrast to the single-objective case, dynamic node selection strategies are rarely applied in the  multi-objective case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The usage of dynamic strategies for the choice of the active node can be seen  in (Stidsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', 2014), (Belotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', 2012) and (Jesus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', 2021), for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Fathoming:  In order to avoid the total enumeration of all feasible solutions, nodes are fathomed  if the respective subproblem is either solved to optimality or does not contain solutions which are  necessary to determine a minimal complete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In particular, there are three different situations in  which a node can be fathomed:  i) Fathoming by infeasibility: If the LP-relaxation of a subproblem is infeasible then the corre-  sponding subproblem is infeasible as well, since the feasible set of the subproblem is a subset  of the feasible set of its relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  ii) Fathoming by optimality: Similar to the single-objective case we can fathom a node by opti-  mality if the lower bound L is equal to the upper bound U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This implies the subproblem is  solved to optimality and the associated node must not be subdiveded further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' However, this  can happen in the multi-objective case only if the lower and upper bound consist of the same  single point, namely the ideal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  iii) Fathoming by dominance: A node can be fathomed by dominance if all feasible solutions of this  subproblem are dominated by points in the incumbent list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In order to check dominance for all  feasible outcome vectors of a subproblem we compare the lower bound L of the corresponding  node to the current upper bound U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' If for all l ∈ L there is a point in the incumbent list u ∈ U  with u ≦ l then all feasible points in the subtree are dominated by the current incumbent list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In other words, if there is no local upper bound defined by U above the computed lower bound  the node can be fathomed by dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Branching:  As mentioned in the beginning of this section, one of the key aspects of Branch and  Bound is iterative subdivision into smaller subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Thereby subproblems are associated with  nodes in a tree, such that the subproblem associated to a child node is obtained by one branching  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Since we consider binary optimization problems (BO01LP), we can divide a (sub)problem into  two new subproblems by fixing a specific variable to 0 and respectively to 1 in the other subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  This results in a binary Branch and Bound tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The branching rule determines which variable is selected as branching variable in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Thereby, one distinguishes between static and dynamic strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Static strategies determine an  order of the variables in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In each iteration of the algorithm the next variable in this list is  used as branching variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' With dynamic strategies the branching variable is selected by consid-  ering information obtained from previous iterations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', from the solution of (linear) relaxations of  (sub)problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The basic idea of static strategies for single-objective problems is to sort the variables, beginning  with the most promising according to the objective function values (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', (Kellerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', 2004)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  However, this cannot be easily extended to the multi-objective case due to conflicting objective  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Nevertheless there are some approaches to extend static strategies to the multi-objective  case (see for example (Ulungu and Teghem, 1997) and (Bazgan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', 2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  6  In contrast to most of the published papers which apply static strategies we use a dynamic strategy  as proposed in Belotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' By solving the linear relaxation of a (sub)problem we obtain  the lower bound set L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' For all extreme points of L we check how often a variable is fractional in the  corresponding solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' As branching variable we choose the one which is most often fractional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  4 Using Objective Space Information in Multi-objective Branch  and Bound  In this section, we propose modifications which improve the computational efficiency of bi-objective  Branch and Bound algorithms in two critical aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  One of the weaknesses of multi-objective  Branch and Bound as compared to its single-objective counterpart is the bounding procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' While  any feasible solution ¯x ∈ X dominates w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' one (linear) objective a half-space in decision space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=',  {x ∈ Rn : c⊤x ≥ c⊤¯x}), the set of feasible solutions which are dominated by a solution ¯x in p ≥ 2  objective functions (C ∈ Rp×n) forms a cone {x ∈ Rn : C x ≧ C ¯x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The cone of dominated solutions  is smaller the more the objective functions are in conflict, leading also to a larger number of efficient  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This implies that a significant part of the Branch and Bound tree has to be enumerated  and only a small number of branches can be pruned by dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Despite of this general problem  in multi-objective optimization, this asks for good bounding procedures to avoid the unnecessary  evaluation of dominated branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This however, requires good solutions in the incumbent list as  well as tight lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In order to achieve this, we suggest a new branching strategy and the hybridization of Branch and  Bound with objective space methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We determine scalarized subproblems adapted to the state of  the Branch and Bound and solve these to integer optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1 Branching Strategy  The branching strategy comprises two subsequent decisions: the choice of the active node and its  branching into subproblems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' the decision on which variable the subproblem is branched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This  second step is denoted as branching rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We discuss these two steps together since the order in  which the nodes are considered has a significant impact on the branched variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Instead of the  static depth- or breadth-first we use a dynamic node selection strategy, while we rely on the most  fractional rule as branching rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The basic idea of our strategy is quite simple and a natural extension of choosing the largest gap  in the single-objective case (see, for example, Dechter and Pearl, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' For every created node we  compute the approximate hypervolume gap between lower and upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We use the definition of  hypervolume proposed in Zitzler and Thiele (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In every iteration we choose the node with the  largest hypervolume gap as active node (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Jesus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Note that when a node is created  during the branching process, the approximated hypervolume gap of the parent node is assigned to  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We distinguish two variants of the hypervolume gap: the total hypervolume gap and the local  hypervolume gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' While the total hypervolume gap measures the volume of the search region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  the volume between lower and upper bound set, the local hypervolume gap approach considers only  the volume of the largest search zone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' the gap between a local upper bound and the lower bound  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' For a more detailed definition of search regions and search zones we refer to Klamroth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Figure 1 illustrates the two different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Here, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' , z4 ∈ K ⊂ U are points of the  incumbent list and lu1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' , lu3 are their corresponding local upper bounds, where K is a subset  of the incumbent list containing just the points above the lower bound of node ¯n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The green line  represents the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Figure 1a shows how to measure the total hypervolume gap of a node ¯n,  in the following denoted by thg(¯n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' For this approach we consider the approximated search region  of the corresponding node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Since there is a natural order in the bi-objective case, it is possible to  consider the approximated search zone of the first local upper bound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' the local upper bound  7  z1  z2  lu1  lu2  lu3  z1  z2  z3  z4  (a)  z1  z2  A  lu1  lu2  lu3  z1  z2  z3  z4  (b)  z1  z2  B  lu1  lu2  lu3  z1  z2  z3  z4  (c)  z1  z2  C  lu1  lu2  lu3  z1  z2  z3  z4  (d)  Figure 1: Example of computation of the two different approximated hypervolume gap approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  8  with the smallest z1-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Therefore we define the two spanning points, which, together with the  corresponding local upper bound, define a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The spanning points of a local upper bound lu  are defined by spi(lu) := {l ∈ L: l3−i = lu3−i}, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' So, the approximate hypervolume gap of lu  is given by  hg(lu) := 1  2  ��sp1(lu)1 − lu1  �� ·  ��sp2(lu)2 − lu2  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  For the remaining local upper bounds we compute the hypervolume of slices as shown in the illus-  tration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The hypervolume of the slice of lui, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' , |K| − 1 is defined as  sl(lui) :=  ��zi  2 − sp2(lui−1)2  �� +  ��lui  2 − sp2(lui)2  ��  2 ��zi  1 − lui  1  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  So, the total (approximated) hypervolume gap, which is assigned to node ¯n, is given by  thg(¯n) := hg(lu1) + sl(lu2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' + sl(lu|K|−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The Figures 1b, 1c and 1d show the computation of the local hypervolume gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The local hyper-  volume gap of a node ¯n is considered as the largest approximated hypervolume gap of a local upper  bound corresponding to points in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Therefore, the the local hypervolume gap, which is assigned  to node ¯n, is defined by  lhg(¯n) :=  max  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=',|K|−1hg(lui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In the given example, B is the largest approximated hypervolume and therefore is assigned to node  ¯n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Note that in our presented algorithms in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2 the local upper bound is initialized with the  point (∞, ∞)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Therefore, it is possible to apply the new branching strategies immediately at the  beginning of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Obviously, this approximation may neglect significantly large parts of  the search regions and search zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' However, the idea of the approximated hypervolume gap eases  computation and saves time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The efficiency of these new dynamic branching strategies is shown in  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2 Augmenting Branch and Bound with IP Scalarizations  In this subsection, we introduce a method to incorporate scalarizations into Branch and Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We  build a hybrid Branch and Bound algorithm combining the partial enumeration of decision space  with objective space information by solving scalarizations to integer optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  An integer optimal solution ¯x of a scalarization can be used to update upper and lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Obviously, the corresponding image point z(¯x) can be added to the incumbent list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Moreover, a  scalarizing function and its optimal solution ¯x define a level set, which can be included in the lower  bound set for all descendant nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In order to utilize these improved lower bounds in all nodes we  solve the IP scalarizations in the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1 Using Weighted Sum Scalarization  During the run of the Branch and Bound algorithm, a strategy triggers the IP solution of weighted  sum scalarizations in the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Thus, we solve problem (WSλ) for for adaptively chosen values  of λ ∈ R2  >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Although we solve the IP scalarization in the root node the parameter λ is gained from  the currently active node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Thereby, λ is determined by the largest approximated local hypervolume  gap in the active node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This gap is spanned by two points in the incumbent list together with their  local upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Note that these points spanning the largest gap are already determined if the  local hypervolume gap branching strategy is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The corresponding value of λ is determined by  computing the normal to the hyperplane that is defined by those two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Once λ is obtained,  we can solve problem (WSλ) with a single-objective integer linear programming solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Let ¯xλ be  9  the optimal solution of the weighted sum scalarization with weighting vector λ, then z(¯xλ) is a  supported non-dominated point of (BO01LP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Thus, we can add this point to the incumbent list (if  it was not found in previous iterations) and filter the resulting list for non-dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Moreover,  the solution of integer scalarizations can also be used to tighten the lower bound set, since the level  set {z ∈ R2 : λ⊤z = WSλ(¯xλ)} provides the valid inequality λ⊤z(x) ≥ WSλ(¯xλ) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  z1  z2  lu1  lu2  lu3  z1  z2  z3  z4  z(¯xλ)  (a)  z1  z2  lu4  lu5  z1  z5  z4  (b)  z1  z2  lu4  lu5  z1  z5  z4  (c)  z1  z2  lu4  lu5  z1  z5  z4  (d)  Figure 2: Example of updating the lower and upper bound with the usage of the weighted sum  scalarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Figure 2 illustrates the update of the lower and upper bound set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In Figure 2a, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' , z4 indicate  points that are currently in the incumbent list U¸ and lu1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' , lu3 are the corresponding local upper  bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The point z(¯xλ) is obtained by solving a weighted sum scalarization (WSλ) to integer  optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Since the new point is not contained in the incumbent list so far, we can update the  upper bound as it is shown in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The new incumbent list then reads as U := {z(¯xλ)} ∪ {z ∈  U : z(¯xλ) � z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Moreover, the lower bound set L can be updated by integrating the blue hyperplane  into the lower bound set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' L := {z ∈ L + R2  ≧ : λ⊤z ≥ WS(¯xλ)}N as it is shown in Figure 2c and  2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In this situation, both —the lower and upper bound— are updated, which is not the case in  10  general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The example illustrates the benefits of hybridizing multi-objective Branch and Bound with IP  scalarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Due to weak bounding, nodes may not be fathomed by dominance even if they do  not contain additional non-dominated points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The tighter upper bound increases the probability of  fathoming a node by dominance in later iterations of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Also, the lower bound might be  improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Since we are solving an IP scalarization in the root node, the obtained optimal level set is  a valid inequality for all subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We combine our new branching strategy and the augmentation  with IP scalarizations to our first hybrid Branch and Bound approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Hybrid Branch and Bound Algorithm using Weighted Sum Scalarization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Lower bound: linear relaxation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Upper bound: incumbent list  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Node selection: node with the largest total/local hypervolume gap  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Branching rule: most fractional  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Adaptively solve weighted sum scalarizations in the root node to integer optimality to improve  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='lower and upper bounds by objective space information  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Instead of using a static depth-first strategy (as in the general Branch and Bound framework in  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Section 3) we apply the dynamic strategy based on the hypervolume gap (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Even  though the extreme points of the lower bound sets might be updated by the weighted sum scalar-  ization, the branching variable is selected based on the original lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This is due to the  fact that the preimages of such intersection points of IP scalarizations and the lower bound set  are in general not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Note that the weighted sum IP scalarizations are included adaptively  into the Branch and Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The description of their algorithmic control, however, is postponed to  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In order to conclude the description of the proposed hybrid Branch and Bound algorithm using  weighted sum scalarizations, we want to briefly discuss its advantages and shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Firstly, it  is easy to determine the scalarization parameter λ and to integrate the hyperplane into the lower  bound set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Its advantage, however, is that the lower bound remains convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Therefore, the check  for fathoming by dominance remains intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Unfortunately, the weighted sum scalarization can  only find supported efficient solutions and the lower bound cannot be improved beyond the convex  hull of YN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This motivates us to consider in the following the augmented weighted Tchebycheff  scalarization, a scalarization approach which can determine also unsupported efficient solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2 Using Augmented Weighted Tchebycheff Scalarization  We start by defining the weighted Tchebycheff norm: Let wi > 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' , p be positive weights  with �p  i=1 wi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Then the weighted Tchebycheff norm of a vector z ∈ Rp is defined by  ∥z∥w  ∞ := max  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=',p  �  wi |zi|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  (1)  So, the weighted Tchebycheff scalarization of a multi-objective optimization problem (MOILP) with  respect to a given reference point s ∈ Rp can be written as:  min  �  ∥z(x) − s∥w  ∞ : x ∈ X  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  (2)  If the reference point is chosen such that s < z(x) for all x ∈ X, every efficient solution can be deter-  mined as optimal solution of the weighted Tchebycheff scalarization (2) by variation of w ∈ Rp  + (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', Miettinen, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Nevertheless, optimal solutions of the weighted Tchebycheff scalarization cor-  respond in general only to weakly efficient solutions of the multi-objective problem (Steuer and Choo,  1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Miettinen, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This shortcoming is compensated by an additive augmentation term in the  augmented weighted Tchebycheff norm  ∥z∥w  τ := ∥z∥w  ∞ + τ ∥z∥1 ,  (3)  11  where ∥z∥1 = |z1| + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' + |zp| denotes the L1-norm, wi ≥ 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' , p, �p  i=1 wi = 1 and τ >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Steuer and Choo (1983) proposed the augmented weighted Tchebycheff scalarization given in  (AWT w  τ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  min AWT w  τ (x) := ∥z(x) − s∥w  τ  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' x ∈ X  (AWT w  τ )  Thereby, the augmentation term makes the augmented weighted Tchebycheff norm a strongly mono-  tone norm and thus the objective function of (AWT w  τ ) a strongly increasing achievement scalar-  izing function (Miettinen, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Consequently, every optimal solution of (AWT w  τ ) is efficient for  (MOILP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Note that an appropriate choice of the parameter τ is difficult in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' On the one hand, too  small values of τ may lead to numerical difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' On the other hand, non-supported efficient  solutions might be suboptimal for (AWT w  τ ) if the value of τ is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' However, for bi-objective  integer programming Dächert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' (2012) propose an adaptive method to determine an optimal  value of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We use this proposed parameters w1, w2 and τ for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  As a reference point s we use the local ideal point of two adjacent non-dominated points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Since the  augmented weighted Tchebycheff scalarization can only determine non-dominated points (and the  corresponding efficient solutions) which are (strictly) dominated by the reference point, we obtain a  non-dominated point in this box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The goal to improve the lower bound set beyond the convex hull of non-dominated points is the  motivation to solve augmented weighted Tchebycheff scalarizations to integer optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Figure  3 shows an example how such an update of the bounds could look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Here, z1 and z2 are two  known non-dominated points (obtained with the weighted sum IP scalarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Point z3 is a  non-supported non-dominated point that has not been found yet in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' By using the local  ideal point of z1 and z2 as the reference point s, Figure 3b illustrates how the non-dominated point  z3 is found by applying the augmented weighted Tchebycheff scalarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In Figure 3c and 3d  the resulting improvements of the lower and upper bound are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Obviously the lower bound  is improved beyond the convex hull of YN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We now define our second hybrid Branch and Bound  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='approach:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Hybrid Branch and Bound Algorithm using augmented weighted Tchebycheff Scalarization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Lower bound: linear relaxation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Upper bound: incumbent list  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Node selection: node with the biggest total/local hypervolume gap  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Branching rule: most fractional  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Adaptively solve weighted sum and augmented weighted Tchebycheff scalarizations in the root  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='node to integer optimality to improve lower and upper bounds by objective space information  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='Additionally to the weighted sum scalarization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' we use the augmented weighted Tchebycheff scalar-  ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Since two adjacent non-dominated points are required as input of the augmented weighted  Tchebycheff scalarization, we cannot rely on points in the incumbent list, which are only non-  dominated so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In fact, we apply augmented weighted Tchebycheff IP scalarizations only to boxes  spanned by points obtained as optimal solutions of the weighted sum scalarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Thus, we do not  rely on parameters from the currently active node, but solve the augmented weighted Tchebycheff  scalarization in the largest area defined by two adjacent known non-dominated points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  When using augmented weighted Tchebycheff IP scalarizations, the lower bound can become  tighter than the convex hull of the set of non-dominated points, which reduces the area where  new non-dominated points can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Additionally, we can find non-supported non-dominated  points in early stages of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This improves the upper bound in the beginning resulting  in a higher chance of fathoming a node by dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' However, this also implies that the lower  12  z1  z2  z1  z2  z3  lu1  (a)  z1  z2  z1  z2  z3  lu1  s  (b)  z1  z2  z1  z2  z3  lu2  lu3  s  (c)  z1  z2  z1  z2  z3  lu2  lu3  (d)  Figure 3: Example of updating the lower and upper bound with the usage of the augmented weighted  Tchebycheff scalarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  bound gets non-convex in general, which makes the fathoming tests significantly harder, and the  lower bound improves only locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='3 Algorithmic Control of IP Scalarizations  In the previous subsections we did not specify when to solve IP scalarizations, which implies a sig-  nificant computational cost itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' However, this might be the most crucial part within the presented  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Obviously, we aim at gaining as much information as possible by solving IP scalarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  More objective space information will lead to tighter bounds that reduce the number of created  nodes, due to a higher probability of fathoming by dominance and smaller search zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Moreover, a  reduced number of created nodes will reduce the total computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' At the same time, solving  overly many IP scalarizations will have a negative impact on the computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Furthermore,  at a certain point the lower and upper bound will not improve anymore when solving additional IP  13  scalarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  So, there exists a trade-off between the reduction of the number of created subproblems and the  decrease of the computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The difficulty is to find an appropriate condition to trigger an IP  scalarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Obviously, solving IP scalarizations more frequently in the beginning of the Branch  and Bound algorithm is very promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The earlier the lower and upper bounds are improved  the more nodes might be fathomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Moreover, solving the IP scalarization when the active node  has weak bounds will lead to stronger improvements than in later stages of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This  is complemented by our adaptive branching strategy, which tends to select subproblems with weak  lower bounds first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The hybrid Branch and Bound algorithm using augmented weighted Tchebycheff scalarization  entails also another problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The augmented weighted Tchebycheff scalarization improves the lower  bound just locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' If we use this scalarization at the beginning of the algorithm instead of the  weighted sum scalarization, this could lead to an increase of created nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Once again, the intuitive  idea is to start with the weighted sum IP scalarization more frequently in the beginning of the  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This ensures that the lower bound improves globally at early stages of the Branch and  Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The augmented weighted Tchebycheff scalarization should be used in later stages of the  algorithm to find non-supported non-dominated points and to improve the lower bound locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The  efficiency of this idea and other approaches will be shown in the next section where we present  numerical test results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  5 Numerical Results  All algorithms were implemented in Julia 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1 and the linear relaxations were solved with Bensolve  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The numerical tests were executed on a single core of a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='20 GHz Intel® Core™ i7-8700 CPU  processor in a computer with 32 GB RAM, running under openSUSE linux Leap 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  We present numerical results of our new approaches and compare them to the general Branch  and Bound framework presented in Section 3 which we use as baseline implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We consider  three different types of problems: knapsack problems, assignment problems and discrete facility  location problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Multiple combinations of parameter settings are used to solve these test problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Thereby, we compare the average number of explored nodes, the average number of solved IPs and  the average computation time for 20 instances per problem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The different evaluated approaches  are  the generic bi-objective Branch and Bouch (BB),  bi-objective Branch and Bound using the local (BS1) respectively global (BS2) hypervolume  gap as node selection criterion,  hybrid Branch and Bound including weighted sum IP scalarizations (WS), and  different combinations of the hybrid Branch and Bound algorithm using weighted sum IP scalar-  ization (M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β) and hybrid Branch and Bound algorithm using weighted sum and augmented  weighted Tchebycheff IP scalarization (M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The parameter α ∈ {1, 2, 3} controls how often IP scalarizations are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Since the algorithmic  control of IP scalarization is selected for each value of α problem dependent, it is described in  detail in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' However, the larger the parameter α is chosen, the fewer IP  scalarizations are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' With β we distinguish between the local (β = 1) and the global (β = 2)  hypervolume gap strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In the hybrid Branch and Bound algorithm using augmented weighted  Tchebycheff scalarization we also distinguish between integrating the objective space information  of the augmented weighted Tchebycheff into the lower bound (γ = 1) or not (γ = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Note that  the tested parameter values are not optimized but have shown to provide good results on our test  instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  14  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1 Bi-objective Multidimensional Knapsack Problems  We consider bi-objective, multidimensional knapsack problems with one, two and three linear re-  strictions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' m = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' For every problem size we randomly generate 20 instances of the form  max  n  �  i=1  ck  i xi  k = 1, 2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  n  �  i=1  wi xi ≤ b  n  �  i=1  vij xi ≤ dj  j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', m − 1  x ∈ {0, 1}n  with ck  i ∈ [50, 100], wi ∈ [5, 15], b = 5 n, vij ∈ [5, 15] and dj =  � r n  2  �  with r ∈ [5, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Depending on  the parameter α we specify when and how often IP scalarizations are solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β and WS  we apply the weighted sum scalarization every 10-th iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β we apply it every 10-th  iteration but only within the first n2 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β we apply the weighted sum scalarization  every 10-th iteration within the first n2/3 iterations, every n-th iteration within the next n2/3  iterations and every 2n-th iteration within the third n2/3 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='γ we apply the  weighted sum scalarization every 10-th iteration and every 50-th iteration the augmented weighted  Tchebycheff scalarization is used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='γ we operate like in M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β but after the first  n2 iterations we apply the augmented weighted Tchebycheff scalarization every 50-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In  M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='γ we operate like in M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β but after the first n2 iterations we apply the augmented weighted  Tchebycheff scalarization every 50-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' If a scalarization cannot be applied or the same IP  scalarization has already been solved before, no IP scalarization is solved in that iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  First of all, we notice that our branching strategies have a huge impact on the number of explored  nodes and the computation time in knapsack problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We observe that in general the local hyper-  volume gap strategy works better than the global hypervolume gap strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' With the local strategy  we can reduce the number of explored nodes by up to 76% (Table 1c and 2b) and the computation  time by up to 73% (Table 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Although the local strategy works better the global hypervolume  gap strategy has also a significant impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The number of explored nodes can be reduced by up to  58% (Table 2c) and the computation time by up to 52% (Table 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The number of nodes and the  computation time is reduced in all our approaches and we can notice that combinations with the  local hypervolume strategy work better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  By limiting the number of solved weighted sum IPs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' in M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β, M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β, M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='γ and  M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='γ) we notice two consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The number of nodes increases while the number of solved  IPs decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Although the number of nodes (and thus the number of considered subproblems) is  increasing, the total computation time decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This implies that the reduced computation time  to solve IP scalarizations compensates the increase of nodes, which results in a trade-off between the  number of explored nodes and the computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Another interesting aspect can be observed in  M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
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+page_content=' The computation time can be reduced if we do not integrate the augmented  weighted Tchebycheff objective level set into the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This can be explained by the fact that  the lower bound improvements of augmented weighted Tchebycheff are only local and do not com-  pensate the computation time needed to integrate the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The intuitive assumption that  the number of explored nodes will then rise significantly is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' So, both our branching strategies  work better, if we do not consider the local updates of the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  We can reach a reduction of the explored nodes by up to 83% (Table 2b) and a reduction of the  computation time by up to 80% (Table 2b) in the best case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The strategies M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
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+page_content='2  seem to work best for knapsack problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In most cases these two strategies have the largest impact  on the number of explored nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
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+page_content='6  (d) Knapsack problem with m = 3 constraints  and n = 80 variables  Table 2: Numerical results of the bi-objective, multidimensional knapsack problems  17  Tchebycheff objective space information into the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Note that with rising numbers of  variables and constraints the hybridization techniques have larger impact on the performance of the  Branch and Bound algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2 Bi-objective Assignment Problems  We consider bi-objective assignment problems having n = ℓ2 variables,  max  ℓ  �  i=1  ℓ  �  j=1  ck  ij xij  k = 1, 2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  ℓ  �  i=1  xij = 1  j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', ℓ  ℓ  �  j=1  xij = 1  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', ℓ  x ∈ {0, 1}ℓ×ℓ  where the cost coefficients ck  ij ∈ [50, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The algorithmic strategy for the solution of IP scalariza-  tions depending on the value of the parameter α is chosen similarly to the previous case of knapsack  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' However, we adapt the boundaries due to the different number of nodes to explore in as-  signment problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β weighted sum scalarizations are solved every 10-th iteration to integer  optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β we apply the weighted sum every 10-th iteration within the first n·ℓ iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β we apply the weighted sum every 10-th iteration within the first n · ℓ/3 iterations, every  ℓ-th iteration in the next n · ℓ/3 iterations and every n-th iteration in the third n · ℓ/3 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  For M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='γ we use the same algorithmic strategy as in hybrid Branch and Bound for knapsack  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' If a scalarization cannot be applied or an IP with identical objective function has been  solved prior, no IP is solved in that iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Due to the total unimodularity of the assignment problem, the weighted sum scalarizations do in  general not improve the lower bound sets of subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' However, in situations where the weighted  sum IP scalarization generates a supported efficient solution, whose corresponding non-dominated  point is not an extreme point of the lower bound set, the local upper bounds move closer to the lower  bound set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' This reduces the gap between upper and lower bound and may lead to a decrease of the  explored subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Note that this update of the upper bound set may also have the contrary  effect (the number of considered subproblems increases), since it can change the order in which the  subproblems are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Nevertheless, the weighted sum IP scalarizations are necessary to find  non-dominated points on which the augmented weighted Tchebycheff scalarization can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Our branching strategies have a significant impact on the number of explored nodes and the  computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Again, the local hypervolume gap performs better than the global hypervolume  gap strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' With the local strategy we can reduce the number of explored nodes by up to 39%  (Table 3c) and the computation time by up to 33% (Table 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Using the global hypervolume gap  strategy we can reduce the number of explored nodes by up to 12% (Table 3b) and the computation  time by up to 12% (Table 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We reach a reduction of the explored nodes by up to 46% (Table 3d)  and a reduction of the computation time by up to 42% (Table 3d), in the best case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Again, the  strategies M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2 seem to work the best for assignment problems in terms of explored  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Nevertheless, M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2 leads to a better computation time which can be explained by the  same argument as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Furthermore, strategy BS1 works very well and is able to compete with  the previously mentioned strategies with respect to number of nodes and computation time without  solving a single IP scalarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  18  assignment problem, n = 100  version  nodes  time (s)  solved  IPs  BB  3117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1507  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='0  BS1  2422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
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+page_content='0  WS  3117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2631  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
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+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
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+page_content='45  (d) Assignment problem with n = 324 variables  Table 3: Numerical results of the bi-objective assignment problems  19  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='3 Bi-objective Discrete Facility Location Problems  We consider discrete facility location problems of the form  max  ℓ  �  i=1  q  �  j=1  ck  ij xij +  q  �  j=1  f k  j yj  k = 1, 2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  q  �  j=1  xij ≤ 1  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', ℓ  xij ≤ yj  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', ℓ, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=', q  x ∈ {0, 1}ℓ×q  y ∈ {0, 1}q  where ℓ is the number of customers and q the number of facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' We randomly generate coordinates  of ℓ customers and q facilities in a square with length 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The costs of the first objective function  correspond to the l1-distances between the customers and facilities, while the costs of the second  objective function are randomly generated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' c2  ij ∈ [1, 200]) and f k  j ∈ [200, 400].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The number  of variables is n = (ℓ + 1) q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  We restrict the numerical tests to problems where the number of  facilities is 20% of the number of customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Again, we need to specify when and how often integer  scalarizations are applied: We use the same methods as before but adapt the boundaries due to the  different number of nodes to explore in discrete facility location problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β we apply the  weighted sum IP scalarization every 10-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β we apply the weighted sum every  10-th iteration within the first n2/4 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β we apply the weighted sum scalarization  every 10-th iteration in the first n2/4 iterations, every n/2-th iteration in the next n2/4 iterations  and every n-th iteration in the third n2/4 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='γ we operate analogous to the  methods used for knapsack and assignment problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' If a scalarization cannot be applied or an IP  with identical objective function has been solved prior, no IP is solved in that iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Again, both new branching strategies have an impact on the number of explored nodes and the  computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The local strategy, once more, leads to better results, namely reduction of the  explored nodes by up to 52% (Table 4d) and reduction of the computation time by up to 45%  (Table 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' With the global hypervolume gap strategy we can reach a reduction of the explored  nodes by up to 24% (Table 4d) and a reduction of the computation time by up to 21% (Table 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  In the best case we can reach a reduction of the explored nodes by up to 57% (Table 4d) and of  the computation time by up to 50% (Table 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Once again, M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='2 seems to be the best choice  with respect to the number of explored nodes and with a rising number of variables it is also the  best choice regarding the computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' With a smaller number of variables, BS1 leads to good  results with respect to both aspects without solving a single IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='4 Summary  In all of the three tested problem classes (knapsack, assignment, discrete facility location) a signif-  icant reduction of the number of explored nodes and the computation time can be realized with  all presented combinations of the hybrid Branch and Bound approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' With increasing problem  size (number of variables) the impact of the presented augmentations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Furthermore, the  approaches perform better on problems where the gap between YN and the solution of the linear  relaxation is larger compared to totally unimodular problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  The local hypervolume gap strategy for the node selection outperforms the global hypervolume gap  strategy in our numerical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' A reason for this is that in the global hypervolume gap strategy many  small search zones can add up to a large gap although the lower bound might be quite close to the non-  dominated points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The local hypervolume gap strategy chooses the node with the largest search zone,  which has the biggest potential to reduce this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Moreover, the local hypervolume gap strategy  aims at an uniform distribution of points in the incumbent list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In our numerical test, M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
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+page_content='1  (d) Facility location problem with 30 customers  and 6 facilities  Table 4: Numerical results of the bi-objective integer facility location problem  21  out to be the best choice in most cases with respect to the number of explored nodes and computation  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' In this version, we use the local hypervolume gap strategy for the choice of the active node,  every 10-th iteration the weighted sum IP scalarization is applied and every 50-th iteration we  apply the augmented weighted Tchebycheff scalarization instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Futhermore, the objective space  information gained by the augmented weighted Tchebycheff scalarization is not used to update the  lower bound set, since its local improvements do not compensate the increased computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Although we need to solve more IPs than in most other approaches, the computation time is the  lowest compared to the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' So, using the augmented weighted Tchebycheff scalarization in the  beginning of the Branch and Bound works best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Due to the likelihood of finding non-supported non-  dominated points in the early stages of the algorithm, the upper bound can be further improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  This results to a higher probability of fathoming a node by dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Nevertheless, with version  BS1 we also achive a remarkable reduction in terms of the number of explored nodes and computation  time by using the local hypervolume gap strategy for node selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  6 Conclusion and Outlook  In this paper, we propose two approaches to incorporate objective space information in bi-objective  Branch and Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' By using the local or global (approximated) hypervolume gap as a node selection  criterion, we adapt the run of the Branch and Bound algorithm to the problem instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Additionally,  we adaptively solve scalarizations to integer optimality to improve the lower and the upper bound  set by the obtained objective space information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Our numerical results show the effectiveness of  both approaches and in particular of their combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' The dynamic branching rule based on the  local (approximated) hypervolume gap has large impact on the number of explored subproblems,  is compuationally efficient and can be easily integrated in other multi-objective Branch and Bound  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  While we tested in this paper the individual contributions of our augmentations on a generic  bi-objective Branch and Bound, we will continue to extend our ideas to multiple dimensions and  integrate them into a competetive multi-objective Branch and Bound framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Particularly in  higher dimensions, it may be promising to combine our approaches with objective space branching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  Acknowledgment  The authors thankfully acknowledge financial support by Deutsche Forschungsgemeinschaft, project  number KL 1076/11-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  References  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
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+page_content=' Hugot, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Vanderpooten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Solving efficiently the 0–1 multi-objective knapsack  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
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+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Belotti, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
+page_content=' Soylu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
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+page_content=' Ehrgott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfCC72/content/2301.11974v1.pdf'}
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diff --git a/BNE3T4oBgHgl3EQfTwqq/content/tmp_files/2301.04445v1.pdf.txt b/BNE3T4oBgHgl3EQfTwqq/content/tmp_files/2301.04445v1.pdf.txt
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+MNRAS 000, 1–10 (2023)
+Preprint 12 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Hi intensity mapping with MeerKAT: forecast for delay power spectrum
+measurement using interferometer mode
+Ming Zhang
+ID1, Yichao Li
+ID1★, Jing-Fei Zhang
+ID1, Xin Zhang
+ID1,2,3†
+1Key Laboratory of Cosmology and Astrophysics (Liaoning Province) & Department of Physics, College of Sciences, Northeastern University, Shenyang 110819, China
+2Key Laboratory of Data Analytics and Optimization for Smart Industry (Ministry of Education), Northeastern University, Shenyang 110819, China
+3National Frontiers Science Center for Industrial Intelligence and Systems Optimization, Northeastern University, Shenyang 110819, China
+12 January 2023
+ABSTRACT
+Neutral hydrogen (Hi) intensity mapping (IM) survey is generally regarded as a promising tool to explore the expansion history
+of the universe. In this work, we investigate the capability of MeerKAT Hi IM observation with interferometric mode to estimate
+the power spectrum and constrain cosmological parameters in typical dark energy models. Besides, a novel approach of ‘delay
+spectrum’ is employed, which can achieve separating the weak Hi signal from the foreground in the frequency space. We find
+that the different survey fields have a great influence on the fractional errors on power spectrum Δ𝑃/𝑃 in a limited observational
+time 10 h. With the integration time increasing from 10 h to 10000 h, Δ𝑃/𝑃 becomes distinctly smaller until the cosmic
+variance begins to dominate. In the total 10000 h observation, the lower Δ𝑃/𝑃 in low 𝑘 can be achieved when tracking 100
+points for MeerKAT L-band and 10 points for MeerKAT UHF-band. Through simulating 10000 h Hi IM survey, we obtain
+𝜎(Ωm) = 0.044 and 𝜎(𝐻0) = 2.8 km s−1 Mpc−1 with MeerKAT L-band, which are worse than the results of 𝜎(Ωm) = 0.028
+and 𝜎(𝐻0) = 2.0 km s−1 Mpc−1 with MeerKAT UHF-band in the ΛCDM model. However, in the 𝑤CDM and CPL models,
+MeerKAT shows a limited capability of constraining dark-energy equation of state, even though combined with Planck data. Our
+analysis is shown to be a useful guide for the near future MeerKAT observations in Hi IM survey.
+Key words: techniques: interferometric – cosmology: large scale structure of Universe – cosmology: cosmological parameters
+– radio lines: general
+1 INTRODUCTION
+In the last two decades, the accurate measurement of the cosmic
+microwave background (CMB) brings us to the era of precision cos-
+mology. Another promising method for reaching precision cosmol-
+ogy is the cosmic large scale structure (LSS) survey. At present, the
+cosmic LSS survey has made significant progress with galaxy red-
+shift survey in constraining cosmological parameters, e.g. the 2dF
+Galaxy Redshift Survey (Colless et al. 2001; Cole et al. 2005), the
+6dF Galaxy Survey (Jones et al. 2009; Beutler et al. 2011), the Wig-
+gleZ Dark Energy Survey (Blake et al. 2011; Drinkwater et al. 2010),
+the Baryon Oscillation Spectroscopic Survey (Alam et al. 2017) and
+the Dark Energy Survey (DES) (Abbott et al. 2018). In addition, the
+next generation galaxy survey targeting at even larger and deeper
+universe, such as Dark Energy Spectroscopic Instrument (Dey et al.
+2019), Large Synoptic Survey Telescope (Ivezić et al. 2019; Chisari
+et al. 2019) and Euclid (Amendola et al. 2018), will significantly
+improve the measurement precision in the near future.
+Besides, neutral hydrogen (Hi) is widely regarded as a promising
+tracer of the underlying dark matter distribution of the late Universe.
+Hi can be detected with radio telescope via its 21 cm line which
+★ E-mail: liyichao@mail.neu.edu.cn
+† E-mail: zhangxin@mail.neu.edu.cn
+arises from the spin-flip transition of ground state hydrogen atom.
+Nevertheless, it is known that detecting Hi signal in individual galaxy
+at higher redshfit requires good angular resolution and sensitivity,
+which relies on large radio interferometer in the near future, such as
+Square Kilometre Array (SKA). However, Hi survey of the cosmic
+LSS can be quickly carried out using existing radio telescopes via
+the intensity mapping (IM) methodology, which observes the total
+Hi intensity of the galaxies in a voxel (Battye et al. 2004; McQuinn
+et al. 2006; Loeb & Wyithe 2008; Chang et al. 2008; Wyithe et al.
+2008; Bagla et al. 2010; Seo et al. 2010; Lidz et al. 2011; Ansari
+et al. 2012). A variety of related studies show that Hi IM survey has a
+great performance in cosmology studies, e.g. constraints on the basic
+and dark-energy cosmological parameters (Pritchard & Loeb 2012;
+Bull et al. 2015; Pourtsidou et al. 2017; Olivari et al. 2018; Obuljen
+et al. 2018; Sprenger et al. 2019; Xu et al. 2018; Cheng et al. 2020;
+Xu & Zhang 2020; Jin et al. 2020; Xiao et al. 2021; Zhang et al.
+2021; Jin et al. 2021; Wu & Zhang 2022; Scelfo et al. 2022; Berti
+et al. 2022; Wu et al. 2022a,b), primordial non-Gaussianity (Camera
+et al. 2013; Xu et al. 2015; Li & Ma 2017; Ballardini et al. 2019;
+Karagiannis et al. 2020; Cunnington et al. 2020; Karagiannis et al.
+2021; Viljoen et al. 2021) and neutrino mass (Villaescusa-Navarro
+et al. 2015; Zhang et al. 2020).
+Hi IM LSS detection was first reported by measuring the cross-
+correlation function between the Hi map observed with Green Bank
+© 2023 The Authors
+arXiv:2301.04445v1  [astro-ph.CO]  11 Jan 2023
+
+2
+M. Zhang et al.
+Telescope (GBT) and the DEEP2 optical redshift survey (Chang
+et al. 2010). The cross-correlation power spectrum between Hi IM
+survey and optical galaxy survey was also detected with the GBT
+and the WiggleZ Dark Energy Survey (Masui et al. 2013). Anderson
+et al. (2018) reported the results of Hi IM maps cross-correlated with
+2dF galaxy survey. Tramonte & Ma (2020) presented the feasibility
+of measuring Hi maps with Parkes radio telescope and 2dF galaxy
+survey. A cross-correlation signal detected with Parkes Hi IM and
+WiggleZ redshift data was discussed in Li et al. (2021a). Wolz et al.
+(2022) gave a joint analysis of GBT IM and eBOSS survey. To date,
+the auto-correlation power spectrum is still not detected because of
+systematics and foreground contamination.
+Several large radio telescopes or interferometers, such as Five
+hundred-meter Aperture Spherical radio Telescope (FAST; Nan
+et al. 2011), Baryon acoustic oscillations In Neutral Gas Obser-
+vations (BINGO; Dickinson 2014; Wuensche et al. 2020) and SKA
+(Maartens et al. 2015; Bacon et al. 2020), are built or planned for
+Hi survey, which are expected to detect the auto-correlation power
+spectrum. In addition, interferometers have inherent advantages of
+being less sensitive to systematics, which can be regarded as a com-
+plementary approach to single-dish IM experiments. The smallest
+𝑘-modes accessible to an interferometer is determined by the short-
+est baselines. Therefore, a compacted radio interferometer array is
+required for probing the cosmic LSS, especially for the scales of
+baryon acoustic oscillation (BAO). To date, several interferometers,
+such as Canadian Hydrogen Intensity Mapping Experiment (CHIME;
+Bandura et al. 2014), Hydrogen Intensity and Real-time Analysis eX-
+periment (HIRAX; Newburgh et al. 2016) and Tianlai (Chen 2012;
+Wu et al. 2021), are designed with short baselines to probe the BAO.
+Besides, the MeerKAT radio telescope array with 64 dish anten-
+nas of 13.5 m diameter is a precursor of SKA, which is located in
+the Northern Cape Province of South Africa. MeerKAT has already
+been operating and producing preliminary results (Pourtsidou 2018;
+Wang et al. 2021; Knowles et al. 2021; Terni de Gregory et al. 2021;
+de Villiers & Cotton 2022). The MeerKAT Large Area Synoptic
+Survey (MeerKLASS; Santos et al. 2017; Irfan et al. 2021) proposed
+Hi IM with single-dish mode (Li et al. 2021b; Wang et al. 2021)
+and reported the cross-correlation power spectrum detection with
+the optical galaxy survey (Cunnington et al. 2022). In addition, the
+small field deep surveys using MeerKAT interferometric mode, such
+as the MeerKAT International GHz Tiered Extragalactic Exploration
+(MIGHTEE; Jarvis et al. 2018; Paul et al. 2021; Maddox, N. et al.
+2021), also provided the Hi cube. In this paper, we investigate the
+performance of Hi IM survey in measuring power spectrum using
+MeerKAT interferometric mode with different survey strategies and
+forecast the constraints on cosmological parameters in typical dark
+energy models including ΛCDM, 𝑤CDM and CPL models (Cheval-
+lier & Polarski 2001; Linder 2003).
+In this work, it is worth mentioning that we employ a novel ap-
+proach, i.e. the ‘delay spectrum’ analysis, which is first applied in the
+PAPER observation (Parsons & Backer 2009). It is known that the
+cosmic Hi signal is contaminated by the bright foreground emissions
+in the same frequency ranges, such as the synchrotron and free-free
+emission from the Galaxy and extra-galactic point sources. Gen-
+erally, the foreground contamination components have the smooth
+frequency spectra, which can be separated from the cosmic Hi fluc-
+tuation in the ‘delay spectrum’ space (Parsons et al. 2012; Liu et al.
+2014a,b; Liu & Shaw 2020).
+This paper is organized as follows. In Section 2, we provide the de-
+tailed description of estimating Hi signal power spectrum, MeerKAT
+survey strategy and its system noise, foregrounds and shot noise. In
+Section 3, we present the constraint results of the power spectrum
+and cosmological parameters. Finally, the conclusion is given in Sec-
+tion 4. In our simulation, we assume a flat ΛCDM model and keep
+all cosmological parameters fixed to Planck 2018 results (Aghanim
+et al. 2020).
+2 METHODOLOGY
+2.1 Hi delay spectrum
+Hi IM observation directly measures Hi brightness temperature. The
+sky brightness temperature can be defined as
+𝑇(𝜽, 𝜈) = ¯𝑇(𝜈)[1 + Δ𝑇(𝜽, 𝜈)] ,
+(1)
+where 𝜽 is the position vector on the sky, 𝜈 is the observation fre-
+quency, ¯𝑇(𝜈) and Δ𝑇(𝜽, 𝜈) denote the isotropic and fluctuating com-
+ponents of the Hi brightness temperature distribution, respectively.
+Radio interferometers detect Hi signals by measuring their visi-
+bilities, which are the cross-correlation signals between each pair of
+antennas. Assuming the flat-sky approximation, the visibility for a
+pair of antennae is given by
+𝑉(𝒖, 𝜈) =
+∫
+𝐴(𝜽, 𝜈)Δ𝑇(𝜽, 𝜈)𝑒−𝑖2𝜋𝒖·𝜽𝑑Ω ,
+(2)
+where 𝒖 = 𝜈𝒃/𝑐 is the baseline vector in units of wavelength, cor-
+responding to each antenna pair, where 𝒃 is the baseline vector in
+physical units and 𝑐 is the speed of light. 𝐴(𝜽, 𝜈) denotes the primary
+beam response of the telescope in the direction of 𝜽 and dΩ repre-
+sents the solid angle element. The visibility function can be Fourier
+transferred to the ‘delay spectrum’ space,
+˜𝑉(𝒖, 𝜏) =
+∫
+𝑉(𝒖, 𝜈)𝑒−𝑖2𝜋𝜈𝜏d𝜈,
+(3)
+where 𝜏 = 1/𝛿𝜈 is the corresponding delay of frequency interval 𝛿𝜈.
+Following McQuinn et al. (2006), Parsons et al. (2012) and Liu &
+Shaw (2020), Hi power spectrum can be obtained from measured
+visibilities in the form of ‘delay spectrum’
+𝑃D(𝑘⊥, 𝑘 ∥) ≡ 𝐴𝑒
+𝜆2𝐵
+𝑟2𝑟𝜈
+𝐵
+�� ˜𝑉(𝒖, 𝜏)
+��2
+� 𝜆2
+2𝑘B
+�2
+,
+(4)
+where 𝐴𝑒 and 𝐵 are the effective antenna area and bandwidth, re-
+spectively, 𝜆 denotes the wavelength at the center of the band, 𝑟 is
+the comoving distance to the redshift 𝑧 corresponding to 𝜆, 𝑟𝜈 is the
+comoving width along the line-of-sight (LoS) corresponding to the
+redshift range determined by 𝐵, and 𝑘B is the Boltzmann constant.
+Here, 𝑘⊥ and 𝑘 ∥ are the Fourier wave vectors perpendicular and par-
+allel to the LoS, respectively. They are related to the interferometric
+variables via
+𝑘⊥ = 2𝜋|𝒖|
+𝑟
+;
+𝑘 ∥ = 2𝜋𝜏𝜈21𝐻(𝑧)
+𝑐(1 + 𝑧)2
+.
+(5)
+where 𝜈21 = 1420 MHz is the rest-frame frequency of the 21 cm line.
+𝐻(𝑧) denotes the Hubble parameter as a function of redshift 𝑧.
+There are various advantages of this ‘delay spectrum’ method
+(Parsons et al. 2012; Vedantham et al. 2012; Paul et al. 2016). The
+different spectral behaviors between Hi signal and foreground make
+it possible to isolate the latter in the Fourier space. In addition, the
+Fourier conjugate variable is associated with the LoS cosmological
+distance, therefore the ‘delay spectrum’ constructed in this method
+can recover the cosmological 3D Hi power spectrum (Parsons et al.
+2012; Liu et al. 2014a,b).
+MNRAS 000, 1–10 (2023)
+
+Hi delay spectrum with MeerKAT interferometer mode
+3
+2.2 Hi signal power spectrum
+The mean sky brightness temperature of Hi 21 cm emission can be
+given by (Santos et al. 2015, 2017)
+¯𝑇𝑏(𝑧) ≈ 566ℎ
+� 𝐻0
+𝐻(𝑧)
+� � ΩHi(𝑧)
+0.003
+�
+(1 + 𝑧)2 𝜇K ,
+(6)
+where 𝐻0 = 100ℎ km s−1 Mpc−1 is the Hubble constant, ΩHi(𝑧) is
+the fractional density of Hi, which can be written as
+ΩHi(𝑧) =
+𝜌Hi(𝑧)
+𝜌𝑐,0(1 + 𝑧)3 ,
+(7)
+where 𝜌𝑐,0 is the critical density today and the proper Hi density is
+calculated by
+𝜌Hi(𝑧) =
+∫ 𝑀max
+𝑀min
+d𝑀 d𝑛
+d𝑀 (𝑀, 𝑧)𝑀Hi(𝑀, 𝑧) ,
+(8)
+where 𝑀 denotes the dark matter halo mass, d𝑛/d𝑀 is the proper
+halo mass function and 𝑀Hi(𝑀, 𝑧) denotes the Hi mass in a halo of
+mass 𝑀 at redshift 𝑧. Throughout this paper, we assume a simple
+power-law model of the halo mass following Santos et al. (2015),
+i.e., 𝑀Hi(𝑀) = 𝐴𝑀 𝛼 with 𝐴 ∼ 220 and 𝛼 = 0.6 that can fit both
+low- and high-redshift observations within reasonable accuracy.
+Considering the redshift space distortion (RSD) effect (Kaiser
+1987), Hi signal power spectrum can be written as
+𝑃Hi(𝑘, 𝜇, 𝑧) = ¯𝑇2
+𝑏 (𝑧)𝐹RSD(𝑘, 𝜇)𝑃(𝑘, 𝑧) ,
+(9)
+where 𝜇 ≡
+𝑘 ∥/𝑘 and the matter power spectrum 𝑃(𝑘, 𝑧)
+=
+𝐷2(𝑧)𝑃(𝑘, 𝑧 = 0) with 𝐷(𝑧) being the growth factor and 𝑃(𝑘, 𝑧 = 0)
+being the matter power spectrum at z = 0 which can be obtained by
+CAMB (Lewis et al. 2000). 𝐹RSD(𝑘, 𝜇) represents the RSD effect, and
+its form can be expressed as
+𝐹RSD(𝑘, 𝜇) =
+�
+𝑏2
+Hi(𝑧) + 𝑓 𝜇2�2
+exp
+�
+−𝑘2𝜇2𝜎2
+NL
+�
+.
+(10)
+Here 𝑏Hi(𝑧) is the Hi bias, written as
+𝑏Hi(𝑧) = 𝜌−1
+Hi (𝑧)
+∫ 𝑀max
+𝑀min
+d𝑀 d𝑛
+d𝑀 𝑀Hi(𝑀, 𝑧)𝑏(𝑀, 𝑧),
+(11)
+where 𝑏(𝑀, 𝑧) is the halo bias. 𝑓 ≡ dln𝐷/dln𝑎 is the linear growth
+rate with 𝑎 being the scale factor. 𝜎NL is the nonlinear dispersion
+scale with a middling value of 𝜎NL = 7Mpc. In this paper, for con-
+venience, ΩHi(𝑧) and 𝑏Hi(𝑧) are employed with the fitting functions
+following Santos et al. (2017).
+2.3 MeerKAT noise power spectrum
+The total thermal noise power spectrum can be written as (Bull et al.
+2015)
+𝑃N(𝑘, 𝜇, 𝑧) = 𝑟2(𝑧)𝑟𝜈(𝑧)
+𝑇2sys𝜆4
+𝑛pol𝜈21𝑡int𝐴2𝑒𝑛(𝒖)
+,
+(12)
+where 𝑛pol = 2 denotes the number of polarization. 𝑡int is the inte-
+gration time. The ratio of MeerKAT effective aperture and system
+temperature, 𝐴𝑒/𝑇sys, is frequency dependent. Currently, there are
+two frequency bands available for observation, i.e., the L-band (900–
+1700 MHz) and the UHF-band (580–1000 MHz). Because of the
+serious RFI contamination in the L-band frequency range, only the
+frequency range of 900–1200 MHz (0.18 < 𝑧 < 0.58) is used in our
+analysis. The full UHF-band is used in this analysis, corresponding
+500
+750
+1000
+1250
+1500
+ν [MHz]
+0
+2
+4
+6
+8
+10
+Ae/T sys [m2/K]
+MeerKAT L-band
+MeerKAT UHF-band
+Figure 1. The sensitivity designs for MeerKAT receivers, shown as 𝐴𝑒/𝑇sys
+for L-band and UHF-band.
+to 0.42 < 𝑧 < 1.45. In Fig. 1, we show 𝐴𝑒/𝑇sys for L-band and
+UHF-band.1
+Here 𝑛(𝒖) is the baseline density referring to the detailed 𝑢𝑣
+coverage of a particular observation in the 𝑢𝑣 plane. We employ the
+actual MeerKAT antenna coordinates and track the COSMOS field
+(RA=10h01m, Dec=+02d12m) following Paul et al. (2021). In a strict
+sense, 𝑛(𝒖) is also a function of frequency. We break the frequency
+range into a couple of Δ𝜈 = 60 MHz sub-bands. The 𝑢𝑣 coverage is
+assumed to be uniform within each sub-band and simulated according
+to the center frequency of each sub-band. The simulated 𝑢𝑣 coverage
+corresponding to the sub-bands in L-band centering at 𝑧 = 0.3 and
+in UHF-band centering at 𝑧 = 1.2 are shown in left and right panels
+of Fig. 2, respectively. For both cases, we assume 10 h tracking
+observation of the COSMOS field spanning over two days (the start
+time is 14:15 and 13:33 at UTC, respectively). The 𝑢𝑣 plane is
+segmented on to a discrete grid with cell size Δ𝑢 = Δ𝑣 = 60𝜆. The
+color represents the number of 𝑢𝑣 points within the grid. It is clear
+that there are more samples in the short 𝑢𝑣 distance region at low
+frequency band than high frequency band.
+Because of the uniform 𝑢𝑣 coverage assumption across the sub-
+band, the baseline density 𝑛(𝒖) and the corresponding total thermal
+noise power spectrum 𝑃N are only the functions of 𝑘⊥. According to
+Eq. (5), 𝑘⊥ is proportional to the 𝑢𝑣 distance, i.e. |𝒖| =
+√
+𝑢2 + 𝑣2. The
+circular averaged 𝑢𝑣 coverage within a |𝒖| shell of width Δ|𝒖| = 100𝜆
+are shown in Fig. 3, where the left panel shows the distribution
+corresponding to the sub-bands centering at 𝑧 = 0.3 and the right
+panel shows the one centering at 𝑧 = 1.2. Since |𝑘⊥| is proportional
+to the 𝑢𝑣 distance, the more densely populated 𝑢𝑣 points at smaller
+distance mean the higher sensitivity at the smaller |𝑘⊥| modes.
+It is known that the 𝑢𝑣 coverage also depends on the pointing direc-
+tion. In order to investigate the influence of the different sky zones,
+we also show in Fig. 3 the numbers of 𝑢𝑣 points with 10 h track-
+ing at different declinations: Dec = +30◦, +02◦, −30◦, −60◦, −90◦
+respectively. The case of Dec = +02◦ is the same as tracking the
+COSMOS field and the case of Dec = −30◦ corresponds to tracking
+a field that the transit line passes near Zenith the MeerKAT site. It
+is obvious that when the field is targeted farther from the zenith, the
+number of short baselines rises substantially for both the L-band and
+1 http://public.ska.ac.za/meerkat/meerkat-schedule
+MNRAS 000, 1–10 (2023)
+
+4
+M. Zhang et al.
+Figure 2. The distribution of baselines on a two-dimensional (2D) 𝑢𝑣 plane for 10 h tracking of the COSMOS field with sub-bands in MeerKAT L-band centering
+at 𝑧 = 0.3 (left panel) and in UHF-band centering at 𝑧 = 1.2 (right panel). The 𝑢𝑣 plane is segmented on to a discrete grid with cell-size Δ𝑢 = Δ𝑣 = 60𝜆. The
+color signifies the number of 𝑢𝑣 points on the grid.
+Figure 3. The average number of baselines as a function of 𝑢𝑣 distance, |𝒖| =
+√
+𝑢2 + 𝑣2, with bin size of Δ|𝒖| = 100𝜆, for 10 h tracking with sub-bands in
+MeerKAT L-band centering at 𝑧 = 0.3 (left panel) and in UHF-band centering at 𝑧 = 1.2 (right panel).
+UHF-band, which potentially increases the sensitivity at the smaller
+|𝑘⊥| modes.
+2.4 The foreground wedge and shot noise
+The foreground contamination, which is several orders of magnitude
+stronger than Hi signal, is the major challenge in recovering the Hi
+LSS. Since foreground spectrum is smooth across frequency chan-
+nels, it only contaminates the power spectrum close to the smallest
+𝑘 ∥. However, the property of the interferometer response function
+will cause foreground leakage into the high-𝑘⊥ modes. Therefore,
+we exclude the 𝑘⊥–𝑘 ∥ space modes within the foreground wedge
+(Datta et al. 2010; Morales et al. 2012; Liu et al. 2014a,b; Pober
+2015; Seo & Hirata 2016) that can be expressed as
+𝑘 ∥ < 𝑟(𝑧)𝐻(𝑧) sin(𝜃)
+𝑐(1 + 𝑧)
+𝑘⊥ ,
+(13)
+where 𝜃 denotes the field of view of the interferometer.
+In addition, shot noise needs to be taken into account in Hi IM
+survey. Because of Poisson fluctuations in halo number, the shot
+noise power spectrum is written as (Bull et al. 2015)
+𝑃shot
+Hi (𝑧) =
+� ¯𝑇𝑏(𝑧)
+𝜌Hi(𝑧)
+�2 ∫ 𝑀max
+𝑀min
+d𝑀 d𝑛
+d𝑀 𝑀2
+Hi(𝑀) .
+(14)
+Here Hi mass model is consistent with the description in the Hi
+signal power spectrum. Since shot noise is very low according to our
+calculation, it makes a very small contribution to the total noise.
+3 RESULTS
+In this section, we present the results of Hi IM survey analysis. In
+Section 3.1, we give a detailed analysis of the power spectrum in the
+different survey strategies. The relative errors on the BAO features
+and the constraints on cosmological parameters in different dark
+energy models are showed in Section 3.2.
+MNRAS 000, 1–10 (2023)
+
+uv coverage at z = 0.3 for RA=10:00:28.60 Dec=2:12:21.0
+104
+10
+103
+5
+(Y)
+0
+102
+101
+-10
+-15
+100
+-10
+0
+10
+u (k入)uv coverage at z = 1.2 for RA=10:00:28.60 Dec=2:12:21.0
+104
+10
+103
+5
+0
+102
+5
+101
+-10
+-15
+100
+-10
+0
+10
+u (k入)5000
+Dec=十30°
+Dec=-90°
+4000
+Number of u points
+Dec=+02°
+Dec=-60°
+3000
+Dec=-30°
+L-band
+2000
+1000
+102
+103
+10
+u distance in ^ （z = 0.312000
+Dec=±30°
+10000
+Dec=-90°
+Dec=+02°
+S
+Dec=-60°
+8000
+Dec=-30°
+UHF-band
+TO
+6000
+4000
+2000
+0
+102
+103
+10
+uv distance in ^ (z = 1.2Hi delay spectrum with MeerKAT interferometer mode
+5
+(a) Hi power spectrum
+(b) Total power spectrum
+Figure 4. 2D power spectrum at 𝑧 = 0.3. Left panel: Hi signal power spectrum 𝑃Hi. Right panel: Total power spectrum 𝑃tot with MeerKAT L-band 10 h
+observation.
+(a) Hi power spectrum
+(b) Total power spectrum
+Figure 5. 2D power spectrum at 𝑧 = 1.2. Left panel: Hi signal power spectrum 𝑃Hi. Right panel: Total power spectrum 𝑃tot with MeerKAT UHF-band 10 h
+observation.
+3.1 Power spectrum estimation
+The Hi LSS carries a significant quantity of cosmic information.
+However, it is extremely weak comparing to the brilliant foreground
+contamination. The 2D Hi power spectrum 𝑃Hi at 𝑧 = 0.3 (for
+MeerKAT L-band) is shown in the left panel of Fig. 4. The scales
+available for Hi IM in interferometric mode observation are limited
+by the detailed configuration. In summary, the scales for Hi IM survey
+are:
+𝑘min
+∥
+= 2𝜋/(𝑟𝜈Δ𝜈/𝜈21),
+𝑘max
+∥
+= 1/𝜎NL,
+𝑘min
+⊥
+= 2𝜋|𝒖|min/𝑟,
+𝑘max
+⊥
+= 2𝜋|𝒖|max/𝑟.
+(15)
+In principle, only the scales between minimum and maximum can be
+probed by a certain instrument, and the sensitivity to scales depends
+on the 𝑢𝑣 coverage. Adopting only 10 h observation, the total power
+spectrum 𝑃tot at the same redshift, which consists of the contributions
+of Hi signal, MeerKAT thermal noise, foregrounds and shot noise,
+is shown in the right panel of Fig. 4.
+The power spectrum at 𝑧 = 1.2 (for MeerKAT UHF-band) are
+shown in Fig. 5, where the Hi power spectrum 𝑃Hi is in the left panel
+and the total power spectrum 𝑃tot is in the right panel, respectively.
+Note that, in the right panel of Fig. 5, the brown part denotes the
+foreground wedge. We find that Hi signal is completely covered by
+the thermal noise and foregrounds for both MeerKAT L-band and
+UHF-band, which makes it difficult to obtain Hi signal directly.
+The Hi detection is quantified with the relative error of the power
+spectrum
+� Δ𝑃
+𝑃
+�2
+=
+�
+1
+8𝜋2𝑉bin
+∫
+𝑘2d𝑘d𝜇
+� 𝑃Hi(𝑘, 𝜇)
+𝑃tot(𝑘, 𝜇)
+�2�−1
+,
+(16)
+where 𝑉bin = 𝑆area𝑟2𝑟𝜈 Δ𝜈
+𝜈21 is the survey volume of each redshift bin
+with the survey area 𝑆area = 𝜋
+�
+1
+2
+𝜆
+13.5𝑚
+�2 �
+180
+𝜋
+�2
+.
+Firstly, we investigate the influence on the 𝑃(𝑘) error when track-
+ing the source at the different declinations. As is shown in Fig. 3,
+when tracking the source at Dec = +30◦, +02◦, −30◦, −60◦, −90◦,
+completely different numbers of 𝑢𝑣 points are obtained, e.g. there are
+more 𝑢𝑣 points in the shorter 𝑢𝑣 distance for the case of Dec = +30◦.
+In the top panel of Fig. 6, the relative errors of power spectrum with
+different tracking declinations are shown in different colors. For all
+the cases, we assume 10 h observation time. The results with L-band
+and UHF-band are shown in solid and dashed lines, respectively.
+Here, we divide the whole range of 𝑘 into 10 logarithmic bins. It is
+clear that the power spectrum uncertainty is reduced by more than
+MNRAS 000, 1–10 (2023)
+
+104
+10-1
+102
+ku [Mpc-1]
+6 × 10-2
+100
+4 × 10-2
+10-2
+3 × 10-2
+100
+10'
+102
+k [Mpc-1]107
+10-1
+105
+ku [Mpc-1]
+6 × 10-2
+103
+4 × 10-2
+3 × 102
+101
+10°
+10
+102
+k [Mpc-1]104
+10-1
+102
+ku [Mpc-1]
+100
+10-2
+10-
+100
+101
+k[Mpc-1]107
+10-1
+105
+ku [Mpc-1]
+103
+100
+101
+10
+10
+k}[Mpc-1]6
+M. Zhang et al.
+10−1
+100
+k [Mpc−1]
+100
+101
+102
+∆P/P
+10 hours
+L-band at Dec=+30◦
+L-band at Dec=+02◦
+L-band at Dec=−30◦
+L-band at Dec=−60◦
+L-band at Dec=−90◦
+UHF-band at Dec=+30◦
+UHF-band at Dec=+02◦
+UHF-band at Dec=−30◦
+UHF-band at Dec=−60◦
+UHF-band at Dec=−90◦
+10−1
+100
+101
+102
+k [Mpc−1]
+10−2
+100
+102
+104
+106
+∆P/P
+1 point
+L-band 10 hours
+L-band 100 hours
+L-band 1000 hours
+L-band 10000 hours
+UHF-band 10 hours
+UHF-band 100 hours
+UHF-band 1000 hours
+UHF-band 10000 hours
+10−1
+100
+101
+102
+k [Mpc−1]
+10−2
+10−1
+100
+101
+102
+103
+104
+∆P/P
+10000 hours
+L-band 1 point
+L-band 10 points
+L-band 100 points
+L-band 1000 points
+UHF-band 1 point
+UHF-band 10 points
+UHF-band 100 points
+UHF-band 1000 points
+Figure 6. Fractional errors on 𝑃(𝑘) obtained with MeerKAT L-band and UHF-band. Top panel: 10 h observations at the different declinations. Middle panel:
+Different observation times of tracking the COSMOS field. Bottom panel: Tracking different numbers of points in a 10000 h observation.
+MNRAS 000, 1–10 (2023)
+
+Hi delay spectrum with MeerKAT interferometer mode
+7
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+z
+10−2
+10−1
+100
+σ(DA)/DA
+L-band 10 points
+UHF-band 100 points
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+z
+10−1
+100
+101
+σ(H)/H
+L-band 10 points
+UHF-band 100 points
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+z
+10−1
+100
+101
+σ(fσ8)/(fσ8)
+L-band 10 points
+UHF-band 100 points
+Figure 7. Fractional errors on 𝐷𝐴(𝑧), 𝐻 (𝑧) and 𝑓 𝜎8(𝑧) obtained with MeerKAT L-band and UHF-band.
+a factor of two when tracking Dec = +30◦ compared to tracking
+Dec = −30◦ or Dec = −60◦. The results show that, with limited
+observation time, the tracking declination has a obvious influence on
+the results of the constraints on the power spectrum.
+Next, in order to assess the influence of integration time, we in-
+crease the integration time by assuming observations on the same
+field at the same local sidereal time as the existing data on different
+days, which means that we obtain the same 𝑢𝑣 points from multiple
+days coherently to increase the sensitivity of the same 𝑘 modes. In
+addition to the current 10 h observation, we further consider 100,
+1000 and 10000 h observations. In the middle panel of Fig. 6, we
+show the fractional errors on the power spectrum 𝑃(𝑘) for 10, 100,
+1000 and 10000 hours observation of tracking the COSMOS field
+with MeerKAT L-band (in blue) and UHF-band (in red). The results
+with different integration times are shown with different color satu-
+rations. It can be seen that, compared to L-band, using the UHF-band
+could measure smaller 𝑘 modes down to ∼ 0.1, which makes it pos-
+sible to detect cosmological LSS on the larger scales. In addition, it
+is expected that the lower Δ𝑃/𝑃 can be obtained with the observa-
+tion time increasing as is shown in the middle panel of Fig. 6. With
+MeerKAT UHF-band 10 h observation, the value of Δ𝑃/𝑃 could
+reach 1 roughly. The values of Δ𝑃/𝑃 are distinctly reduced when
+tracking 1000 h, approximately reaching 0.1. However, we find that
+when the integration time increases from 1000 h to 10000 h, the
+reduction of Δ𝑃/𝑃 is not significant at low 𝑘. It is mainly because
+the cosmic variance, which is limited by the survey volume, plays
+the dominating role.
+Therefore, we consider tracking multiple points equally in the total
+10000 h observation. In this case, compared to tracking one point
+with 10000 h, the survey volume 𝑉bin and the thermal noise power
+spectrum 𝑃N are increased by a factor of the number of points 𝑁.
+In our analysis, we calculate the additional fractional error on 𝑃(𝑘)
+for 𝑁 = 10, 100 and 1000 in the 10000 h observation, as shown
+in the bottom panel of Fig. 6. In order to constrain cosmological
+parameters, we expect to obtain lower Δ𝑃/𝑃 in low 𝑘. We find that
+the lower values of Δ𝑃/𝑃 in low 𝑘 are obtained when tracking 100
+points for MeerKAT L-band in the total 10000 h observation, while
+tracking 10 points for MeerKAT UHF-band. Therefore, we employ
+these two survey strategies for MeerKAT L-band and UHF-band,
+respectively, in the next subsection.
+3.2 Cosmological parameters
+In this subsection, we explore the capability of MeerKAT Hi IM
+survey with interferometer mode of constraining cosmological pa-
+rameters using the Fisher matrix method. Given the power spectrum
+measurement at a given redshift, the Fisher matrix for a set of ob-
+servables {𝑝} can be written as
+𝐹𝑖 𝑗 =
+1
+8𝜋2𝑉bin
+∫ 1
+−1
+d𝜇
+∫ 𝑘max
+𝑘min
+𝑘2d𝑘 𝜕𝑃tot
+𝜕𝑝𝑖
+𝜕𝑃tot
+𝜕𝑝 𝑗
+.
+(17)
+Here, we take the set of observables as {𝐷 𝐴(𝑧𝑖), 𝐻(𝑧𝑖), 𝑓 𝜎8(𝑧𝑖),
+𝑏𝜎8(𝑧𝑖), 𝜎NL} in each redshift bin 𝑧𝑖. The nuisance parameters
+𝑏𝜎8(𝑧𝑖) and 𝜎NL can be marginalized by selecting the submatrix of
+𝐹−1
+𝑖 𝑗 with only the appropriate columns and rows. Therefore, we can
+derive the measurement errors on 𝐷 𝐴(𝑧), 𝐻(𝑧) and 𝑓 𝜎8(𝑧).
+For MeerKAT L-band (900-1200 MHz) and UHF-band (580-1000
+MHz), we divide these frequency bands into some bins with equal
+bandwidth Δ𝜈 = 60 MHz and then obtain the estimates for the
+measurement errors on observables in the corresponding redshift
+bins. We plot the fractional measurement errors on 𝐷 𝐴(𝑧), 𝐻(𝑧)
+and 𝑓 𝜎8(𝑧) with MeerKAT 10000 h observation in Fig. 7. We find
+that the survey with interferometer mode has a better measurement
+on 𝐷 𝐴(𝑧), of which the fractional errors can reach roughly 10%
+for MeerKAT L-band and UHF-band. Comparatively speaking, the
+fractional measurement errors on 𝐻(𝑧) and 𝑓 𝜎8(𝑧) seem slightly
+larger, though MeerKAT UHF-band preforms slightly better than
+MeerKAT L-band.
+Next, from the cosmological measurements on 𝐷 𝐴(𝑧), 𝐻(𝑧) and
+𝑓 𝜎8(𝑧), we can constrain the various dark enenrgy models, including
+the ΛCDM, 𝑤CDM and CPL models, by performing a Markov Chain
+Monte Carlo (MCMC) analysis. The 1𝜎 errors of the cosmological
+parameters are summarized in Table 1. In addition, The 1𝜎 and
+2𝜎 posterior distribution contours for cosmological parameters are
+shown in Figs. 8–10.
+In the flat ΛCDM model, we obtain 𝜎(Ωm) = 0.044 and 𝜎(𝐻0) =
+2.8 km s−1 Mpc−1 with MeerKAT L-band and 𝜎(Ωm) = 0.028
+and 𝜎(𝐻0) = 2.0 km s−1 Mpc−1 with MeerKAT UHF-band. We
+find that UHF-band performs better than L-band in constraining
+Ωm and 𝐻0. Recently, Cunnington (2022) gave a result of 𝐻0 =
+69.1+8.4
+−5.7 km s−1 Mpc−1 with MeerKAT UHF-band 4000 h survey
+with single-dish mode. It is found that we give a better constraint
+on 𝐻0 with MeerKAT UHF-band interferometric mode although we
+use a longer observational time of 10000 h. In comparison with other
+radio telescopes, MeerKAT L-band and BINGO perform similarly,
+while MeerKAT UHF-band performs nearly as well as FAST in
+constraining Ωm and 𝐻0 in the flat ΛCDM model (Wu & Zhang
+2022). Compared to the Stage-III dark energy experiments, such as
+DES, we find that MeerKAT UHF-band gives a smaller error on Ωm
+than DES with Ωm = 0.339+0.032
+−0.031 in the ΛCDM model (Abbott et al.
+2020).
+In the 𝑤CDM model, in order to help break the parameter degen-
+MNRAS 000, 1–10 (2023)
+
+8
+M. Zhang et al.
+Table 1. The 1𝜎 errors of the cosmological parameters in the ΛCDM, 𝑤CDM, and CPL models using MeerKAT L-band and UHF-band or in combination
+with Planck data. Note that here 𝐻0 is in units of km s−1 Mpc−1.
+Error
+ΛCDM
+𝑤CDM
+CPL
+L-band
+UHF-band
+Planck+L-band
+Planck+UHF-band
+Planck+L-band
+Planck+UHF-band
+𝜎(Ωm)
+0.044
+0.028
+0.030
+0.024
+0.092
+0.046
+𝜎(𝐻0)
+2.8
+2.0
+3.5
+2.6
+6.1
+4.1
+𝜎(𝑤)
+−
+−
+0.12
+0.08
+−
+−
+𝜎(𝑤0)
+−
+−
+−
+−
+1.1
+0.6
+𝜎(𝑤𝑎)
+−
+−
+−
+−
+4.3
+2.0
+0.2
+0.3
+0.4
+0.5
+Ωm
+60
+65
+70
+75
+H0 [km s−1 Mpc−1]
+L-band
+UHF-band
+Figure 8. Constraints on Ωm and 𝐻0 with MeerKAT L-band and UHF-band
+in the ΛCDM model.
+eracy, we combine the BAO data from MeerKAT IM with Planck
+TT,TE,EE+lowE power spectrum (Aghanim et al. 2020) in the
+MCMC analysis. The 1𝜎 and 2𝜎 measurement error contours for
+Ωm, 𝐻0 and dark-energy equation of state parameter 𝑤 are shown in
+Fig. 9. We obtain 𝜎(Ωm) = 0.030, 𝜎(𝐻0) = 3.5 km s−1 Mpc−1
+and 𝜎(𝑤) = 0.12 with Planck+L-band and 𝜎(Ωm) = 0.024,
+𝜎(𝐻0) = 2.6 km s−1 Mpc−1 and 𝜎(𝑤) = 0.08 with Planck+UHF-
+band. It can be seen that MeerKAT UHF-band combined with Planck
+data gives tighter constraints on cosmological parameters in the
+𝑤CDM model, with the conclusion the same as in the ΛCDM model.
+For dark energy, we find that MeerKAT has a very limited capability
+of constraining 𝑤, and the error on 𝑤 is still larger even though in
+combination with Planck data.
+Finally, we forecast the constraints on cosmological parameters
+in the CPL model. The 1𝜎 and 2𝜎 measurement error contours are
+shown in Fig. 10. We focus on the DE equation of state parameters 𝑤0
+and 𝑤𝑎. We obtain 𝜎(𝑤0) = 1.1 and 𝜎(𝑤𝑎) = 4.3 with Planck+L-
+band, and 𝜎(𝑤0) = 0.6 and 𝜎(𝑤𝑎) = 2.0 with Planck+MeerKAT
+UHF-band. Note that dark energy dominates the evolution of the
+universe in the redshift range of 𝑧 ≲ 0.4. MeerKAT L-band has
+a very limited constraining power for dark energy at the range of
+0.18 < 𝑧 < 0.58, and MeerKAT UHF-band only surveys at the range
+of 0.42 < 𝑧 < 1.45. Therefore, MeerKAT in interferometer mode
+−1.5
+−1.0
+w
+0.2
+0.3
+0.4
+Ωm
+60
+70
+80
+H0 [km s−1 Mpc−1]
+60
+70
+80
+H0 [km s−1 Mpc−1]
+0.3
+0.4
+Ωm
+Planck+L-band
+Planck+UHF-band
+Figure 9. Constraints on Ωm, 𝐻0 and 𝑤 with MeerKAT L-band and UHF-
+band in combination with Planck data in the 𝑤CDM model.
+cannot give stringent constraints on dark energy. But we still keep
+optimistic since the precise measurements on dark energy would be
+achieved by the future larger radio telescopes, such as HIRAX and
+SKA (Wu & Zhang 2022; Wu et al. 2022a,b).
+4 CONCLUSIONS
+In this work, we give a detailed analysis on measuring the Hi IM
+delay power spectrum using the MeerKAT interferometer mode. We
+also discuss the capability of MeerKAT interferometer mode of con-
+straining cosmological parameters.
+We use the Fisher matrix method to estimate the Hi power spec-
+trum with MeerKAT IM observation. We find that the different survey
+fields have the distinct impacts on determining the power spectrum
+errors in the limited observational time of 10 hours. As the obser-
+vational time increases from 10 h to 10000 h, the power spectrum
+errors are reduced evidently until the cosmic variance begins to dom-
+inate. We also discuss the different survey strategies and find that the
+lower fractional errors on power spectrum at low 𝑘 are obtained when
+tracking 100 points for L-band and tracking 10 points for UHF-band
+in a total 10000 h observation.
+We obtain the measurement errors on 𝐷 𝐴(𝑧), 𝐻(𝑧) and 𝑓 𝜎8(𝑧)
+MNRAS 000, 1–10 (2023)
+
+Hi delay spectrum with MeerKAT interferometer mode
+9
+−2
+0
+2
+w0
+0.2
+0.4
+0.6
+Ωm
+50
+60
+70
+80
+H0 [km s−1 Mpc−1]
+−10
+−5
+0
+wa
+−10
+−5
+0
+wa
+50
+60
+70
+80
+H0 [km s−1 Mpc−1]
+0.4
+0.6
+Ωm
+Planck+L-band
+Planck+UHF-band
+Figure 10. Constraints on Ωm, 𝐻0, 𝑤0 and 𝑤𝑎 with MeerKAT L-band and
+UHF-band in combination with Planck data in the CPL model.
+by using the Fisher matrix, and then use these measurements to
+constrain cosmological parameters in typical dark energy models, in-
+cluding ΛCDM, 𝑤CDM and CPL models, by performing the MCMC
+analysis. We obtain 𝜎(Ωm) = 0.028 and 𝜎(𝐻0) = 2.0 km s−1
+Mpc−1 with MeerKAT UHF-band which are better than the results
+of 𝜎(Ωm) = 0.044 and 𝜎(𝐻0) = 2.8 km s−1 Mpc−1 with MeerKAT
+L-band in the ΛCDM model. However, MeerKAT has a very limited
+constraining power for dark-energy equation of state, such as 𝑤 in
+the 𝑤CDM model and 𝑤0 and 𝑤𝑎 in the CPL model, even though in
+combination with Planck data.
+Though MeerKAT L-band and UHF-band Hi IM surveys in in-
+terferometer mode have a very limited constraining power for dark
+energy, our analysis still provide a useful guide for the near future
+MeerKAT survey. It is expected that the future larger radio telescope
+arrays, such as SKA, will have a much better and powerful perfor-
+mance on cosmological research. In addition, MeerKAT baselines
+are not short enough for detecting large cosmological scales, but the
+measurements with MeerKAT interferometer mode on these scales
+are still very useful in detecting Hi content of galaxies, obtaining the
+cross-correlation between Hi content and star formation rates (Wolz
+et al. 2016), constraining warm dark matter (Carucci et al. 2015) and
+breaking the degeneracy between ΩHi and 𝑏Hi (Chen et al. 2021).
+These aspects deserve further detailed investigations in the future.
+ACKNOWLEDGEMENTS
+We thank Peng-Ju Wu and Li-Yang Gao for helpful discussions.
+This work was supported by the National SKA Program of China
+(Grants Nos. 2022SKA0110200 and 2022SKA0110203) and the Na-
+tional Natural Science Foundation of China (Grants Nos. 11975072,
+11875102, and 11835009).
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable request
+to the corresponding author.
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+
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+page_content=' China  3National Frontiers Science Center for Industrial Intelligence and Systems Optimization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Northeastern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Shenyang 110819,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' China  12 January 2023  ABSTRACT  Neutral hydrogen (Hi) intensity mapping (IM) survey is generally regarded as a promising tool to explore the expansion history  of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In this work, we investigate the capability of MeerKAT Hi IM observation with interferometric mode to estimate  the power spectrum and constrain cosmological parameters in typical dark energy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Besides, a novel approach of ‘delay  spectrum’ is employed, which can achieve separating the weak Hi signal from the foreground in the frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We find  that the different survey fields have a great influence on the fractional errors on power spectrum Δ𝑃/𝑃 in a limited observational  time 10 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' With the integration time increasing from 10 h to 10000 h, Δ𝑃/𝑃 becomes distinctly smaller until the cosmic  variance begins to dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In the total 10000 h observation, the lower Δ𝑃/𝑃 in low 𝑘 can be achieved when tracking 100  points for MeerKAT L-band and 10 points for MeerKAT UHF-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Through simulating 10000 h Hi IM survey, we obtain  𝜎(Ωm) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='044 and 𝜎(𝐻0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='8 km s−1 Mpc−1 with MeerKAT L-band, which are worse than the results of 𝜎(Ωm) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='028  and 𝜎(𝐻0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='0 km s−1 Mpc−1 with MeerKAT UHF-band in the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' However, in the 𝑤CDM and CPL models,  MeerKAT shows a limited capability of constraining dark-energy equation of state, even though combined with Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Our  analysis is shown to be a useful guide for the near future MeerKAT observations in Hi IM survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Key words: techniques: interferometric – cosmology: large scale structure of Universe – cosmology: cosmological parameters  – radio lines: general  1 INTRODUCTION  In the last two decades, the accurate measurement of the cosmic  microwave background (CMB) brings us to the era of precision cos-  mology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Another promising method for reaching precision cosmol-  ogy is the cosmic large scale structure (LSS) survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' At present, the  cosmic LSS survey has made significant progress with galaxy red-  shift survey in constraining cosmological parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' the 2dF  Galaxy Redshift Survey (Colless et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Cole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2005), the  6dF Galaxy Survey (Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Beutler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2011), the Wig-  gleZ Dark Energy Survey (Blake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Drinkwater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2010),  the Baryon Oscillation Spectroscopic Survey (Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2017) and  the Dark Energy Survey (DES) (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In addition, the  next generation galaxy survey targeting at even larger and deeper  universe, such as Dark Energy Spectroscopic Instrument (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  2019), Large Synoptic Survey Telescope (Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Chisari  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2019) and Euclid (Amendola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2018), will significantly  improve the measurement precision in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Besides, neutral hydrogen (Hi) is widely regarded as a promising  tracer of the underlying dark matter distribution of the late Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Hi can be detected with radio telescope via its 21 cm line which  ★ E-mail: liyichao@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='cn  † E-mail: zhangxin@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='cn  arises from the spin-flip transition of ground state hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Nevertheless, it is known that detecting Hi signal in individual galaxy  at higher redshfit requires good angular resolution and sensitivity,  which relies on large radio interferometer in the near future, such as  Square Kilometre Array (SKA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' However, Hi survey of the cosmic  LSS can be quickly carried out using existing radio telescopes via  the intensity mapping (IM) methodology, which observes the total  Hi intensity of the galaxies in a voxel (Battye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' McQuinn  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Loeb & Wyithe 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Wyithe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Bagla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Seo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Lidz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Ansari  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' A variety of related studies show that Hi IM survey has a  great performance in cosmology studies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' constraints on the basic  and dark-energy cosmological parameters (Pritchard & Loeb 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Bull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Pourtsidou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Olivari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Obuljen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Sprenger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Xu & Zhang 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Wu & Zhang 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Scelfo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Berti  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2022a,b), primordial non-Gaussianity (Camera  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Li & Ma 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Ballardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Karagiannis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Cunnington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Karagiannis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Viljoen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2021) and neutrino mass (Villaescusa-Navarro  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Hi IM LSS detection was first reported by measuring the cross-  correlation function between the Hi map observed with Green Bank  © 2023 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='04445v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='CO]  11 Jan 2023  2  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Telescope (GBT) and the DEEP2 optical redshift survey (Chang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The cross-correlation power spectrum between Hi IM  survey and optical galaxy survey was also detected with the GBT  and the WiggleZ Dark Energy Survey (Masui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Anderson  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' (2018) reported the results of Hi IM maps cross-correlated with  2dF galaxy survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Tramonte & Ma (2020) presented the feasibility  of measuring Hi maps with Parkes radio telescope and 2dF galaxy  survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' A cross-correlation signal detected with Parkes Hi IM and  WiggleZ redshift data was discussed in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Wolz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  (2022) gave a joint analysis of GBT IM and eBOSS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' To date,  the auto-correlation power spectrum is still not detected because of  systematics and foreground contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Several large radio telescopes or interferometers, such as Five  hundred-meter Aperture Spherical radio Telescope (FAST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Nan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2011), Baryon acoustic oscillations In Neutral Gas Obser-  vations (BINGO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Dickinson 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Wuensche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2020) and SKA  (Maartens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2020), are built or planned for  Hi survey, which are expected to detect the auto-correlation power  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In addition, interferometers have inherent advantages of  being less sensitive to systematics, which can be regarded as a com-  plementary approach to single-dish IM experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The smallest  𝑘-modes accessible to an interferometer is determined by the short-  est baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Therefore, a compacted radio interferometer array is  required for probing the cosmic LSS, especially for the scales of  baryon acoustic oscillation (BAO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' To date, several interferometers,  such as Canadian Hydrogen Intensity Mapping Experiment (CHIME;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Bandura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2014), Hydrogen Intensity and Real-time Analysis eX-  periment (HIRAX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Newburgh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2016) and Tianlai (Chen 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2021), are designed with short baselines to probe the BAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Besides, the MeerKAT radio telescope array with 64 dish anten-  nas of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='5 m diameter is a precursor of SKA, which is located in  the Northern Cape Province of South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' MeerKAT has already  been operating and producing preliminary results (Pourtsidou 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Knowles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Terni de Gregory et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  de Villiers & Cotton 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The MeerKAT Large Area Synoptic  Survey (MeerKLASS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Irfan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2021) proposed  Hi IM with single-dish mode (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2021)  and reported the cross-correlation power spectrum detection with  the optical galaxy survey (Cunnington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In addition, the  small field deep surveys using MeerKAT interferometric mode, such  as the MeerKAT International GHz Tiered Extragalactic Exploration  (MIGHTEE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Jarvis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Paul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Maddox, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  2021), also provided the Hi cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In this paper, we investigate the  performance of Hi IM survey in measuring power spectrum using  MeerKAT interferometric mode with different survey strategies and  forecast the constraints on cosmological parameters in typical dark  energy models including ΛCDM, 𝑤CDM and CPL models (Cheval-  lier & Polarski 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Linder 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  In this work, it is worth mentioning that we employ a novel ap-  proach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' the ‘delay spectrum’ analysis, which is first applied in the  PAPER observation (Parsons & Backer 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' It is known that the  cosmic Hi signal is contaminated by the bright foreground emissions  in the same frequency ranges, such as the synchrotron and free-free  emission from the Galaxy and extra-galactic point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Gen-  erally, the foreground contamination components have the smooth  frequency spectra, which can be separated from the cosmic Hi fluc-  tuation in the ‘delay spectrum’ space (Parsons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  2014a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Liu & Shaw 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In Section 2, we provide the de-  tailed description of estimating Hi signal power spectrum, MeerKAT  survey strategy and its system noise, foregrounds and shot noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In  Section 3, we present the constraint results of the power spectrum  and cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Finally, the conclusion is given in Sec-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In our simulation, we assume a flat ΛCDM model and keep  all cosmological parameters fixed to Planck 2018 results (Aghanim  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  2 METHODOLOGY  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='1 Hi delay spectrum  Hi IM observation directly measures Hi brightness temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The  sky brightness temperature can be defined as  𝑇(𝜽, 𝜈) = ¯𝑇(𝜈)[1 + Δ𝑇(𝜽, 𝜈)] ,  (1)  where 𝜽 is the position vector on the sky, 𝜈 is the observation fre-  quency, ¯𝑇(𝜈) and Δ𝑇(𝜽, 𝜈) denote the isotropic and fluctuating com-  ponents of the Hi brightness temperature distribution, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Radio interferometers detect Hi signals by measuring their visi-  bilities, which are the cross-correlation signals between each pair of  antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Assuming the flat-sky approximation, the visibility for a  pair of antennae is given by  𝑉(𝒖, 𝜈) =  ∫  𝐴(𝜽, 𝜈)Δ𝑇(𝜽, 𝜈)𝑒−𝑖2𝜋𝒖·𝜽𝑑Ω ,  (2)  where 𝒖 = 𝜈𝒃/𝑐 is the baseline vector in units of wavelength, cor-  responding to each antenna pair, where 𝒃 is the baseline vector in  physical units and 𝑐 is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 𝐴(𝜽, 𝜈) denotes the primary  beam response of the telescope in the direction of 𝜽 and dΩ repre-  sents the solid angle element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The visibility function can be Fourier  transferred to the ‘delay spectrum’ space,  ˜𝑉(𝒖, 𝜏) =  ∫  𝑉(𝒖, 𝜈)𝑒−𝑖2𝜋𝜈𝜏d𝜈,  (3)  where 𝜏 = 1/𝛿𝜈 is the corresponding delay of frequency interval 𝛿𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Following McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' (2006), Parsons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' (2012) and Liu &  Shaw (2020),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Hi power spectrum can be obtained from measured  visibilities in the form of ‘delay spectrum’  𝑃D(𝑘⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 𝑘 ∥) ≡ 𝐴𝑒  𝜆2𝐵  𝑟2𝑟𝜈  𝐵  �� ˜𝑉(𝒖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 𝜏)  ��2  � 𝜆2  2𝑘B  �2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  (4)  where 𝐴𝑒 and 𝐵 are the effective antenna area and bandwidth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' re-  spectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 𝜆 denotes the wavelength at the center of the band,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 𝑟 is  the comoving distance to the redshift 𝑧 corresponding to 𝜆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 𝑟𝜈 is the  comoving width along the line-of-sight (LoS) corresponding to the  redshift range determined by 𝐵,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' and 𝑘B is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Here, 𝑘⊥ and 𝑘 ∥ are the Fourier wave vectors perpendicular and par-  allel to the LoS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' They are related to the interferometric  variables via  𝑘⊥ = 2𝜋|𝒖|  𝑟  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  𝑘 ∥ = 2𝜋𝜏𝜈21𝐻(𝑧)  𝑐(1 + 𝑧)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  (5)  where 𝜈21 = 1420 MHz is the rest-frame frequency of the 21 cm line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  𝐻(𝑧) denotes the Hubble parameter as a function of redshift 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  There are various advantages of this ‘delay spectrum’ method  (Parsons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Vedantham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Paul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The  different spectral behaviors between Hi signal and foreground make  it possible to isolate the latter in the Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In addition, the  Fourier conjugate variable is associated with the LoS cosmological  distance, therefore the ‘delay spectrum’ constructed in this method  can recover the cosmological 3D Hi power spectrum (Parsons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2014a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  MNRAS 000, 1–10 (2023)  Hi delay spectrum with MeerKAT interferometer mode  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2 Hi signal power spectrum  The mean sky brightness temperature of Hi 21 cm emission can be  given by (Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2015, 2017)  ¯𝑇𝑏(𝑧) ≈ 566ℎ  � 𝐻0  𝐻(𝑧)  � � ΩHi(𝑧)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='003  �  (1 + 𝑧)2 𝜇K ,  (6)  where 𝐻0 = 100ℎ km s−1 Mpc−1 is the Hubble constant, ΩHi(𝑧) is  the fractional density of Hi, which can be written as  ΩHi(𝑧) =  𝜌Hi(𝑧)  𝜌𝑐,0(1 + 𝑧)3 ,  (7)  where 𝜌𝑐,0 is the critical density today and the proper Hi density is  calculated by  𝜌Hi(𝑧) =  ∫ 𝑀max  𝑀min  d𝑀 d𝑛  d𝑀 (𝑀, 𝑧)𝑀Hi(𝑀, 𝑧) ,  (8)  where 𝑀 denotes the dark matter halo mass, d𝑛/d𝑀 is the proper  halo mass function and 𝑀Hi(𝑀, 𝑧) denotes the Hi mass in a halo of  mass 𝑀 at redshift 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Throughout this paper, we assume a simple  power-law model of the halo mass following Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' (2015),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=', 𝑀Hi(𝑀) = 𝐴𝑀 𝛼 with 𝐴 ∼ 220 and 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='6 that can fit both  low- and high-redshift observations within reasonable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Considering the redshift space distortion (RSD) effect (Kaiser  1987), Hi signal power spectrum can be written as  𝑃Hi(𝑘, 𝜇, 𝑧) = ¯𝑇2  𝑏 (𝑧)𝐹RSD(𝑘, 𝜇)𝑃(𝑘, 𝑧) ,  (9)  where 𝜇 ≡  𝑘 ∥/𝑘 and the matter power spectrum 𝑃(𝑘, 𝑧)  =  𝐷2(𝑧)𝑃(𝑘, 𝑧 = 0) with 𝐷(𝑧) being the growth factor and 𝑃(𝑘, 𝑧 = 0)  being the matter power spectrum at z = 0 which can be obtained by  CAMB (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 𝐹RSD(𝑘, 𝜇) represents the RSD effect, and  its form can be expressed as  𝐹RSD(𝑘, 𝜇) =  �  𝑏2  Hi(𝑧) + 𝑓 𝜇2�2  exp  �  −𝑘2𝜇2𝜎2  NL  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  (10)  Here 𝑏Hi(𝑧) is the Hi bias, written as  𝑏Hi(𝑧) = 𝜌−1  Hi (𝑧)  ∫ 𝑀max  𝑀min  d𝑀 d𝑛  d𝑀 𝑀Hi(𝑀, 𝑧)𝑏(𝑀, 𝑧),  (11)  where 𝑏(𝑀, 𝑧) is the halo bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 𝑓 ≡ dln𝐷/dln𝑎 is the linear growth  rate with 𝑎 being the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 𝜎NL is the nonlinear dispersion  scale with a middling value of 𝜎NL = 7Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In this paper, for con-  venience, ΩHi(𝑧) and 𝑏Hi(𝑧) are employed with the fitting functions  following Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3 MeerKAT noise power spectrum  The total thermal noise power spectrum can be written as (Bull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  2015)  𝑃N(𝑘, 𝜇, 𝑧) = 𝑟2(𝑧)𝑟𝜈(𝑧)  𝑇2sys𝜆4  𝑛pol𝜈21𝑡int𝐴2𝑒𝑛(𝒖)  ,  (12)  where 𝑛pol = 2 denotes the number of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 𝑡int is the inte-  gration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The ratio of MeerKAT effective aperture and system  temperature, 𝐴𝑒/𝑇sys, is frequency dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Currently, there are  two frequency bands available for observation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=', the L-band (900–  1700 MHz) and the UHF-band (580–1000 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Because of the  serious RFI contamination in the L-band frequency range, only the  frequency range of 900–1200 MHz (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='18 < 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='58) is used in our  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The full UHF-band is used in this analysis, corresponding  500  750  1000  1250  1500  ν [MHz]  0  2  4  6  8  10  Ae/T sys [m2/K]  MeerKAT L-band  MeerKAT UHF-band  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The sensitivity designs for MeerKAT receivers, shown as 𝐴𝑒/𝑇sys  for L-band and UHF-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='42 < 𝑧 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 1, we show 𝐴𝑒/𝑇sys for L-band and  UHF-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='1  Here 𝑛(𝒖) is the baseline density referring to the detailed 𝑢𝑣  coverage of a particular observation in the 𝑢𝑣 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We employ the  actual MeerKAT antenna coordinates and track the COSMOS field  (RA=10h01m, Dec=+02d12m) following Paul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In a strict  sense, 𝑛(𝒖) is also a function of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We break the frequency  range into a couple of Δ𝜈 = 60 MHz sub-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The 𝑢𝑣 coverage is  assumed to be uniform within each sub-band and simulated according  to the center frequency of each sub-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The simulated 𝑢𝑣 coverage  corresponding to the sub-bands in L-band centering at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3 and  in UHF-band centering at 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2 are shown in left and right panels  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' For both cases, we assume 10 h tracking  observation of the COSMOS field spanning over two days (the start  time is 14:15 and 13:33 at UTC, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The 𝑢𝑣 plane is  segmented on to a discrete grid with cell size Δ𝑢 = Δ𝑣 = 60𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The  color represents the number of 𝑢𝑣 points within the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' It is clear  that there are more samples in the short 𝑢𝑣 distance region at low  frequency band than high frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Because of the uniform 𝑢𝑣 coverage assumption across the sub-  band, the baseline density 𝑛(𝒖) and the corresponding total thermal  noise power spectrum 𝑃N are only the functions of 𝑘⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' According to  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' (5), 𝑘⊥ is proportional to the 𝑢𝑣 distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' |𝒖| =  √  𝑢2 + 𝑣2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The  circular averaged 𝑢𝑣 coverage within a |𝒖| shell of width Δ|𝒖| = 100𝜆  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 3, where the left panel shows the distribution  corresponding to the sub-bands centering at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3 and the right  panel shows the one centering at 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Since |𝑘⊥| is proportional  to the 𝑢𝑣 distance, the more densely populated 𝑢𝑣 points at smaller  distance mean the higher sensitivity at the smaller |𝑘⊥| modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  It is known that the 𝑢𝑣 coverage also depends on the pointing direc-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In order to investigate the influence of the different sky zones,  we also show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 3 the numbers of 𝑢𝑣 points with 10 h track-  ing at different declinations: Dec = +30◦, +02◦, −30◦, −60◦, −90◦  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The case of Dec = +02◦ is the same as tracking the  COSMOS field and the case of Dec = −30◦ corresponds to tracking  a field that the transit line passes near Zenith the MeerKAT site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' It  is obvious that when the field is targeted farther from the zenith, the  number of short baselines rises substantially for both the L-band and  1 http://public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='ska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='za/meerkat/meerkat-schedule  MNRAS 000, 1–10 (2023)  4  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The distribution of baselines on a two-dimensional (2D) 𝑢𝑣 plane for 10 h tracking of the COSMOS field with sub-bands in MeerKAT L-band centering  at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3 (left panel) and in UHF-band centering at 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The 𝑢𝑣 plane is segmented on to a discrete grid with cell-size Δ𝑢 = Δ𝑣 = 60𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The  color signifies the number of 𝑢𝑣 points on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The average number of baselines as a function of 𝑢𝑣 distance, |𝒖| =  √  𝑢2 + 𝑣2, with bin size of Δ|𝒖| = 100𝜆, for 10 h tracking with sub-bands in  MeerKAT L-band centering at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3 (left panel) and in UHF-band centering at 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  UHF-band, which potentially increases the sensitivity at the smaller  |𝑘⊥| modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='4 The foreground wedge and shot noise  The foreground contamination, which is several orders of magnitude  stronger than Hi signal, is the major challenge in recovering the Hi  LSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Since foreground spectrum is smooth across frequency chan-  nels, it only contaminates the power spectrum close to the smallest  𝑘 ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' However, the property of the interferometer response function  will cause foreground leakage into the high-𝑘⊥ modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Therefore,  we exclude the 𝑘⊥–𝑘 ∥ space modes within the foreground wedge  (Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Morales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2014a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Pober  2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Seo & Hirata 2016) that can be expressed as  𝑘 ∥ < 𝑟(𝑧)𝐻(𝑧) sin(𝜃)  𝑐(1 + 𝑧)  𝑘⊥ ,  (13)  where 𝜃 denotes the field of view of the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  In addition, shot noise needs to be taken into account in Hi IM  survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Because of Poisson fluctuations in halo number, the shot  noise power spectrum is written as (Bull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2015)  𝑃shot  Hi (𝑧) =  � ¯𝑇𝑏(𝑧)  𝜌Hi(𝑧)  �2 ∫ 𝑀max  𝑀min  d𝑀 d𝑛  d𝑀 𝑀2  Hi(𝑀) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  (14)  Here Hi mass model is consistent with the description in the Hi  signal power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Since shot noise is very low according to our  calculation, it makes a very small contribution to the total noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  3 RESULTS  In this section, we present the results of Hi IM survey analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='1, we give a detailed analysis of the power spectrum in the  different survey strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The relative errors on the BAO features  and the constraints on cosmological parameters in different dark  energy models are showed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  MNRAS 000, 1–10 (2023)  uv coverage at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3 for RA=10:00:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='60 Dec=2:12:21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='0  104  10  103  5  (Y)  0  102  101  10  15  100  10  0  10  u (k入)uv coverage at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2 for RA=10:00:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='60 Dec=2:12:21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='0  104  10  103  5  0  102  5  101  10  15  100  10  0  10  u (k入)5000  Dec=十30°  Dec=-90°  4000  Number of u points  Dec=+02°  Dec=-60°  3000  Dec=-30°  L-band  2000  1000  102  103  10  u distance in ^ （z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='312000  Dec=±30°  10000  Dec=-90°  Dec=+02°  S  Dec=-60°  8000  Dec=-30°  UHF-band  TO  6000  4000  2000  0  102  103  10  uv distance in ^ (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2Hi delay spectrum with MeerKAT interferometer mode  5  (a) Hi power spectrum  (b) Total power spectrum  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2D power spectrum at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Left panel: Hi signal power spectrum 𝑃Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Right panel: Total power spectrum 𝑃tot with MeerKAT L-band 10 h  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  (a) Hi power spectrum  (b) Total power spectrum  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2D power spectrum at 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Left panel: Hi signal power spectrum 𝑃Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Right panel: Total power spectrum 𝑃tot with MeerKAT UHF-band 10 h  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='1 Power spectrum estimation  The Hi LSS carries a significant quantity of cosmic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  However, it is extremely weak comparing to the brilliant foreground  contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The 2D Hi power spectrum 𝑃Hi at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3 (for  MeerKAT L-band) is shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The scales  available for Hi IM in interferometric mode observation are limited  by the detailed configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In summary, the scales for Hi IM survey  are:  𝑘min  ∥  = 2𝜋/(𝑟𝜈Δ𝜈/𝜈21),  𝑘max  ∥  = 1/𝜎NL,  𝑘min  ⊥  = 2𝜋|𝒖|min/𝑟,  𝑘max  ⊥  = 2𝜋|𝒖|max/𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  (15)  In principle, only the scales between minimum and maximum can be  probed by a certain instrument, and the sensitivity to scales depends  on the 𝑢𝑣 coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Adopting only 10 h observation, the total power  spectrum 𝑃tot at the same redshift, which consists of the contributions  of Hi signal, MeerKAT thermal noise, foregrounds and shot noise,  is shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  The power spectrum at 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2 (for MeerKAT UHF-band) are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 5, where the Hi power spectrum 𝑃Hi is in the left panel  and the total power spectrum 𝑃tot is in the right panel, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Note that, in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 5, the brown part denotes the  foreground wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We find that Hi signal is completely covered by  the thermal noise and foregrounds for both MeerKAT L-band and  UHF-band, which makes it difficult to obtain Hi signal directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  The Hi detection is quantified with the relative error of the power  spectrum  � Δ𝑃  𝑃  �2  =  �  1  8𝜋2𝑉bin  ∫  𝑘2d𝑘d𝜇  � 𝑃Hi(𝑘, 𝜇)  𝑃tot(𝑘, 𝜇)  �2�−1  ,  (16)  where 𝑉bin = 𝑆area𝑟2𝑟𝜈 Δ𝜈  𝜈21 is the survey volume of each redshift bin  with the survey area 𝑆area = 𝜋  �  1  2  𝜆  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='5𝑚  �2 �  180  𝜋  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Firstly, we investigate the influence on the 𝑃(𝑘) error when track-  ing the source at the different declinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' As is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 3,  when tracking the source at Dec = +30◦, +02◦, −30◦, −60◦, −90◦,  completely different numbers of 𝑢𝑣 points are obtained, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' there are  more 𝑢𝑣 points in the shorter 𝑢𝑣 distance for the case of Dec = +30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  In the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 6, the relative errors of power spectrum with  different tracking declinations are shown in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' For all  the cases, we assume 10 h observation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The results with L-band  and UHF-band are shown in solid and dashed lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Here, we divide the whole range of 𝑘 into 10 logarithmic bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=" It is  clear that the power spectrum uncertainty is reduced by more than  MNRAS 000, 1–10 (2023)  104  10-1  102  ku [Mpc-1]  6 × 10-2  100  4 × 10-2  10-2  3 × 10-2  100  10'  102  k [Mpc-1]107  10-1  105  ku [Mpc-1]  6 × 10-2  103  4 × 10-2  3 × 102  101  10°  10  102  k [Mpc-1]104  10-1  102  ku [Mpc-1]  100  10-2  10-  100  101  k[Mpc-1]107  10-1  105  ku [Mpc-1]  103  100  101  10  10  k}[Mpc-1]6  M." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='10−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='k [Mpc−1]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='101  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='∆P/P  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='10 hours  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band at Dec=+30◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band at Dec=+02◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band at Dec=−30◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band at Dec=−60◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band at Dec=−90◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band at Dec=+30◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band at Dec=+02◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band at Dec=−30◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band at Dec=−60◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band at Dec=−90◦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='10−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='101  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='k [Mpc−1]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='10−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='104  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='106  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='∆P/P  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='1 point  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band 10 hours  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band 100 hours  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band 1000 hours  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band 10000 hours  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band 10 hours  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band 100 hours  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band 1000 hours  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band 10000 hours  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='10−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='101  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='k [Mpc−1]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='10−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='10−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='101  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='103  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='104  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='∆P/P  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='10000 hours  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band 1 point  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band 10 points  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band 100 points  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='L-band 1000 points  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band 1 point  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band 10 points  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band 100 points  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='UHF-band 1000 points  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Fractional errors on 𝑃(𝑘) obtained with MeerKAT L-band and UHF-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Top panel: 10 h observations at the different declinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Middle panel:  Different observation times of tracking the COSMOS field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Bottom panel: Tracking different numbers of points in a 10000 h observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  MNRAS 000, 1–10 (2023)  Hi delay spectrum with MeerKAT interferometer mode  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='50  z  10−2  10−1  100  σ(DA)/DA  L-band 10 points  UHF-band 100 points  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='50  z  10−1  100  101  σ(H)/H  L-band 10 points  UHF-band 100 points  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='50  z  10−1  100  101  σ(fσ8)/(fσ8)  L-band 10 points  UHF-band 100 points  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Fractional errors on 𝐷𝐴(𝑧), 𝐻 (𝑧) and 𝑓 𝜎8(𝑧) obtained with MeerKAT L-band and UHF-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  a factor of two when tracking Dec = +30◦ compared to tracking  Dec = −30◦ or Dec = −60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The results show that, with limited  observation time, the tracking declination has a obvious influence on  the results of the constraints on the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Next, in order to assess the influence of integration time, we in-  crease the integration time by assuming observations on the same  field at the same local sidereal time as the existing data on different  days, which means that we obtain the same 𝑢𝑣 points from multiple  days coherently to increase the sensitivity of the same 𝑘 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In  addition to the current 10 h observation, we further consider 100,  1000 and 10000 h observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 6, we  show the fractional errors on the power spectrum 𝑃(𝑘) for 10, 100,  1000 and 10000 hours observation of tracking the COSMOS field  with MeerKAT L-band (in blue) and UHF-band (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The results  with different integration times are shown with different color satu-  rations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' It can be seen that, compared to L-band, using the UHF-band  could measure smaller 𝑘 modes down to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='1, which makes it pos-  sible to detect cosmological LSS on the larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In addition, it  is expected that the lower Δ𝑃/𝑃 can be obtained with the observa-  tion time increasing as is shown in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' With  MeerKAT UHF-band 10 h observation, the value of Δ𝑃/𝑃 could  reach 1 roughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The values of Δ𝑃/𝑃 are distinctly reduced when  tracking 1000 h, approximately reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' However, we find that  when the integration time increases from 1000 h to 10000 h, the  reduction of Δ𝑃/𝑃 is not significant at low 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' It is mainly because  the cosmic variance, which is limited by the survey volume, plays  the dominating role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Therefore, we consider tracking multiple points equally in the total  10000 h observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In this case, compared to tracking one point  with 10000 h, the survey volume 𝑉bin and the thermal noise power  spectrum 𝑃N are increased by a factor of the number of points 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  In our analysis, we calculate the additional fractional error on 𝑃(𝑘)  for 𝑁 = 10, 100 and 1000 in the 10000 h observation, as shown  in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In order to constrain cosmological  parameters, we expect to obtain lower Δ𝑃/𝑃 in low 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We find that  the lower values of Δ𝑃/𝑃 in low 𝑘 are obtained when tracking 100  points for MeerKAT L-band in the total 10000 h observation, while  tracking 10 points for MeerKAT UHF-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Therefore, we employ  these two survey strategies for MeerKAT L-band and UHF-band,  respectively, in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2 Cosmological parameters  In this subsection, we explore the capability of MeerKAT Hi IM  survey with interferometer mode of constraining cosmological pa-  rameters using the Fisher matrix method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Given the power spectrum  measurement at a given redshift, the Fisher matrix for a set of ob-  servables {𝑝} can be written as  𝐹𝑖 𝑗 =  1  8𝜋2𝑉bin  ∫ 1  −1  d𝜇  ∫ 𝑘max  𝑘min  𝑘2d𝑘 𝜕𝑃tot  𝜕𝑝𝑖  𝜕𝑃tot  𝜕𝑝 𝑗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  (17)  Here, we take the set of observables as {𝐷 𝐴(𝑧𝑖), 𝐻(𝑧𝑖), 𝑓 𝜎8(𝑧𝑖),  𝑏𝜎8(𝑧𝑖), 𝜎NL} in each redshift bin 𝑧𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The nuisance parameters  𝑏𝜎8(𝑧𝑖) and 𝜎NL can be marginalized by selecting the submatrix of  𝐹−1  𝑖 𝑗 with only the appropriate columns and rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Therefore, we can  derive the measurement errors on 𝐷 𝐴(𝑧), 𝐻(𝑧) and 𝑓 𝜎8(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  For MeerKAT L-band (900-1200 MHz) and UHF-band (580-1000  MHz), we divide these frequency bands into some bins with equal  bandwidth Δ𝜈 = 60 MHz and then obtain the estimates for the  measurement errors on observables in the corresponding redshift  bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We plot the fractional measurement errors on 𝐷 𝐴(𝑧), 𝐻(𝑧)  and 𝑓 𝜎8(𝑧) with MeerKAT 10000 h observation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We find  that the survey with interferometer mode has a better measurement  on 𝐷 𝐴(𝑧), of which the fractional errors can reach roughly 10%  for MeerKAT L-band and UHF-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Comparatively speaking, the  fractional measurement errors on 𝐻(𝑧) and 𝑓 𝜎8(𝑧) seem slightly  larger, though MeerKAT UHF-band preforms slightly better than  MeerKAT L-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Next, from the cosmological measurements on 𝐷 𝐴(𝑧), 𝐻(𝑧) and  𝑓 𝜎8(𝑧), we can constrain the various dark enenrgy models, including  the ΛCDM, 𝑤CDM and CPL models, by performing a Markov Chain  Monte Carlo (MCMC) analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The 1𝜎 errors of the cosmological  parameters are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In addition, The 1𝜎 and  2𝜎 posterior distribution contours for cosmological parameters are  shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 8–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  In the flat ΛCDM model, we obtain 𝜎(Ωm) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='044 and 𝜎(𝐻0) =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='8 km s−1 Mpc−1 with MeerKAT L-band and 𝜎(Ωm) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='028  and 𝜎(𝐻0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='0 km s−1 Mpc−1 with MeerKAT UHF-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We  find that UHF-band performs better than L-band in constraining  Ωm and 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Recently, Cunnington (2022) gave a result of 𝐻0 =  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='1+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='4  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='7 km s−1 Mpc−1 with MeerKAT UHF-band 4000 h survey  with single-dish mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' It is found that we give a better constraint  on 𝐻0 with MeerKAT UHF-band interferometric mode although we  use a longer observational time of 10000 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In comparison with other  radio telescopes, MeerKAT L-band and BINGO perform similarly,  while MeerKAT UHF-band performs nearly as well as FAST in  constraining Ωm and 𝐻0 in the flat ΛCDM model (Wu & Zhang  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Compared to the Stage-III dark energy experiments, such as  DES, we find that MeerKAT UHF-band gives a smaller error on Ωm  than DES with Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='339+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='032  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='031 in the ΛCDM model (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  In the 𝑤CDM model, in order to help break the parameter degen-  MNRAS 000, 1–10 (2023)  8  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The 1𝜎 errors of the cosmological parameters in the ΛCDM, 𝑤CDM, and CPL models using MeerKAT L-band and UHF-band or in combination  with Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Note that here 𝐻0 is in units of km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Error  ΛCDM  𝑤CDM  CPL  L-band  UHF-band  Planck+L-band  Planck+UHF-band  Planck+L-band  Planck+UHF-band  𝜎(Ωm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='044  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='024  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='092  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='046  𝜎(𝐻0)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='1  𝜎(𝑤)  −  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='08  −  −  𝜎(𝑤0)  −  −  −  −  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='6  𝜎(𝑤𝑎)  −  −  −  −  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='5  Ωm  60  65  70  75  H0 [km s−1 Mpc−1]  L-band  UHF-band  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Constraints on Ωm and 𝐻0 with MeerKAT L-band and UHF-band  in the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  eracy, we combine the BAO data from MeerKAT IM with Planck  TT,TE,EE+lowE power spectrum (Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2020) in the  MCMC analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The 1𝜎 and 2𝜎 measurement error contours for  Ωm, 𝐻0 and dark-energy equation of state parameter 𝑤 are shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We obtain 𝜎(Ωm) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='030, 𝜎(𝐻0) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='5 km s−1 Mpc−1  and 𝜎(𝑤) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='12 with Planck+L-band and 𝜎(Ωm) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='024,  𝜎(𝐻0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='6 km s−1 Mpc−1 and 𝜎(𝑤) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='08 with Planck+UHF-  band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' It can be seen that MeerKAT UHF-band combined with Planck  data gives tighter constraints on cosmological parameters in the  𝑤CDM model, with the conclusion the same as in the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  For dark energy, we find that MeerKAT has a very limited capability  of constraining 𝑤, and the error on 𝑤 is still larger even though in  combination with Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Finally, we forecast the constraints on cosmological parameters  in the CPL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' The 1𝜎 and 2𝜎 measurement error contours are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We focus on the DE equation of state parameters 𝑤0  and 𝑤𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We obtain 𝜎(𝑤0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='1 and 𝜎(𝑤𝑎) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3 with Planck+L-  band, and 𝜎(𝑤0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='6 and 𝜎(𝑤𝑎) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='0 with Planck+MeerKAT  UHF-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Note that dark energy dominates the evolution of the  universe in the redshift range of 𝑧 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' MeerKAT L-band has  a very limited constraining power for dark energy at the range of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='18 < 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='58, and MeerKAT UHF-band only surveys at the range  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='42 < 𝑧 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Therefore, MeerKAT in interferometer mode  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='0  w  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='4  Ωm  60  70  80  H0 [km s−1 Mpc−1]  60  70  80  H0 [km s−1 Mpc−1]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='4  Ωm  Planck+L-band  Planck+UHF-band  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Constraints on Ωm, 𝐻0 and 𝑤 with MeerKAT L-band and UHF-  band in combination with Planck data in the 𝑤CDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  cannot give stringent constraints on dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' But we still keep  optimistic since the precise measurements on dark energy would be  achieved by the future larger radio telescopes, such as HIRAX and  SKA (Wu & Zhang 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  4 CONCLUSIONS  In this work, we give a detailed analysis on measuring the Hi IM  delay power spectrum using the MeerKAT interferometer mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We  also discuss the capability of MeerKAT interferometer mode of con-  straining cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  We use the Fisher matrix method to estimate the Hi power spec-  trum with MeerKAT IM observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We find that the different survey  fields have the distinct impacts on determining the power spectrum  errors in the limited observational time of 10 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' As the obser-  vational time increases from 10 h to 10000 h, the power spectrum  errors are reduced evidently until the cosmic variance begins to dom-  inate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We also discuss the different survey strategies and find that the  lower fractional errors on power spectrum at low 𝑘 are obtained when  tracking 100 points for L-band and tracking 10 points for UHF-band  in a total 10000 h observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  We obtain the measurement errors on 𝐷 𝐴(𝑧), 𝐻(𝑧) and 𝑓 𝜎8(𝑧)  MNRAS 000, 1–10 (2023)  Hi delay spectrum with MeerKAT interferometer mode  9  −2  0  2  w0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='6  Ωm  50  60  70  80  H0 [km s−1 Mpc−1]  −10  −5  0  wa  −10  −5  0  wa  50  60  70  80  H0 [km s−1 Mpc−1]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='6  Ωm  Planck+L-band  Planck+UHF-band  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' Constraints on Ωm, 𝐻0, 𝑤0 and 𝑤𝑎 with MeerKAT L-band and  UHF-band in combination with Planck data in the CPL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  by using the Fisher matrix, and then use these measurements to  constrain cosmological parameters in typical dark energy models, in-  cluding ΛCDM, 𝑤CDM and CPL models, by performing the MCMC  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' We obtain 𝜎(Ωm) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='028 and 𝜎(𝐻0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='0 km s−1  Mpc−1 with MeerKAT UHF-band which are better than the results  of 𝜎(Ωm) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='044 and 𝜎(𝐻0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='8 km s−1 Mpc−1 with MeerKAT  L-band in the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' However, MeerKAT has a very limited  constraining power for dark-energy equation of state, such as 𝑤 in  the 𝑤CDM model and 𝑤0 and 𝑤𝑎 in the CPL model, even though in  combination with Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  Though MeerKAT L-band and UHF-band Hi IM surveys in in-  terferometer mode have a very limited constraining power for dark  energy, our analysis still provide a useful guide for the near future  MeerKAT survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' It is expected that the future larger radio telescope  arrays, such as SKA, will have a much better and powerful perfor-  mance on cosmological research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' In addition, MeerKAT baselines  are not short enough for detecting large cosmological scales, but the  measurements with MeerKAT interferometer mode on these scales  are still very useful in detecting Hi content of galaxies, obtaining the  cross-correlation between Hi content and star formation rates (Wolz  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2016), constraining warm dark matter (Carucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2015) and  breaking the degeneracy between ΩHi and 𝑏Hi (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  These aspects deserve further detailed investigations in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  We thank Peng-Ju Wu and Li-Yang Gao for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  This work was supported by the National SKA Program of China  (Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 2022SKA0110200 and 2022SKA0110203) and the Na-  tional Natural Science Foundation of China (Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content=' 11975072,  11875102, and 11835009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
+page_content='  DATA AVAILABILITY  The data underlying this article will be shared on reasonable request  to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfTwqq/content/2301.04445v1.pdf'}
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diff --git a/ENFRT4oBgHgl3EQfAze2/content/tmp_files/2301.13463v1.pdf.txt b/ENFRT4oBgHgl3EQfAze2/content/tmp_files/2301.13463v1.pdf.txt
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@@ -0,0 +1,974 @@
+arXiv:2301.13463v1  [hep-ph]  31 Jan 2023
+Higgs production at next generation e+e− colliders
+Deniz YILMAZ1∗, Mehmet SAHIN2, Dogukan Hazar YAVUZ1
+1Physics Engineering Department, Ankara University, Ankara,Turkey
+2Department of Computer Engineering, Usak University, Usak,Turkey
+February 1, 2023
+Abstract
+In this study, Higgs production processes, Higgsstrahlung and vector boson (W and Z) fusion
+processes, were investigated for four different future lepton colliders (CEPC, ILC, CLIC, and FCC-
+ee). The cross sections for each production process and corresponding backgrounds were calculated
+considering the ISR and beamstrahlung effects. Various cuts and the b-tagging method were used
+to reduce the background.
+Finally, the number of events for each collider was determined, and
+significance calculations were performed. In our calculations, high event numbers were obtained for
+all four colliders for the Higgsstrahlung, W, and Z fusion process. This shows that electron-positron
+colliders will play an important role in future Higgs physics research.
+1
+Introduction
+The discovery of the Higgs boson at the Large Hadron Collider (LHC) [1, 2] confirmed the electroweak
+symmetry breaking mechanism of the Standard Model (SM) [3, 4, 5, 6]. However, there is still some
+unknown about the observed Higgs boson: is it the fundamental scalar of the SM, or a more complex
+object, or part of an extended Higgs sector? Studying the properties of the Higgs boson at the LHC
+and in future colliders is crucial to understanding its true nature. Up to now, some properties of the
+Higgs boson have been measured at the LHC with an accuracy of about 10% [7, 8, 9, 10]. Although the
+LHC Run 2 to be developed will examine it with higher data, because of the complexity of the internal
+structure of the proton, the LHC will not be sensitive enough to examine the properties of Higgs.
+Electron-positron colliders, which will be installed to precisely measure the properties of the Higgs
+particle, have unique capabilities for the measurement of Higgs boson parameters, including the Higgs
+total cross section, decay width, branching ratios, Higgs width, and determination of Higgs couplings.
+Therefore, today, four e+e− colliders are being designed to study the properties of the Higgs boson
+and other standard model (SM) particles with high precision: the International Linear Collider (ILC)
+[11], with a center of mass energy of 250 – 500 GeV, Compact Linear Collider (CLIC) [12] with center
+of mass energies of 380 – 1500 – 3000 GeV, Circular Electron Positron Collider (CEPC) with center of
+mass energies between 90 and 250 GeV [13] and the Future e+e− Circular Collider (FCC-ee) [14], which
+will be located in a new tunnel at CERN at 240 GeV center of mass energy. The main beam parameters
+of these colliders [11, 12, 13, 14] are given in Table 1. The integrated luminosities given here are annual
+values.
+Table 1: The main collider parameters
+Parameters
+CEPC
+FCC-ee
+ILC
+CLIC
+Center of mass energy (GeV)
+240
+240
+250
+500
+380
+1500
+3000
+Number of particles per bunch (1010)
+15
+18
+2
+2
+0.52
+0.37
+0.37
+Horizontal beam size at IP (σx) (µm)
+20.9
+13.7
+0.516
+0.474
+0.149
+0.06
+0.04
+Vertical beam size at IP (σy) (nm)
+60
+36
+7.66
+5.86
+2.9
+1.5
+1
+Bunch length (mm)
+4.4
+5.3
+0.3
+0.3
+0.07
+0.044
+0.044
+Luminosity (105pb−1 )
+6
+17
+1.35
+1.8
+1.5
+3.7
+5.9
+In the electron-positron collider, Higgs bosons are produced by the Higgsstrahlung and vector boson
+(W and Z) fusion processes [15, 16, 17, 18, 19, 20, 21]. In this study, these three processes were examined
+∗dyilmaz@eng.ankara.edu.tr
+1
+
+Figure 1: The Feynmann diagrams of the Higgs production processes
+and calculations were performed using CalcHEP [22, 23]. In the electron-positron collider, it is important
+to consider the effects of ISR and Beamstrahlung [24, 25]. The parameters listed in Table 1 were used
+to calculate the ISR and the beamstraghlung effects. In section 2 cross sections are given for these three
+processes. Section 3 provides signal and background analyses, the number of events for each collider, and
+the significance calculations. Finally, conclusion is provided in the section 4.
+2
+Higgs Production at the electron - positron colliders
+The main production processes of Higgs at the e+e− colliders are the Higgsstrahlung and W/Z fusion
+mechanisms given below, as shown in Figure 1.
+Higgs − strahlung
+e+e− → ZH
+Wfusion
+e+e− → νeνeH
+Zfusion
+e+e− → e+e−H
+The cross section for the Higgsstrahlung process can be written as
+σ(e+e− → ZH) = G2
+F M 4
+Z
+96πs (η2
+e + a2
+e)κ1/2 κ + 12M 2
+Z/s
+(1 − M 2
+Z/s)2
+(1)
+where ae = −1, ηe = −1 + 4sin2θW are the Z charges of the electron and κ = (1 − (MH + MZ)2/s)(1 −
+(MH − MZ)2/s) is the usual two particle phase space function. The total cross section for the vector
+boson fusion mechanism is
+σ(e+e− → V V → llH) = G2
+F m4
+V
+64
+√
+2π3
+� 1
+xH
+dx
+� 1
+x
+dyT (x, y)
+[1 + (y − x)/xV ]2 .
+(2)
+T (x, y) = (2x
+y3 − 3x + 1
+y2
++ x + 2
+y
+− 1)[
+z
+z + 1 − log(z + 1)] + xz2(1 − y)
+y3(z + 1) ,
+where V denotes the vector bosons W or Z and xH = m2
+H/s, xV = m2
+V /s and z = y(x − xH)/xxV (√s
+is the center-of-mass energy) [26].
+200
+250
+300
+350
+400
+450
+500
+√s [GeV]
+0
+0,05
+0,1
+0,15
+0,2
+0,25
+σ [pb]
+Higgsstrahlung
+W Fusion
+Z Fusion
+Figure 2: The cross sections of the Higgs production mechanisms as a function of center-of-mass energy.
+2
+
+200
+250
+300
+350
+400
+450
+500
+√s [GeV]
+0,05
+0,1
+0,15
+0,2
+0,25
+σ [pb]
+CEPC
+ILC
+FCC-ee
+CLIC
+Higgsstrahlung
+200
+250
+300
+350
+400
+450
+500
+√s [GeV]
+0,01
+0,02
+0,03
+0,04
+0,05
+0,06
+0,07
+σ [pb]
+CEPC
+ILC
+FCC-ee
+CLIC
+W Fusion
+200
+250
+300
+350
+400
+450
+500
+√s [GeV]
+0,001
+0,002
+0,003
+0,004
+0,005
+0,006
+0,007
+σ [pb]
+CEPC
+ILC
+FCC-ee
+CLIC
+Z Fusion
+Figure 3: The cross section comparison for Higgsstrahlung, W Fusion and Z Fusion processes for four
+e+e−colliders.
+The behavior of the production cross-sections of the Higgs boson calculated by the Higgsstrahlung
+and the W/Z fusion mechanisms using the CalcHEP simulation program, depending on the center of
+mass energy, are shown in Figure 2 and Figure 3. The relevant production cross sections as a function of
+the center of mass energy are shown in Figure 2. As shown in Figure 2, the Higgsstrahlung suppresses the
+vector boson production processes for moderate values of the energy due to the additional electroweak
+coupling. With the increase in energy, the cross sections of the vector boson procecesses increase log-
+arithmically and become dominant. At a center of mass energy of about 250 GeV, Higgs bosons are
+predominantly produced from the ZH process as seen in the same figure. In the Figure 3, the cross
+sections are shown as a function of the center of mass energy for each production mechanisms for four
+electron-positron colliders with the ISR and the beamstrahlung effects of each colliders.
+3
+Signal and Background Analyses
+Because the Higgs boson’s decay rate to bb is greater than the decay rate to other quarks and leptons
+[27, 28, 29, 30, 31], the bb decay mode of Higgs (H −→ bb) is considered in all production processes in
+this study. Since the cross sections of the background processes corresponding to the leptonic decays of
+the Z boson are less than the background cross sections corresponding to the other decays, the leptonic
+decays of the Z boson in the Higgsstrahlung process are taken into account. The signal processes are
+given below.
+Signal 1:
+Higgsstrahlung
+e+e− → ZH → llbb
+Zfusion
+e+e− → e+e−bb
+Signal 2:
+Wfusion
+e+e− → νeνebb
+3
+
+Here, l and l are e−, µ− and e+, µ+, respectively. The corresponding background processes analysed here
+are as follows:
+For signal 1:
+i)
+e+e− → ZZ → llJJ,
+ii)
+e+e− → e+e−Z → e+e−JJ,
+iii)
+e+e− → tt → W +JW −J → llJJνlνl,
+For signal 2:
+e+e− → JJ,
+here, J represents the quark and antiquark: J = d, d, u, u, s, s, c, c, b, b. The transverse momentum (PT ),
+pseudo rapidity (η) and invariant mass (Minv) distributions of the final state particles were investigated
+by using CalcHEP program in order to find the cut values to distinguish the signal from the background
+in the FCC-ee collider with a center of mass energy of 240 GeV. The background iii process corresponding
+to Signal 1 is not included in the calculations for 240 GeV, as it starts to contribute at 350 GeV and
+greater center of mass energies. Because the transverse momentum, pseudo rapidity, and invariant mass
+distributions of the final state particles in the signal and background processes will exhibit similar behavior
+for other colliders, the cut values obtained can be used for CEPC, ILC, and CLIC. Transverse momentum
+distribution plots for the final state particles of signal 1 and the corresponding background processes i
+and ii are shown in Figure 4, while the graphs of signal 2 are shown in Figure 5. As can be seen from
+Figure 4 and 5, when a transverse momentum cut of 35 GeV is applied to the e−, e+, µ−, µ+, and two
+jets (J) in the final state particles of signal 1 and signal 2 and the corresponding background processes,
+the signal will almost not change, but the background will be significantly reduced.
+Pseudorapidity plots for signal 1, signal 2, and the corresponding backgrounds are shown in Figure
+6 and 7. As can be seen from the figures, cut regions of −2.5 < ηJ,J < 2.5, −2.5 < ηe−,e+ < 2.5,
+−2.5 < ηµ−,µ+ < 2.5 will be appropriate for e−, e+, µ−, µ+ and two jets (J) in the final state particles
+of signal 1 and signal 2 and the corresponding background processes.
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+√s=240 GeV
+(1/σ)dσ/dpT (1/GeV)
+pT (GeV)
+signal 1
+background i
+background ii
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+√s=240 GeV
+(1/σ)dσ/dpT (1/GeV)
+pT (GeV)
+signal 1
+background i
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+√s=240 GeV
+(1/σ)dσ/dpT (1/GeV)
+pT (GeV)
+signal 1
+background i
+background ii
+Figure 4: Transverse momentum distribution plots for the e−/e+ (upper left ), µ−/µ+ (upper right)
+and J/J (bottom) final state particles of signal 1 and the corresponding bacground processes in FCC-ee
+collider with 240 GeV center of mass energy.
+4
+
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+√s=240 GeV
+(1/σ)dσ/dpT (1/GeV)
+pT (GeV)
+signal 2
+background
+Figure 5: Transverse momentum distribution plots for the b/b and J/J final state particles of signal 2
+and the corresponding bacground processes in FCC-ee collider with 240 GeV center of mass energy.
+An Emiss
+T
+cut value of >15 GeV was also used for neutrinos in our calculations.
+Invariant mass distribution plots for signal 1, signal 2 and their corresponding background processes
+are shown in Figure 8. As can be seen from the figures, in the calculations, it would be appropriate to
+exclude the 80 GeV < Minv(e−, e+) <100 GeV and 80 GeV < Minv(µ−, µ+) < 100 GeV regions for the
+ll final states in signal and background processes. In addition, only the 115 GeV < Minv(J, J) <135
+GeV region was included in the calculations for two final jet states in the signal and background pro-
+cesses. These included and excluded invariant mass regions allow the signal to be distinguished from the
+background.
+In addition to these cut values, the separation cuts of ∆R(l, J) >0.5 and ∆R(l, J) >0.5 distinguish
+the final state leptons and antileptons from the jets, while the ∆R(J, J) >0.5 separation cut was used to
+distinguish the final state jets from each other.
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+-6
+-4
+-2
+ 0
+ 2
+ 4
+ 6
+√s=240 GeV
+(1/σ)dσ/dη
+ηe
+signal 1
+background i
+background ii
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+-6
+-4
+-2
+ 0
+ 2
+ 4
+ 6
+√s=240 GeV
+(1/σ)dσ/dη
+ηµ
+signal 1
+background i
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+-6
+-4
+-2
+ 0
+ 2
+ 4
+ 6
+√s=240 GeV
+(1/σ)dσ/dη
+ηJ
+signal 1
+background i
+background ii
+Figure 6: Pseudorapidity distribution plots for the e−/e+ (upper left ), µ−/µ+ (upper right) and J/J
+(bottom) final state particles of signal 1 and the corresponding bacground processes in FCC-ee collider
+with 240 GeV center of mass energy.
+5
+
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+-6
+-4
+-2
+ 0
+ 2
+ 4
+ 6
+√s=240 GeV
+(1/σ)dσ/dη
+ηJ
+signal 2
+background 
+Figure 7: Pseudorapidity distribution plots for the b/b and J/J final state particles of signal 2 and the
+corresponding bacground processes in FCC-ee collider with 240 GeV center of mass energy.
+All the cut values obtained are listed in Table 2, and these cut values were used in the calculations for
+the four colliders. In addition to the cut values in Table 2, because the Higgs boson decays to bb in our
+signal processes, it is possible to further reduce the background cross section value using the b-tagging
+method [27]: 68% is used for the b-tagging identification rate, and a 1% ratio is used for misidentification
+rate with light quarks as b quarks.
+The following equation is used to calculate the significance of the
+10-8
+10-6
+10-4
+10-2
+100
+ 0
+ 25
+ 50
+ 75
+ 100
+ 125
+ 150
+ 175
+ 200
+ 225
+ 250
+√s=240 GeV
+dσ/dMinv (pb/GeV)
+Minv (GeV)
+signal Z(e-,e+)
+signal h(b, b-)
+background Z(e-,e+)
+background γ(e-,e+)
+background Z(J,J)
+10-8
+10-6
+10-4
+10-2
+100
+ 0
+ 25
+ 50
+ 75
+ 100
+ 125
+ 150
+ 175
+ 200
+ 225
+ 250
+√s=240 GeV
+(1/σ)dσ/dMinv (1/GeV)
+Minv (GeV)
+signal h(b,b-)
+background Z(J,J))
+Figure 8: Invariant mass plots for the signal 1 (left) and signal 2 (right) and the corresponding bacground
+processes in FCC-ee collider with 240 GeV center of mass energy.
+obtained data:
+S =
+�
+2((s + b) ln(1 + s/b) − s)
+(3)
+where s and b represent signal and background events, respectively [32]. Cross-sections, event rates, and
+significance values were calculated for the signal and background processes using the cut values in Table 2,
+Table 2: Cut values
+Emiss
+T
+(νl, νl) > 15 GeV
+PT (l, l) > 35 GeV
+PT (J) > 35 GeV
+-2.5 < η(l, l) < 2.5
+-2.5 < η(J) < 2.5
+80 GeV < Minv(l, l) < 100 GeV region is excluded
+115 GeV< Minv(J, J) < 135 GeV region is included
+∆R(l, J) > 0.5
+∆R(l, J) > 0.5
+∆R(J, J) > 0.5
+6
+
+Table 3: Cross sections, number of events and the significance values for CEPC.
+Colliders
+Processes
+Sg CS
+(pb)
+Bg CS
+(pb)
+L
+(pb−1)
+No. SgE
+No.BgE
+S
+CEPC
+(240 GeV)
+Signal 1
+9.91×10−5
+2.38×10−4
+6×105
+59.5
+142.8
+4.7
+Signal 1
+(with b-tagging)
+6.72×10−5
+3.68×10−5
+40.32
+22.08
+7
+Signal 2
+1.24×10−2
+5.22×10−1
+7440
+313200
+13.24
+Signal 2
+(with b-tagging)
+8.45×10−3
+7.17×10−2
+5070
+43020
+23.98
+b-tagging method, and nominal integrated luminosity given in Table 1. The event rates and significance
+values of the signals and corresponding backgrounds are obtained for four future lepton colliders. The
+numerical results are given in Table 3-6. The abbreviations used in the tables are: Sg CS (signal cross-
+section), Bg CS ( background cross-section),L (integrated luminosity), No. SgE (number of signal events)
+and No. BgE (number of background events).
+4
+Conclusion
+After the discovery of the Higgs particle, precise measurements of the Higgs properties became an im-
+portant step forward for future research in particle physics. Electron positron colliders to be installed
+for this purpose have unique capabilities for the measurement of Higgs boson parameters, including the
+Higgs total cross section of production processes, decay width, branching rates and determination of
+Higgs couplings. In this study, the Higgsstrahlung and W and Z fusion processes were examined, and the
+data obtained are presented in graphs and tables for four different electron-positron colliders. The pro-
+duction cross-sections for each process and additionally cross-sections for various final state backgrounds
+were calculated. In the calculations, we attempted to reduce the background by transverse momentum,
+pseudo rapidity, invariant mass, cone-angle constraints, and the b-tagging method. Significance calcula-
+tions were performed by determining the number of events related to the production processes and the
+background for each collider. The values are listed in Table 3-6.
+When the results are examined in Table 5 , it is seen that the desired significance value for Signal 1
+cannot be reached at the luminosity value given for ILC – 250 GeV. For Signal 1 processes to be observed
+in the ILC-250 GeV, the collider needs to accumulate data for a longer period of time. Again, at the end
+of one year, it was seen that the statistical significance value of 5σ would be reached after the b-tagging
+method for the Signal 1 processes in the CEPC collider. Therefore, the CEPC collider will enable the
+properties of the Higgs boson to be investigated precisely through Signal 1 processes. It is seen that at
+the end of 1 year in the FCC-ee collider, a significance value of 7.95 will be reached without b-tagging and
+a high significance value of 11.9 can be reached by using b-tagging. This shows that FCC-ee will be more
+advantageous than ILC-250 GeV and CEPC 250 GeV colliders for investigating Higgs boson properties
+through the Signal 1 group around these center of mass energies (240-250 GeV). In the ILC-500 GeV
+and CLIC-380-1500-3000 GeV colliders, results well above the desired significance value can be obtained
+for signal 1 processes, even without the b-tagging. Therefore, the properties of the Higgs boson through
+Signal 1 processes can be studied with precision in colliders other than the ILC-250 GeV collider. Since
+the results obtained for the Signal 2 process are greater than 5 significance values, the properties of the
+Table 4: Cross sections, number of events and the significance values for FCC-ee.
+Colliders
+Processes
+Sg CS
+(pb)
+Bg CS
+(pb)
+L
+(pb−1)
+No. SgE
+No.BgE
+S
+FCC-ee
+(240 GeV)
+Signal 1
+9.98×10−5
+2.36×10−4
+1.7×106
+169.7
+401.2
+7.95
+Signal 1
+(with b-tagging)
+6.79×10−5
+3.67×10−5
+115.4
+62.4
+11.9
+Signal 2
+1.25×10−2
+5.37×10−1
+21250
+912900
+22.15
+Signal 2
+(with b-tagging)
+8.53×10−3
+7.38×10−2
+14501
+125460
+40.18
+7
+
+Table 5: Cross sections, number of events and the significance values for ILC.
+Colliders
+Processes
+Sg CS
+(pb)
+Bg CS
+(pb)
+L
+(pb−1)
+No. SgE
+No.BgE
+S
+ILC
+(250 GeV)
+Signal 1
+1.33×10−4
+2.9×10−4
+1.35×105
+17.95
+39.15
+2.68
+Signal 1
+(with b-tagging)
+9.04×10−5
+4.34×10−5
+12.2
+5.86
+4.03
+Signal 2
+1.3×10−2
+5.33×10−1
+1755
+71955
+6.51
+Signal 2
+(with b-tagging)
+8.82×10−3
+7.32×10−2
+1190
+9882
+11.74
+ILC
+(500 GeV)
+Signal 1
+1.41×10−3
+1.7×10−3
+1.8×105
+253.8
+306
+12.98
+Signal 1
+(with b-tagging)
+9.57×10−4
+3.32×10−4
+172.3
+59.8
+16.88
+Signal 2
+2.86×10−2
+1.13×10−1
+5148
+20340
+34.71
+Signal 2
+(with b-tagging)
+1.94×10−2
+1.56×10−2
+3492
+2808
+56.54
+Table 6: Cross sections, number of events and the significance values for CLIC.
+Colliders
+Processes
+Sg CS
+(pb)
+Bg CS
+(pb)
+L
+(pb−1)
+No. SgE
+No.BgE
+S
+CLIC
+(380 GeV)
+Signal 1
+6.44×10−4
+1.08×10−3
+1.5×105
+96.6
+162
+6.98
+Signal 1
+(with b-tagging)
+4.38×10−4
+2.53×10−4
+65.7
+37.95
+8.76
+Signal 2
+1.7×10−2
+2.11×10−1
+2550
+31650
+14.14
+Signal 2
+(with b-tagging)
+1.16×10−2
+2.9×10−2
+1740
+4350
+24.86
+CLIC
+(1500 GeV)
+Signal 1
+1.95×10−3
+1.47×10−3
+3.7×105
+721.5
+543.9
+26.34
+Signal 1
+(with b-tagging)
+1.32×10−3
+2.34×10−4
+488.4
+86.58
+36.64
+Signal 2
+1.03×10−1
+9.22×10−3
+38110
+3411
+362
+Signal 2
+(with b-tagging)
+6.99×10−2
+1.27×10−3
+25863
+470
+400
+CLIC
+(3000 GeV)
+Signal 1
+6.01×10−4
+8.47×10−4
+5.9×105
+355
+499.7
+14.38
+Signal 1
+(with b-tagging)
+4.09×10−4
+1.32×10−4
+241
+77.8
+20.44
+Signal 2
+1.61×10−1
+2.72×10−3
+94990
+1605
+776
+Signal 2
+(with b-tagging)
+1.09×10−1
+3.74×10−4
+64310
+221
+777
+Higgs boson can be studied precisely for all colliders through this channel.
+As a result, in future lepton colliders, the Higgs boson can be observed with high event rates via
+Higgsstrahlung, W and Z fusion. Thus, electron–positron colliders can precisely measure the properties
+of the Higgs boson.
+Acknowledgment
+We would like to thank Professor Dr Inanc Sahin for his suggestions.
+8
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf,len=562
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='13463v1  [hep-ph]  31 Jan 2023  Higgs production at next generation e+e− colliders  Deniz YILMAZ1∗, Mehmet SAHIN2, Dogukan Hazar YAVUZ1  1Physics Engineering Department, Ankara University, Ankara,Turkey  2Department of Computer Engineering, Usak University, Usak,Turkey  February 1, 2023  Abstract  In this study, Higgs production processes, Higgsstrahlung and vector boson (W and Z) fusion  processes, were investigated for four different future lepton colliders (CEPC, ILC, CLIC, and FCC-  ee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The cross sections for each production process and corresponding backgrounds were calculated  considering the ISR and beamstrahlung effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Various cuts and the b-tagging method were used  to reduce the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Finally, the number of events for each collider was determined, and  significance calculations were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' In our calculations, high event numbers were obtained for  all four colliders for the Higgsstrahlung, W, and Z fusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' This shows that electron-positron  colliders will play an important role in future Higgs physics research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  1  Introduction  The discovery of the Higgs boson at the Large Hadron Collider (LHC) [1, 2] confirmed the electroweak  symmetry breaking mechanism of the Standard Model (SM) [3, 4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' However, there is still some  unknown about the observed Higgs boson: is it the fundamental scalar of the SM, or a more complex  object, or part of an extended Higgs sector?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Studying the properties of the Higgs boson at the LHC  and in future colliders is crucial to understanding its true nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Up to now, some properties of the  Higgs boson have been measured at the LHC with an accuracy of about 10% [7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Although the  LHC Run 2 to be developed will examine it with higher data, because of the complexity of the internal  structure of the proton, the LHC will not be sensitive enough to examine the properties of Higgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Electron-positron colliders, which will be installed to precisely measure the properties of the Higgs  particle, have unique capabilities for the measurement of Higgs boson parameters, including the Higgs  total cross section, decay width, branching ratios, Higgs width, and determination of Higgs couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' today,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' four e+e− colliders are being designed to study the properties of the Higgs boson  and other standard model (SM) particles with high precision: the International Linear Collider (ILC)  [11],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' with a center of mass energy of 250 – 500 GeV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Compact Linear Collider (CLIC) [12] with center  of mass energies of 380 – 1500 – 3000 GeV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Circular Electron Positron Collider (CEPC) with center of  mass energies between 90 and 250 GeV [13] and the Future e+e− Circular Collider (FCC-ee) [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' which  will be located in a new tunnel at CERN at 240 GeV center of mass energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The main beam parameters  of these colliders [11, 12, 13, 14] are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The integrated luminosities given here are annual  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Table 1: The main collider parameters  Parameters  CEPC  FCC-ee  ILC  CLIC  Center of mass energy (GeV)  240  240  250  500  380  1500  3000  Number of particles per bunch (1010)  15  18  2  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='37  Horizontal beam size at IP (σx) (µm)  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='9  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='516  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='474  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='149  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='04  Vertical beam size at IP (σy) (nm)  60  36  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='66  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='86  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5  1  Bunch length (mm)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='044  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='044  Luminosity (105pb−1 )  6  17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='9  In the electron-positron collider, Higgs bosons are produced by the Higgsstrahlung and vector boson  (W and Z) fusion processes [15, 16, 17, 18, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' In this study, these three processes were examined  ∗dyilmaz@eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='ankara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='tr  1  Figure 1: The Feynmann diagrams of the Higgs production processes  and calculations were performed using CalcHEP [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' In the electron-positron collider, it is important  to consider the effects of ISR and Beamstrahlung [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The parameters listed in Table 1 were used  to calculate the ISR and the beamstraghlung effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' In section 2 cross sections are given for these three  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Section 3 provides signal and background analyses, the number of events for each collider, and  the significance calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Finally, conclusion is provided in the section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  2  Higgs Production at the electron - positron colliders  The main production processes of Higgs at the e+e− colliders are the Higgsstrahlung and W/Z fusion  mechanisms given below, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Higgs − strahlung  e+e− → ZH  Wfusion  e+e− → νeνeH  Zfusion  e+e− → e+e−H  The cross section for the Higgsstrahlung process can be written as  σ(e+e− → ZH) = G2  F M 4  Z  96πs (η2  e + a2  e)κ1/2 κ + 12M 2  Z/s  (1 − M 2  Z/s)2  (1)  where ae = −1, ηe = −1 + 4sin2θW are the Z charges of the electron and κ = (1 − (MH + MZ)2/s)(1 −  (MH − MZ)2/s) is the usual two particle phase space function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The total cross section for the vector  boson fusion mechanism is  σ(e+e− → V V → llH) = G2  F m4  V  64  √  2π3  � 1  xH  dx  � 1  x  dyT (x, y)  [1 + (y − x)/xV ]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  (2)  T (x, y) = (2x  y3 − 3x + 1  y2  + x + 2  y  − 1)[  z  z + 1 − log(z + 1)] + xz2(1 − y)  y3(z + 1) ,  where V denotes the vector bosons W or Z and xH = m2  H/s, xV = m2  V /s and z = y(x − xH)/xxV (√s  is the center-of-mass energy) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  200  250  300  350  400  450  500  √s [GeV]  0  0,05  0,1  0,15  0,2  0,25  σ [pb]  Higgsstrahlung  W Fusion  Z Fusion  Figure 2: The cross sections of the Higgs production mechanisms as a function of center-of-mass energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  2  200  250  300  350  400  450  500  √s [GeV]  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='05  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='1  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='15  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='2  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='25  σ [pb]  CEPC  ILC  FCC-ee  CLIC  Higgsstrahlung  200  250  300  350  400  450  500  √s [GeV]  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='01  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='02  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='03  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='04  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='05  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='06  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='07  σ [pb]  CEPC  ILC  FCC-ee  CLIC  W Fusion  200  250  300  350  400  450  500  √s [GeV]  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='001  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='002  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='003  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='004  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='005  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='006  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='007  σ [pb]  CEPC  ILC  FCC-ee  CLIC  Z Fusion  Figure 3: The cross section comparison for Higgsstrahlung,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' W Fusion and Z Fusion processes for four  e+e−colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  The behavior of the production cross-sections of the Higgs boson calculated by the Higgsstrahlung  and the W/Z fusion mechanisms using the CalcHEP simulation program, depending on the center of  mass energy, are shown in Figure 2 and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The relevant production cross sections as a function of  the center of mass energy are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' As shown in Figure 2, the Higgsstrahlung suppresses the  vector boson production processes for moderate values of the energy due to the additional electroweak  coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' With the increase in energy, the cross sections of the vector boson procecesses increase log-  arithmically and become dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' At a center of mass energy of about 250 GeV, Higgs bosons are  predominantly produced from the ZH process as seen in the same figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' In the Figure 3, the cross  sections are shown as a function of the center of mass energy for each production mechanisms for four  electron-positron colliders with the ISR and the beamstrahlung effects of each colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  3  Signal and Background Analyses  Because the Higgs boson’s decay rate to bb is greater than the decay rate to other quarks and leptons  [27, 28, 29, 30, 31], the bb decay mode of Higgs (H −→ bb) is considered in all production processes in  this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Since the cross sections of the background processes corresponding to the leptonic decays of  the Z boson are less than the background cross sections corresponding to the other decays, the leptonic  decays of the Z boson in the Higgsstrahlung process are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The signal processes are  given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Signal 1:  Higgsstrahlung  e+e− → ZH → llbb  Zfusion  e+e− → e+e−bb  Signal 2:  Wfusion  e+e− → νeνebb  3  Here, l and l are e−, µ− and e+, µ+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The corresponding background processes analysed here  are as follows:  For signal 1:  i)  e+e− → ZZ → llJJ,  ii)  e+e− → e+e−Z → e+e−JJ,  iii)  e+e− → tt → W +JW −J → llJJνlνl,  For signal 2:  e+e− → JJ,  here, J represents the quark and antiquark: J = d, d, u, u, s, s, c, c, b, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The transverse momentum (PT ),  pseudo rapidity (η) and invariant mass (Minv) distributions of the final state particles were investigated  by using CalcHEP program in order to find the cut values to distinguish the signal from the background  in the FCC-ee collider with a center of mass energy of 240 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The background iii process corresponding  to Signal 1 is not included in the calculations for 240 GeV, as it starts to contribute at 350 GeV and  greater center of mass energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Because the transverse momentum, pseudo rapidity, and invariant mass  distributions of the final state particles in the signal and background processes will exhibit similar behavior  for other colliders, the cut values obtained can be used for CEPC, ILC, and CLIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Transverse momentum  distribution plots for the final state particles of signal 1 and the corresponding background processes i  and ii are shown in Figure 4, while the graphs of signal 2 are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' As can be seen from  Figure 4 and 5, when a transverse momentum cut of 35 GeV is applied to the e−, e+, µ−, µ+, and two  jets (J) in the final state particles of signal 1 and signal 2 and the corresponding background processes,  the signal will almost not change, but the background will be significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Pseudorapidity plots for signal 1, signal 2, and the corresponding backgrounds are shown in Figure  6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' As can be seen from the figures, cut regions of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5 < ηJ,J < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5 < ηe−,e+ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5,  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5 < ηµ−,µ+ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5 will be appropriate for e−, e+, µ−, µ+ and two jets (J) in the final state particles  of signal 1 and signal 2 and the corresponding background processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
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+page_content='√s=240 GeV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='(1/σ)dσ/dpT (1/GeV)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='pT (GeV)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='signal 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='background i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='background ii  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='Figure 4: Transverse momentum distribution plots for the e−/e+ (upper left ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' µ−/µ+ (upper right)  and J/J (bottom) final state particles of signal 1 and the corresponding bacground processes in FCC-ee  collider with 240 GeV center of mass energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  4  10-5  10-4  10-3  10-2  10-1  100  101  0  20  40  60  80  100  120  140  √s=240 GeV  (1/σ)dσ/dpT (1/GeV)  pT (GeV)  signal 2  background  Figure 5: Transverse momentum distribution plots for the b/b and J/J final state particles of signal 2  and the corresponding bacground processes in FCC-ee collider with 240 GeV center of mass energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  An Emiss  T  cut value of >15 GeV was also used for neutrinos in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Invariant mass distribution plots for signal 1, signal 2 and their corresponding background processes  are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' As can be seen from the figures, in the calculations, it would be appropriate to  exclude the 80 GeV < Minv(e−, e+) <100 GeV and 80 GeV < Minv(µ−, µ+) < 100 GeV regions for the  ll final states in signal and background processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' In addition, only the 115 GeV < Minv(J, J) <135  GeV region was included in the calculations for two final jet states in the signal and background pro-  cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' These included and excluded invariant mass regions allow the signal to be distinguished from the  background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  In addition to these cut values, the separation cuts of ∆R(l, J) >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5 and ∆R(l, J) >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5 distinguish  the final state leptons and antileptons from the jets, while the ∆R(J, J) >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5 separation cut was used to  distinguish the final state jets from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  10-5  10-4  10-3  10-2  10-1  100  101  6  4  2  0  2  4  6  √s=240 GeV  (1/σ)dσ/dη  ηe  signal 1  background i  background ii  10-5  10-4  10-3  10-2  10-1  100  101  6  4  2  0  2  4  6  √s=240 GeV  (1/σ)dσ/dη  ηµ  signal 1  background i  10-5  10-4  10-3  10-2  10-1  100  101  6  4  2  0  2  4  6  √s=240 GeV  (1/σ)dσ/dη  ηJ  signal 1  background i  background ii  Figure 6: Pseudorapidity distribution plots for the e−/e+ (upper left ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' µ−/µ+ (upper right) and J/J  (bottom) final state particles of signal 1 and the corresponding bacground processes in FCC-ee collider  with 240 GeV center of mass energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  5  10-5  10-4  10-3  10-2  10-1  100  101  6  4  2  0  2  4  6  √s=240 GeV  (1/σ)dσ/dη  ηJ  signal 2  background  Figure 7: Pseudorapidity distribution plots for the b/b and J/J final state particles of signal 2 and the  corresponding bacground processes in FCC-ee collider with 240 GeV center of mass energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  All the cut values obtained are listed in Table 2, and these cut values were used in the calculations for  the four colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' In addition to the cut values in Table 2, because the Higgs boson decays to bb in our  signal processes, it is possible to further reduce the background cross section value using the b-tagging  method [27]: 68% is used for the b-tagging identification rate, and a 1% ratio is used for misidentification  rate with light quarks as b quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  The following equation is used to calculate the significance of the  10-8  10-6  10-4  10-2  100  0  25  50  75  100  125  150  175  200  225  250  √s=240 GeV  dσ/dMinv (pb/GeV)  Minv (GeV)  signal Z(e-,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='e+)  signal h(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' b-)  background Z(e-,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='e+)  background γ(e-,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='e+)  background Z(J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='J)  10-8  10-6  10-4  10-2  100  0  25  50  75  100  125  150  175  200  225  250  √s=240 GeV  (1/σ)dσ/dMinv (1/GeV)  Minv (GeV)  signal h(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='b-)  background Z(J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='J))  Figure 8: Invariant mass plots for the signal 1 (left) and signal 2 (right) and the corresponding bacground  processes in FCC-ee collider with 240 GeV center of mass energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  obtained data:  S =  �  2((s + b) ln(1 + s/b) − s)  (3)  where s and b represent signal and background events, respectively [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Cross-sections, event rates, and  significance values were calculated for the signal and background processes using the cut values in Table 2,  Table 2: Cut values  Emiss  T  (νl, νl) > 15 GeV  PT (l, l) > 35 GeV  PT (J) > 35 GeV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5 < η(l, l) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5 < η(J) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5  80 GeV < Minv(l, l) < 100 GeV region is excluded  115 GeV< Minv(J, J) < 135 GeV region is included  ∆R(l, J) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5  ∆R(l, J) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5  ∆R(J, J) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5  6  Table 3: Cross sections, number of events and the significance values for CEPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Colliders  Processes  Sg CS  (pb)  Bg CS  (pb)  L  (pb−1)  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' SgE  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='BgE  S  CEPC  (240 GeV)  Signal 1  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='91×10−5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='38×10−4  6×105  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5  142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='7  Signal 1  (with b-tagging)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='72×10−5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='68×10−5  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='32  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='08  7  Signal 2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='24×10−2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='22×10−1  7440  313200  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='24  Signal 2  (with b-tagging)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='45×10−3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='17×10−2  5070  43020  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='98  b-tagging method, and nominal integrated luminosity given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The event rates and significance  values of the signals and corresponding backgrounds are obtained for four future lepton colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The  numerical results are given in Table 3-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The abbreviations used in the tables are: Sg CS (signal cross-  section), Bg CS ( background cross-section),L (integrated luminosity), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' SgE (number of signal events)  and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' BgE (number of background events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  4  Conclusion  After the discovery of the Higgs particle, precise measurements of the Higgs properties became an im-  portant step forward for future research in particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Electron positron colliders to be installed  for this purpose have unique capabilities for the measurement of Higgs boson parameters, including the  Higgs total cross section of production processes, decay width, branching rates and determination of  Higgs couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' In this study, the Higgsstrahlung and W and Z fusion processes were examined, and the  data obtained are presented in graphs and tables for four different electron-positron colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The pro-  duction cross-sections for each process and additionally cross-sections for various final state backgrounds  were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' In the calculations, we attempted to reduce the background by transverse momentum,  pseudo rapidity, invariant mass, cone-angle constraints, and the b-tagging method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Significance calcula-  tions were performed by determining the number of events related to the production processes and the  background for each collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' The values are listed in Table 3-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  When the results are examined in Table 5 , it is seen that the desired significance value for Signal 1  cannot be reached at the luminosity value given for ILC – 250 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' For Signal 1 processes to be observed  in the ILC-250 GeV, the collider needs to accumulate data for a longer period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Again, at the end  of one year, it was seen that the statistical significance value of 5σ would be reached after the b-tagging  method for the Signal 1 processes in the CEPC collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Therefore, the CEPC collider will enable the  properties of the Higgs boson to be investigated precisely through Signal 1 processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' It is seen that at  the end of 1 year in the FCC-ee collider, a significance value of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='95 will be reached without b-tagging and  a high significance value of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='9 can be reached by using b-tagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' This shows that FCC-ee will be more  advantageous than ILC-250 GeV and CEPC 250 GeV colliders for investigating Higgs boson properties  through the Signal 1 group around these center of mass energies (240-250 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' In the ILC-500 GeV  and CLIC-380-1500-3000 GeV colliders, results well above the desired significance value can be obtained  for signal 1 processes, even without the b-tagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Therefore, the properties of the Higgs boson through  Signal 1 processes can be studied with precision in colliders other than the ILC-250 GeV collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Since  the results obtained for the Signal 2 process are greater than 5 significance values, the properties of the  Table 4: Cross sections, number of events and the significance values for FCC-ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Colliders  Processes  Sg CS  (pb)  Bg CS  (pb)  L  (pb−1)  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' SgE  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='BgE  S  FCC-ee  (240 GeV)  Signal 1  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='98×10−5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='36×10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='7×106  169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='7  401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='2  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='95  Signal 1  (with b-tagging)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='79×10−5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='67×10−5  115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='4  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='4  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='9  Signal 2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='25×10−2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='37×10−1  21250  912900  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='15  Signal 2  (with b-tagging)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='53×10−3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='38×10−2  14501  125460  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='18  7  Table 5: Cross sections, number of events and the significance values for ILC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Colliders  Processes  Sg CS  (pb)  Bg CS  (pb)  L  (pb−1)  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' SgE  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='BgE  S  ILC  (250 GeV)  Signal 1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='33×10−4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='9×10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='35×105  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='95  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='68  Signal 1  (with b-tagging)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='04×10−5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='34×10−5  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='86  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='03  Signal 2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='3×10−2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='33×10−1  1755  71955  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='51  Signal 2  (with b-tagging)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='82×10−3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='32×10−2  1190  9882  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='74  ILC  (500 GeV)  Signal 1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='41×10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='7×10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='8×105  253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='8  306  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='98  Signal 1  (with b-tagging)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='57×10−4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='32×10−4  172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='3  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='8  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='88  Signal 2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='86×10−2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='13×10−1  5148  20340  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='71  Signal 2  (with b-tagging)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='94×10−2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='56×10−2  3492  2808  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='54  Table 6: Cross sections, number of events and the significance values for CLIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Colliders  Processes  Sg CS  (pb)  Bg CS  (pb)  L  (pb−1)  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' SgE  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='BgE  S  CLIC  (380 GeV)  Signal 1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='44×10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='08×10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5×105  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='6  162  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='98  Signal 1  (with b-tagging)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='38×10−4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='53×10−4  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='7  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='95  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='76  Signal 2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='7×10−2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='11×10−1  2550  31650  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='14  Signal 2  (with b-tagging)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='16×10−2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='9×10−2  1740  4350  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='86  CLIC  (1500 GeV)  Signal 1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='95×10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='47×10−3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='7×105  721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='5  543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='9  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='34  Signal 1  (with b-tagging)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='32×10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='34×10−4  488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='4  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='58  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='64  Signal 2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='03×10−1  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='22×10−3  38110  3411  362  Signal 2  (with b-tagging)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='99×10−2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='27×10−3  25863  470  400  CLIC  (3000 GeV)  Signal 1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='01×10−4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='47×10−4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='9×105  355  499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='7  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='38  Signal 1  (with b-tagging)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='09×10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='32×10−4  241  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='8  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='44  Signal 2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='61×10−1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='72×10−3  94990  1605  776  Signal 2  (with b-tagging)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='09×10−1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='74×10−4  64310  221  777  Higgs boson can be studied precisely for all colliders through this channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  As a result, in future lepton colliders, the Higgs boson can be observed with high event rates via  Higgsstrahlung, W and Z fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content=' Thus, electron–positron colliders can precisely measure the properties  of the Higgs boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
+page_content='  Acknowledgment  We would like to thank Professor Dr Inanc Sahin for his suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
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+page_content=' Higgs Properties,” ,arXiv:hep-ph/1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
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+page_content='data-an].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfAze2/content/2301.13463v1.pdf'}
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+Complete identification of spin-wave eigenmodes excited by parametric pumping in YIG microdisks
+T. Srivastava*,1, ∗ H. Merbouche*,2, † I. Ngouagnia Yemeli,1 N. Beaulieu,3 J. Ben Youssef,3 M.
+Mu˜noz,4 P. Che,5 P. Bortolotti,5 V. Cros,5 O. Klein,6 S. Sangiao,7 J. M. De Teresa,7 S. O.
+Demokritov,2 V. E. Demidov,2 A. Anane,5 C. Serpico,8 M. d’Aquino,8 and G. de Loubens1, ‡
+1SPEC, CEA, CNRS, Universit´e Paris-Saclay, Gif-sur-Yvette, France
+2Institute for Applied Physics, University of Muenster, Germany
+3LabSTICC, CNRS, Universit´e de Bretagne Occidentale, Brest, France
+4Instituto de Tecnolog´ıas F´ısicas y de la Informaci´on (CSIC), Madrid, Spain
+5Unit´e Mixte de Physique, CNRS, Thales, Universit´e Paris-Saclay, Palaiseau, France
+6Universit´e Grenoble Alpes, CEA, CNRS, Grenoble INP, Spintec, Grenoble, France
+7Instituto de Nanociencia y Materiales de Arag´on (INMA) and Laboratorio de
+Microscop´ıas Avanzadas (LMA), Universidad de Zaragoza, Zaragoza, Spain
+8Department of Electrical Engineering and ICT, University of Naples Federico II, Italy
+We present the parametric excitation of spin-wave modes in YIG microdisks via parallel pumping. Their
+spectroscopy is performed using magnetic resonance force microscopy (MRFM), while their spatial profiles
+are determined by micro-focus Brillouin light scattering (BLS). We observe that almost all the fundamental
+eigenmodes of an in-plane magnetized YIG microdisk, calculated using a micromagnetic eigenmode solver,
+can be excited using the parallel pumping scheme, as opposed to the transverse one. The comparison between
+the MRFM and BLS data on one side, and the simulations on the other side, provides the complete spectro-
+scopic labeling of over 40 parametrically excited modes. Our findings could be promising for spin-wave-based
+computation schemes, in which the amplitudes of a large number of spin-wave modes have to be controlled.
+I.
+INTRODUCTION
+Novel proposals for spin-wave-based computing schemes
+necessitate the generation and control of multiple spin-wave
+(SW) modes1–5. The most standard way to excite SW modes
+in a magnetic microstructure is by direct inductive coupling.
+There, the quasi-uniform microwave field, produced on the
+magnetic volume by an rf antenna, couples to the transverse
+dynamical component of the magnetization associated with
+the SW mode, with a maximal efficiency when the applied
+rf frequency coincides with the eigenfrequency of the mode.
+However, this method is not adapted to excite modes with anti-
+symmetric spatial profiles, as their overlap integral with the
+excitation field is zero6, nor short-wavelength modes, as their
+excitation efficiency quickly decreases with their wavevector.
+Yet, these two categories of modes make up a significant part
+of the SW k-space. In order to excite a large number of modes
+irrespective of their spatial profiles, parametric parallel pump-
+ing, which does not suffer from these limitations, becomes the
+ideal choice7. In this case, the microwave magnetic field cre-
+ated by the rf antenna is aligned parallel to the static field. As
+a result, it does not couple to the SW modes directly. Instead,
+it interacts with the dynamic component of magnetization os-
+cillating at 2ω in the static field direction, which arises due to
+the elliptical trajectory of magnetization precession at ω. An
+rf field at 2ω can therefore excite SW modes at ω. A quantum
+mechanical picture of this process is a photon generating two
+magnons of opposite momenta at half its frequency8. Since
+this is a nonlinear process, SWs are excited only if the am-
+plitude of the excitation field exceeds a parametric threshold,
+which depends on the mode relaxation, and on the mode ellip-
+ticity. The threshold power is lower for lower relaxation rates
+and higher ellipticities.
+Parallel pumping has been employed to generate SW modes
+in extended films9–13 and micro- and nano-waveguides14,15
+of yttrium iron garnet (YIG), as well as in magnetic
+nanocontacts16, magnetic tunnel junctions17, and micro- and
+nano-dots of Permalloy18–20. It has also been used for SW
+amplification21. All these studies have been limited to a hand-
+ful number of modes.
+The excitation and identification of
+many modes in an adequate system would pave the way to-
+wards simultaneous control and manipulation of a large num-
+ber of SW modes for different applications in magnonics.
+In this study, we present the excitation and identification of
+multiple SW modes in YIG microdisks via parametric pump-
+ing. The scheme of the experiments is shown in Fig. 1. The
+SW modes are excited in YIG disks of diameters 1 µm, 3 µm
+and 5 µm through an integrated rf antenna and detected us-
+ing a magnetic resonance force microscope (MRFM). Their
+spatial profiles can also be recorded using micro-focus Bril-
+louin light scattering spectroscopy (µ-BLS). We observe that
+almost all the SW eigenmodes are accessible by parametric
+pumping. As expected, these eigenmodes become fewer in
+number as the size of the disk decreases. For the 3 µm disk,
+we label over 40 eigenmodes by comparing its MRFM para-
+metric spectroscopy to micromagnetic simulations, and con-
+firm the identification of as many as 10 of them through their
+profiles thanks to µ-BLS. Our results could be instrumental in
+designing basic units for unconventional computing schemes
+like neuromorphic computing using hyperconnected popula-
+tions of a large number of eigen-excitations in a single mi-
+crostructure.
+II.
+RESULTS
+A.
+Sample
+We use 50 nm thick YIG grown on 0.5 mm thick GGG
+substrate by liquid phase epitaxy22.
+The characteristics of
+arXiv:2301.13468v1  [physics.app-ph]  31 Jan 2023
+
+2
+FIG. 1. Schematics of the experimental setup. Parallel pump-
+ing of a YIG microdisk using an rf antenna deposited on top. The
+spectroscopy of the parametrically excited modes is achieved using
+a magnetic resonance force microscope (MRFM) positioned above
+the sample. Their spatial profiles are measured by micro-focus Bril-
+louin light scattering (µ-BLS) using a separate experimental setup,
+the laser beam being focused to the bottom of the sample, through
+the transparent GGG substrate.
+the extended film are measured by standard magnetometry
+and broadband FMR techniques.
+These yield a saturation
+magnetization Ms = 140.7 kA/m, a gyromagnetic ratio γ =
+28.28 GHz/T, a Gilbert damping parameter α = 7.5 × 10−5,
+and a weak inhomogeneous broadening of the FMR linewidth,
+found to be 0.1 mT. These parameters are typical of the YIG
+material; the exchange constant, which has not been specifi-
+cally determined on this film, is assumed to be A = 3.7 pJ/m, a
+standard value from literature23. The YIG layer was patterned
+into disks of diameters 1 µm, 3 µm and 5 µm using e-beam
+lithography. A 220 nm thick Ti/Au antenna, of width equal to
+8 µm, was then deposited on top of the disks. Injecting an rf
+current in the antenna generates an rf in-plane magnetic field
+that is orthogonal to the long axis of the antenna.
+B.
+Parallel pumping spectroscopy
+The SW mode spectroscopy is done using MRFM. It em-
+ploys a very soft cantilever, at the end of which a submicronic
+magnetic spherical probe made of cobalt is attached24, to me-
+chanically detect the magnetization dynamics in the sample
+placed underneath25. When SWs are excited in the sample
+by the microwave field, the (static) longitudinal component of
+magnetization is reduced and so is the dipolar force on the
+MRFM probe, resulting in a displacement of the cantilever
+beam, which is detected optically. The rf excitation applied
+to the sample via the antenna is modulated at the mechanical
+resonance frequency of the cantilever to improve the quality
+factor and the signal-to-noise ratio.
+In these measurements, the dc magnetic field is applied in-
+plane at an angle of 45° with respect to the direction of the rf
+magnetic field, as displayed in Fig. 1. Therefore the rf field ex-
+citation has both a transverse and a parallel component relative
+to the magnetization direction. The parallel pumped SW spec-
+trum is studied for different-sized disks as a function of the
+applied microwave power. Figure 2 shows the results of the
+MRFM parametric spectroscopy performed at a constant mi-
+crowave frequency of 4 GHz for the three disks (color-coded
+intensity maps), together with the corresponding transverse
+excitation spectra measured at fixed frequency of 2 GHz and
+power of −5 dBm (continuous white curves). Only a few SW
+modes are detected in the latter regime. In contrast, we ob-
+serve that a large number of modes can be excited by parallel
+pumping at 4 GHz for all the disks, in the range of applied dc
+field corresponding to the direct excitation of modes at 2 GHz,
+because parametrically excited modes are generated at half the
+pumping frequency. As expected, this occurs only above a
+minimum power level, that ranges from about −4 dBm for the
+5 µm disk to −2 dBm for the 1 µm disk. The fact that the para-
+metric threshold increases and that the density of the excited
+modes decreases as the lateral size decreases can be explained
+by geometrical confinement effects, as reported earlier20.
+In the following, we will mainly focus on the 3 µm disk
+where the SW modes are quite abundant but at the same time
+discernible (not too closely spaced). We perform similar mea-
+surements on this disk, this time fixing the value of the dc field
+to 27 mT, and scanning the parallel pumping frequency as a
+function of the microwave power. Fig. 3b shows the intensity
+map of the parametrically excited modes in these conditions,
+as a function of half the pumping frequency fp/2 and the rf
+power P, varied along the horizontal and vertical axes, respec-
+tively. We note that the threshold power increases with the
+frequency in a non-monotonic way, which can be explained
+as follows. The threshold excitation field of each mode can
+indeed be computed as the ratio between the relaxation rate
+ωr(k) to a coupling coefficient V(k), that is related to the mode
+ellipticity7: the more elliptical a mode is, the larger its V(k)
+and the lower will its threshold be. Both the terms depend
+non-monotonously on the wavevector k and on the mode fre-
+quency. However, on a wide range, when k increases, so does
+the mode frequency and its relaxation rate, while its ellipticity
+and its coupling V(k) tends to decrease7,21. This leads to the
+clear but not monotonous increase of the experimental thresh-
+old power with frequency seen in Fig. 3b.
+C.
+Simulations
+In order to identify these parametrically excited modes, mi-
+cromagnetic simulations using the eigenmode solver imple-
+mented in the micromagnetic code MaGICo26 have been per-
+formed to calculate the SW spectrum. The magnetic ground
+state is first computed for the specific geometry and applied
+magnetic field. Once the magnetic ground state is known,
+the equation describing magnetization dynamics, the Landau-
+Lifshitz-Gilbert equation, is linearized around the ground state
+and small-amplitude spatial profiles of the modes are com-
+
+G
+rf antenna
+GGG3
+FIG. 2. MRFM parametric spectroscopy. Intensity maps of the parametrically excited modes in the field-power coordinates excited by the
+microwave field of frequency 4 GHz, measured by MRFM on the 5 µm (a), 3 µm (b) and 1 µm (c) diameter YIG disks. In each panel, the
+continuous white curve corresponds to the direct excitation spectrum at the frequency of 2 GHz. The rf pumping field hrf is at 45° from the dc
+field H.
+puted. This problem can be formulated as a generalized eigen-
+value problem as described in ref.27.
+The solution of the
+eigenvalue problem allows to the determination of the SW
+spectrum of the magnetic sample under investigation. Here,
+the geometry of the body, a 50 nm thick disk of 3 µm in di-
+ameter, was discretized using 300 × 300 × 5 cubic cells (mesh
+size of 10 × 10 × 10 nm3), and the values of the magnetic pa-
+rameters used in the simulation were those determined exper-
+imentally. As in the experimental case, the applied field lies
+in the plane of the disk and is set to 27 mT. The implemen-
+tation of suitable matrix-free large-scale methods described
+in ref.28 allowed the calculation of hundreds of eigenmodes
+for such an extended structure (353440 computational cells,
+eigenvalue problem size 706880 × 706880) in a few hours.
+Figure 3a displays the computed spatial profiles of the first
+100 eigenmodes. The 10 lowest frequency modes (first row)
+correspond to edge modes, where the precession of the mag-
+netization is strongly confined at the boundaries of the disk,
+in the (horizontal) direction of the applied dc field due to the
+demagnetizing field29,30. The following modes correspond to
+standing SW modes, which can be labelled by the number of
+precession lobes in the horizontal (nx) and vertical (ny) direc-
+tions. For instance, mode 20 (second row, last column) can be
+labelled by nx = 2 and ny = 1, i.e., it is the (2,1) mode. Mode
+40 (fourth row, last column) is the (11,3) mode. The most uni-
+form mode, usually referred as the FMR mode, is mode 29, or
+mode (1,1).
+Figure 3b presents the comparison between the experimen-
+tal spectroscopy and the computed eigenfrequencies of modes
+11 to 70, shown as red ticks on top of the intensity map of
+the parametrically excited modes. We observe a good agree-
+ment between the computed mode frequencies and the exper-
+imental mode frequencies (at half-pumping frequencies fp/2)
+observed at the bottom of the parametric instability regions
+(elongated yellow-green triangles extending downwards on
+the intensity map). From this comparison, it is possible to
+state that almost all, if not all SW eigenmodes, can be para-
+metrically excited, irrespective of their spatial profile. Due to
+the high density of modes in the investigated frequency range,
+we will only focus on a few modes, to emphasize the good
+agreement noted above. The lowest-lying computed modes in
+Fig. 3b are the pair of modes 11 and 12 with respective fre-
+quencies 1.938 and 1.9383 GHz, which correspond rather well
+to the measured parametric instability region with a threshold
+power of 2 dBm at around 1.95 GHz. The small disagree-
+ment of 10 MHz between the computed and measured fre-
+quencies is not unexpected, since these modes belong to the
+category of edge modes, whose characteristics are very sen-
+sitive to imperfections at the periphery of the disk30,31, which
+are not taken into account in the simulations. If we move to
+the next parametrically excited modes, which have the low-
+est power threshold and have frequencies around 1.975 GHz,
+the comparison with computed frequencies shows that they
+correspond to two pairs of modes: modes 13 and 14 with re-
+spective frequencies 1.973 and 1.9731 GHz, and modes 15
+and 16, at 1.9796 and 1.9809 GHz. The next excited modes
+in the experimental spectroscopy map are at around 2 GHz,
+and they correspond to mode 17 at 1.998 GHz and mode 18
+at 2.001 GHz. As a matter of fact, a detailed inspection of the
+data shows that indeed, the parametric instability region has
+two nearby minima with frequencies equally spaced around
+2 GHz. This good agreement between experimental and com-
+puted mode frequencies continues over the full range of inves-
+tigated frequencies. We note that among the 60 modes whose
+frequencies have been plotted in ig. 3b, only 44 modes have
+discernible frequencies and spatial profiles, a few of them be-
+ing pairs of modes with very similar characteristics (e.g., pairs
+of modes 11 and 12, 13 and 14, 15 and 16, 21 and 22, etc.).
+D.
+Spatial profiles with µ-BLS
+To push further the comparison between computed SW
+modes and experiments, it is possible to take advantage of
+µ-BLS, to map the spatial profiles of dynamic magnetization
+in micro-structures32. A probing laser light (λ = 473 nm and
+Plaser = 0.1 mW) is focused into a diffraction-limited spot on
+the surface of a similar 3 µm YIG disk (Fig. 1) and the mod-
+
+a
+C
+10.0
+10.0
+10.0
+7.5 
+7.5 
+7.5
+5.0 -
+5.0 
+5.0
+(dBm)
+2.5 
+(dBm)
+2.5
+(dBm)
+2.5
+0.0 -
+0.0 
+0.0
+P
+P
+P
+-2.5 -
+2.5
+2.5
+-5.0
+5.0 
+5.0
+7.5
+5
+wn
+7.5 
+μm
+7.5
+1 μm
+10.0-
+-10.0-
+10.0
+10
+15
+20
+25
+30
+35
+10
+15
+20
+25
+30
+35
+10
+15
+20
+25
+30
+35
+μoH (mT)
+μoH (mT)
+μoH (mT)4
+FIG. 3. Computed spin-wave eigenmodes. (a) Computed spatial
+profiles of the 100 lowest frequency modes of the 3 µm diameter YIG
+disk, in-plane magnetized by a field of 27 mT. The color code refers
+to the oscillation amplitude of the local magnetization, from blue
+(minimum) to red (maximum). (b) Comparison between parametric
+spectroscopy MRFM data (color-coded intensity map) and computed
+eigenfrequencies (red vertical ticks) of modes 11 to 70 (surrounded
+by the red rectangle in panel (a)).
+ulation of this probing light by the magnetization oscillations
+is analysed using a high-contrast optical spectrometer. The
+obtained signal – the BLS intensity – is proportional to the
+intensity of the magnetic oscillations at a given frequency. In
+this BLS measurement, the in-plane bias field is set at 30 mT.
+To compare the experimental mode profiles with the computed
+mode profiles, the micromagnetic simulations have therefore
+been repeated at 30 mT as well. To avoid nonlinear distor-
+tions of the mode profiles, known to occur when the mode
+amplitude increases too much, the BLS mapping of the mode
+profiles is performed at microwave power only slightly above
+threshold. By sweeping the laser spot position across the disk,
+a dozen of different modes are imaged, Fig. 4 presents the
+comparison between the experimental and computed profiles
+of 10 modes. Overall, the measured profiles are in good agree-
+ment with the computed ones, taking into account the exper-
+imental spatial resolution (≃ 250 nm) and the long duration
+of these measurements, which are subjected to experimental
+drifts. Similarly to the analysis performed in Fig. 3b, we ob-
+serve that the mode frequencies obtained by BLS correspond
+very well to the computed mode frequencies, with a mismatch
+that remains under 13 MHz for all modes. In particular, we
+observe well defined modes up to nx = 7 and ny = 3, which
+validates the agreement between experiment and simulations
+for a large number of modes.
+III.
+CONCLUSION
+Thanks to the comparison between parametric spectroscopy
+and mode imaging respectively performed by MRFM and
+BLS on one side, and micromagnetic simulations on the other
+side, we have successfully excited, detected and identified a
+large number (> 40) of SW eigenmodes in a 3 µm YIG disk,
+where the mode density is large due to the large lateral dimen-
+sions. The computed spatial profiles provide a direct way to
+label those modes, using the numbers of precession nodes in
+the directions parallel (nx) and transverse (ny) to the applied
+magnetic field. This study opens up the possibility to per-
+form experiments where many parametric modes are simulta-
+neously excited while using the normal mode approach33,34 to
+understand and harness the complex dynamics in the modal
+space of confined magnetic structures.
+ACKNOWLEDGEMENTS
+This work was supported by the Horizon2020 Research
+Framework Programme of the European Commission under
+grant no. 899646 (k-NET). It is also supported by a pub-
+lic grant overseen by the Agence Nationale de la Recherche
+as part of the “Investissements d’Avenir” program (Labex
+NanoSaclay, reference: ANR-10-LABX-0035).
+I.N.Y. ac-
+knowledges support from the ANR grant no. ANR-18-CE24-
+0021 (Maestro).
+
+0983GH2
+2.1275GHzf.
+n=2.1381GH
+Q0000
+2.1493 GH
+.1892GHzf
+.1992GHz
+2583GHz
+4GHz
+2GHzfa.2.3457GH2
+.3505GHz
+3519 GHz fgs =2.3557 GHz fa4
+9 GHz
+2.3774GHz fag-2.392GHzfgo-2.3949GHz5
+FIG. 4. BLS imaging of mode profiles. The central graph displays the BLS detected frequencies at 30 mT for 10 modes of the 3 µm disk
+(blue lines) and the corresponding computed frequencies of eigenmodes (red line). These frequencies are matched (dotted dark lines) by
+associating the mode profiles measured in the experiment (above the graph) to the ones computed in the simulation (below). The experimental
+and simulated mode frequencies (in GHz) and their difference (in MHz) are given in the table.
+∗ titiksha.srivastava@cea.fr
+† hugo.merbouche@uni-muenster.de
+‡ gregoire.deloubens@cea.fr
+1 T. Br¨acher and P. Pirro, “An analog magnon adder for all-
+magnonic neurons,” J. Appl. Phys. 124, 152119 (2018).
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+2.093
+2.117
+2.128
+2.145
+2.149
+2.173
+2.184
+2.217
+2.238
+2.252
+fexp - fsimu
+5
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+13
+2
+13
+0
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+page_content=' Bortolotti,5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' Serpico,8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' d’Aquino,8 and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' de Loubens1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' ‡  1SPEC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' CEA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Gif-sur-Yvette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' France  2Institute for Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' University of Muenster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Germany  3LabSTICC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Universit´e de Bretagne Occidentale,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Brest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' France  4Instituto de Tecnolog´ıas F´ısicas y de la Informaci´on (CSIC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Spain  5Unit´e Mixte de Physique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Thales,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Palaiseau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' France  6Universit´e Grenoble Alpes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' CEA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Grenoble INP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Spintec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Grenoble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' France  7Instituto de Nanociencia y Materiales de Arag´on (INMA) and Laboratorio de  Microscop´ıas Avanzadas (LMA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Universidad de Zaragoza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Zaragoza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Spain  8Department of Electrical Engineering and ICT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' University of Naples Federico II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Italy  We present the parametric excitation of spin-wave modes in YIG microdisks via parallel pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Their  spectroscopy is performed using magnetic resonance force microscopy (MRFM), while their spatial profiles  are determined by micro-focus Brillouin light scattering (BLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' We observe that almost all the fundamental  eigenmodes of an in-plane magnetized YIG microdisk, calculated using a micromagnetic eigenmode solver,  can be excited using the parallel pumping scheme, as opposed to the transverse one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The comparison between  the MRFM and BLS data on one side, and the simulations on the other side, provides the complete spectro-  scopic labeling of over 40 parametrically excited modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Our findings could be promising for spin-wave-based  computation schemes, in which the amplitudes of a large number of spin-wave modes have to be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  INTRODUCTION  Novel proposals for spin-wave-based computing schemes  necessitate the generation and control of multiple spin-wave  (SW) modes1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The most standard way to excite SW modes  in a magnetic microstructure is by direct inductive coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  There, the quasi-uniform microwave field, produced on the  magnetic volume by an rf antenna, couples to the transverse  dynamical component of the magnetization associated with  the SW mode, with a maximal efficiency when the applied  rf frequency coincides with the eigenfrequency of the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  However, this method is not adapted to excite modes with anti-  symmetric spatial profiles, as their overlap integral with the  excitation field is zero6, nor short-wavelength modes, as their  excitation efficiency quickly decreases with their wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  Yet, these two categories of modes make up a significant part  of the SW k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' In order to excite a large number of modes  irrespective of their spatial profiles, parametric parallel pump-  ing, which does not suffer from these limitations, becomes the  ideal choice7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' In this case, the microwave magnetic field cre-  ated by the rf antenna is aligned parallel to the static field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' As  a result, it does not couple to the SW modes directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Instead,  it interacts with the dynamic component of magnetization os-  cillating at 2ω in the static field direction, which arises due to  the elliptical trajectory of magnetization precession at ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' An  rf field at 2ω can therefore excite SW modes at ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' A quantum  mechanical picture of this process is a photon generating two  magnons of opposite momenta at half its frequency8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Since  this is a nonlinear process, SWs are excited only if the am-  plitude of the excitation field exceeds a parametric threshold,  which depends on the mode relaxation, and on the mode ellip-  ticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The threshold power is lower for lower relaxation rates  and higher ellipticities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  Parallel pumping has been employed to generate SW modes  in extended films9–13 and micro- and nano-waveguides14,15  of yttrium iron garnet (YIG), as well as in magnetic  nanocontacts16, magnetic tunnel junctions17, and micro- and  nano-dots of Permalloy18–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' It has also been used for SW  amplification21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' All these studies have been limited to a hand-  ful number of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  The excitation and identification of  many modes in an adequate system would pave the way to-  wards simultaneous control and manipulation of a large num-  ber of SW modes for different applications in magnonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  In this study, we present the excitation and identification of  multiple SW modes in YIG microdisks via parametric pump-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The scheme of the experiments is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The  SW modes are excited in YIG disks of diameters 1 µm, 3 µm  and 5 µm through an integrated rf antenna and detected us-  ing a magnetic resonance force microscope (MRFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Their  spatial profiles can also be recorded using micro-focus Bril-  louin light scattering spectroscopy (µ-BLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' We observe that  almost all the SW eigenmodes are accessible by parametric  pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' As expected, these eigenmodes become fewer in  number as the size of the disk decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' For the 3 µm disk,  we label over 40 eigenmodes by comparing its MRFM para-  metric spectroscopy to micromagnetic simulations, and con-  firm the identification of as many as 10 of them through their  profiles thanks to µ-BLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Our results could be instrumental in  designing basic units for unconventional computing schemes  like neuromorphic computing using hyperconnected popula-  tions of a large number of eigen-excitations in a single mi-  crostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  Sample  We use 50 nm thick YIG grown on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5 mm thick GGG  substrate by liquid phase epitaxy22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  The characteristics of  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='13468v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='app-ph]  31 Jan 2023  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Schematics of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Parallel pump-  ing of a YIG microdisk using an rf antenna deposited on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The  spectroscopy of the parametrically excited modes is achieved using  a magnetic resonance force microscope (MRFM) positioned above  the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Their spatial profiles are measured by micro-focus Bril-  louin light scattering (µ-BLS) using a separate experimental setup,  the laser beam being focused to the bottom of the sample, through  the transparent GGG substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  the extended film are measured by standard magnetometry  and broadband FMR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  These yield a saturation  magnetization Ms = 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='7 kA/m, a gyromagnetic ratio γ =  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='28 GHz/T, a Gilbert damping parameter α = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5 × 10−5,  and a weak inhomogeneous broadening of the FMR linewidth,  found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='1 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' These parameters are typical of the YIG  material;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' the exchange constant, which has not been specifi-  cally determined on this film, is assumed to be A = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='7 pJ/m, a  standard value from literature23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The YIG layer was patterned  into disks of diameters 1 µm, 3 µm and 5 µm using e-beam  lithography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' A 220 nm thick Ti/Au antenna, of width equal to  8 µm, was then deposited on top of the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Injecting an rf  current in the antenna generates an rf in-plane magnetic field  that is orthogonal to the long axis of the antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  Parallel pumping spectroscopy  The SW mode spectroscopy is done using MRFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' It em-  ploys a very soft cantilever, at the end of which a submicronic  magnetic spherical probe made of cobalt is attached24, to me-  chanically detect the magnetization dynamics in the sample  placed underneath25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' When SWs are excited in the sample  by the microwave field, the (static) longitudinal component of  magnetization is reduced and so is the dipolar force on the  MRFM probe, resulting in a displacement of the cantilever  beam, which is detected optically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The rf excitation applied  to the sample via the antenna is modulated at the mechanical  resonance frequency of the cantilever to improve the quality  factor and the signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  In these measurements, the dc magnetic field is applied in-  plane at an angle of 45° with respect to the direction of the rf  magnetic field, as displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Therefore the rf field ex-  citation has both a transverse and a parallel component relative  to the magnetization direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The parallel pumped SW spec-  trum is studied for different-sized disks as a function of the  applied microwave power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Figure 2 shows the results of the  MRFM parametric spectroscopy performed at a constant mi-  crowave frequency of 4 GHz for the three disks (color-coded  intensity maps), together with the corresponding transverse  excitation spectra measured at fixed frequency of 2 GHz and  power of −5 dBm (continuous white curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Only a few SW  modes are detected in the latter regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' In contrast, we ob-  serve that a large number of modes can be excited by parallel  pumping at 4 GHz for all the disks, in the range of applied dc  field corresponding to the direct excitation of modes at 2 GHz,  because parametrically excited modes are generated at half the  pumping frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' As expected, this occurs only above a  minimum power level, that ranges from about −4 dBm for the  5 µm disk to −2 dBm for the 1 µm disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The fact that the para-  metric threshold increases and that the density of the excited  modes decreases as the lateral size decreases can be explained  by geometrical confinement effects, as reported earlier20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  In the following, we will mainly focus on the 3 µm disk  where the SW modes are quite abundant but at the same time  discernible (not too closely spaced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' We perform similar mea-  surements on this disk, this time fixing the value of the dc field  to 27 mT, and scanning the parallel pumping frequency as a  function of the microwave power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 3b shows the intensity  map of the parametrically excited modes in these conditions,  as a function of half the pumping frequency fp/2 and the rf  power P, varied along the horizontal and vertical axes, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' We note that the threshold power increases with the  frequency in a non-monotonic way, which can be explained  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The threshold excitation field of each mode can  indeed be computed as the ratio between the relaxation rate  ωr(k) to a coupling coefficient V(k), that is related to the mode  ellipticity7: the more elliptical a mode is, the larger its V(k)  and the lower will its threshold be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Both the terms depend  non-monotonously on the wavevector k and on the mode fre-  quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' However, on a wide range, when k increases, so does  the mode frequency and its relaxation rate, while its ellipticity  and its coupling V(k) tends to decrease7,21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' This leads to the  clear but not monotonous increase of the experimental thresh-  old power with frequency seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  Simulations  In order to identify these parametrically excited modes, mi-  cromagnetic simulations using the eigenmode solver imple-  mented in the micromagnetic code MaGICo26 have been per-  formed to calculate the SW spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The magnetic ground  state is first computed for the specific geometry and applied  magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Once the magnetic ground state is known,  the equation describing magnetization dynamics, the Landau-  Lifshitz-Gilbert equation, is linearized around the ground state  and small-amplitude spatial profiles of the modes are com-  G  rf antenna  GGG3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' MRFM parametric spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Intensity maps of the parametrically excited modes in the field-power coordinates excited by the  microwave field of frequency 4 GHz, measured by MRFM on the 5 µm (a), 3 µm (b) and 1 µm (c) diameter YIG disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' In each panel, the  continuous white curve corresponds to the direct excitation spectrum at the frequency of 2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The rf pumping field hrf is at 45° from the dc  field H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  puted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' This problem can be formulated as a generalized eigen-  value problem as described in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  The solution of the  eigenvalue problem allows to the determination of the SW  spectrum of the magnetic sample under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Here,  the geometry of the body, a 50 nm thick disk of 3 µm in di-  ameter, was discretized using 300 × 300 × 5 cubic cells (mesh  size of 10 × 10 × 10 nm3), and the values of the magnetic pa-  rameters used in the simulation were those determined exper-  imentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' As in the experimental case, the applied field lies  in the plane of the disk and is set to 27 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The implemen-  tation of suitable matrix-free large-scale methods described  in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='28 allowed the calculation of hundreds of eigenmodes  for such an extended structure (353440 computational cells,  eigenvalue problem size 706880 × 706880) in a few hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  Figure 3a displays the computed spatial profiles of the first  100 eigenmodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The 10 lowest frequency modes (first row)  correspond to edge modes, where the precession of the mag-  netization is strongly confined at the boundaries of the disk,  in the (horizontal) direction of the applied dc field due to the  demagnetizing field29,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The following modes correspond to  standing SW modes, which can be labelled by the number of  precession lobes in the horizontal (nx) and vertical (ny) direc-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' For instance, mode 20 (second row, last column) can be  labelled by nx = 2 and ny = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=', it is the (2,1) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Mode  40 (fourth row, last column) is the (11,3) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The most uni-  form mode, usually referred as the FMR mode, is mode 29, or  mode (1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  Figure 3b presents the comparison between the experimen-  tal spectroscopy and the computed eigenfrequencies of modes  11 to 70, shown as red ticks on top of the intensity map of  the parametrically excited modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' We observe a good agree-  ment between the computed mode frequencies and the exper-  imental mode frequencies (at half-pumping frequencies fp/2)  observed at the bottom of the parametric instability regions  (elongated yellow-green triangles extending downwards on  the intensity map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' From this comparison, it is possible to  state that almost all, if not all SW eigenmodes, can be para-  metrically excited, irrespective of their spatial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Due to  the high density of modes in the investigated frequency range,  we will only focus on a few modes, to emphasize the good  agreement noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The lowest-lying computed modes in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 3b are the pair of modes 11 and 12 with respective fre-  quencies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='938 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='9383 GHz, which correspond rather well  to the measured parametric instability region with a threshold  power of 2 dBm at around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='95 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The small disagree-  ment of 10 MHz between the computed and measured fre-  quencies is not unexpected, since these modes belong to the  category of edge modes, whose characteristics are very sen-  sitive to imperfections at the periphery of the disk30,31, which  are not taken into account in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' If we move to  the next parametrically excited modes, which have the low-  est power threshold and have frequencies around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='975 GHz,  the comparison with computed frequencies shows that they  correspond to two pairs of modes: modes 13 and 14 with re-  spective frequencies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='973 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='9731 GHz, and modes 15  and 16, at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='9796 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='9809 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The next excited modes  in the experimental spectroscopy map are at around 2 GHz,  and they correspond to mode 17 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='998 GHz and mode 18  at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='001 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' As a matter of fact, a detailed inspection of the  data shows that indeed, the parametric instability region has  two nearby minima with frequencies equally spaced around  2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' This good agreement between experimental and com-  puted mode frequencies continues over the full range of inves-  tigated frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' We note that among the 60 modes whose  frequencies have been plotted in ig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 3b, only 44 modes have  discernible frequencies and spatial profiles, a few of them be-  ing pairs of modes with very similar characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=', pairs  of modes 11 and 12, 13 and 14, 15 and 16, 21 and 22, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  Spatial profiles with µ-BLS  To push further the comparison between computed SW  modes and experiments, it is possible to take advantage of  µ-BLS, to map the spatial profiles of dynamic magnetization  in micro-structures32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' A probing laser light (λ = 473 nm and  Plaser = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='1 mW) is focused into a diffraction-limited spot on  the surface of a similar 3 µm YIG disk (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 1) and the mod-  a  C  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0 -  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0  (dBm)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5  (dBm)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5  (dBm)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0  P  P  P  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5 -  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5  5  wn  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5  μm  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='5  1 μm  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0-  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0-  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='0  10  15  20  25  30  35  10  15  20  25  30  35  10  15  20  25  30  35  μoH (mT)  μoH (mT)  μoH (mT)4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Computed spin-wave eigenmodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' (a) Computed spatial  profiles of the 100 lowest frequency modes of the 3 µm diameter YIG  disk, in-plane magnetized by a field of 27 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The color code refers  to the oscillation amplitude of the local magnetization, from blue  (minimum) to red (maximum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' (b) Comparison between parametric  spectroscopy MRFM data (color-coded intensity map) and computed  eigenfrequencies (red vertical ticks) of modes 11 to 70 (surrounded  by the red rectangle in panel (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  ulation of this probing light by the magnetization oscillations  is analysed using a high-contrast optical spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The  obtained signal – the BLS intensity – is proportional to the  intensity of the magnetic oscillations at a given frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' In  this BLS measurement, the in-plane bias field is set at 30 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  To compare the experimental mode profiles with the computed  mode profiles, the micromagnetic simulations have therefore  been repeated at 30 mT as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' To avoid nonlinear distor-  tions of the mode profiles, known to occur when the mode  amplitude increases too much, the BLS mapping of the mode  profiles is performed at microwave power only slightly above  threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' By sweeping the laser spot position across the disk,  a dozen of different modes are imaged, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 4 presents the  comparison between the experimental and computed profiles  of 10 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Overall, the measured profiles are in good agree-  ment with the computed ones, taking into account the exper-  imental spatial resolution (≃ 250 nm) and the long duration  of these measurements, which are subjected to experimental  drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Similarly to the analysis performed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 3b, we ob-  serve that the mode frequencies obtained by BLS correspond  very well to the computed mode frequencies, with a mismatch  that remains under 13 MHz for all modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' In particular, we  observe well defined modes up to nx = 7 and ny = 3, which  validates the agreement between experiment and simulations  for a large number of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  CONCLUSION  Thanks to the comparison between parametric spectroscopy  and mode imaging respectively performed by MRFM and  BLS on one side, and micromagnetic simulations on the other  side, we have successfully excited, detected and identified a  large number (> 40) of SW eigenmodes in a 3 µm YIG disk,  where the mode density is large due to the large lateral dimen-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The computed spatial profiles provide a direct way to  label those modes, using the numbers of precession nodes in  the directions parallel (nx) and transverse (ny) to the applied  magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' This study opens up the possibility to per-  form experiments where many parametric modes are simulta-  neously excited while using the normal mode approach33,34 to  understand and harness the complex dynamics in the modal  space of confined magnetic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This work was supported by the Horizon2020 Research  Framework Programme of the European Commission under  grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 899646 (k-NET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' It is also supported by a pub-  lic grant overseen by the Agence Nationale de la Recherche  as part of the “Investissements d’Avenir” program (Labex  NanoSaclay, reference: ANR-10-LABX-0035).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' ac-  knowledges support from the ANR grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' ANR-18-CE24-  0021 (Maestro).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  0983GH2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='1275GHzf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='1381GH  Q0000  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='1493 GH  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='1892GHzf  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='1992GHz  2583GHz  4GHz  2GHzfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='3457GH2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='3505GHz  3519 GHz fgs =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='3557 GHz fa4  9 GHz  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='3774GHz fag-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='392GHzfgo-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='3949GHz5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' BLS imaging of mode profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The central graph displays the BLS detected frequencies at 30 mT for 10 modes of the 3 µm disk  (blue lines) and the corresponding computed frequencies of eigenmodes (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' These frequencies are matched (dotted dark lines) by  associating the mode profiles measured in the experiment (above the graph) to the ones computed in the simulation (below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The experimental  and simulated mode frequencies (in GHz) and their difference (in MHz) are given in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  ∗ titiksha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='srivastava@cea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='fr  † hugo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='merbouche@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='de  ‡ gregoire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='deloubens@cea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='fr  1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Br¨acher and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' Faini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Grollier, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Hurdequint, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Locatelli,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Pigeau, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Slavin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Tiberkevich, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Ulysse, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Valet,  and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Klein, “Identification and selection rules of the spin-wave  eigenmodes in a normally magnetized nanopillar,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content='  7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Gurevich and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Melkov, Magnetization Oscillations and  Waves (CRC Press, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  8 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' White and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Sparks, “Ferromagnetic Relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' The-  ory of instabilities,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' Kurebayashi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  Ferguson, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' Hillebrands, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Kreisel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Kopietz,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' Jungfleisch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Agrawal, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Kobljanskyj,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Melkov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Dubs, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Hillebrands, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Chumak, “Spin-  transfer torque based damping control of parametrically excited  spin waves in a magnetic insulator,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 108, 012402  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  14 Morteza Mohseni, Martin Kewenig, Roman Verba, Qi Wang,  Michael Schneider, Bj¨orn Heinz, Felix Kohl, Carsten Dubs, Bert  L¨agel, Alexander A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Serga, Burkard Hillebrands, Andrii V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Chu-  mak, and Philipp Pirro, “Parametric generation of propagating  spin waves in ultrathin yttrium iron garnet waveguides,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Sta-  tus Solidi RRL 14, 2000011 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  15 Bj¨orn Heinz, Morteza Mohseni, Akira Lentfert, Roman Verba,  Michael Schneider, Bert L¨agel, Khrystyna Levchenko, Thomas  Br¨acher, Carsten Dubs, Andrii V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Chumak,  and Philipp Pirro,  “Parametric generation of spin waves in nanoscaled magnonic  conduits,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content='  16 Sergei Urazhdin, Vasil Tiberkevich, and Andrei Slavin, “Paramet-  ric excitation of a magnetic nanocontact by a microwave field,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' 105, 237204 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='  17 Yu-Jin Chen, Han Kyu Lee, Roman Verba, Jordan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Katine, Igor  Barsukov, Vasil Tiberkevich, John Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Xiao, Andrei N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Slavin, and  Profile  00  (experiment)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='15  Frequency,GHz  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='20  Profile  00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='00  (simulation)  fexp (GHz)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='098  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='124  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='141  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='147  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='162  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='173  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='192  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='226  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='236  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='253  fsimu (GHz)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='093  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='117  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='128  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='145  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='149  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='173  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='184  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='217  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='238  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content='252  fexp - fsimu  5  7  13  2  13  0  8  9  2  1  (MHz)6  Ilya N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Krivorotov, “Parametric resonance of magnetization ex-  cited by electric field,” Nano Letters 17, 572–577 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' Demidov, Sergej O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
+page_content=' Demokritov,  and Sergei Urazhdin, “Parametric excitation of eigenmodes in mi-  croscopic magnetic dots,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' Demokritov,  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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+page_content=' Klein, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfCTcy/content/2301.13468v1.pdf'}
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diff --git a/MtAyT4oBgHgl3EQf6vrQ/content/tmp_files/2301.00828v1.pdf.txt b/MtAyT4oBgHgl3EQf6vrQ/content/tmp_files/2301.00828v1.pdf.txt
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@@ -0,0 +1,1347 @@
+MNRAS 000, 1–13 (2020)
+Preprint 4 January 2023
+Compiled using MNRAS LATEX style file v3.0
+From dark matter halos to pre-stellar cores: High resolution follow-up of
+cosmological Lyman-Werner simulations.
+Lewis R. Prole,1★ Anna T. P. Schauer,2 Paul C. Clark,1 Simon C. O. Glover,3 Felix D. Priestley,1 Ralf S. Klessen,3,4
+1Cardiff University School of Physics and Astronomy
+2Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA
+3Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Straße 2, D-69120 Heidelberg, Germany
+4Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+Molecular hydrogen allows cooling in primordial gas, facilitating its collapse into Population III stars within primordial halos.
+Lyman-Werner (LW) radiation from these stars can escape the halo and delay further star formation by destroying H2 in other
+halos. As cosmological simulations show that increasing the background LW field strength increases the average halo mass
+required for star formation, we perform follow-up simulations of selected halos to investigate the knock-on effects this has on
+the Population III IMF. We follow 5 halos for each of the 𝐽21 = 0, 0.01 and 0.1 LW field strengths, resolving the pre-stellar core
+density of 10−6 g cm−3 (1018 cm−3) before inserting sink particles and following the fragmentation behaviour for hundreds of
+years further. We find that the mass accreted onto sinks by the end of the simulations is proportional to the mass within the
+∼ 10−2 pc molecular core, which is not correlated to the initial mass of the halo. As such, the IMF shows little dependence on
+the LW strength. As the range of background LW field strengths tested here covers the most likely values from literature, we
+conclude that the IMF for so-called Pop III.2 stars is not significantly different from the initial population of Pop III.1 stars. The
+primordial IMF therefore likely remains unchanged until the formation of the next generation of Population II stars.
+Key words: stars: Population III – dark ages, reionization, first stars – hydrodynamics – stars: luminosity function, mass function
+– software: simulations
+1 INTRODUCTION
+This first stars are able to form because pristine baryonic gas can
+collapse within the gravitational potential well of dark matter (DM)
+halos (Couchman & Rees 1986; Haiman et al. 1996a; Tegmark et al.
+1997), heating it to ∼ 5000 K and facilitating the formation of H2
+(Bromm et al. 2002). H2 primarily forms via the radiative association
+reaction forming H−
+H + e− → H− + 𝛾,
+(1)
+followed by the fast associative detachment reaction forming H2
+H− + H → H2 + e−.
+(2)
+Radiative cooling from the molecular hydrogen renders the gas grav-
+itationally unstable, which allows it to decouple from the DM and
+collapse to form the first stars, known as Population III (Pop III)
+stars. The necessity of H2 renders any process of H2 destruction as a
+mechanism to delay or prevent Pop III star formation.
+While so-called Pop III.1 stars form from purely cosmological
+initial conditions, the radiation they produce affects Pop III.2 stars
+that form in its presence. As the masses of Pop III stars are predicted
+to be larger than present-day counterparts (due to the lack of cooling
+from dust and metals), they are expected to emit large amounts of
+★ E-mail: Prolel@cardiff.ac.uk
+ionizing radiation (e.g. Schaerer 2002). Ionizing photons above the
+Lyman limit (13.6 eV) create H ii regions around the stars up to the
+boundary of their Strömgren spheres (Whalen et al. 2004; Kitayama
+et al. 2004; Alvarez et al. 2006; Abel et al. 2007; Yoshida et al.
+2007a; Jaura et al. 2022) while photons below the Lyman limit are
+free to escape their Strömgren sphere. Lyman-Werner (LW) photons
+between 11.2 and 13.6 eV can dissociate H2 via the two-step Solomon
+process (Field et al. 1966; Stecher & Williams 1967)
+H2 + 𝛾 → H∗
+2 → 2H,
+(3)
+where H∗
+2 represents an electronically excited state of H2. Photons
+with energy above 0.76 eV can also photodissociate H− via
+H− + 𝛾 → H + e−,
+(4)
+reducing the H− abundance and hence the rate at which H2 can
+form via reaction 2 (e.g. Chuzhoy et al. 2007). This stellar feedback
+provides a potential obstacle for Pop III.2 stars to overcome during
+formation. Investigations into the effects of these far UV fields typ-
+ically categorise the field strength by the intensity in the LW band
+𝐽21, in units of 10−21 erg s−1 cm−2 Hz−1 sr−1.
+Calculations by Haiman et al. (1997) found that before the Ström-
+gren spheres of Pop III stars overlap, the UV background below the
+ionization threshold was able to penetrate large clouds and suppress
+their H2 abundance. They also found that the flux necessary for H2
+photodissociation is several orders of magnitude smaller than the flux
+© 2020 The Authors
+arXiv:2301.00828v1  [astro-ph.GA]  2 Jan 2023
+
+2
+L. R. Prole
+needed to reionize the universe. Haiman et al. (2000) showed that
+this photodissociation of H2 suppresses further Pop III star formation
+inside small halos and delays reionization until larger halos form.
+Collapse is not impossible without sufficient H2 for cooling.
+Omukai (2001) showed that if the LW field is sufficient to keep a
+halo free of molecular hydrogen, the gas can nevertheless collapse
+via atomic hydrogen line cooling if the halo has a virial temperature
+𝑇vir > 8000 K. The collapse occurs almost isothermally, possibly
+resulting in the formation of a direct collapse black hole (DCBH)
+(e.g. Bromm & Loeb 2003; Spaans & Silk 2006; Latif et al. 2013), a
+possible progenitor of the supermassive black holes observed at high
+redshifts (e.g. Mortlock et al. 2011; Matsuoka et al. 2019). However,
+for a T = 105 K blackbody spectrum expected from Pop III stars, a
+field strength of 𝐽21 ∼ 104 is required to keep the gas atomic during
+the collapse (Glover 2015; Agarwal & Khochfar 2015; Agarwal et al.
+2016; Sugimura et al. 2014), while the average exposure is expected
+to be 𝐽21 < 0.1 at 𝑧 ∼ 15 (Ahn et al. 2009; Trenti & Stiavelli 2009;
+Wise et al. 2012; Agarwal et al. 2012; Skinner & Wise 2020). Dijk-
+stra et al. (2008) showed that only a fraction of 10−8 − 10−6 of DM
+halos with virial temperatures > 104 K have a close luminous neigh-
+bour within < 10 kpc, and are exposed to an LW flux 𝐽21 > 103.
+The occurrence of atomically cooled halos is therefore expected to
+be rare.
+Studies have shown that values of 𝐽21 orders of magnitude lower
+than the critical intensity required to completely suppress H2 cooling
+in massive halos can still drastically affect halo collapse. Typically,
+the critical mass for efficient molecular hydrogen cooling and sub-
+sequent star formation increases with increasing 𝐽21 (e.g. Machacek
+et al. 2001; Yoshida et al. 2003; O’Shea & Norman 2008; Vis-
+bal et al. 2014; Schauer et al. 2021). Cosmological simulations by
+Yoshida et al. (2003) found gas cooling was suppressed for 𝐽21 = 0.1,
+leading them to predict that star formation would not occur in halos
+with 𝑇vir < 8000 K for LW field strengths greater than this. Con-
+versely, O’Shea & Norman (2008) found that for field strengths as
+high as 𝐽21 = 1, H2 cooling leads to collapse despite the depressed
+core molecular hydrogen fractions. They also noted that higher LW
+background fluxes lead to higher accretion rates. High resolution
+cosmological simulations by Schauer et al. (2021) (hereafter AS21)
+examined the impact of different values of the LW field strength on
+a large sample of minihalos. They showed that both Mav, the aver-
+age minihalo mass required for efficient H2 cooling (i.e. the mass
+above which more than 50% of minihalos of that mass can cool), and
+Mmin, the minimum minihalo mass required for efficient cooling, in-
+creased with increasing 𝐽21. An increase in the critical halo mass for
+star formation with increasing 𝐽21 was also found by Kulkarni et al.
+(2021), although they find a significant effect only for 𝐽21 > 1. In
+contrast, cosmological simulations by Skinner & Wise (2020) found
+no relationship between the LW intensity and host halo mass.
+The true average 𝐽21 intensity is expected to vary with redshift.
+Hirano et al. (2015) followed the formation and evolution of 1540 star-
+forming gas clouds. They found that in their models, the characteristic
+mass of Pop III stars shifted to lower masses with decreasing redshift
+due to the radiative feedback of previous generations of stars. For
+𝑧 > 20, half of the star-forming gas clouds were exposed to intense
+FUV radiation, with an average exposure of 𝐽21 ∼ 0.07. Due to
+smaller stellar masses and the expanding distance between stars, the
+FUV background became weaker at lower redshifts. For 15 < 𝑧 < 25,
+almost all the clouds had nonzero intensity 𝐽21 > 0.01. The average
+LW intensity in Skinner & Wise (2020) increased stochastically from
+10−3 at 𝑧 ∼ 25 to 10 at 𝑧 ∼ 10. For redshifts above ∼12, 𝐽21 remained
+> 0.1.
+Self-shielding is a process that occurs when large column den-
+sity of molecular hydrogen protects the inner regions against pho-
+todissociation because one photon can only photodissociate one H2
+molecule. This self-shielding allows further H2 production and H2
+cooling (Shang et al. 2010; Agarwal et al. 2014; Regan et al. 2014;
+Hartwig et al. 2015a). The large nonequilibrium abundance of elec-
+trons in gas cooling from above T > 104 K also boosts H2 formation
+(Oh & Haiman 2002). Early attempts to model self shielding (e.g.
+Shang et al. 2010) multiplied the intensity in the LW band by a self-
+shielding factor given by Draine & Bertoldi (1996). Wolcott-Green
+et al. (2011) showed that this method underestimated the numerically
+calculated self-shielding rate by more than an order of magnitude in
+low-density regions, by overestimating shielding by a large factor at
+temperatures above a few hundred kelvin. They modified the method
+of Draine & Bertoldi by estimating the shielding factor based on
+the Sobolev length, using local properties of the gas. This modifica-
+tion was computationally inexpensive and used in many subsequent
+investigations into the aforementioned critical intensity required to
+form atomic halos, typically producing values an order of magnitude
+lower than those using the original Draine & Bertoldi shielding (e.g.
+Glover 2015; Agarwal & Khochfar 2015; Agarwal et al. 2016). Clark
+et al. (2012) improved on this method further with their introduction
+of the TreeCol algorithm, which calculates maps of the column den-
+sity distribution seen by each computational element in a simulation
+in a computationally efficient fashion with the help of an oct-tree.
+Hartwig et al. (2015b) took this approach further by accounting for
+the relative velocities between different computational elements. This
+Doppler-shifts the spectral lines, reducing the effectiveness of self-
+shielding (since molecules shifted by more than the linewidth do not
+contribute to the effective column density).
+In addition to a background LW field, primordial star formation is
+complicated further by streaming velocities between the the gas and
+DM. Prior to recombination, baryons were tightly coupled to photons.
+As DM does not experience Thomson scattering, there should have
+been a relative velocity between the DM and baryons (e.g. Ma &
+Bertschinger 1995). At recombination, the relative velocity was ∼ 30
+km s−1 and was coherent over several comoving Mpc. Recombination
+resulted in a drop in the sound speed to ∼ 6 km s−1 as the gas
+transitioned from plasma to a neutral state, meaning the relative
+velocities were highly supersonic. Tseliakhovich & Hirata (2010)
+showed that the presence of these large-scale streaming velocities
+suppresses the abundance of the first bound objects by advecting
+small-scale perturbations near the baryonic Jeans scale. Moving-
+mesh calculations by Greif et al. (2011) found that the additional
+momentum and energy from the streaming velocities reduces the gas
+fractions and central densities of halos, increasing the typical virial
+mass required for efficient cooling by a factor of three. They also
+noted that the turbulent velocity dispersion increased in the presence
+of streaming velocities. The simulations of AS21 found that the
+increase in the average and minimum halo mass from increasing
+streaming velocities is additive on top of the effect of a LW field,
+with streaming velocities having the larger impact of the two.
+While it was initally believed that Pop III stars formed in isolation
+(Haiman et al. 1996b) and were massive (Abel et al. 2002; Bromm
+et al. 2002), more recent studies show that primordial gas fragments
+to give rise to a larger populations of lower mass stars (Clark et al.
+2011; Smith et al. 2011; Greif et al. 2012; Stacy & Bromm 2013;
+Machida & Doi 2013; Stacy et al. 2014; Susa 2019; Wollenberg
+et al. 2020). In Prole et al. (2022a) (hereafter LP22), we used high
+resolution simulations of idealised, purely hydrodynamical Pop III
+star formation to show that a number of cores are ejected from the
+system with masses capable of surviving until the present day. As
+small-scale primordial magnetic fields do not appear to prevent disc
+MNRAS 000, 1–13 (2020)
+
+Cosmological Lyman-Werner simulations.
+3
+fragmentation (Prole et al. 2022b) and accretion of metals onto the
+surface of these stars during their lifetime is unlikely (e.g. Johnson
+& Khochfar 2011; Tanaka et al. 2017), the question is raised about
+why these stars have not been found within archeological surveys
+(see e.g. Beers & Christlieb 2005; Frebel & Norris 2015; Starken-
+burg et al. 2017). Since most high resolution simulations of Pop III
+star formation have considered only the Pop III.1 case, one possible
+explanation could be that Pop III.2 star formation yields a different
+IMF, i.e. that Pop III stars forming in the presence of a LW back-
+ground have systematically larger masses than those forming in the
+absence of a background.
+In this paper, we aim to test this hypothesis by producing the
+most accurate prediction of the Pop III.1 and Pop III.2 initial mass
+functions (IMF) to date. We investigate how the increase in halo
+masses due to increasing LW field intensity changes star formation
+within them, by performing high resolution follow-up simulations
+of cosmological halos drawn from the simulations of AS21. The
+structure of our paper is as follows. In Section 2, we describe the
+cosmological simulations of AS21, our selection criteria for the halos
+chosen for follow-up simulations, the chemical model we use and our
+use of sink particles. In Section 3 we review the characteristics of the
+halos as they are taken from AS21, before presenting the results of the
+zoom-in simulations in Section 4, where we probe the density regime
+of the molecular core. In Section 5, we compare the fragmentation
+behaviour once sink particles have formed and present the IMFs at
+the end of the simulations. We discuss caveats in Section 6 before
+concluding in Section 7.
+2 NUMERICAL METHOD
+2.1 Arepo
+The simulations presented here were performed with the moving
+mesh code Arepo (Springel 2010) with a primordial chemistry set-
+up. Arepo combines the advantages of AMR and smoothed particle
+hydrodynamics (SPH: Monaghan 1992) with a mesh made up of a
+moving, unstructured, Voronoi tessellation of discrete points. Arepo
+solves hyperbolic conservation laws of ideal hydrodynamics with a
+finite volume approach, based on a second-order unsplit Godunov
+scheme with an exact Riemann solver. Automatic and continuous
+refinement overcome the challenge of structure growth associated
+with AMR (e.g. Heitmann et al. 2008).
+2.2 Cosmological simulations
+The cosmological simulations performed by AS21 assumed a ΛCDM
+cosmology with parameters ℎ = 0.6774, Ω0 = 0.3089, Ωb =
+0.04864, ΩΛ = 0.6911, 𝑛 = 0.96 and 𝜎8 = 0.8159 as derived by
+the Planck Collaboration et al. (2020). The simulations were ini-
+tialised at 𝑧 = 200 with an initial DM distribution created by MUSIC
+(Hahn & Abel 2011) using the transfer functions of Eisenstein &
+Hu (1998) and the gas distribution initially followed the DM. The
+DM was represented by 10243 particles and the gas was initially
+modelled with 10243 grid cells, all contained within a box with side
+length 1 ℎ−1 Mpc in comoving units. During the simulation, an ad-
+ditional Jeans refinement criterion was applied: cells were refined
+whenever necessary so as to ensure that the Jeans length was always
+resolved with at least 16 cells. This refinement was carried out until
+the gas reached a threshold density of ∼ 10−19 g cm−3. Above this
+threshold density, gravitationally bound and collapsing gas was con-
+verted into collisionless sink particles, as explained at greater length
+in AS21. The simulations were carried out with four different values
+of the baryonic streaming velocity (𝑣str = 0, 1, 2 and 3, in units of
+𝜎rms, the large-scale root mean squared value) and three different
+values for the LW field strength (𝐽21 = 0, 0.01 and 0.1). In this study,
+we make use of the three simulations with 𝑣str = 1𝜎rms because this
+is the most representative value available, as the volume fraction of
+streaming velocities peaks at 0.8𝜎rms (AS21).
+2.3 Halo selection
+Given snapshots at 𝑧 = 15 from the three simulations with 𝑣str = 1
+presented in AS21, we have selected 5 halos for each value of 𝐽21.
+Halo positions and masses were provided by the friends-of-friends
+(FoF) algorithm as described in that study. The selection criteria for
+the halos was as follows:
+• The halos identified by the FoF algorithm were sorted by their
+mass difference with Mav, the average minihalo mass above which
+minihalos become capable of cooling and forming stars. This average
+mass was defined in AS21, following Schauer et al. (2019), to be the
+minihalo mass above which more than 50% of minihalos can cool
+and form stars.1 AS21 report Mav at a range of redshifts between
+𝑧 = 22 and 𝑧 = 14; here, we adopt their values at 𝑧 = 15. By sorting
+the halos in this way, we ensure that the halos that we eventually
+select will have masses close to Mav, i.e. that they are representative
+of the common case at that redshift.
+• We considered a halo if it contained a cell denser than 𝜌th =
+10−22 g cm−3 within a search radius Rsearch = 1 kpc in physical
+units from the halo’s central coordinate. Note that checking the H2
+abundance is not necessary, as collapse to this density is not possible
+without a high H2 abundance outside of the atomically cooling halo
+scenario.
+• We check that the halo is sufficiently resolved i.e. that the 16
+cells per Jeans length criteria has not decreased as the cell sizes
+approach the maximum resolution of the cosmological simulations.
+This process begins when the density reaches ∼ 10−20 g cm−3, so
+we reject halos including a cell above this density. Note that this also
+ensures that a sink particle has not yet been formed within Rsearch.
+• Lastly, we removed the halo from consideration if there were
+other dense objects in the vicinity by checking there were no cells
+above 𝜌th within the radius Rsearch/2 < r < Rsearch.
+We performed this search throughout the simulation cube until 5
+suitable halos were selected for each 𝐽21 value. The selected halo
+masses from each simulation are shown in Table 1. Mass-weighted
+mean temperatures and H2 abundances are plotted as a function of
+density for each halo in Figure 1 and radial distributions of enclosed
+gas and DM along with velocity information are shown in Figure
+2. Projections of the initial density, temperature and H2 abundance
+distributions are visualised in Figures 3, 4 and 5. Note that all of
+these figures show the state of the halos at the point at which we
+extract them from the AS21 simulations, i.e. prior to our zoom-in
+calculation. Our choice of selection criteria means that the same
+halos are not chosen for direct comparison between each simulation.
+Rather, they are chosen to be most representative of the Universe at
+the given background LW field strength. Direct comparison also has
+the drawback that individual halos will collapse at different redshifts
+depending on the LW field strength, whereas we exclusively select
+halos that become able to form stars at 𝑧 = 15.
+1 Note that Mav is typically around a factor of three larger than 𝑀min, the
+mass of the least massive minihalo with cool gas.
+MNRAS 000, 1–13 (2020)
+
+4
+L. R. Prole
+10
+2
+10
+1
+100
+101
+102
+103
+n [cm
+3]
+101
+102
+103
+T [K]
+J21=0
+10
+2
+10
+1
+100
+101
+102
+103
+n [cm
+3]
+J21=0.01
+10
+2
+10
+1
+100
+101
+102
+103
+n [cm
+3]
+J21=0.1
+halo 1
+halo 2
+halo 3
+halo 4
+halo 5
+10
+26 10
+25 10
+24 10
+23 10
+22 10
+21
+ [g cm
+3]
+10
+12
+10
+10
+10
+8
+10
+6
+10
+4
+H2
+10
+26 10
+25 10
+24 10
+23 10
+22 10
+21
+ [g cm
+3]
+10
+26 10
+25 10
+24 10
+23 10
+22 10
+21
+ [g cm
+3]
+Figure 1. Summaries of the initial conditions for our simulations. Mass-weighted density profiles of temperature and H2 abundances.
+102
+104
+106
+Mgas [M
+]
+J21=0
+J21=0.01
+J21=0.1
+halo 1
+halo 2
+halo 3
+halo 4
+halo 5
+103
+105
+107
+MDM [M
+]
+0.25
+0.50
+0.75
+1.00
+v /v
+100
+101
+102
+103
+R [pc]
+100
+101
+v/cs
+100
+101
+102
+103
+R [pc]
+100
+101
+102
+103
+R [pc]
+Figure 2. Summaries of the initial conditions for our simulations. Cumulative radial profiles of gas and DM mass and mass-weighted radial profiles of the ratio
+of rotational to total velocity (note the dotted line represents the value above which rotational component dominates the velocity) and ratio of velocity to sound
+speed i.e. Mach number.
+2.4 Chemistry
+Collapse of primordial gas is closely linked to the chemistry involved
+(e.g. Glover et al. 2006; Yoshida et al. 2007b; Glover & Abel 2008;
+Turk et al. 2011). We therefore use a fully time-dependent chemical
+network to model the gas. We use the same chemistry and cooling
+as Wollenberg et al. (2020), which is described in the appendix of
+Clark et al. (2011), but with updated rate coefficients, as summarised
+in Schauer et al. (2017). The network has 45 chemical reactions
+to model primordial gas made up of 12 species: H, H+, H−, H+
+2 ,
+H2, He, He+, He++, D, D+, HD and free electrons. Included in the
+network are: H2 cooling (including an approximate treatment of the
+effects of opacity), collisionally-induced H2 emission, HD cooling,
+ionisation and recombination, heating and cooling from changes in
+the chemical make-up of the gas and from shocks, compression and
+expansion of the gas, three-body H2 formation and heating from
+accretion luminosity. For reasons of computational efficiency, the
+MNRAS 000, 1–13 (2020)
+
+Cosmological Lyman-Werner simulations.
+5
+500 pc
+Figure 3. Column-weighted density projections of the 2 kpc region around the 5 cosmological halos for each 𝐽21 value, taken from Schauer et al. (2021),
+which serve as the initial conditions of the high resolution follow-up simulations presented in this work. The halo numbers to compare with upcoming plots are
+indicated at the top of the figure.
+500 pc
+Figure 4. Column-weighted temperature projections of the 2 kpc region around the 5 cosmological halos for each 𝐽21 value, taken from Schauer et al. (2021),
+which serve as the initial conditions of the high resolution follow-up simulations presented in this work.
+network switches off tracking of deuterium chemistry2 at densities
+above 10−16 g cm−3, instead assuming that the ratio of HD to H2 at
+these densities is given by the cosmological D to H ratio of 2.6×10−5.
+The adiabatic index of the gas is computed as a function of chemical
+composition and temperature with the Arepo HLLD Riemann solver.
+As in Schauer et al. (2021), we use a radiation field expected from
+massive Pop III stars. We use a blackbody spectrum at temperature
+2 Note that HD cooling continues to be included in the model.
+105 K for energies below 13.6 eV. Above this energy, the flux is
+expected to drop due to absorption in the intergalactic medium, so
+we set the value of the radiation field to 0 above 13.6 eV. We model
+the effects of H2 self-shielding using the TreeCol algorithm (Clark
+et al. 2012).
+MNRAS 000, 1–13 (2020)
+
+6
+L. R. Prole
+500 pc
+Figure 5. Column-weighted H2 abundance projections of the 2 kpc region around the 5 cosmological halos for each 𝐽21 value, taken from Schauer et al. (2021),
+which serve as the initial conditions of the high resolution follow-up simulations presented in this work.
+Table 1. Total mass (DM and gas) as detected by the FoF algorithm within
+the selected halos from Schauer et al. (2021).
+𝐽21=0
+𝐽21=0.01
+𝐽21=0.1
+halo
+Mtot [106 M⊙]
+1
+3.50
+4.76
+8.70
+2
+3.65
+4.84
+8.30
+3
+2.81
+4.30
+6.70
+4
+2.76
+4.02
+4.15
+5
+2.96
+3.91
+3.90
+2.5 Sink particles
+If the local Jeans length falls below the minimum cell size of the
+mesh, artificial collapse occurs. We insert sink particles into the sim-
+ulations at a threshold density to prevent artificial collapse when the
+simulation reaches its maximum refinement level. Our sink particle
+implementation was introduced in Wollenberg et al. (2020) and Tress
+et al. (2020). A cell is converted into a sink particle if it satisfies three
+criteria: 1) it reaches a threshold density; 2) it is sufficiently far away
+from pre-existing sink particles so that their accretion radii do not
+overlap; 3) the gas occupying the region inside the sink is gravitation-
+ally bound and collapsing. Likewise, for the sink particle to accrete
+mass from surrounding cells it must meet two criteria: 1) the cell lies
+within the accretion radius; 2) it is gravitationally bound to the sink
+particle. A sink particle can accrete up to 90% of a cell’s mass, above
+which the cell is removed and the total cell mass is transferred to the
+sink.
+Increasing the threshold density for sink particle creation dras-
+tically increases the degree of fragmentation, reducing the masses
+of subsequent secondary protostars (LP22). Ideally, sink particles
+would be introduced when the gas becomes adiabatic at ∼ 10−4 g
+cm−3. This is currently computationally challenging. However, the
+zero metallicity protostellar model of Machida & Nakamura (2015)
+suggests that stellar feedback kicks in to halt collapse at ∼ 10−6 g
+cm−3 (1018 cm−3), so we choose this as our sink particle creation
+density.
+The initial accretion radius of a sink particle 𝑅sink is chosen to
+be the Jeans length 𝜆J corresponding to the sink particle creation
+density and corresponding temperature. At 10−6 g cm−3, we take
+the temperature value from LP22 of 4460 K to give a Jeans length
+of 1.67 × 1012 cm. We set the minimum cell length to fit 8 cells
+across the sink particle in compliance with the Truelove condition,
+by setting a minimum cell volume 𝑉min = (𝑅sink/4)3. The minimum
+gravitational softening length for cells and sink particles 𝐿soft is set
+to 𝑅sink/4.
+The increasing radius of a Pop III protostar is dependent on both
+its mass and accretion rate (e.g. Omukai & Palla 2003; Hosokawa
+& Omukai 2009; Hosokawa et al. 2012; Hirano et al. 2014). We
+allow the sink particle accretion radius 𝑅sink to vary throughout its
+accretion history, using on-the-fly calculations of the stellar radius
+using an approximate analytic formulae originally derrived by Stahler
+et al. (1986):
+𝑅sink = 26𝑅⊙
+� 𝑀
+𝑀⊙
+�0.27 �
+�𝑀
+10−3𝑀⊙yr−1
+�0.41
+,
+(5)
+where we smooth �𝑀 by taking the average over the time taken to
+accrete 0.1M⊙.
+The sink particle treatment also includes the accretion luminosity
+feedback from Smith et al. (2011), as implemented in Arepo by
+Wollenberg et al. (2020). Stellar internal luminosity is not included
+in this work because the Kelvin-Helmholtz times of the protostars
+formed in our simulations are much longer than the period simulated,
+meaning that none will have yet begun nuclear burning. We also
+include the treatment of sink particle mergers used in LP22.
+3 INITIAL HALO CHARACTERISTICS
+Figures 1 and 2 show some of the characteristics of the halos at the
+time when they were selected for zoom-in follow-up simulations.
+MNRAS 000, 1–13 (2020)
+
+Cosmological Lyman-Werner simulations.
+7
+The top panel of Figure 1 shows that the gas has already been shock
+heated as it fell into the gravitational potential well of the DM halo,
+allowing it to produce the necessary H2 to cool and collapse to higher
+densities. The bottom panel shows the destructive impact of the LW
+radiation field on the H2 abundance in the outer regions of the halo.
+The H2 abundance in the central regions reaches the same peak value
+of 4 × 10−4 due to self-shielding from the radiation field, however
+this happens at increasingly higher densities for larger 𝐽21 values.
+Figure 2 shows that within ∼ 100 pc, the gas velocity is dominated
+by its rotational component. The gas surrounding the halos is highly
+supersonic due to large-scale streaming, which cascades down to
+become subsonic at scales smaller than ∼ 10 pc, although we show
+in Section 4 that the velocities do not remain subsonic once the gas
+collapses further. From the density projections of Figure 3, halo sizes
+range from ∼ 100−200 pc, their shapes range from roughly spherical
+to structurally complex, and each is embedded within a network of
+filamentary structures. Figure 4 shows that these webs of filaments
+are a few hundred kelvin hotter than the ∼ 10 K gas that surrounds
+them, with the central halo reaching ∼ 1000 − 2000 K. Figure 5
+shows how the background H2 abundance is reduced drastically by
+the LW background, with significant levels of H2 only appearing in
+the central regions of the halo.
+4 FURTHER COLLAPSE
+We continue the collapse down to densities of 10−6 g cm−3 before
+inserting sink particles. Figure 6 shows temperature and chemical
+abundance profiles as a function of density, just before the formation
+of the first sink particle. At densities above 10−15 g cm−3, three-body
+H2 formation raises its abundance to over 0.1 within the molecular
+core. The abundance of free electrons falls off with density as the gas
+recombines. Figure 7 shows radial profiles of density and velocity.
+The rotational component of the gas velocity remains dominated by
+rotation only down to scales of ∼ 1 pc, below which infall begins to
+dominate. The velocities remain supersonic down to scales of 10−6
+pc (0.2 au). Halos experiencing a LW field are capable of achieving
+higher velocities, likely due to the higher halo mass.
+The left hand side of Figure 8 compares the temperature, density,
+accretion timescale and H2 abundance radial profiles for the differ-
+ent 𝐽21 values just before the formation of the first sink particle.
+Stronger LW fields require higher mass halos for star formation, as
+their stronger gravitational potential is capable of shock-heating the
+gas to higher temperatures, which increases the H2 formation rate
+enough to build up a high column density of H2 in order to self-shield
+the collapsing regions. Despite larger halo masses and shock-heating
+to higher initial temperatures, the density profile of the gas remains
+unaffected. Following Abel et al. (2002) and O’Shea & Norman
+(2008), we have also estimated the accretion timescale (𝑡acc = M/ �M)
+where
+�M = 4𝜋𝑅2𝜌(𝑅)𝑣rad(𝑅)
+(6)
+is our estimate of the mass inflow rate at radius 𝑅 and 𝜌(𝑅) and
+𝑣rad(𝑅) are the mass-weighted density and radial velocity within
+shells at radius 𝑅. For 𝑡acc > 104 yr, there is very little difference
+between the runs, suggesting that the accretion rate at early times is
+not influenced by the LW field. For larger 𝑡acc, we do see a difference
+between different runs, but this manifests as an increased scatter in
+𝑡acc at a given 𝑅 rather than any systematic dependence on the LW
+field strength.
+The right hand side of Figure 8 shows the gas as it transitions into a
+fully molecular state within the inner ∼ 10−2 pc core corresponding
+to the density regime above 10−15 g cm−3. Halos illuminated gy
+a LW background have higher gas kinetic energies owing to their
+larger masses, which promotes a larger molecular core. However,
+the increasing photodissociation rate with increasing 𝐽21 acts against
+this mechanism, reducing the H2 formation rate and shrinking the
+molecular core. This results in the 𝐽21 = 0.01 halos having molecular
+cores that extend to larger radii than the 𝐽21 = 0.1 halos, despite both
+having larger molecular cores than the 𝐽21 = 0 halos. As we only
+have access to these 3 values of 𝐽21, the value where the core size is
+maximised could lie anywhere between 0 < 𝐽21 < 0.1.
+The importance of the molecular core is shown in Figure 9, which
+shows the total mass in sink particles at the end of the simulations
+as functions of the halo mass, virial temperature and mass within
+different regimes of the collapse, just before the maximum density
+was reached. The mass in sink particles grows almost linearly with
+the mass within the inner molecular core with H2 abundances above
+10−1 as
+log10(Msinks) = (0.85±0.11)log10(MH2>10−1) + (0.14±0.24). (7)
+Due to competing effects between halo mass and H2 photodissoci-
+ation rate, the mass in sink particles is not correlated with the total
+halo mass or the subsequent mass that initially falls into its potential
+well with H2 abundances > 10−4. However, we have only followed
+the accretion for 300 yr, which corresponds to a free-fall time for gas
+at density 10−14 g cm−3. Gas below this density will not have been
+accreted within the simulation time, while gas above this density re-
+sides within the molecular core (see Figure 6). It is therefore unclear
+if the relationship between accretion and mass of the molecular core
+would remain if the simulations ran for a longer time. If the relation-
+ship does hold, we speculate that increasing the size of the streaming
+velocities between gas and DM may have a greater effect on the IMF,
+since this also increases halo masses without the counteracting ef-
+fects of H2 dissociation (Tseliakhovich & Hirata 2010; Greif et al.
+2011; Hirano et al. 2017; Schauer et al. 2019, AS21).
+5 FRAGMENTATION AND THE IMF
+The IMF of Pop III stars is determined by the fragmentation be-
+haviour of the disc around the initial central object and the subse-
+quent accretion onto fragments. Density projections of the inner 650
+AU of the halos are shown in Figure 10, while Figure 11 shows the
+evolution of the total number of sink particles formed, the total mass
+in sink particles and highest mass sink particle in each halo as a
+function of time. While the halos from the simulation without a LW
+background typically yield less fragmentation, the overall fragmen-
+tation behaviour is stochastic, as expected. The total mass accreted
+onto sinks is typically higher in the halos illuminated by a LW field
+due to their higher halo mass.
+Figure 12 shows the IMF at a time 300 yr after the formation
+of the first sink particle. The peak of the IMF positioned at ∼0.2-
+0.5 M⊙ shows little dependence on the LW strength, likely because
+the positive influence of larger halo masses on the molecular core
+is regulated by the increasing photodissociation rate. We also show
+the evolution of the cumulative IMFs in time, which converge by the
+end of the simulations for the 𝐽21=0.01 and 0.1 suites. The left side
+of Figure 13 compares the combined cumulative IMFs for different
+𝐽21 values. While the high mass end of the IMFs are nearly identical
+between the different 𝐽21 values, the low mass end of the IMFs show
+variance due to the random and stochastic nature of ejection events
+for low mass objects. Assuming these low mass objects (< 0.075M⊙)
+do not go on to accrete significant mass, they will remain as brown
+MNRAS 000, 1–13 (2020)
+
+8
+L. R. Prole
+105
+108
+1011
+1014
+1017
+n [cm
+3]
+103
+T [K]
+J21=0
+105
+108
+1011
+1014
+1017
+n [cm
+3]
+J21=0.01
+105
+108
+1011
+1014
+1017
+n [cm
+3]
+J21=0.1
+halo 1
+halo 2
+halo 3
+halo 4
+halo 5
+10
+3
+10
+2
+10
+1
+100
+H2
+10
+7
+10
+6
+10
+5
+HD
+10
+19
+10
+16
+10
+13
+10
+10
+10
+7
+10
+4
+ [g cm
+3]
+10
+11
+10
+9
+10
+7
+10
+5
+H +
+10
+19
+10
+16
+10
+13
+10
+10
+10
+7
+10
+4
+ [g cm
+3]
+10
+19
+10
+16
+10
+13
+10
+10
+10
+7
+10
+4
+ [g cm
+3]
+Figure 6. Mass-weighted temperature, H2, HD and H+ abundances versus density at a time shortly after the formation of the first sink particle. (Note that the
+HD abundance is only tracked self-consistently up to 𝜌 = 10−16 g cm−3; above this, we simply fix the HD/H2 ratio at 2.6×10−5, as explained in Section 2.4).
+10
+22
+10
+16
+10
+10
+10
+4
+ [g cm
+3]
+J21=0
+J21=0.01
+J21=0.1
+halo 1
+halo 2
+halo 3
+halo 4
+halo 5
+5
+10
+15
+|v| [km s
+1]
+10
+5
+0
+vr [km s
+1]
+0.50
+0.75
+1.00
+|v /v|
+10
+7
+10
+5
+10
+3
+10
+1
+101
+103
+R [pc]
+10
+1
+100
+101
+|v|/cs
+10
+7
+10
+5
+10
+3
+10
+1
+101
+103
+R [pc]
+10
+7
+10
+5
+10
+3
+10
+1
+101
+103
+R [pc]
+Figure 7. Radial distribution of cumulative gas mass and mass-weighted radial profiles of radial velocity, ratio of rotational to total velocity (note the dotted line
+represents the value above which rotational component dominates the velocity) and ratio of velocity to sound speed, taken at a time shortly after the formation
+of the first sink particle.
+MNRAS 000, 1–13 (2020)
+
+Cosmological Lyman-Werner simulations.
+9
+Figure 8. Left: Comparison of the mass-weighted temperature, density and and H2 abundance profiles between the 𝐽21 values. Right: Zoom-in of the transition
+to fully molecular core, showing the total kinetic energy within radial shells, mass weighted H2 photodissociation heating rate and H2 abundance.
+4
+6
+8
+Mhalo [M
+]
+1e6
+5
+10
+15
+20
+25
+30
+35
+40
+Msinks [M
+]
+2000
+3000
+4000
+Tvir [K]
+50000
+100000
+150000
+MH2 > 10
+4 [M
+]
+J21=0
+J21=0.01
+J21=0.1
+10
+20
+30
+40
+MH2 > 10
+1 [M
+]
+M0.85
+Figure 9. Total mass in sinks at the end of the simulations versus halo mass, virial temperature, mass within the halo with H2 abundance > 10−4 and mass
+within the molecular core with H2 abundance > 10−1. The size of the markers is scaled with the halo mass.
+dwarfs and never sustain nuclear fusion. The right side of the plot
+shows the cumulative IMFs if the brown dwarfs are ignored. Here,
+the IMFs fit well within each other’s regions of uncertainty, which
+are the standard deviations from the cumulative IMFs of the 5 halos
+individually. The overlap between the regions of uncertainty indicates
+that the LW strength does not significantly affect the primordial IMF.
+As the range of background LW field strengths tested here covers the
+most likely values from literature, we infer that the IMF for Pop III.2
+stars is not significantly different from the initial population of Pop
+III.1 stars.
+We also show the IMF from LP22 as a dotted line, which was
+produced from idealised halos as opposed the cosmological initial
+conditions. The cosmological halos produced a bi-modal distribution
+with a significant population of brown dwarfs that the idealised halos
+are missing. Even when ignoring the brown dwarfs, the cosmological
+initial conditions have yielded distributions tending to lower mass
+pre-stellar cores.
+MNRAS 000, 1–13 (2020)
+
+10
+L. R. Prole
+250 AU
+Figure 10. Column-weighted density projections of the inner 650 AU of the halos at a time 300 yr after the formation of the first sink particle. Sink particles are
+represented as red dots.
+0
+10
+20
+30
+Nsink
+J21=0
+J21=0.01
+J21=0.1
+halo 1
+halo 2
+halo 3
+halo 4
+halo 5
+100
+101
+Mtot [M
+]
+0
+100
+200
+300
+t [yr]
+10
+1
+100
+101
+Mmax [M
+]
+100
+200
+300
+t [yr]
+100
+200
+300
+t [yr]
+Figure 11. Number of sink particles formed, total mass in sinks, largest mass sink particle and median mass of sink particles as a function of time.
+6 CAVEATS
+Aside from the obvious uncertainties in the ΛCDM model on which
+this work and the work of AS21 are based on, there are a number of
+caveats to note.
+The LW fields we have used in this study assume a population
+of massive Pop III stars already exists. While previous studies do
+suggest that Pop III stars were massive, more recent work suggests
+that they may only grow to a few M⊙ (e.g. Stacy & Bromm 2013;
+Wollenberg et al. 2020; Prole et al. 2022a; Jaura et al. 2022). In this
+scenario, significantly less LW radiation would be produced, however
+radiation from solar mass stars can inhibit H2 formation through the
+destruction of H−. The fields used in this study therefore represent the
+maximum effect Pop III stars can have on the the Pop III.2 stars that
+follow. Since the effects of the LW fields appear to be insignificant, it
+is likely that this result also represents the outcome for weaker fields.
+We have assumed that the pre-stellar core radius grows as Equation
+5. This process begins immediately after sink particle formation.
+While this is predicted by stellar theory, it is unclear at what point
+the pre-stellar core would begin to expand, which may affect the
+accretion behaviour.
+MNRAS 000, 1–13 (2020)
+
+Cosmological Lyman-Werner simulations.
+11
+0
+1
+2
+3
+4
+J21=0
+0.5
+1.0
+0
+5
+10
+15
+20
+25
+Nsink
+J21=0.01
+0.5
+1.0
+M/Mtot
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+M [M
+]
+0
+2
+4
+6
+8
+J21=0.1
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+M [M
+]
+0.0
+0.5
+1.0
+10 yr
+50 yr
+100 yr
+200 yr
+300 yr
+Figure 12. Left: combined IMFs of sink particles for the different the 𝐽21 val-
+ues, Right: Cumulative IMFs of the combined sink particles for the different
+the 𝐽21 values.
+We have neglected the presence of primordial magnetic fields in
+this study. While the findings of Prole et al. (2022b) suggest that
+primordial magnetic fields make little difference to the primordial
+IMF due to their small-scale structure, the field is still active, with
+other recent studies finding that magnetic fields indeed lead to higher
+mass Pop III stars (e.g. Saad et al. 2022; Hirano & Machida 2022;
+Stacy et al. 2022).
+7 CONCLUSIONS
+The results of cosmological simulations by AS21 show that increas-
+ing the background LW field strength increases the average halo mass
+required for star formation. These simulations ran up to a maximum
+density of 10−19 g cm−3, hence the knock-on effects on the Pop III
+IMF were unclear. In this investigation, we have performed follow-up
+simulations of 5 halos for each of the 𝐽21 = 0, 0.01 and 0.1 LW field
+strengths, resolving the pre-stellar core density of 10−6 g cm−3 before
+inserting sink particles and following the fragmentation behaviour
+for hundreds of years further. We have found that the mass accreted
+onto sinks by the end of the simulations is proportional to the mass
+within the ∼ 10−2 pc molecular core, which is not correlated to the
+initial mass of the halo. As such, the IMF shows little dependence
+on the LW field strength. We also find no clear relationship between
+the estimated accretion time of gas lying further out within the halo
+and the LW field strength, suggesting that the LW field is unlikely
+to influence the development of the IMF at later times. As the range
+of background LW field strengths tested here covers the most likely
+values from literature, we conclude that the IMF for so-called Pop
+III.2 stars is not significantly different from that of the initial popu-
+lation of Pop III.1 stars, although we cannot rule out greater effects
+in the small subset of halos that are illuminated by LW fields much
+stronger than the average value (see e.g Latif et al. 2014). In a future
+paper, we will explore the effects of increasing the streaming veloci-
+ties between the gas and dark matter on the Pop III IMF, as this has
+been shown to increase halo masses through a different mechanism.
+ACKNOWLEDGEMENTS
+This work used the DiRAC@Durham facility managed by the
+Institute for Computational Cosmology on behalf of the STFC
+DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded
+by BEIS capital funding via STFC capital grants ST/P002293/1,
+ST/R002371/1 and ST/S002502/1, Durham University and STFC
+operations grant ST/R000832/1. DiRAC is part of the National e-
+Infrastructure.
+The authors gratefully acknowledge the Gauss Centre for Super-
+computing e.V. (www.gauss-centre.eu) for supporting this project by
+providing computing time on the GCS Supercomputer SuperMUC at
+Leibniz Supercomputing Centre (www.lrz.de) under project pr53ka.
+AS was partially supported by NSF grant AST-1752913.
+RSK and SCOG acknowledge computing resources provided by
+the Ministry of Science, Research and the Arts (MWK) of the State
+of Baden-Württemberg through bwHPC and the German Research
+Foundation (DFG) through grant INST 35/1134-1 FUGG and for
+data storage at SDS@hd through grant INST 35/1314-1 FUGG.
+RSK and SCOG acknowledge financial support from DFG via the
+collaborative research center (SFB 881, Project-ID 138713538) “The
+Milky Way System” (subprojects A1, B1, B2 and B8), from the Hei-
+delberg Cluster of Excellence “STRUCTURES” in the framework
+of Germany’s Excellence Strategy (grant EXC-2181/1, Project-ID
+390900948) and from the European Research Council (ERC) via the
+ERC Synergy Grant “ECOGAL” (grant 855130). RSK furthermore
+thanks the German Ministry for Economic Affairs and Climate Ac-
+tion for funding in the project “MAINN” (funding ID 50OO2206).
+We also acknowledge the support of the Supercomputing Wales
+project, which is part-funded by the European Regional Development
+Fund (ERDF) via Welsh Government.
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable request
+to the corresponding author.
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+
diff --git a/MtAyT4oBgHgl3EQf6vrQ/content/tmp_files/load_file.txt b/MtAyT4oBgHgl3EQf6vrQ/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf,len=1079
+page_content='MNRAS 000, 1–13 (2020)  Preprint 4 January 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='0  From dark matter halos to pre-stellar cores: High resolution follow-up of  cosmological Lyman-Werner simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Lewis R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Prole,1★ Anna T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Schauer,2 Paul C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Clark,1 Simon C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Glover,3 Felix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Priestley,1 Ralf S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Klessen,3,4  1Cardiff University School of Physics and Astronomy  2Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA  3Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Straße 2, D-69120 Heidelberg, Germany  4Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  Molecular hydrogen allows cooling in primordial gas, facilitating its collapse into Population III stars within primordial halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Lyman-Werner (LW) radiation from these stars can escape the halo and delay further star formation by destroying H2 in other  halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' As cosmological simulations show that increasing the background LW field strength increases the average halo mass  required for star formation, we perform follow-up simulations of selected halos to investigate the knock-on effects this has on  the Population III IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We follow 5 halos for each of the 𝐽21 = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 LW field strengths, resolving the pre-stellar core  density of 10−6 g cm−3 (1018 cm−3) before inserting sink particles and following the fragmentation behaviour for hundreds of  years further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We find that the mass accreted onto sinks by the end of the simulations is proportional to the mass within the  ∼ 10−2 pc molecular core, which is not correlated to the initial mass of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' As such, the IMF shows little dependence on  the LW strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' As the range of background LW field strengths tested here covers the most likely values from literature, we  conclude that the IMF for so-called Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='2 stars is not significantly different from the initial population of Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The  primordial IMF therefore likely remains unchanged until the formation of the next generation of Population II stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Key words: stars: Population III – dark ages, reionization, first stars – hydrodynamics – stars: luminosity function, mass function  – software: simulations  1 INTRODUCTION  This first stars are able to form because pristine baryonic gas can  collapse within the gravitational potential well of dark matter (DM)  halos (Couchman & Rees 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Haiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 1996a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Tegmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  1997), heating it to ∼ 5000 K and facilitating the formation of H2  (Bromm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' H2 primarily forms via the radiative association  reaction forming H−  H + e− → H− + 𝛾,  (1)  followed by the fast associative detachment reaction forming H2  H− + H → H2 + e−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  (2)  Radiative cooling from the molecular hydrogen renders the gas grav-  itationally unstable, which allows it to decouple from the DM and  collapse to form the first stars, known as Population III (Pop III)  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The necessity of H2 renders any process of H2 destruction as a  mechanism to delay or prevent Pop III star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  While so-called Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 stars form from purely cosmological  initial conditions, the radiation they produce affects Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='2 stars  that form in its presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' As the masses of Pop III stars are predicted  to be larger than present-day counterparts (due to the lack of cooling  from dust and metals), they are expected to emit large amounts of  ★ E-mail: Prolel@cardiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='uk  ionizing radiation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Schaerer 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Ionizing photons above the  Lyman limit (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='6 eV) create H ii regions around the stars up to the  boundary of their Strömgren spheres (Whalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Kitayama  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Alvarez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Abel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  2007a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Jaura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2022) while photons below the Lyman limit are  free to escape their Strömgren sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Lyman-Werner (LW) photons  between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='2 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='6 eV can dissociate H2 via the two-step Solomon  process (Field et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Stecher & Williams 1967)  H2 + 𝛾 → H∗  2 → 2H,  (3)  where H∗  2 represents an electronically excited state of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Photons  with energy above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='76 eV can also photodissociate H− via  H− + 𝛾 → H + e−,  (4)  reducing the H− abundance and hence the rate at which H2 can  form via reaction 2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Chuzhoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' This stellar feedback  provides a potential obstacle for Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='2 stars to overcome during  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Investigations into the effects of these far UV fields typ-  ically categorise the field strength by the intensity in the LW band  𝐽21, in units of 10−21 erg s−1 cm−2 Hz−1 sr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Calculations by Haiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (1997) found that before the Ström-  gren spheres of Pop III stars overlap, the UV background below the  ionization threshold was able to penetrate large clouds and suppress  their H2 abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' They also found that the flux necessary for H2  photodissociation is several orders of magnitude smaller than the flux  © 2020 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='00828v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='GA]  2 Jan 2023  2  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Prole  needed to reionize the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Haiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2000) showed that  this photodissociation of H2 suppresses further Pop III star formation  inside small halos and delays reionization until larger halos form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Collapse is not impossible without sufficient H2 for cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Omukai (2001) showed that if the LW field is sufficient to keep a  halo free of molecular hydrogen, the gas can nevertheless collapse  via atomic hydrogen line cooling if the halo has a virial temperature  𝑇vir > 8000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The collapse occurs almost isothermally, possibly  resulting in the formation of a direct collapse black hole (DCBH)  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Bromm & Loeb 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Spaans & Silk 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Latif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2013), a  possible progenitor of the supermassive black holes observed at high  redshifts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Mortlock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Matsuoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' However,  for a T = 105 K blackbody spectrum expected from Pop III stars, a  field strength of 𝐽21 ∼ 104 is required to keep the gas atomic during  the collapse (Glover 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Agarwal & Khochfar 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Sugimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2014), while the average exposure is expected  to be 𝐽21 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 at 𝑧 ∼ 15 (Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Trenti & Stiavelli 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Wise et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Skinner & Wise 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Dijk-  stra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2008) showed that only a fraction of 10−8 − 10−6 of DM  halos with virial temperatures > 104 K have a close luminous neigh-  bour within < 10 kpc, and are exposed to an LW flux 𝐽21 > 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The occurrence of atomically cooled halos is therefore expected to  be rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Studies have shown that values of 𝐽21 orders of magnitude lower  than the critical intensity required to completely suppress H2 cooling  in massive halos can still drastically affect halo collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Typically,  the critical mass for efficient molecular hydrogen cooling and sub-  sequent star formation increases with increasing 𝐽21 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Machacek  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' O’Shea & Norman 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Vis-  bal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Schauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Cosmological simulations by  Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2003) found gas cooling was suppressed for 𝐽21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1,  leading them to predict that star formation would not occur in halos  with 𝑇vir < 8000 K for LW field strengths greater than this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Con-  versely, O’Shea & Norman (2008) found that for field strengths as  high as 𝐽21 = 1, H2 cooling leads to collapse despite the depressed  core molecular hydrogen fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' They also noted that higher LW  background fluxes lead to higher accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' High resolution  cosmological simulations by Schauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2021) (hereafter AS21)  examined the impact of different values of the LW field strength on  a large sample of minihalos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' They showed that both Mav, the aver-  age minihalo mass required for efficient H2 cooling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' the mass  above which more than 50% of minihalos of that mass can cool), and  Mmin, the minimum minihalo mass required for efficient cooling, in-  creased with increasing 𝐽21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' An increase in the critical halo mass for  star formation with increasing 𝐽21 was also found by Kulkarni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  (2021), although they find a significant effect only for 𝐽21 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' In  contrast, cosmological simulations by Skinner & Wise (2020) found  no relationship between the LW intensity and host halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The true average 𝐽21 intensity is expected to vary with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Hirano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2015) followed the formation and evolution of 1540 star-  forming gas clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' They found that in their models, the characteristic  mass of Pop III stars shifted to lower masses with decreasing redshift  due to the radiative feedback of previous generations of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' For  𝑧 > 20, half of the star-forming gas clouds were exposed to intense  FUV radiation, with an average exposure of 𝐽21 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Due to  smaller stellar masses and the expanding distance between stars, the  FUV background became weaker at lower redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' For 15 < 𝑧 < 25,  almost all the clouds had nonzero intensity 𝐽21 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The average  LW intensity in Skinner & Wise (2020) increased stochastically from  10−3 at 𝑧 ∼ 25 to 10 at 𝑧 ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' For redshifts above ∼12, 𝐽21 remained  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Self-shielding is a process that occurs when large column den-  sity of molecular hydrogen protects the inner regions against pho-  todissociation because one photon can only photodissociate one H2  molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' This self-shielding allows further H2 production and H2  cooling (Shang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Regan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Hartwig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The large nonequilibrium abundance of elec-  trons in gas cooling from above T > 104 K also boosts H2 formation  (Oh & Haiman 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Early attempts to model self shielding (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Shang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2010) multiplied the intensity in the LW band by a self-  shielding factor given by Draine & Bertoldi (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Wolcott-Green  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2011) showed that this method underestimated the numerically  calculated self-shielding rate by more than an order of magnitude in  low-density regions, by overestimating shielding by a large factor at  temperatures above a few hundred kelvin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' They modified the method  of Draine & Bertoldi by estimating the shielding factor based on  the Sobolev length, using local properties of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' This modifica-  tion was computationally inexpensive and used in many subsequent  investigations into the aforementioned critical intensity required to  form atomic halos, typically producing values an order of magnitude  lower than those using the original Draine & Bertoldi shielding (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Glover 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Agarwal & Khochfar 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Clark  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2012) improved on this method further with their introduction  of the TreeCol algorithm, which calculates maps of the column den-  sity distribution seen by each computational element in a simulation  in a computationally efficient fashion with the help of an oct-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Hartwig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2015b) took this approach further by accounting for  the relative velocities between different computational elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' This  Doppler-shifts the spectral lines, reducing the effectiveness of self-  shielding (since molecules shifted by more than the linewidth do not  contribute to the effective column density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  In addition to a background LW field, primordial star formation is  complicated further by streaming velocities between the the gas and  DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Prior to recombination, baryons were tightly coupled to photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  As DM does not experience Thomson scattering, there should have  been a relative velocity between the DM and baryons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Ma &  Bertschinger 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' At recombination, the relative velocity was ∼ 30  km s−1 and was coherent over several comoving Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Recombination  resulted in a drop in the sound speed to ∼ 6 km s−1 as the gas  transitioned from plasma to a neutral state, meaning the relative  velocities were highly supersonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Tseliakhovich & Hirata (2010)  showed that the presence of these large-scale streaming velocities  suppresses the abundance of the first bound objects by advecting  small-scale perturbations near the baryonic Jeans scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Moving-  mesh calculations by Greif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2011) found that the additional  momentum and energy from the streaming velocities reduces the gas  fractions and central densities of halos, increasing the typical virial  mass required for efficient cooling by a factor of three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' They also  noted that the turbulent velocity dispersion increased in the presence  of streaming velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The simulations of AS21 found that the  increase in the average and minimum halo mass from increasing  streaming velocities is additive on top of the effect of a LW field,  with streaming velocities having the larger impact of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  While it was initally believed that Pop III stars formed in isolation  (Haiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 1996b) and were massive (Abel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Bromm  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2002), more recent studies show that primordial gas fragments  to give rise to a larger populations of lower mass stars (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Greif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Stacy & Bromm 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Machida & Doi 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Stacy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Susa 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Wollenberg  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' In Prole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2022a) (hereafter LP22), we used high  resolution simulations of idealised, purely hydrodynamical Pop III  star formation to show that a number of cores are ejected from the  system with masses capable of surviving until the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' As  small-scale primordial magnetic fields do not appear to prevent disc  MNRAS 000, 1–13 (2020)  Cosmological Lyman-Werner simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  3  fragmentation (Prole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2022b) and accretion of metals onto the  surface of these stars during their lifetime is unlikely (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Johnson  & Khochfar 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2017), the question is raised about  why these stars have not been found within archeological surveys  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Beers & Christlieb 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Frebel & Norris 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Starken-  burg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Since most high resolution simulations of Pop III  star formation have considered only the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 case, one possible  explanation could be that Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='2 star formation yields a different  IMF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' that Pop III stars forming in the presence of a LW back-  ground have systematically larger masses than those forming in the  absence of a background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  In this paper, we aim to test this hypothesis by producing the  most accurate prediction of the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 and Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='2 initial mass  functions (IMF) to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We investigate how the increase in halo  masses due to increasing LW field intensity changes star formation  within them, by performing high resolution follow-up simulations  of cosmological halos drawn from the simulations of AS21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The  structure of our paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' In Section 2, we describe the  cosmological simulations of AS21, our selection criteria for the halos  chosen for follow-up simulations, the chemical model we use and our  use of sink particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' In Section 3 we review the characteristics of the  halos as they are taken from AS21, before presenting the results of the  zoom-in simulations in Section 4, where we probe the density regime  of the molecular core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' In Section 5, we compare the fragmentation  behaviour once sink particles have formed and present the IMFs at  the end of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We discuss caveats in Section 6 before  concluding in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  2 NUMERICAL METHOD  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 Arepo  The simulations presented here were performed with the moving  mesh code Arepo (Springel 2010) with a primordial chemistry set-  up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Arepo combines the advantages of AMR and smoothed particle  hydrodynamics (SPH: Monaghan 1992) with a mesh made up of a  moving, unstructured, Voronoi tessellation of discrete points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Arepo  solves hyperbolic conservation laws of ideal hydrodynamics with a  finite volume approach, based on a second-order unsplit Godunov  scheme with an exact Riemann solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Automatic and continuous  refinement overcome the challenge of structure growth associated  with AMR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Heitmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='2 Cosmological simulations  The cosmological simulations performed by AS21 assumed a ΛCDM  cosmology with parameters ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='6774, Ω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='3089, Ωb =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='04864, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='6911, 𝑛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='96 and 𝜎8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='8159 as derived by  the Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The simulations were ini-  tialised at 𝑧 = 200 with an initial DM distribution created by MUSIC  (Hahn & Abel 2011) using the transfer functions of Eisenstein &  Hu (1998) and the gas distribution initially followed the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The  DM was represented by 10243 particles and the gas was initially  modelled with 10243 grid cells, all contained within a box with side  length 1 ℎ−1 Mpc in comoving units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' During the simulation, an ad-  ditional Jeans refinement criterion was applied: cells were refined  whenever necessary so as to ensure that the Jeans length was always  resolved with at least 16 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' This refinement was carried out until  the gas reached a threshold density of ∼ 10−19 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Above this  threshold density, gravitationally bound and collapsing gas was con-  verted into collisionless sink particles, as explained at greater length  in AS21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The simulations were carried out with four different values  of the baryonic streaming velocity (𝑣str = 0, 1, 2 and 3, in units of  𝜎rms, the large-scale root mean squared value) and three different  values for the LW field strength (𝐽21 = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' In this study,  we make use of the three simulations with 𝑣str = 1𝜎rms because this  is the most representative value available, as the volume fraction of  streaming velocities peaks at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='8𝜎rms (AS21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='3 Halo selection  Given snapshots at 𝑧 = 15 from the three simulations with 𝑣str = 1  presented in AS21, we have selected 5 halos for each value of 𝐽21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Halo positions and masses were provided by the friends-of-friends  (FoF) algorithm as described in that study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The selection criteria for  the halos was as follows:  The halos identified by the FoF algorithm were sorted by their  mass difference with Mav, the average minihalo mass above which  minihalos become capable of cooling and forming stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' This average  mass was defined in AS21, following Schauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2019), to be the  minihalo mass above which more than 50% of minihalos can cool  and form stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 AS21 report Mav at a range of redshifts between  𝑧 = 22 and 𝑧 = 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' here, we adopt their values at 𝑧 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' By sorting  the halos in this way, we ensure that the halos that we eventually  select will have masses close to Mav, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' that they are representative  of the common case at that redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  We considered a halo if it contained a cell denser than 𝜌th =  10−22 g cm−3 within a search radius Rsearch = 1 kpc in physical  units from the halo’s central coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Note that checking the H2  abundance is not necessary, as collapse to this density is not possible  without a high H2 abundance outside of the atomically cooling halo  scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  We check that the halo is sufficiently resolved i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' that the 16  cells per Jeans length criteria has not decreased as the cell sizes  approach the maximum resolution of the cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  This process begins when the density reaches ∼ 10−20 g cm−3, so  we reject halos including a cell above this density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Note that this also  ensures that a sink particle has not yet been formed within Rsearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Lastly, we removed the halo from consideration if there were  other dense objects in the vicinity by checking there were no cells  above 𝜌th within the radius Rsearch/2 < r < Rsearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  We performed this search throughout the simulation cube until 5  suitable halos were selected for each 𝐽21 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The selected halo  masses from each simulation are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Mass-weighted  mean temperatures and H2 abundances are plotted as a function of  density for each halo in Figure 1 and radial distributions of enclosed  gas and DM along with velocity information are shown in Figure  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Projections of the initial density, temperature and H2 abundance  distributions are visualised in Figures 3, 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Note that all of  these figures show the state of the halos at the point at which we  extract them from the AS21 simulations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' prior to our zoom-in  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Our choice of selection criteria means that the same  halos are not chosen for direct comparison between each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Rather, they are chosen to be most representative of the Universe at  the given background LW field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Direct comparison also has  the drawback that individual halos will collapse at different redshifts  depending on the LW field strength, whereas we exclusively select  halos that become able to form stars at 𝑧 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  1 Note that Mav is typically around a factor of three larger than 𝑀min, the  mass of the least massive minihalo with cool gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2020)  4  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Prole  10  2  10  1  100  101  102  103  n [cm  3]  101  102  103  T [K]  J21=0  10  2  10  1  100  101  102  103  n [cm  3]  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01  10  2  10  1  100  101  102  103  n [cm  3]  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1  halo 1  halo 2  halo 3  halo 4  halo 5  10  26 10  25 10  24 10  23 10  22 10  21  [g cm  3]  10  12  10  10  10  8  10  6  10  4  H2  10  26 10  25 10  24 10  23 10  22 10  21  [g cm  3]  10  26 10  25 10  24 10  23 10  22 10  21  [g cm  3]  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Summaries of the initial conditions for our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Mass-weighted density profiles of temperature and H2 abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  102  104  106  Mgas [M  ]  J21=0  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1  halo 1  halo 2  halo 3  halo 4  halo 5  103  105  107  MDM [M  ]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='00  v /v  100  101  102  103  R [pc]  100  101  v/cs  100  101  102  103  R [pc]  100  101  102  103  R [pc]  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Summaries of the initial conditions for our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Cumulative radial profiles of gas and DM mass and mass-weighted radial profiles of the ratio  of rotational to total velocity (note the dotted line represents the value above which rotational component dominates the velocity) and ratio of velocity to sound  speed i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Mach number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='4 Chemistry  Collapse of primordial gas is closely linked to the chemistry involved  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Glover et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2007b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Glover & Abel 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Turk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We therefore use a fully time-dependent chemical  network to model the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We use the same chemistry and cooling  as Wollenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2020), which is described in the appendix of  Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2011), but with updated rate coefficients, as summarised  in Schauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The network has 45 chemical reactions  to model primordial gas made up of 12 species: H, H+, H−, H+  2 ,  H2, He, He+, He++, D, D+, HD and free electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Included in the  network are: H2 cooling (including an approximate treatment of the  effects of opacity), collisionally-induced H2 emission, HD cooling,  ionisation and recombination, heating and cooling from changes in  the chemical make-up of the gas and from shocks, compression and  expansion of the gas, three-body H2 formation and heating from  accretion luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' For reasons of computational efficiency, the  MNRAS 000, 1–13 (2020)  Cosmological Lyman-Werner simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  5  500 pc  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Column-weighted density projections of the 2 kpc region around the 5 cosmological halos for each 𝐽21 value, taken from Schauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2021),  which serve as the initial conditions of the high resolution follow-up simulations presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The halo numbers to compare with upcoming plots are  indicated at the top of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  500 pc  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Column-weighted temperature projections of the 2 kpc region around the 5 cosmological halos for each 𝐽21 value, taken from Schauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2021),  which serve as the initial conditions of the high resolution follow-up simulations presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  network switches off tracking of deuterium chemistry2 at densities  above 10−16 g cm−3, instead assuming that the ratio of HD to H2 at  these densities is given by the cosmological D to H ratio of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='6×10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The adiabatic index of the gas is computed as a function of chemical  composition and temperature with the Arepo HLLD Riemann solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  As in Schauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2021), we use a radiation field expected from  massive Pop III stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We use a blackbody spectrum at temperature  2 Note that HD cooling continues to be included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  105 K for energies below 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Above this energy, the flux is  expected to drop due to absorption in the intergalactic medium, so  we set the value of the radiation field to 0 above 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We model  the effects of H2 self-shielding using the TreeCol algorithm (Clark  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2020)  6  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Prole  500 pc  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Column-weighted H2 abundance projections of the 2 kpc region around the 5 cosmological halos for each 𝐽21 value, taken from Schauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2021),  which serve as the initial conditions of the high resolution follow-up simulations presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Total mass (DM and gas) as detected by the FoF algorithm within  the selected halos from Schauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  𝐽21=0  𝐽21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01  𝐽21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1  halo  Mtot [106 M⊙]  1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='50  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='76  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='70  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='65  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='84  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='30  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='81  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='30  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='70  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='76  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='02  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='15  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='96  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='91  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='90  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='5 Sink particles  If the local Jeans length falls below the minimum cell size of the  mesh, artificial collapse occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We insert sink particles into the sim-  ulations at a threshold density to prevent artificial collapse when the  simulation reaches its maximum refinement level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Our sink particle  implementation was introduced in Wollenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2020) and Tress  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' A cell is converted into a sink particle if it satisfies three  criteria: 1) it reaches a threshold density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2) it is sufficiently far away  from pre-existing sink particles so that their accretion radii do not  overlap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 3) the gas occupying the region inside the sink is gravitation-  ally bound and collapsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Likewise, for the sink particle to accrete  mass from surrounding cells it must meet two criteria: 1) the cell lies  within the accretion radius;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2) it is gravitationally bound to the sink  particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' A sink particle can accrete up to 90% of a cell’s mass, above  which the cell is removed and the total cell mass is transferred to the  sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Increasing the threshold density for sink particle creation dras-  tically increases the degree of fragmentation, reducing the masses  of subsequent secondary protostars (LP22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Ideally, sink particles  would be introduced when the gas becomes adiabatic at ∼ 10−4 g  cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' This is currently computationally challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' However, the  zero metallicity protostellar model of Machida & Nakamura (2015)  suggests that stellar feedback kicks in to halt collapse at ∼ 10−6 g  cm−3 (1018 cm−3), so we choose this as our sink particle creation  density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The initial accretion radius of a sink particle 𝑅sink is chosen to  be the Jeans length 𝜆J corresponding to the sink particle creation  density and corresponding temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' At 10−6 g cm−3, we take  the temperature value from LP22 of 4460 K to give a Jeans length  of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='67 × 1012 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We set the minimum cell length to fit 8 cells  across the sink particle in compliance with the Truelove condition,  by setting a minimum cell volume 𝑉min = (𝑅sink/4)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The minimum  gravitational softening length for cells and sink particles 𝐿soft is set  to 𝑅sink/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The increasing radius of a Pop III protostar is dependent on both  its mass and accretion rate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Omukai & Palla 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Hosokawa  & Omukai 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Hosokawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Hirano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We  allow the sink particle accretion radius 𝑅sink to vary throughout its  accretion history, using on-the-fly calculations of the stellar radius  using an approximate analytic formulae originally derrived by Stahler  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (1986):  𝑅sink = 26𝑅⊙  � 𝑀  𝑀⊙  �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='27 �  �𝑀  10−3𝑀⊙yr−1  �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='41  ,  (5)  where we smooth �𝑀 by taking the average over the time taken to  accrete 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The sink particle treatment also includes the accretion luminosity  feedback from Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2011), as implemented in Arepo by  Wollenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Stellar internal luminosity is not included  in this work because the Kelvin-Helmholtz times of the protostars  formed in our simulations are much longer than the period simulated,  meaning that none will have yet begun nuclear burning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We also  include the treatment of sink particle mergers used in LP22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  3 INITIAL HALO CHARACTERISTICS  Figures 1 and 2 show some of the characteristics of the halos at the  time when they were selected for zoom-in follow-up simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2020)  Cosmological Lyman-Werner simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  7  The top panel of Figure 1 shows that the gas has already been shock  heated as it fell into the gravitational potential well of the DM halo,  allowing it to produce the necessary H2 to cool and collapse to higher  densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The bottom panel shows the destructive impact of the LW  radiation field on the H2 abundance in the outer regions of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The H2 abundance in the central regions reaches the same peak value  of 4 × 10−4 due to self-shielding from the radiation field, however  this happens at increasingly higher densities for larger 𝐽21 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Figure 2 shows that within ∼ 100 pc, the gas velocity is dominated  by its rotational component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The gas surrounding the halos is highly  supersonic due to large-scale streaming, which cascades down to  become subsonic at scales smaller than ∼ 10 pc, although we show  in Section 4 that the velocities do not remain subsonic once the gas  collapses further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' From the density projections of Figure 3, halo sizes  range from ∼ 100−200 pc, their shapes range from roughly spherical  to structurally complex, and each is embedded within a network of  filamentary structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Figure 4 shows that these webs of filaments  are a few hundred kelvin hotter than the ∼ 10 K gas that surrounds  them, with the central halo reaching ∼ 1000 − 2000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Figure 5  shows how the background H2 abundance is reduced drastically by  the LW background, with significant levels of H2 only appearing in  the central regions of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  4 FURTHER COLLAPSE  We continue the collapse down to densities of 10−6 g cm−3 before  inserting sink particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Figure 6 shows temperature and chemical  abundance profiles as a function of density, just before the formation  of the first sink particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' At densities above 10−15 g cm−3, three-body  H2 formation raises its abundance to over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 within the molecular  core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The abundance of free electrons falls off with density as the gas  recombines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Figure 7 shows radial profiles of density and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The rotational component of the gas velocity remains dominated by  rotation only down to scales of ∼ 1 pc, below which infall begins to  dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The velocities remain supersonic down to scales of 10−6  pc (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='2 au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Halos experiencing a LW field are capable of achieving  higher velocities, likely due to the higher halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The left hand side of Figure 8 compares the temperature, density,  accretion timescale and H2 abundance radial profiles for the differ-  ent 𝐽21 values just before the formation of the first sink particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Stronger LW fields require higher mass halos for star formation, as  their stronger gravitational potential is capable of shock-heating the  gas to higher temperatures, which increases the H2 formation rate  enough to build up a high column density of H2 in order to self-shield  the collapsing regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Despite larger halo masses and shock-heating  to higher initial temperatures, the density profile of the gas remains  unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Following Abel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2002) and O’Shea & Norman  (2008), we have also estimated the accretion timescale (𝑡acc = M/ �M)  where  �M = 4𝜋𝑅2𝜌(𝑅)𝑣rad(𝑅)  (6)  is our estimate of the mass inflow rate at radius 𝑅 and 𝜌(𝑅) and  𝑣rad(𝑅) are the mass-weighted density and radial velocity within  shells at radius 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' For 𝑡acc > 104 yr, there is very little difference  between the runs, suggesting that the accretion rate at early times is  not influenced by the LW field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' For larger 𝑡acc, we do see a difference  between different runs, but this manifests as an increased scatter in  𝑡acc at a given 𝑅 rather than any systematic dependence on the LW  field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The right hand side of Figure 8 shows the gas as it transitions into a  fully molecular state within the inner ∼ 10−2 pc core corresponding  to the density regime above 10−15 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Halos illuminated gy  a LW background have higher gas kinetic energies owing to their  larger masses, which promotes a larger molecular core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' However,  the increasing photodissociation rate with increasing 𝐽21 acts against  this mechanism, reducing the H2 formation rate and shrinking the  molecular core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' This results in the 𝐽21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01 halos having molecular  cores that extend to larger radii than the 𝐽21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 halos, despite both  having larger molecular cores than the 𝐽21 = 0 halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' As we only  have access to these 3 values of 𝐽21, the value where the core size is  maximised could lie anywhere between 0 < 𝐽21 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The importance of the molecular core is shown in Figure 9, which  shows the total mass in sink particles at the end of the simulations  as functions of the halo mass, virial temperature and mass within  different regimes of the collapse, just before the maximum density  was reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The mass in sink particles grows almost linearly with  the mass within the inner molecular core with H2 abundances above  10−1 as  log10(Msinks) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='11)log10(MH2>10−1) + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (7)  Due to competing effects between halo mass and H2 photodissoci-  ation rate, the mass in sink particles is not correlated with the total  halo mass or the subsequent mass that initially falls into its potential  well with H2 abundances > 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' However, we have only followed  the accretion for 300 yr, which corresponds to a free-fall time for gas  at density 10−14 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Gas below this density will not have been  accreted within the simulation time, while gas above this density re-  sides within the molecular core (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' It is therefore unclear  if the relationship between accretion and mass of the molecular core  would remain if the simulations ran for a longer time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' If the relation-  ship does hold, we speculate that increasing the size of the streaming  velocities between gas and DM may have a greater effect on the IMF,  since this also increases halo masses without the counteracting ef-  fects of H2 dissociation (Tseliakhovich & Hirata 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Greif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Hirano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Schauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2019, AS21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  5 FRAGMENTATION AND THE IMF  The IMF of Pop III stars is determined by the fragmentation be-  haviour of the disc around the initial central object and the subse-  quent accretion onto fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Density projections of the inner 650  AU of the halos are shown in Figure 10, while Figure 11 shows the  evolution of the total number of sink particles formed, the total mass  in sink particles and highest mass sink particle in each halo as a  function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' While the halos from the simulation without a LW  background typically yield less fragmentation, the overall fragmen-  tation behaviour is stochastic, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The total mass accreted  onto sinks is typically higher in the halos illuminated by a LW field  due to their higher halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Figure 12 shows the IMF at a time 300 yr after the formation  of the first sink particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The peak of the IMF positioned at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='2-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='5 M⊙ shows little dependence on the LW strength, likely because  the positive influence of larger halo masses on the molecular core  is regulated by the increasing photodissociation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We also show  the evolution of the cumulative IMFs in time, which converge by the  end of the simulations for the 𝐽21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The left side  of Figure 13 compares the combined cumulative IMFs for different  𝐽21 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' While the high mass end of the IMFs are nearly identical  between the different 𝐽21 values, the low mass end of the IMFs show  variance due to the random and stochastic nature of ejection events  for low mass objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Assuming these low mass objects (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='075M⊙)  do not go on to accrete significant mass, they will remain as brown  MNRAS 000, 1–13 (2020)  8  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Prole  105  108  1011  1014  1017  n [cm  3]  103  T [K]  J21=0  105  108  1011  1014  1017  n [cm  3]  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01  105  108  1011  1014  1017  n [cm  3]  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1  halo 1  halo 2  halo 3  halo 4  halo 5  10  3  10  2  10  1  100  H2  10  7  10  6  10  5  HD  10  19  10  16  10  13  10  10  10  7  10  4  [g cm  3]  10  11  10  9  10  7  10  5  H +  10  19  10  16  10  13  10  10  10  7  10  4  [g cm  3]  10  19  10  16  10  13  10  10  10  7  10  4  [g cm  3]  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Mass-weighted temperature, H2, HD and H+ abundances versus density at a time shortly after the formation of the first sink particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (Note that the  HD abundance is only tracked self-consistently up to 𝜌 = 10−16 g cm−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' above this, we simply fix the HD/H2 ratio at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='6×10−5, as explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  10  22  10  16  10  10  10  4  [g cm  3]  J21=0  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1  halo 1  halo 2  halo 3  halo 4  halo 5  5  10  15  |v| [km s  1]  10  5  0  vr [km s  1]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='00  |v /v|  10  7  10  5  10  3  10  1  101  103  R [pc]  10  1  100  101  |v|/cs  10  7  10  5  10  3  10  1  101  103  R [pc]  10  7  10  5  10  3  10  1  101  103  R [pc]  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Radial distribution of cumulative gas mass and mass-weighted radial profiles of radial velocity, ratio of rotational to total velocity (note the dotted line  represents the value above which rotational component dominates the velocity) and ratio of velocity to sound speed, taken at a time shortly after the formation  of the first sink particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2020)  Cosmological Lyman-Werner simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  9  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Left: Comparison of the mass-weighted temperature, density and and H2 abundance profiles between the 𝐽21 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Right: Zoom-in of the transition  to fully molecular core, showing the total kinetic energy within radial shells, mass weighted H2 photodissociation heating rate and H2 abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  4  6  8  Mhalo [M  ]  1e6  5  10  15  20  25  30  35  40  Msinks [M  ]  2000  3000  4000  Tvir [K]  50000  100000  150000  MH2 > 10  4 [M  ]  J21=0  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1  10  20  30  40  MH2 > 10  1 [M  ]  M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='85  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Total mass in sinks at the end of the simulations versus halo mass, virial temperature, mass within the halo with H2 abundance > 10−4 and mass  within the molecular core with H2 abundance > 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The size of the markers is scaled with the halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  dwarfs and never sustain nuclear fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The right side of the plot  shows the cumulative IMFs if the brown dwarfs are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Here,  the IMFs fit well within each other’s regions of uncertainty, which  are the standard deviations from the cumulative IMFs of the 5 halos  individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The overlap between the regions of uncertainty indicates  that the LW strength does not significantly affect the primordial IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  As the range of background LW field strengths tested here covers the  most likely values from literature, we infer that the IMF for Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='2  stars is not significantly different from the initial population of Pop  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  We also show the IMF from LP22 as a dotted line, which was  produced from idealised halos as opposed the cosmological initial  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The cosmological halos produced a bi-modal distribution  with a significant population of brown dwarfs that the idealised halos  are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Even when ignoring the brown dwarfs, the cosmological  initial conditions have yielded distributions tending to lower mass  pre-stellar cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2020)  10  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Prole  250 AU  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Column-weighted density projections of the inner 650 AU of the halos at a time 300 yr after the formation of the first sink particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Sink particles are  represented as red dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  0  10  20  30  Nsink  J21=0  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1  halo 1  halo 2  halo 3  halo 4  halo 5  100  101  Mtot [M  ]  0  100  200  300  t [yr]  10  1  100  101  Mmax [M  ]  100  200  300  t [yr]  100  200  300  t [yr]  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Number of sink particles formed, total mass in sinks, largest mass sink particle and median mass of sink particles as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  6 CAVEATS  Aside from the obvious uncertainties in the ΛCDM model on which  this work and the work of AS21 are based on, there are a number of  caveats to note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The LW fields we have used in this study assume a population  of massive Pop III stars already exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' While previous studies do  suggest that Pop III stars were massive, more recent work suggests  that they may only grow to a few M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Stacy & Bromm 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Wollenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Prole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Jaura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' In this  scenario, significantly less LW radiation would be produced, however  radiation from solar mass stars can inhibit H2 formation through the  destruction of H−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The fields used in this study therefore represent the  maximum effect Pop III stars can have on the the Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='2 stars that  follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Since the effects of the LW fields appear to be insignificant, it  is likely that this result also represents the outcome for weaker fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  We have assumed that the pre-stellar core radius grows as Equation  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' This process begins immediately after sink particle formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  While this is predicted by stellar theory, it is unclear at what point  the pre-stellar core would begin to expand, which may affect the  accretion behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2020)  Cosmological Lyman-Werner simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  11  0  1  2  3  4  J21=0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='0  0  5  10  15  20  25  Nsink  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='0  M/Mtot  10  4  10  3  10  2  10  1  100  101  M [M  ]  0  2  4  6  8  J21=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1  10  4  10  3  10  2  10  1  100  101  M [M  ]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='0  10 yr  50 yr  100 yr  200 yr  300 yr  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Left: combined IMFs of sink particles for the different the 𝐽21 val-  ues, Right: Cumulative IMFs of the combined sink particles for the different  the 𝐽21 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  We have neglected the presence of primordial magnetic fields in  this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' While the findings of Prole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (2022b) suggest that  primordial magnetic fields make little difference to the primordial  IMF due to their small-scale structure, the field is still active, with  other recent studies finding that magnetic fields indeed lead to higher  mass Pop III stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Saad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' Hirano & Machida 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  Stacy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  7 CONCLUSIONS  The results of cosmological simulations by AS21 show that increas-  ing the background LW field strength increases the average halo mass  required for star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' These simulations ran up to a maximum  density of 10−19 g cm−3, hence the knock-on effects on the Pop III  IMF were unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' In this investigation, we have performed follow-up  simulations of 5 halos for each of the 𝐽21 = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 LW field  strengths, resolving the pre-stellar core density of 10−6 g cm−3 before  inserting sink particles and following the fragmentation behaviour  for hundreds of years further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We have found that the mass accreted  onto sinks by the end of the simulations is proportional to the mass  within the ∼ 10−2 pc molecular core, which is not correlated to the  initial mass of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' As such, the IMF shows little dependence  on the LW field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' We also find no clear relationship between  the estimated accretion time of gas lying further out within the halo  and the LW field strength, suggesting that the LW field is unlikely  to influence the development of the IMF at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' As the range  of background LW field strengths tested here covers the most likely  values from literature, we conclude that the IMF for so-called Pop  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='2 stars is not significantly different from that of the initial popu-  lation of Pop III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='1 stars, although we cannot rule out greater effects  in the small subset of halos that are illuminated by LW fields much  stronger than the average value (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='g Latif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' In a future  paper, we will explore the effects of increasing the streaming veloci-  ties between the gas and dark matter on the Pop III IMF, as this has  been shown to increase halo masses through a different mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This work used the DiRAC@Durham facility managed by the  Institute for Computational Cosmology on behalf of the STFC  DiRAC HPC Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' The equipment was funded  by BEIS capital funding via STFC capital grants ST/P002293/1,  ST/R002371/1 and ST/S002502/1, Durham University and STFC  operations grant ST/R000832/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' DiRAC is part of the National e-  Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  The authors gratefully acknowledge the Gauss Centre for Super-  computing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='gauss-centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='eu) for supporting this project by  providing computing time on the GCS Supercomputer SuperMUC at  Leibniz Supercomputing Centre (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='lrz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='de) under project pr53ka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  AS was partially supported by NSF grant AST-1752913.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  RSK and SCOG acknowledge computing resources provided by  the Ministry of Science, Research and the Arts (MWK) of the State  of Baden-Württemberg through bwHPC and the German Research  Foundation (DFG) through grant INST 35/1134-1 FUGG and for  data storage at SDS@hd through grant INST 35/1314-1 FUGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  RSK and SCOG acknowledge financial support from DFG via the  collaborative research center (SFB 881, Project-ID 138713538) “The  Milky Way System” (subprojects A1, B1, B2 and B8), from the Hei-  delberg Cluster of Excellence “STRUCTURES” in the framework  of Germany’s Excellence Strategy (grant EXC-2181/1, Project-ID  390900948) and from the European Research Council (ERC) via the  ERC Synergy Grant “ECOGAL” (grant 855130).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content=' RSK furthermore  thanks the German Ministry for Economic Affairs and Climate Ac-  tion for funding in the project “MAINN” (funding ID 50OO2206).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  We also acknowledge the support of the Supercomputing Wales  project, which is part-funded by the European Regional Development  Fund (ERDF) via Welsh Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
+page_content='  DATA AVAILABILITY  The data underlying this article will be shared on reasonable request  to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAyT4oBgHgl3EQf6vrQ/content/2301.00828v1.pdf'}
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+arXiv:2301.01997v1  [cs.LG]  5 Jan 2023
+1
+Data-Driven Inverse Reinforcement Learning for
+Expert-Learner Zero-Sum Games
+Wenqian Xue, Bosen Lian, Jialu Fan, Tianyou Chai, and Frank L. Lewis
+Abstract—In this paper, we formulate inverse reinforcement
+learning (IRL) as an expert-learner interaction whereby the
+optimal performance intent of an expert or target agent is
+unknown to a learner agent. The learner observes the states
+and controls of the expert and hence seeks to reconstruct the
+expert’s cost function intent and thus mimics the expert’s optimal
+response. Next, we add non-cooperative disturbances that seek to
+disrupt the learning and stability of the learner agent. This leads
+to the formulation of a new interaction we call zero-sum game
+IRL. We develop a framework to solve the zero-sum game IRL
+problem that is a modified extension of RL policy iteration (PI)
+to allow unknown expert performance intentions to be computed
+and non-cooperative disturbances to be rejected. The framework
+has two parts: a value function and control action update based
+on an extension of PI, and a cost function update based on
+standard inverse optimal control. Then, we eventually develop
+an off-policy IRL algorithm that does not require knowledge
+of the expert and learner agent dynamics and performs single-
+loop learning. Rigorous proofs and analyses are given. Finally,
+simulation experiments are presented to show the effectiveness
+of the new approach.
+I. INTRODUCTION
+For an agent or system suffering from disturbances, its con-
+trol input, as a defender, desires to complete a specified control
+mission by determining control policy to reject the influences
+of antagonistic input, i.e., non-cooperative disturbances that
+intend to disrupt the mission. This is known as zero-sum games
+or min-max problems [1], [2]. In real-world applications, agent
+dynamics may be unknown. In order to make such an agent
+perform in the target trajectories exhibited by a target agent
+with optimal policy, optimal control theory assumes that the
+performance cost function is known, and RL [3], [4] based
+optimal tracking control methods [5]–[7] compute optimal
+policy by observing states and control actions without knowing
+the system dynamics, where a standard iterative form for RL
+is known as policy iteration (PI) [3], [4], [8], [9]. However,
+in real interactions, operators may not know the appropriate
+specified cost functions, i.e., the weights on states and inputs.
+As a result, these optimal control methods may not obtain the
+expected control performance or even be used.
+Wenqian Xue, Jialu Fan, Tianyou Chai are with the State Key Laboratory of
+Synthetical Automation for Process Industries and International Joint Research
+Laboratory of Integrated Automation, Northeastern University, Shenyang
+110819, China. (e-mail: xuewenqian23@163.com, fanjialu@gmail.com, ty-
+chai@mail.neu.edu.cn).
+Bosen Lian and Frank L. Lewis are with the UTA Research Insti-
+tute, the University of Texas at Arlington, Texas 76118, USA. (email:
+bosen.lian@mavs.uta.edu; lewis@uta.edu).
+Instead of manually selecting cost function weights, many
+efforts have been made on constructing cost function weights.
+Inverse optimal control (IOC) and inverse RL (IRL) construct
+cost function weights given system control behaviors. Some-
+times they are referred to as the same thing [10]–[12], but they
+may differ in structure and how they are applied [13].
+Assuming a stable control system, IOC constructs a cost
+function concerning which the system behavior is optimal.
+The cost function is constructed in the framework of Lyapunov
+stability condition for continuous-time (CT) systems [14]–[17]
+and discrete-time (DT) systems [18]–[20] where [20] considers
+finite horizon. Online IOC methods to determine cost function
+in the infinite and finite horizon are studied in [21], [22]. IOC
+is also used to verify the effectiveness of the proposed control
+laws in [23], [24]. These works do not consider min-max
+or zero-sum games, but [25] does. They all require system
+dynamics, which cannot be applied directly to systems with
+unknown dynamics.
+IRL generally reconstructs reward and cost functions from
+expert demonstrations of the optimal policy. It is usually
+applied to apprenticeship learning and imitation learning prob-
+lems of Markov decision processes (MDPs) [12], [26]–[29]
+where a learner seeks to imitate the demonstrations by learning
+the unknown expert’s reward function from the observed
+demonstrations. IRL methods construct reward function since
+reward function is a more succinct, robust, and transferable
+definition for the task than the policy mapping from states to
+actions. Lyapunov stability is not necessarily considered here.
+IRL has also been developed for trajectory tracking and
+imitation problems of differential systems in [30], [31], where
+[30] uses a bilevel structure (also see [26], [27], [32]). That is,
+an optimal control problem is solved repeatedly in the inner
+loop. This two-loop iteration is computationally expensive. All
+of these works are model-based and do not consider min-max
+or zero-sum games. The work [33] makes an effort for data-
+driven control by estimating model parameters before adopt-
+ing the model-based IRL method. Unlike them, without the
+need for model identification, our previous studies [34], [35]
+propose completely model-free IRL methods that use merely
+system data, but not for zero-sum games. Our work [36]
+considers zero-sum games but propose an IRL method using
+a two-loop iteration structure and partial system dynamics.
+This paper considers an expert-learner zero-sum game, that
+is, a learner agent suffering from non-cooperative disturbances
+with unknown dynamics expects to mimic the behaviors of the
+expert agent of optimal policy. As the solution, we propose a
+new interaction called zero-sum game IRL, namely a novel
+data-driven off-policy IRL algorithm for expert-learner zero-
+
+2
+sum games of differential systems. It consists of a game
+solution correction modified from the standard RL and a cost
+function weight reconstruction using the standard IOC. Using
+only the behavior data of the expert and learner, a learner
+agent learns the unknown cost function objective and the
+optimal control policy to mimic the expert’s behavior. This
+algorithm does not need to know or identify system models
+and performs a single-loop learning procedure without solving
+optimal problems repeatedly in inner loops. Moreover, no
+initial stabilizing policy is needed to start the iteration. The
+properties and effectiveness of the proposed data-driven are
+well-analyzed.
+Notations. ∥·∥ is the Euclidean norm. In is n×n identity ma-
+trix, and diag{a,b,..} is diagonal matrix with a,b,.. in diagonal
+line. For a vector x = [x1, ··· , xn]T ∈ Rn, ˆx ≜ [x2
+1, 2x1x2, ···,
+2x1xn, x2
+2, ···, 2xn−1xn, x2
+n]T. For a matrix A = {aij} ∈ Rm×n,
+vec(A) ≜ [a11, a12, ···, a1n, a21, ···, am,n−1, amn]T.
+II. IRL PROBLEM FORMULATION
+We consider two dynamical agents. A target expert agent
+exhibits the demonstrations optimally associated with an ex-
+pert performance cost function. A learner agent attempts to
+determine the unknown cost function objective of the expert
+agent and mimic its behavior. The learner agent only knows
+the target agent’s control actions and state behavior but does
+not know its performance cost function and system dynamics.
+A. Target optimal control
+Consider a target expert agent
+˙xT = AxT + BuT + DdT,
+(1)
+where xT ∈ Rn is the target state, uT ∈ Rm is the target input
+and dT ∈ Rz is the non-cooperative disturbance. Matrices A, B,
+and D have appropriate dimensions. The pair (A,B) is assumed
+to be controllable.
+According to [2], the input of target (1) is uT = −KTxT
+that minimizes the following target performance cost function
+against dT
+VT(xT) =
+� ∞
+t (xT
+T QTxT + uT
+TRTuT − γ2
+TdT
+T dT)dτ,
+(2)
+where QT = QT
+T > 0 and RT = RT
+T > 0 are weights, γT > 0 is
+attenuation factor. The input uT and the worst disturbance d∗
+T
+are given by
+uT = −KTxT = −R−1
+T BTPTxT,
+(3a)
+d∗
+T = LTxT = 1
+γ2
+T
+DTPTxT,
+(3b)
+and form the Nash equilibrium
+VT(xT)∗ = min
+uT max
+dT
+� ∞
+t
+rTdτ = xT
+TPTxT
+(4)
+where rT = xT
+TQTxT + uT
+TRTuT − γ2
+TdT
+T dT, and PT = PT
+T > 0
+satisfies the target Bellman equation
+(AxT + BuT + DdT)TPTxT + xT
+TPT(AxT + BuT + DdT)
++ xT
+TQTxT + uT
+TRTuT − γ2
+TdT
+T dT = 0,
+(5)
+and the target game algebraic Riccati equation (GARE)
+ATPT + PTA+ QT − PTBR−1
+T BTPT + 1
+γ2
+T
+PTDDTPT = 0.
+(6)
+The GARE guarantees the uniqueness of stabilizing optimal
+control policy (3a) given the performance cost function (2).
+B. Learner dynamics and IRL control problem
+Consider a learner agent to be controlled
+˙x = Ax+ Bu + Dd,
+(7)
+where x ∈ Rn, u ∈ Rm, d ∈ Rz are the learner’s state, control
+input and disturbance, respectively.
+Assumption 1. The target cost function VT (2) and the target
+control policy in (3a) are unknown to the learner (7). That is,
+the weights QT, RT, γT in VT, the optimal strategy KT in (3a),
+the worst disturbance d∗
+T, LT and GARE solution PT are all
+unknown.
+Assumption 2. The learner knows the target behaviour data
+xT, uT and dT.
+Definition 1. Expert-Learner Zero-Sum Game. By using
+the behaviour data of the expert (1) and the learner itself (7),
+the learner desires to reconstruct the unknown performance
+cost function (2) to exhibit the same control actions uT in
+(3a) and states xT as the expert (1).
+□
+The learner (7) will be stabilized and perform the same way
+as the target (1) does if KT in (3a) is applied to the learner (7)
+with bounded dT = d. Hence, our control goal is to determine
+the unknown cost function objective (2) to produce the optimal
+control input u∗ = −K∗x with K∗ = KT using only target data
+of xT,uT,dT and learner data of x,u,d.
+III. MODEL-BASED IRL FRAMEWORK
+In this section, we develop a novel model-based IRL
+framework for learner (7) to determine the cost function (2)
+and use this knowledge to compute control input u(t), such
+that its behavior trajectories of u(t),x(t) mimic the observed
+target trajectories of uT(t),xT(t). In Section 5, we will finally
+propose a data-driven IRL algorithm that does not need any
+system dynamics.
+A. Learner optimal control
+Let us take a review of the zero-sum game [2] of the learner
+(7) with an arbitrary given cost function
+V(x) =
+� ∞
+t (xT Qx+ uTRu − γ2dTd)dτ,
+(8)
+where Q = QT > 0, R = RT > 0, and γ > 0. The optimal input
+uo and the worst dw are given by
+uo = −Kx = −R−1BTPx,
+(9)
+dw = Lx = 1
+γ2 DTPx,
+(10)
+
+3
+and (uo,dw) forms the Nash equilibrium
+V ∗(x) = min
+u max
+d
+� ∞
+t
+rdτ = xTPx,
+(11)
+where r = xTQx+uTRu−γ2dTd, and P = PT > 0 satisfies the
+following learner GARE
+ATP+ PA+ Q− PBR−1BTP+ 1
+γ2 PDDTP = 0.
+(12)
+B. Expert-learner zero-sum game solution
+We now present a theorem to show the conditions that the
+solution to the expert-learner zero-sum game must satisfy.
+Theorem 1. If the cost weights Q,R,γ and the solution P that
+satisfy the GARE (12) also satisfy the following equation
+(A− BKT)T P+P(A− BKT)
++Q+ 1
+γ2 PDDTP+ KT
+T RKT = 0,
+(13)
+then, the corresponding control strategy K in (9) equals the
+target strategy KT in (3a).
+Proof: Rewrite (12) with KT in (3a) and K in (9) as
+(A− BKT)TP+ P(A− BKT)+ Q+ 1
+γ2 PDDTP
++ KT
+T RK + KTRKT − KTRK = 0.
+(14)
+By subtracting (14) from (13), we have
+KT
+T RKT − KT
+T RK − KTRKT + KTRK
+= (KT − K)TR(KT − K) = 0.
+(15)
+Since R > 0, (15) concludes that K = KT .
+□
+C. Learning rules for cost function and game control policy
+To find the Q,R,γ,P satisfying Theorem 1, we select γ > 0
+and R > 0 and propose an iterative procedure based on
+Theorem 1 so as to learn the weight Q, the game control
+policy P, and consequently K in (9) and L in (10).
+First, we apply a modified PI to correct P using (13). Set
+current iteration step as i,i = 0,1,··· , and give the estimates Qi
+and Li. Then the iterative form of (13), i.e., (16) in Algorithm
+1 is presented below to obtain Pi. Then optimal control ((9)
+and (10)) is used to update the strategy Ki+1 and Li+1 based
+on the corrected Pi by (17a) and (17b), respectively.
+Now we must update the cost function weight estimate Qi+1
+based on the corrected Pi. By IOC [15], taking the iterative
+form of GARE (12) yields (18) in Algorithm 1 presented as
+follows.
+Remark 1. In Algorithm 1, step 4 is the standard IOC
+computation based on Lyapunov stability condition. Note also
+that if we set Qi = QT and KT = Ki in (16), then steps 2-3 are
+the standard RL PI. As such, Algorithm 1 combines modified
+PI with IOC to solve the IRL problem.
+Algorithm 1 Model-based IRL algorithm for expert-learner
+zero-sum games.
+Step 1: Initialize with R > 0 and γ > 0, Q0 > 0, and L0 = 0,
+and set i = 0.
+Step 2: (Game policy correction) Update policy Pi by
+(A− BKT)T Pi + Pi(A− BKT)
+= −Qi − KT
+T RKT − γ2(Li)TLi.
+(16)
+Step 3: (Input update) Update control input and disturbance
+gain based on (9) and (10)
+ui+1 = −Ki+1x = −R−1BTPix,
+(17a)
+di+1 = Li+1x = 1
+γ2 DTPix.
+(17b)
+Step 4: (Cost function weight construction) Update Qi+1 by
+Qi+1 =− ATPi − PiA
++ (Ki+1)TRKi+1 − γ2(Li+1)T Li+1.
+(18)
+Step 5: Stop if it converges. Otherwise, set i = i+1 and repeat
+steps 2 to 4.
+IV. ANALYSIS OF ALGORITHM 1
+The convergence, stability, and optimality of the proposed
+Algorithm 1 are analyzed here. It is also shown that Algorithm
+1 may not converge to a unique solution (Pi,Qi) even if all
+solutions give the correct target strategy Ki = KT .
+A. Convergence analysis
+Theorem 2. i). With an initial Q0 such that 0 < Q0 ≤ ˆQ and
+L0 = 0 where ˆQ > 0 is a solution to Theorem 1 associated with
+the R > 0 and γ > 0, Algorithm 1 converges. ii). As i → ∞, the
+solutions Qi,Pi,Ki,Li converge to Q∗,P∗,K∗,L∗ that satisfy
+ATP∗ + P∗A+ Q∗−P∗BR−1BTP∗+ 1
+γ2 P∗DDTP∗ = 0,
+(19a)
+K∗ = KT = R−1BTP∗,
+(19b)
+L∗ = 1
+γ2 DTP∗.
+(19c)
+iii). The solutions Q∗,P∗,K∗,L∗ satisfy Theorem 1.
+Proof: i). Convergence proof. First, we prove that Algorithm
+1 solves an increasing sequence Qi for all i = 0,1,···. Substi-
+tuting (17a) for Ki into (18) for Qi yields
+ATPi−1 + Pi−1A = (Ki)T RKi − Qi − γ2(Li)TLi,
+(20)
+which can be rewritten as
+(A− BKT)TPi−1 + Pi−1(A− BKT)
+= (Ki)TRKi − Qi − γ2(Li)T Li − (Ki)T RKT − KT
+T RKi.
+(21)
+Subtracting (16) from (21) gives
+(A− BKT)T (Pi−1 − Pi)+ (Pi−1 − Pi)(A− BKT)
+= (Ki − KT)TR(Ki − KT).
+(22)
+It follows from (Ki − KT )TR(Ki − KT ) ≥ 0 and Hurwitz
+A − BKT that Pi−1 ≤ Pi holds for all iterations. Each pair of
+
+4
+(Pi,Qi+1) satisfies (18), and they uniquely correspond to each
+other. This is followed by the known fact that Pi−1 ≤ Pi if
+Qi ≤ Qi+1 [37]. Therefore, Qi+1 ≥ Qi > 0 holds for i = 0,1,···,
+and Qi+1 = Qi holds if and only if KT = Ki+1. Note that
+achieving KT = Ki+1 is the goal of the algorithm.
+Now we show that Qi is bounded by an upper bound. Let
+ˆQ > 0 and ˆP > 0 be a group of solution to Theorem 1. That
+is
+AT ˆP+ ˆPA+ ˆQ− ˆPBR−1BT ˆP+ 1
+γ2 ˆPDDT ˆP = 0,
+(23a)
+KT = R−1BT ˆP, ˆL = 1
+γ2 DT ˆP.
+(23b)
+Rewriting (23a) using (23b) yields
+(A− BKT)T ˆP+ ˆP(A− BKT)+ ˆQ+ KT
+T RKT + γ2ˆLT ˆL = 0.
+(24)
+If Qi + γ2(Li)TLi ≤ ˆQ+ γ2 ˆLT ˆL holds, then (16) and (24) will
+solve 0 < Pi ≤ ˆP with Hurwitz A−BKT. With (3a), (17a) and
+(23b), AREs (18) and (23a) can be rewritten as
+(A− BKi+1)TPi + Pi(A− BKi+1)
+= −Qi+1 − (Ki+1)TRKi+1 − γ2(Li+1)T(Li+1),
+(25a)
+(A− BKi+1)T ˆP+ ˆP(A− BKi+1)
+= KT
+T RKT − ˆQ− γ2ˆLT ˆL− (Ki+1)TRKT − KT
+T RKi+1,
+(25b)
+respectively. Since (18) ensures Hurwitz A− BKi+1, subtract-
+ing (25a) from (25b) and using 0 < Pi ≤ ˆP obtains
+(A− BKi+1)T ( ˆP− Pi)+ ( ˆP− Pi)(A− BKi+1)
+= (Qi+1 + γ2(Li+1)TLi+1)− ( ˆQ+ γ2 ˆLT ˆL)
++ (Ki+1 − KT)TR(Ki+1 − KT) ≤ 0.
+(26)
+Therefore, Qi+1 + γ2(Li+1)T Li+1 ≤ ˆQ+ γ2ˆLT ˆL holds.
+By deduction, it is inferred that initializing Algorithm 1 with
+a Q0 such that 0 < Q0 ≤ ˆQ and L0 = 0, then Qi > 0,i = 0,1,···
+will be increasing with an upper bound. Therefore, Algorithm
+1 converges.
+ii). Converged solutions satisfy (19a), (19b), (19c).
+Substituting (18) into (16) yields
+ATPi + PiA− PiBR−1BTPi
+= ATPi+1 + Pi+1A− Pi+1BKT −KT
+T BTPi+1 + KT
+T RKT.
+(27)
+Taking Pi+1 = Pi = P∗ as converged value, (27) becomes
+(KT − K∗)R(KT − K∗) = 0,
+(28)
+where K∗ = R−1BTP∗. Since R > 0, (28) implies KT = K∗,
+which is exactly (19b). The converged P∗ produces the con-
+verged L∗ using (17b) as shown in (19c) and the converged
+Q∗ using (18) as shown in (19a).
+iii). Converged solutions satisfy Theorem 1.
+Rewriting (19a) with (19b) yields
+(A− BKT)T P∗ + P∗(A− BKT)+ Q∗
++ KT
+T RKT + 1
+γ2 P∗DDTP∗ = 0,
+(29)
+which is exactly (13). Obviously, (19a) is exactly (12). There-
+fore, Q∗,P∗,K∗ satisfy Theorem 1.
+□
+B. Stability and optimality analysis
+We now prove the stability of Algorithm 1 in Theorem 3,
+and optimality and Nash equilibrium in Theorem 4
+Theorem 3. Each iteration of Algorithm 1 exponentially
+stabilizes the learner agent (7) with d = 0.
+Proof: Rewriting (16) with (3a) and (17a) yields
+ATPi + PiA+ ˜Qi − PiBR−1BTPi + γ2(Li)TLi = 0.
+(30)
+where ˜Qi = Qi + (Ki+1 − KT)TR(Ki+1 − KT ). With (Ki+1 −
+KT)T R(Ki+1 −KT) ≥ 0 and Qi > 0, then ˜Qi > 0. It is obvious
+that Pi solved by (30) or equivalently (16) is a symmetrical
+positive definite matrix satisfying
+ATPi + PiA− PiBR−1BTPi + γ2(Li)T Li < 0,
+(31)
+and one has
+˙V i(x,Pi) = xT(A− BKi+1)TPix+ xTPi(A− BKi+1)x
+< −xTPiBR−1BTPix− γ2xT(Li)TLix < 0.
+(32)
+That is, Pi yields stabilizing Ki+1 by (17a) for learner (7) with
+d = 0.
+Considering (17a) and (31), Qi+1 in (18) satisfies
+Qi+1 + γ2(Li+1)T Li+1
+= −(ATPi + PiA− PiBR−1BTPi) > 0.
+(33)
+Using Qi+1 +γ2(Li+1)TLi+1 > 0 in (16) would still make (32)
+hold for the next iteration. Therefore, provided Q0 > 0 (32)
+will hold for all i = 0,1,···.
+□
+Before optimality analysis of Algorithm 1, we now give a
+lemma of importance which extends the idea of classic IOC
+[15] to two-player zero-sum games.
+Lemma 1. Consider the two-player learner agent (7) with
+x(t0) = x0, t ≥ t0 and (8) and (11). Assume there exists a
+positive definite symmetric matrix P ∈ Rn×n such that
+ATP+ PA− PBR−1BTP+ 1
+γ2 PDDTP < 0.
+(34)
+Then, with the optimal feedback control input uo and the worst
+disturbance dw such that
+uo = −R−1BTPx, dw = 1
+γ2 DTPx,
+(35)
+and the cost function weight
+Q = −(ATP+ PA− PBR−1BTP+ 1
+γ2 PDDTP),
+(36)
+the saddle point (uo,dw) makes the cost value function (8)
+reach the Nash equilibrium
+V(x0,uo,d) ≤ V(x0,uo,dw) ≤ V(x0,u,dw).
+(37)
+Proof: First, V(x) in (8) can be represented with P > 0 as
+V(x) = xTPx ≥ 0, V(0) = 0.
+(38)
+It follows from (35) and (36) that H(x,uo,dw) = 0 [2] where
+the Hamiltonian is
+H(x,u,d) =xTQx+ uTRu − γ2dTd + (Ax+ Bu + Dd)TPx
++ xTP(Ax+ Bu + Dd).
+(39)
+
+5
+One writes
+H(x,u,d) = H(x,u,d)− H(x,uo,dw)
+= (u − uo)R(u − uo)− γ2(d − dw)T (d − dw),
+(40)
+and hence
+H(x0,uo,d) ≤ H(x0,uo,dw) ≤ H(x0,u,dw).
+(41)
+Based on [2], we obtain the conclusion (37).
+□
+Theorem 4. The converged solutions Q∗,P∗,K∗,L∗ obtained
+by Algorithm 1 shown in Theorem 2 yield Nash equilibrium
+of the value function V(x) in (8) such that
+V(x0,u∗,d) ≤ V(x0,u∗,d∗) ≤ V(x0,u,d∗),
+(42)
+where u∗ = −K∗x and d∗ = L∗x.
+Proof: It follows from Theorem 1 that Qi > 0 holds for all
+i = 0,1,···. Thus one has the converged Q∗ > 0 and
+ATP∗ + P∗A− P∗BR−1BTP∗ + 1
+γ2 (P∗)TDDTP∗ < 0,
+(43)
+from (19a), which means that the converged P∗ satisfies (34)
+in Lemma 1. Also, the converged control strategy K∗ in (19b)
+and disturbance gain L∗ in (19c) satisfy (35) in Lemma 1. This
+indicates that (41) also holds for u∗ and d∗, namely the Nash
+equilibrium (42) holds.
+□
+C. Non-uniqueness of solution
+In fact, the Q∗,R,γ,P∗ satisfying (19a)-(19c) that explain
+the same strategy K∗ = KT may not be unique and can be
+different from the actual target values QT,RT,γT,PT shown
+in (3b) and (6). This multi-solution phenomenon is known
+as the ill-posedness property, which is well-analyzed for DT
+ARE in [38] and coupled ARE in [39]. In the next result,
+we characterize the relationship between QT,RT,γT ,PT, and
+Q∗,R,γ,P∗ for CT GARE and show the conditions for the
+occurrence of this phenomenon.
+Theorem 5. Recall QT, RT, γT, PT satisfying (6) and (3b),
+and let Qo, Ro, Po satisfy
+BTPo = RoR−1
+T BTPT,
+(44a)
+Qo + ATPo + PoA− KT
+T RoKT + 1
+γ2
+T
+PTDDTPT − 1
+γ2 P∗DDTP∗
+= 0,
+(44b)
+where Ro = RT −R. Then any Q∗ = QT −Qo and P∗ = PT −Po
+satisfy (19a)-(19c).
+Proof: Subtracting (44b) from (6) and using (3b) yields
+AT(PT − Po)+ (PT − Po)A+ (QT − Qo)
+− KT
+T (RT − Ro)KT + 1
+γ2 P∗DDTP∗ = 0.
+(45)
+Using P∗ = PT − Po, Ro = RT − R, and R > 0 in (44a) gives
+K∗ = R−1BTP∗ = R−1
+T BTPT = KT,
+(46)
+which is (19b). Substituting it into (45) yields (19a) and (19c).
+This proves the relationship between the obtained solution
+Q∗,R,γ,P∗ and the expert’s QT,RT,γT ,PT. We observe that
+Po, Ro, Qo satisfying (44a) and (44b) can be nonzero. That
+is, Q∗,R,γ,P∗ associate optimally with the same strategy as
+QT,RT,γT,PT, i.e., KT = K∗, but QT ̸= Q∗, RT ̸= R, γT ̸= γ.
+Therefore, there could be multiple solutions to (19a) to gen-
+erate a K∗ in (19b) equal to the target KT in (3a).
+□
+The following corollary shows a special case of the Q∗,R,γ
+of Theorem 5 which gives V(x)∗ = cVT(xT)∗.
+corollary 1. With scalar c > 0, any Q∗ = cQT, R = cRT, γ =
+√cγT would yield V(x)∗ = cVT(xT)∗ in (4) and (11), and they
+optimally associate with the same K∗ as the expert such that
+K∗ = KT in (19b) and (3a).
+Proof: Bring such Q∗,R,γ into (19a), since QT,RT,γT satisfy
+(6), then one has P∗ = cPT and K∗ = KT . Then, using this
+result in (11) for V(x)∗ and comparing it with VT(xT)∗ in (4)
+shows that V(x)∗ = cVT(xT)∗.
+□
+V. DATA-DRIVEN OFF-POLICY IRL ALGORITHM
+Algorithm 1 relies on the system dynamics A,B,D and the
+target strategy KT . To remove this requirement, we develop
+here a data-driven IRL algorithm for expert-learner zero-sum
+games based on Algorithm 1, which only requires the data
+xT,uT,dT of the target agent (1) and x,u,d of the learner
+agent (7). To accomplish this, we use two techniques similar
+to integral RL [6], [8] and off-policy RL [6], [40]. The end
+result is Algorithm 2.
+A. Data-driven game policy correction
+In order to update Pi, Ki+1 and Li+1 in (16)-(17b) using
+only target data xT,uT,dT, inspired by the idea of off-policy
+integral RL technique [6], [40], rewrite (1) as
+˙xT = AxT − BKixT + DdT + B(uT + KixT).
+(47)
+Using (47) and (16) one writes
+˙xT
+T PixT + xT
+TPi ˙xT
+= (AxT − BKixT)T PixT + xT
+TPi(AxT − BKixT)
++ 2(uT + KixT)TBTPixT + 2dT
+T DTPixT
+= −xT
+TQixT − xT
+TKT
+T RKTxT − γ2xT
+T(Li)T LixT
++ 2xT
+TKT
+T BTPixT + 2uT
+TBTPixT + 2dT
+T DTPixT.
+(48)
+Using (17a), (17b), (3a) in (48) and integrating both sides from
+t to t + T, where T > 0 is the integral time period, obtains
+(50) in Algorithm 2 to be presented, by which Pi, Ki+1 and
+Li+1 are updated simultaneously. Similar to [6], probing noise
+e is added to uT, i.e., uT = −KTxT + e, for the persistence
+of excitation condition in merely learning process. It is not
+needed anymore when solutions converge. Unlike (16)-(17b),
+(50) does not need the knowledge of agent dynamics A,B,D
+or the strategy KT in (3a).
+
+6
+B. Data-driven cost function weight reconstruction
+In order to update Qi+1 in (18) using only data x,u,d,
+inspired by the integral RL technique [8], multiplying both
+sides of (18) by x and adding and subtracting terms uTBTPix
+and dTDTPix, (18) can be rewritten as
+xTQi+1x =− [(Ax+ Bu + Dd)TPix+ xTPi(Ax+ Bu + Dd)
+− xT(Ki+1)TRKi+1x+ γ2xT(Li+1)T Li+1x
+− 2uTBTPix− 2dTDTPix],
+(49)
+where u can be generated by any stabilizing policy and d
+can be random and different from dT in the learning process.
+Substituting (7), (17a) and (17b) into (49) and integrating it
+gives (51) in Algorithm 2 below. Using Pi, Ki+1 and Li+1
+obtained by (50), (51) equivalently replaces (18) in Algorithm
+1 to calculate Qi+1 without knowing any system dynamics.
+Algorithm 2 Data-driven off-policy IRL algorithm for expert-
+learner zero-sum games.
+Step 1: Initialize with R > 0, γ > 0, Q0 > 0, and L0 = 0,
+and collect system data generated by any stabilizing
+control input u. Set i = 0.
+Step 2: (Game policy correction) Update policy Pi, control
+strategy Ki+1 and disturbance gain Li+1 by
+xT(t + T)TPixT(t + T)− xT(t)TPixT(t)
+− 2
+� t+T
+t
+eTRKi+1xTdτ − 2γ2
+� t+T
+t
+dT
+T Li+1xTdτ
+= −
+� t+T
+t
+�
+xT
+TQixT + (uT − e)TR(uT − e)
++ γ2xT
+T (Li)TLixT
+�
+dτ.
+(50)
+Step 3: (Cost function weight construction) Update cost
+function weight Qi+1 by
+� t+T
+t
+xT Qi+1xdτ
+= −[x(t + T)TPix(t + T)− x(t)TPix(t)
+−
+� t+T
+t
+(2uTRKi+1x+ xT(Ki+1)TRKi+1x)dτ
+−γ2
+� t+T
+t
+(2dTLi+1x− xT(Li+1)TLi+1x)dτ].
+(51)
+Step 4: Stop if it converges. Otherwise, set i = i+1 and repeat
+steps 2 to 4, .
+Remark 2. Algorithm 2 does not need system dynamics.
+Moreover, it iterates in single loop indicated by i, no inner-
+loop iteration is needed.
+C. Implementation and Analysis of Algorithm 2
+In order to show how to implement data-driven IRL Al-
+gorithm 2 using only data, first, consider Kronecker product
+aTWb = (bT ⊗ aT)vec(W) for (50) and define the following
+operators,
+OxT xT = [x2
+T1,2xT1xT2,...,x2
+T2,2xT2xT3,...,x2
+Tn]T;
+dxT xT = [OxT xT (t + T)− OxT xT (t),...,
+OxT xT (t + lT)− OxT xT (t + (l − 1)T)]T;
+IxT xT = [
+� t+T
+t
+OxT xT dτ,...,
+� t+lT
+t+(l−1)T OxT xT dτ]T;
+IxT uT = [
+� t+T
+t
+xT ⊗ uTdτ,...,
+� t+lT
+t+(l−1)T xT ⊗ uTdτ]T ;
+IxT dT = [
+� t+T
+t
+xT ⊗ dTdτ,...,
+� t+lT
+t+(l−1)T xT ⊗ dTdτ]T ;
+IxT e = [
+� t+T
+t
+xT ⊗ edτ,...,
+� t+lT
+t+(l−1)T xT ⊗ edτ]T;
+Φp = [dxT xT ,−2IxT e(In ⊗ R),−2γ2IxT dT ]T;
+ri = xT
+TQixT + (uT − e)TR(uT − e)+ γ2xT
+T(Li)T LixT;
+Ψi
+p = −[
+� t+T
+t
+ridτ,...,
+� t+lT
+t+(l−1)T ridτ]T;
+ˆPi = [Pi
+11,Pi
+12,...,Pi
+22,Pi
+23,...,Pi
+nn]T,
+(52)
+where l is the group number of collected data and should be
+l ≥ n(n+1)
+2
++ nm + nz. Using batch least squares method [8],
+[40], [41], ˆPi, Ki+1, Li+1 can be calculated by
+�
+( ˆPi)T ,vec(Ki+1)T,vec(Li+1)T �T=(ΦT
+pΦp)−1ΦT
+pΨi
+p.
+(53)
+Similarly, for (51), we define
+Oxx = [x2
+1,2x1x2,...,x2
+2,2x2x3,...,x2
+n]T;
+dxx = [Oxx(t + T)− Oxx(t),...,
+Oxx(t + kT)− Oxx(t + (k − 1)T)]T ;
+ˆqi = [Qi
+11,Qi
+12,...,Qi
+22,Qi
+23,...,Qi
+nn]T;
+Φq = Ixx = [
+� t+T
+t
+Oxxdτ,...,
+� t+kT
+t+(k−1)T Oxxdτ]T ;
+Ii+1
+q
+(t) =
+� t+T
+t
+(2uTRKi+1x+ xT(Ki+1)TRKi+1x)dτ
++ γ2
+� t+T
+t
+(2dTLi+1x− xT(Li+1)TLi+1x)dτ
+Ψi+1
+q
+= −dxx ˆPi + [Ii+1
+q
+(t),...,Ii+1
+q
+(t + (k − 1)T)]T ,
+(54)
+where k is the group number of collected data and should be
+k ≥ n(n+1)
+2
+. Then, Qi+1 can be uniquely solved by
+ˆqi+1 = (ΦT
+q Φq)−1ΦT
+q Ψi+1
+q
+.
+(55)
+By using (53) and (55), we solve Pi, Qi+1, Ki+1, Li+1 in a
+data-driven mode.
+Each step of Algorithm 1 yields a unique solution. Algo-
+rithm 2 is developed based on Algorithm 1. To illustrate that
+the solution obtained by Algorithm 2 estimates the solution
+obtained by Algorithm 1, we show that equations (53) and
+(55) yield unique solutions in the next result.
+
+7
+Theorem 6. If there exist lo > 0, ko > 0, for all l ≥ lo and
+k ≥ ko,
+rank([IxT xT ,IxT uT ,IxT dT ]) = n(n + 1)
+2
++ nm+ nz,
+(56a)
+rank(Ixx) = n(n + 1)
+2
+,
+(56b)
+then, (53) and (55) solve unique solution, respectively.
+Proof: First, we show that (53) solves unique solution. This
+is aiming to show that
+ΦpΩ = 0
+(57)
+has only the trivial solution Ω = 0. Now, we prove Ω = 0
+by contradiction. Assume Ω = [XT
+v ,Y T
+v ,ZT
+v ]T ∈ R
+n(n+1)
+2
++nm+nz
+is a nonzero solution of (57), where Xv ∈ R
+n(n+1)
+2
+, Yv ∈ Rmn,
+Zv ∈ Rnz. Then, Xv,Yv,Zv uniquely determine matrices X,Y,Z
+by Xv = ˆX, Yv = vec(Y) and Zv = vec(Z), respectively, where
+X is a symmetrical matrix.
+Define
+Inx = [
+� t+T
+t
+xT ⊗ xTdτ,...,
+� t+kT
+t+(k−1)T xT ⊗ xTdτ]T.
+(58)
+Integrating (48) from t to t + T gives
+ΦpΩ = Inxvec(E)+ 2IxT uT vec(G)+ 2IxTdT vec(F)= 0, (59)
+where
+E = ATX + XA− KT
+T RY −YT RKT,
+(60a)
+G = BTX − RY,
+(60b)
+F = DTX − γ2Z.
+(60c)
+Since E is a symmetrical matrix, one has Inxvec(E) = IxT xT ˆE.
+Using this in (59) yields
+ΦpΩ = [IxT xT ,2IxT uT ,2IxT dT ]
+
+
+ˆE
+vec(G)
+vec(F)
+
+ = 0.
+(61)
+Under (56a), we know that [Ixx,Ixu,Ixd] has full column rank,
+and thus (61) has only the solution ˆE = 0, vec(G) = 0 and
+vec(F) = 0. That is,
+ATX + XA− KT
+T RY −Y TRKT = 0,
+(62a)
+BTX = RY,
+(62b)
+DTX = γ2Z.
+(62c)
+Since A − BKT is Hurwitz, substituting (62b) and (62c) into
+(62a) gives X = 0. This implies that Y = 0 and Z = 0 due to
+R > 0 and γ2 > 0. In summary, we have Ω = 0. However, this
+conflicts with the assumption that Ω is nonzero. Therefore, it
+concludes that under (56a), (55) solves unique solution.
+Second, from the integral RL work [8], we conclude that
+when (56b) holds for (51), Qi+1 in (51) can be uniquely
+determined by (55) with collected data.
+□
+VI. SIMULATION
+We show three simulation experiments, a first one of the
+data-driven Algorithm 2 to show its performance, a com-
+parison simulation with the bilevel IRL method in [36] to
+show the reduction of iteration steps of Algorithm 2, and
+a second comparison simulation with the RL-based optimal
+tracking control method in [6] to show the improvement of
+control performance with the cost function weights correction
+of Algorithm 2.
+A. Simulation result of Algorithm 2
+The system dynamics information of the target (1) and the
+learner (7) for simulation is
+A =
+�
+−1
+2
+2.2
+1.7
+�
+,B =
+�
+0
+3
+�
+,D =
+�
+1
+0
+�
+.
+(63)
+For the target agent (1), the actual target cost function objec-
+tive (2) consists of QT = diag{8,12},RT = 2I1,γT = 3. The
+disturbance is dT = 0.003rand(1). The expert’s LT, KT and PT
+are
+KT = [1.9869,3.5779],LT = [0.4162,0.1472],
+PT =
+� 3.7459
+1.3246
+1.3246
+2.3853
+�
+.
+(64)
+For the learner, the behaviour strategy to generate data
+is Kb = [1.2129,
+2.2812], and the disturbance is d =
+0.003rand(1). To start Algorithm 2, the initial weights for cost
+function, initial disturbance gain L0, and integral time period
+T are given by
+Q0 = diag{1,0.5},R = I1,γ = 40,L0 = [0, 0],T = 0.008s.
+(65)
+The Fig. 1(a) captures the iteration process from the initial
+spot to the spot on ∥Ki+1−KT∥ ≤ 0.01. The final values of Ki,
+Qi, Li, and Pi are K∗, Q∗, L∗, and P∗, respectively, as follows
+K∗ =
+� 1.9827
+3.5839 �
+,
+Q∗ =
+�
+2.2796
+2.6670
+2.6670
+6.0151
+�
+,
+L∗ = 10−3 ×
+�
+0.4021
+0.4131
+�
+,
+P∗ =
+� 0.6441
+0.6622
+0.6622
+1.1968
+�
+,
+(66)
+where K∗ closely approximates the target KT in (64) with
+∥K∗ − KT∥ = 0.0073, while Q∗, L∗ and P∗ are not equal to
+QT,LT and PT in (64), respectively. This is the multiple-
+solution phenomenon. In Fig. 1(b), the learner’s state x can
+mimic the trajectories of the target xT very well under the
+learned K∗. Therefore, the proposed Algorithm 2 can learn an
+appropriate cost function and optimal policy for the learner to
+mimic the target trajectories.
+B. Comparison simulation case 1
+This subsection shows the simulation results of the bilevel
+IRL method in [36] that iterates in two-loop to show the
+reduction of computational complexity in terms of iteration
+steps. The same expert-learner system, initial parameters for
+
+8
+0
+10
+20
+30
+40
+50
+0
+1
+2
+||Ki − KT||
+0
+10
+20
+30
+40
+50
+9
+10
+11
+12
+||Qi − QT||
+0
+10
+20
+30
+40
+50
+0.44095
+0.441
+0.44105
+0.4411
+||Li − LT||
+0
+10
+20
+30
+40
+50
+Update steps i
+(a)
+3.4
+3.6
+||P i − PT||
+0
+0.5
+1
+1.5
+2
+Time(s)
+(b)
+-20
+-10
+0
+10
+20
+x1
+x2
+xT1
+xT2
+Fig. 1. Convergence and imitation performance using Algorithm 2
+Algorithm 2 in (63)-(65) are used for this comparison method
+1.
+Fig. 2 also captures the iteration process from the initial
+spot to the spot on ∥Ki+1 −KT∥ ≤ 0.01 as Fig. 1 to show the
+difference in iteration steps of the two methods. The inner-
+loop iteration figures are omitted since they are too many to
+put here. In Fig. 2, the final values of K j,Qj,Lj, and P j of
+the outer-loop iterations are
+K∗ =
+�
+1.9822
+3.5691
+�
+,
+Q∗ =
+� 2.3186
+2.6974
+2.6974
+6.0506
+�
+,
+L∗ = 10−3 ×
+� 0.4053
+0.4130 �
+,
+P∗ =
+� 0.6486
+0.6607
+0.6607
+1.1897
+�
+.
+(67)
+where K∗ approximates the target KT in (64) with ∥K∗−KT∥ =
+0.01. Table I shows that the total iteration steps of the method
+is 3370, including 587 outer-loop updates (See Fig. 2) and
+2783 inner-loop updates, while Algorithm 2 iterates 51 steps
+in total (See Fig. 1(a)). The time of the learning process of
+Algorithm 2 is 4.08s, while that of the comparison method 1 is
+169.936s. It is proportional to the amount of utilized collected
+data. Algorithm 2 uses 510 groups of data, and comparison
+method 1 uses 21242 groups of data. Therefore, Algorithm 2
+costs much fewer data and time than the bilevel comparison
+method 1.
+0
+100
+200
+300
+400
+500
+600
+0
+1
+2
+||Kj − KT||
+0
+100
+200
+300
+400
+500
+600
+8.5
+9
+9.5
+10
+||Qj − QT||
+0
+100
+200
+300
+400
+500
+600
+0.441
+0.4411
+||Lj − LT||
+0
+100
+200
+300
+400
+500
+600
+Outer-loop update steps j
+3.4
+3.6
+||P j − PT||
+Fig. 2. Convergence of K j,Qj,Lj, and Pj using the comparison method 1
+TABLE I
+ITERATION STEPS AND LEARNING TIME OF ALGORITHM 2 AND THE
+COMPARISON METHOD 1
+Methods
+Total updates
+Learning time
+Algorithm 2
+51
+4.08s
+Comparison method 1
+3370
+169.936s
+C. Comparison simulation case 2
+In this subsection, the typical RL-based optimal tracking
+control method [6] for linear disturbed systems, which com-
+putes optimal control policy given cost function weights, is
+simulated to show the advantage of Algorithm 2 in control
+performance by computing both optimal control policy and
+cost function weights.
+The same system and cost weights shown in (63)-(65) and
+discount factor α = 0.9 are used. The obtained optimal control
+law is u∗ = −[1.1760 1.9139 1.0044 1.7639][xT,rT]T, and the
+corresponding imitation performance in Fig. 3 is not as good
+as that of Algorithm 2 in Fig. 1(b). By evenly sampling the
+trajectory data, the imitation performance of the two methods
+is quantified by the error index defined as follows
+Te = 1
+n
+n
+∑
+i=1
+�
+1
+a
+a
+∑
+k=1
+|xi(kT)− xTi(kT)|2
+where n = 2, a = 250, and T = 0.008s. As shown in Table II,
+Te of Algorithm 2 is much smaller than that of the comparison
+method 2. The reason is that Algorithm 2 can correct the
+given cost function weights when it is inappropriate, but the
+comparison method 2 cannot. Algorithm 2 thus obtains much
+better imitation performance.
+TABLE II
+IMITATION INDEX OF ALGORITHM 2 AND THE COMPARISON METHOD 2
+Methods
+Error index Te
+Algorithm 2
+0.0162
+Comparison method 2
+1.4461
+
+9
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+-20
+-10
+0
+10
+20
+Fig. 3. Imitation performance of learner’s x to target xT using the comparison
+method 2
+VII. CONCLUSION
+This paper proposes a novel data-driven off-policy IRL
+approach to determine both cost function and optimal con-
+trol policy to stabilize a learner agent suffering from non-
+cooperative disturbances by mimicking a target agent’s tra-
+jectories using data of both agents. The proposed approach
+does not need any system dynamics and guarantees stability,
+Nash optimality, and imitation performance with single-loop
+iteration. The rigorous theoretical proofs and simulation ex-
+periments verify its effectiveness.
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diff --git a/O9A0T4oBgHgl3EQfC_9-/content/tmp_files/load_file.txt b/O9A0T4oBgHgl3EQfC_9-/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf,len=800
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='01997v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='LG]  5 Jan 2023  1  Data-Driven Inverse Reinforcement Learning for  Expert-Learner Zero-Sum Games  Wenqian Xue, Bosen Lian, Jialu Fan, Tianyou Chai, and Frank L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Lewis  Abstract—In this paper, we formulate inverse reinforcement  learning (IRL) as an expert-learner interaction whereby the  optimal performance intent of an expert or target agent is  unknown to a learner agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The learner observes the states  and controls of the expert and hence seeks to reconstruct the  expert’s cost function intent and thus mimics the expert’s optimal  response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Next, we add non-cooperative disturbances that seek to  disrupt the learning and stability of the learner agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' This leads  to the formulation of a new interaction we call zero-sum game  IRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' We develop a framework to solve the zero-sum game IRL  problem that is a modified extension of RL policy iteration (PI)  to allow unknown expert performance intentions to be computed  and non-cooperative disturbances to be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The framework  has two parts: a value function and control action update based  on an extension of PI, and a cost function update based on  standard inverse optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Then, we eventually develop  an off-policy IRL algorithm that does not require knowledge  of the expert and learner agent dynamics and performs single-  loop learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Rigorous proofs and analyses are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Finally,  simulation experiments are presented to show the effectiveness  of the new approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' INTRODUCTION  For an agent or system suffering from disturbances, its con-  trol input, as a defender, desires to complete a specified control  mission by determining control policy to reject the influences  of antagonistic input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=', non-cooperative disturbances that  intend to disrupt the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' This is known as zero-sum games  or min-max problems [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' In real-world applications, agent  dynamics may be unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' In order to make such an agent  perform in the target trajectories exhibited by a target agent  with optimal policy, optimal control theory assumes that the  performance cost function is known, and RL [3], [4] based  optimal tracking control methods [5]–[7] compute optimal  policy by observing states and control actions without knowing  the system dynamics, where a standard iterative form for RL  is known as policy iteration (PI) [3], [4], [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' However,  in real interactions, operators may not know the appropriate  specified cost functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=', the weights on states and inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  As a result, these optimal control methods may not obtain the  expected control performance or even be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Wenqian Xue, Jialu Fan, Tianyou Chai are with the State Key Laboratory of  Synthetical Automation for Process Industries and International Joint Research  Laboratory of Integrated Automation, Northeastern University, Shenyang  110819, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' (e-mail: xuewenqian23@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='com, fanjialu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='com, ty-  chai@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Bosen Lian and Frank L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Lewis are with the UTA Research Insti-  tute, the University of Texas at Arlington, Texas 76118, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' (email:  bosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='lian@mavs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='uta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' lewis@uta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Instead of manually selecting cost function weights, many  efforts have been made on constructing cost function weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Inverse optimal control (IOC) and inverse RL (IRL) construct  cost function weights given system control behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Some-  times they are referred to as the same thing [10]–[12], but they  may differ in structure and how they are applied [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Assuming a stable control system, IOC constructs a cost  function concerning which the system behavior is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  The cost function is constructed in the framework of Lyapunov  stability condition for continuous-time (CT) systems [14]–[17]  and discrete-time (DT) systems [18]–[20] where [20] considers  finite horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Online IOC methods to determine cost function  in the infinite and finite horizon are studied in [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' IOC  is also used to verify the effectiveness of the proposed control  laws in [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' These works do not consider min-max  or zero-sum games, but [25] does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' They all require system  dynamics, which cannot be applied directly to systems with  unknown dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  IRL generally reconstructs reward and cost functions from  expert demonstrations of the optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' It is usually  applied to apprenticeship learning and imitation learning prob-  lems of Markov decision processes (MDPs) [12], [26]–[29]  where a learner seeks to imitate the demonstrations by learning  the unknown expert’s reward function from the observed  demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' IRL methods construct reward function since  reward function is a more succinct, robust, and transferable  definition for the task than the policy mapping from states to  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Lyapunov stability is not necessarily considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  IRL has also been developed for trajectory tracking and  imitation problems of differential systems in [30], [31], where  [30] uses a bilevel structure (also see [26], [27], [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' That is,  an optimal control problem is solved repeatedly in the inner  loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' This two-loop iteration is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' All  of these works are model-based and do not consider min-max  or zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The work [33] makes an effort for data-  driven control by estimating model parameters before adopt-  ing the model-based IRL method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Unlike them, without the  need for model identification, our previous studies [34], [35]  propose completely model-free IRL methods that use merely  system data, but not for zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Our work [36]  considers zero-sum games but propose an IRL method using  a two-loop iteration structure and partial system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  This paper considers an expert-learner zero-sum game, that  is, a learner agent suffering from non-cooperative disturbances  with unknown dynamics expects to mimic the behaviors of the  expert agent of optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' As the solution, we propose a  new interaction called zero-sum game IRL, namely a novel  data-driven off-policy IRL algorithm for expert-learner zero-  2  sum games of differential systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' It consists of a game  solution correction modified from the standard RL and a cost  function weight reconstruction using the standard IOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Using  only the behavior data of the expert and learner, a learner  agent learns the unknown cost function objective and the  optimal control policy to mimic the expert’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' This  algorithm does not need to know or identify system models  and performs a single-loop learning procedure without solving  optimal problems repeatedly in inner loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Moreover, no  initial stabilizing policy is needed to start the iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The  properties and effectiveness of the proposed data-driven are  well-analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' ∥·∥ is the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' In is n×n identity ma-  trix, and diag{a,b,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='.} is diagonal matrix with a,b,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='. in diagonal  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' For a vector x = [x1, ··· , xn]T ∈ Rn, ˆx ≜ [x2  1, 2x1x2, ···,  2x1xn, x2  2, ···, 2xn−1xn, x2  n]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' For a matrix A = {aij} ∈ Rm×n,  vec(A) ≜ [a11, a12, ···, a1n, a21, ···, am,n−1, amn]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' IRL PROBLEM FORMULATION  We consider two dynamical agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' A target expert agent  exhibits the demonstrations optimally associated with an ex-  pert performance cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' A learner agent attempts to  determine the unknown cost function objective of the expert  agent and mimic its behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The learner agent only knows  the target agent’s control actions and state behavior but does  not know its performance cost function and system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Target optimal control  Consider a target expert agent  ˙xT = AxT + BuT + DdT,  (1)  where xT ∈ Rn is the target state, uT ∈ Rm is the target input  and dT ∈ Rz is the non-cooperative disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Matrices A, B,  and D have appropriate dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The pair (A,B) is assumed  to be controllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  According to [2], the input of target (1) is uT = −KTxT  that minimizes the following target performance cost function  against dT  VT(xT) =  � ∞  t (xT  T QTxT + uT  TRTuT − γ2  TdT  T dT)dτ,  (2)  where QT = QT  T > 0 and RT = RT  T > 0 are weights, γT > 0 is  attenuation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The input uT and the worst disturbance d∗  T  are given by  uT = −KTxT = −R−1  T BTPTxT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (3a)  d∗  T = LTxT = 1  γ2  T  DTPTxT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (3b)  and form the Nash equilibrium  VT(xT)∗ = min  uT max  dT  � ∞  t  rTdτ = xT  TPTxT  (4)  where rT = xT  TQTxT + uT  TRTuT − γ2  TdT  T dT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' and PT = PT  T > 0  satisfies the target Bellman equation  (AxT + BuT + DdT)TPTxT + xT  TPT(AxT + BuT + DdT)  + xT  TQTxT + uT  TRTuT − γ2  TdT  T dT = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (5)  and the target game algebraic Riccati equation (GARE)  ATPT + PTA+ QT − PTBR−1  T BTPT + 1  γ2  T  PTDDTPT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (6)  The GARE guarantees the uniqueness of stabilizing optimal  control policy (3a) given the performance cost function (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Learner dynamics and IRL control problem  Consider a learner agent to be controlled  ˙x = Ax+ Bu + Dd,  (7)  where x ∈ Rn, u ∈ Rm, d ∈ Rz are the learner’s state, control  input and disturbance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The target cost function VT (2) and the target  control policy in (3a) are unknown to the learner (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' That is,  the weights QT, RT, γT in VT, the optimal strategy KT in (3a),  the worst disturbance d∗  T, LT and GARE solution PT are all  unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The learner knows the target behaviour data  xT, uT and dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Expert-Learner Zero-Sum Game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' By using  the behaviour data of the expert (1) and the learner itself (7),  the learner desires to reconstruct the unknown performance  cost function (2) to exhibit the same control actions uT in  (3a) and states xT as the expert (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  □  The learner (7) will be stabilized and perform the same way  as the target (1) does if KT in (3a) is applied to the learner (7)  with bounded dT = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Hence, our control goal is to determine  the unknown cost function objective (2) to produce the optimal  control input u∗ = −K∗x with K∗ = KT using only target data  of xT,uT,dT and learner data of x,u,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' MODEL-BASED IRL FRAMEWORK  In this section, we develop a novel model-based IRL  framework for learner (7) to determine the cost function (2)  and use this knowledge to compute control input u(t), such  that its behavior trajectories of u(t),x(t) mimic the observed  target trajectories of uT(t),xT(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' In Section 5, we will finally  propose a data-driven IRL algorithm that does not need any  system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Learner optimal control  Let us take a review of the zero-sum game [2] of the learner  (7) with an arbitrary given cost function  V(x) =  � ∞  t (xT Qx+ uTRu − γ2dTd)dτ,  (8)  where Q = QT > 0, R = RT > 0, and γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The optimal input  uo and the worst dw are given by  uo = −Kx = −R−1BTPx,  (9)  dw = Lx = 1  γ2 DTPx,  (10)  3  and (uo,dw) forms the Nash equilibrium  V ∗(x) = min  u max  d  � ∞  t  rdτ = xTPx,  (11)  where r = xTQx+uTRu−γ2dTd, and P = PT > 0 satisfies the  following learner GARE  ATP+ PA+ Q− PBR−1BTP+ 1  γ2 PDDTP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (12)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Expert-learner zero-sum game solution  We now present a theorem to show the conditions that the  solution to the expert-learner zero-sum game must satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' If the cost weights Q,R,γ and the solution P that  satisfy the GARE (12) also satisfy the following equation  (A− BKT)T P+P(A− BKT)  +Q+ 1  γ2 PDDTP+ KT  T RKT = 0,  (13)  then, the corresponding control strategy K in (9) equals the  target strategy KT in (3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Proof: Rewrite (12) with KT in (3a) and K in (9) as  (A− BKT)TP+ P(A− BKT)+ Q+ 1  γ2 PDDTP  + KT  T RK + KTRKT − KTRK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (14)  By subtracting (14) from (13), we have  KT  T RKT − KT  T RK − KTRKT + KTRK  = (KT − K)TR(KT − K) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (15)  Since R > 0, (15) concludes that K = KT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  □  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Learning rules for cost function and game control policy  To find the Q,R,γ,P satisfying Theorem 1, we select γ > 0  and R > 0 and propose an iterative procedure based on  Theorem 1 so as to learn the weight Q, the game control  policy P, and consequently K in (9) and L in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  First, we apply a modified PI to correct P using (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Set  current iteration step as i,i = 0,1,··· , and give the estimates Qi  and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Then the iterative form of (13), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=', (16) in Algorithm  1 is presented below to obtain Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Then optimal control ((9)  and (10)) is used to update the strategy Ki+1 and Li+1 based  on the corrected Pi by (17a) and (17b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Now we must update the cost function weight estimate Qi+1  based on the corrected Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' By IOC [15], taking the iterative  form of GARE (12) yields (18) in Algorithm 1 presented as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' In Algorithm 1, step 4 is the standard IOC  computation based on Lyapunov stability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Note also  that if we set Qi = QT and KT = Ki in (16), then steps 2-3 are  the standard RL PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' As such, Algorithm 1 combines modified  PI with IOC to solve the IRL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Algorithm 1 Model-based IRL algorithm for expert-learner  zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Step 1: Initialize with R > 0 and γ > 0, Q0 > 0, and L0 = 0,  and set i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Step 2: (Game policy correction) Update policy Pi by  (A− BKT)T Pi + Pi(A− BKT)  = −Qi − KT  T RKT − γ2(Li)TLi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (16)  Step 3: (Input update) Update control input and disturbance  gain based on (9) and (10)  ui+1 = −Ki+1x = −R−1BTPix,  (17a)  di+1 = Li+1x = 1  γ2 DTPix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (17b)  Step 4: (Cost function weight construction) Update Qi+1 by  Qi+1 =− ATPi − PiA  + (Ki+1)TRKi+1 − γ2(Li+1)T Li+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (18)  Step 5: Stop if it converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Otherwise, set i = i+1 and repeat  steps 2 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' ANALYSIS OF ALGORITHM 1  The convergence, stability, and optimality of the proposed  Algorithm 1 are analyzed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' It is also shown that Algorithm  1 may not converge to a unique solution (Pi,Qi) even if all  solutions give the correct target strategy Ki = KT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Convergence analysis  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' With an initial Q0 such that 0 < Q0 ≤ ˆQ and  L0 = 0 where ˆQ > 0 is a solution to Theorem 1 associated with  the R > 0 and γ > 0, Algorithm 1 converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' As i → ∞, the  solutions Qi,Pi,Ki,Li converge to Q∗,P∗,K∗,L∗ that satisfy  ATP∗ + P∗A+ Q∗−P∗BR−1BTP∗+ 1  γ2 P∗DDTP∗ = 0,  (19a)  K∗ = KT = R−1BTP∗,  (19b)  L∗ = 1  γ2 DTP∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (19c)  iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The solutions Q∗,P∗,K∗,L∗ satisfy Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Proof: i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Convergence proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' First, we prove that Algorithm  1 solves an increasing sequence Qi for all i = 0,1,···.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Substi-  tuting (17a) for Ki into (18) for Qi yields  ATPi−1 + Pi−1A = (Ki)T RKi − Qi − γ2(Li)TLi,  (20)  which can be rewritten as  (A− BKT)TPi−1 + Pi−1(A− BKT)  = (Ki)TRKi − Qi − γ2(Li)T Li − (Ki)T RKT − KT  T RKi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (21)  Subtracting (16) from (21) gives  (A− BKT)T (Pi−1 − Pi)+ (Pi−1 − Pi)(A− BKT)  = (Ki − KT)TR(Ki − KT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (22)  It follows from (Ki − KT )TR(Ki − KT ) ≥ 0 and Hurwitz  A − BKT that Pi−1 ≤ Pi holds for all iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Each pair of  4  (Pi,Qi+1) satisfies (18), and they uniquely correspond to each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' This is followed by the known fact that Pi−1 ≤ Pi if  Qi ≤ Qi+1 [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Therefore, Qi+1 ≥ Qi > 0 holds for i = 0,1,···,  and Qi+1 = Qi holds if and only if KT = Ki+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Note that  achieving KT = Ki+1 is the goal of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Now we show that Qi is bounded by an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Let  ˆQ > 0 and ˆP > 0 be a group of solution to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' That  is  AT ˆP+ ˆPA+ ˆQ− ˆPBR−1BT ˆP+ 1  γ2 ˆPDDT ˆP = 0,  (23a)  KT = R−1BT ˆP, ˆL = 1  γ2 DT ˆP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (23b)  Rewriting (23a) using (23b) yields  (A− BKT)T ˆP+ ˆP(A− BKT)+ ˆQ+ KT  T RKT + γ2ˆLT ˆL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (24)  If Qi + γ2(Li)TLi ≤ ˆQ+ γ2 ˆLT ˆL holds, then (16) and (24) will  solve 0 < Pi ≤ ˆP with Hurwitz A−BKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' With (3a), (17a) and  (23b), AREs (18) and (23a) can be rewritten as  (A− BKi+1)TPi + Pi(A− BKi+1)  = −Qi+1 − (Ki+1)TRKi+1 − γ2(Li+1)T(Li+1),  (25a)  (A− BKi+1)T ˆP+ ˆP(A− BKi+1)  = KT  T RKT − ˆQ− γ2ˆLT ˆL− (Ki+1)TRKT − KT  T RKi+1,  (25b)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Since (18) ensures Hurwitz A− BKi+1, subtract-  ing (25a) from (25b) and using 0 < Pi ≤ ˆP obtains  (A− BKi+1)T ( ˆP− Pi)+ ( ˆP− Pi)(A− BKi+1)  = (Qi+1 + γ2(Li+1)TLi+1)− ( ˆQ+ γ2 ˆLT ˆL)  + (Ki+1 − KT)TR(Ki+1 − KT) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (26)  Therefore, Qi+1 + γ2(Li+1)T Li+1 ≤ ˆQ+ γ2ˆLT ˆL holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  By deduction, it is inferred that initializing Algorithm 1 with  a Q0 such that 0 < Q0 ≤ ˆQ and L0 = 0, then Qi > 0,i = 0,1,···  will be increasing with an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Therefore, Algorithm  1 converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Converged solutions satisfy (19a), (19b), (19c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Substituting (18) into (16) yields  ATPi + PiA− PiBR−1BTPi  = ATPi+1 + Pi+1A− Pi+1BKT −KT  T BTPi+1 + KT  T RKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (27)  Taking Pi+1 = Pi = P∗ as converged value, (27) becomes  (KT − K∗)R(KT − K∗) = 0,  (28)  where K∗ = R−1BTP∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Since R > 0, (28) implies KT = K∗,  which is exactly (19b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The converged P∗ produces the con-  verged L∗ using (17b) as shown in (19c) and the converged  Q∗ using (18) as shown in (19a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Converged solutions satisfy Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Rewriting (19a) with (19b) yields  (A− BKT)T P∗ + P∗(A− BKT)+ Q∗  + KT  T RKT + 1  γ2 P∗DDTP∗ = 0,  (29)  which is exactly (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Obviously, (19a) is exactly (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' There-  fore, Q∗,P∗,K∗ satisfy Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  □  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Stability and optimality analysis  We now prove the stability of Algorithm 1 in Theorem 3,  and optimality and Nash equilibrium in Theorem 4  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Each iteration of Algorithm 1 exponentially  stabilizes the learner agent (7) with d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Proof: Rewriting (16) with (3a) and (17a) yields  ATPi + PiA+ ˜Qi − PiBR−1BTPi + γ2(Li)TLi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (30)  where ˜Qi = Qi + (Ki+1 − KT)TR(Ki+1 − KT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' With (Ki+1 −  KT)T R(Ki+1 −KT) ≥ 0 and Qi > 0, then ˜Qi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' It is obvious  that Pi solved by (30) or equivalently (16) is a symmetrical  positive definite matrix satisfying  ATPi + PiA− PiBR−1BTPi + γ2(Li)T Li < 0,  (31)  and one has  ˙V i(x,Pi) = xT(A− BKi+1)TPix+ xTPi(A− BKi+1)x  < −xTPiBR−1BTPix− γ2xT(Li)TLix < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (32)  That is, Pi yields stabilizing Ki+1 by (17a) for learner (7) with  d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Considering (17a) and (31), Qi+1 in (18) satisfies  Qi+1 + γ2(Li+1)T Li+1  = −(ATPi + PiA− PiBR−1BTPi) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (33)  Using Qi+1 +γ2(Li+1)TLi+1 > 0 in (16) would still make (32)  hold for the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Therefore, provided Q0 > 0 (32)  will hold for all i = 0,1,···.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  □  Before optimality analysis of Algorithm 1, we now give a  lemma of importance which extends the idea of classic IOC  [15] to two-player zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Consider the two-player learner agent (7) with  x(t0) = x0, t ≥ t0 and (8) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Assume there exists a  positive definite symmetric matrix P ∈ Rn×n such that  ATP+ PA− PBR−1BTP+ 1  γ2 PDDTP < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (34)  Then, with the optimal feedback control input uo and the worst  disturbance dw such that  uo = −R−1BTPx, dw = 1  γ2 DTPx,  (35)  and the cost function weight  Q = −(ATP+ PA− PBR−1BTP+ 1  γ2 PDDTP),  (36)  the saddle point (uo,dw) makes the cost value function (8)  reach the Nash equilibrium  V(x0,uo,d) ≤ V(x0,uo,dw) ≤ V(x0,u,dw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (37)  Proof: First, V(x) in (8) can be represented with P > 0 as  V(x) = xTPx ≥ 0, V(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (38)  It follows from (35) and (36) that H(x,uo,dw) = 0 [2] where  the Hamiltonian is  H(x,u,d) =xTQx+ uTRu − γ2dTd + (Ax+ Bu + Dd)TPx  + xTP(Ax+ Bu + Dd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (39)  5  One writes  H(x,u,d) = H(x,u,d)− H(x,uo,dw)  = (u − uo)R(u − uo)− γ2(d − dw)T (d − dw),  (40)  and hence  H(x0,uo,d) ≤ H(x0,uo,dw) ≤ H(x0,u,dw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (41)  Based on [2], we obtain the conclusion (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  □  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The converged solutions Q∗,P∗,K∗,L∗ obtained  by Algorithm 1 shown in Theorem 2 yield Nash equilibrium  of the value function V(x) in (8) such that  V(x0,u∗,d) ≤ V(x0,u∗,d∗) ≤ V(x0,u,d∗),  (42)  where u∗ = −K∗x and d∗ = L∗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Proof: It follows from Theorem 1 that Qi > 0 holds for all  i = 0,1,···.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Thus one has the converged Q∗ > 0 and  ATP∗ + P∗A− P∗BR−1BTP∗ + 1  γ2 (P∗)TDDTP∗ < 0,  (43)  from (19a), which means that the converged P∗ satisfies (34)  in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Also, the converged control strategy K∗ in (19b)  and disturbance gain L∗ in (19c) satisfy (35) in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' This  indicates that (41) also holds for u∗ and d∗, namely the Nash  equilibrium (42) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  □  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Non-uniqueness of solution  In fact, the Q∗,R,γ,P∗ satisfying (19a)-(19c) that explain  the same strategy K∗ = KT may not be unique and can be  different from the actual target values QT,RT,γT,PT shown  in (3b) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' This multi-solution phenomenon is known  as the ill-posedness property, which is well-analyzed for DT  ARE in [38] and coupled ARE in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' In the next result,  we characterize the relationship between QT,RT,γT ,PT, and  Q∗,R,γ,P∗ for CT GARE and show the conditions for the  occurrence of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Recall QT, RT, γT, PT satisfying (6) and (3b),  and let Qo, Ro, Po satisfy  BTPo = RoR−1  T BTPT,  (44a)  Qo + ATPo + PoA− KT  T RoKT + 1  γ2  T  PTDDTPT − 1  γ2 P∗DDTP∗  = 0,  (44b)  where Ro = RT −R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Then any Q∗ = QT −Qo and P∗ = PT −Po  satisfy (19a)-(19c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Proof: Subtracting (44b) from (6) and using (3b) yields  AT(PT − Po)+ (PT − Po)A+ (QT − Qo)  − KT  T (RT − Ro)KT + 1  γ2 P∗DDTP∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (45)  Using P∗ = PT − Po, Ro = RT − R, and R > 0 in (44a) gives  K∗ = R−1BTP∗ = R−1  T BTPT = KT,  (46)  which is (19b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Substituting it into (45) yields (19a) and (19c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  This proves the relationship between the obtained solution  Q∗,R,γ,P∗ and the expert’s QT,RT,γT ,PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' We observe that  Po, Ro, Qo satisfying (44a) and (44b) can be nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' That  is, Q∗,R,γ,P∗ associate optimally with the same strategy as  QT,RT,γT,PT, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=', KT = K∗, but QT ̸= Q∗, RT ̸= R, γT ̸= γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Therefore, there could be multiple solutions to (19a) to gen-  erate a K∗ in (19b) equal to the target KT in (3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  □  The following corollary shows a special case of the Q∗,R,γ  of Theorem 5 which gives V(x)∗ = cVT(xT)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' With scalar c > 0, any Q∗ = cQT, R = cRT, γ =  √cγT would yield V(x)∗ = cVT(xT)∗ in (4) and (11), and they  optimally associate with the same K∗ as the expert such that  K∗ = KT in (19b) and (3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Proof: Bring such Q∗,R,γ into (19a), since QT,RT,γT satisfy  (6), then one has P∗ = cPT and K∗ = KT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Then, using this  result in (11) for V(x)∗ and comparing it with VT(xT)∗ in (4)  shows that V(x)∗ = cVT(xT)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  □  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' DATA-DRIVEN OFF-POLICY IRL ALGORITHM  Algorithm 1 relies on the system dynamics A,B,D and the  target strategy KT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' To remove this requirement, we develop  here a data-driven IRL algorithm for expert-learner zero-sum  games based on Algorithm 1, which only requires the data  xT,uT,dT of the target agent (1) and x,u,d of the learner  agent (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' To accomplish this, we use two techniques similar  to integral RL [6], [8] and off-policy RL [6], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The end  result is Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Data-driven game policy correction  In order to update Pi, Ki+1 and Li+1 in (16)-(17b) using  only target data xT,uT,dT, inspired by the idea of off-policy  integral RL technique [6], [40], rewrite (1) as  ˙xT = AxT − BKixT + DdT + B(uT + KixT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (47)  Using (47) and (16) one writes  ˙xT  T PixT + xT  TPi ˙xT  = (AxT − BKixT)T PixT + xT  TPi(AxT − BKixT)  + 2(uT + KixT)TBTPixT + 2dT  T DTPixT  = −xT  TQixT − xT  TKT  T RKTxT − γ2xT  T(Li)T LixT  + 2xT  TKT  T BTPixT + 2uT  TBTPixT + 2dT  T DTPixT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (48)  Using (17a), (17b), (3a) in (48) and integrating both sides from  t to t + T, where T > 0 is the integral time period, obtains  (50) in Algorithm 2 to be presented, by which Pi, Ki+1 and  Li+1 are updated simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Similar to [6], probing noise  e is added to uT, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=', uT = −KTxT + e, for the persistence  of excitation condition in merely learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' It is not  needed anymore when solutions converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Unlike (16)-(17b),  (50) does not need the knowledge of agent dynamics A,B,D  or the strategy KT in (3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  6  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Data-driven cost function weight reconstruction  In order to update Qi+1 in (18) using only data x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  inspired by the integral RL technique [8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' multiplying both  sides of (18) by x and adding and subtracting terms uTBTPix  and dTDTPix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' (18) can be rewritten as  xTQi+1x =− [(Ax+ Bu + Dd)TPix+ xTPi(Ax+ Bu + Dd)  − xT(Ki+1)TRKi+1x+ γ2xT(Li+1)T Li+1x  − 2uTBTPix− 2dTDTPix],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (49)  where u can be generated by any stabilizing policy and d  can be random and different from dT in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Substituting (7), (17a) and (17b) into (49) and integrating it  gives (51) in Algorithm 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Using Pi, Ki+1 and Li+1  obtained by (50), (51) equivalently replaces (18) in Algorithm  1 to calculate Qi+1 without knowing any system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Algorithm 2 Data-driven off-policy IRL algorithm for expert-  learner zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Step 1: Initialize with R > 0, γ > 0, Q0 > 0, and L0 = 0,  and collect system data generated by any stabilizing  control input u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Set i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Step 2: (Game policy correction) Update policy Pi, control  strategy Ki+1 and disturbance gain Li+1 by  xT(t + T)TPixT(t + T)− xT(t)TPixT(t)  − 2  � t+T  t  eTRKi+1xTdτ − 2γ2  � t+T  t  dT  T Li+1xTdτ  = −  � t+T  t  �  xT  TQixT + (uT − e)TR(uT − e)  + γ2xT  T (Li)TLixT  �  dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (50)  Step 3: (Cost function weight construction) Update cost  function weight Qi+1 by  � t+T  t  xT Qi+1xdτ  = −[x(t + T)TPix(t + T)− x(t)TPix(t)  −  � t+T  t  (2uTRKi+1x+ xT(Ki+1)TRKi+1x)dτ  −γ2  � t+T  t  (2dTLi+1x− xT(Li+1)TLi+1x)dτ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (51)  Step 4: Stop if it converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Otherwise, set i = i+1 and repeat  steps 2 to 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Algorithm 2 does not need system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Moreover, it iterates in single loop indicated by i, no inner-  loop iteration is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Implementation and Analysis of Algorithm 2  In order to show how to implement data-driven IRL Al-  gorithm 2 using only data, first, consider Kronecker product  aTWb = (bT ⊗ aT)vec(W) for (50) and define the following  operators,  OxT xT = [x2  T1,2xT1xT2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',x2  T2,2xT2xT3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',x2  Tn]T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  dxT xT = [OxT xT (t + T)− OxT xT (t),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',  OxT xT (t + lT)− OxT xT (t + (l − 1)T)]T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  IxT xT = [  � t+T  t  OxT xT dτ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',  � t+lT  t+(l−1)T OxT xT dτ]T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  IxT uT = [  � t+T  t  xT ⊗ uTdτ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',  � t+lT  t+(l−1)T xT ⊗ uTdτ]T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  IxT dT = [  � t+T  t  xT ⊗ dTdτ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',  � t+lT  t+(l−1)T xT ⊗ dTdτ]T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  IxT e = [  � t+T  t  xT ⊗ edτ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',  � t+lT  t+(l−1)T xT ⊗ edτ]T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Φp = [dxT xT ,−2IxT e(In ⊗ R),−2γ2IxT dT ]T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  ri = xT  TQixT + (uT − e)TR(uT − e)+ γ2xT  T(Li)T LixT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Ψi  p = −[  � t+T  t  ridτ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',  � t+lT  t+(l−1)T ridτ]T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  ˆPi = [Pi  11,Pi  12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',Pi  22,Pi  23,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',Pi  nn]T,  (52)  where l is the group number of collected data and should be  l ≥ n(n+1)  2  + nm + nz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Using batch least squares method [8],  [40], [41], ˆPi, Ki+1, Li+1 can be calculated by  �  ( ˆPi)T ,vec(Ki+1)T,vec(Li+1)T �T=(ΦT  pΦp)−1ΦT  pΨi  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (53)  Similarly, for (51), we define  Oxx = [x2  1,2x1x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',x2  2,2x2x3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',x2  n]T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  dxx = [Oxx(t + T)− Oxx(t),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',  Oxx(t + kT)− Oxx(t + (k − 1)T)]T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  ˆqi = [Qi  11,Qi  12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',Qi  22,Qi  23,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',Qi  nn]T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Φq = Ixx = [  � t+T  t  Oxxdτ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',  � t+kT  t+(k−1)T Oxxdτ]T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Ii+1  q  (t) =  � t+T  t  (2uTRKi+1x+ xT(Ki+1)TRKi+1x)dτ  + γ2  � t+T  t  (2dTLi+1x− xT(Li+1)TLi+1x)dτ  Ψi+1  q  = −dxx ˆPi + [Ii+1  q  (t),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',Ii+1  q  (t + (k − 1)T)]T ,  (54)  where k is the group number of collected data and should be  k ≥ n(n+1)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Then, Qi+1 can be uniquely solved by  ˆqi+1 = (ΦT  q Φq)−1ΦT  q Ψi+1  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (55)  By using (53) and (55), we solve Pi, Qi+1, Ki+1, Li+1 in a  data-driven mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Each step of Algorithm 1 yields a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Algo-  rithm 2 is developed based on Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' To illustrate that  the solution obtained by Algorithm 2 estimates the solution  obtained by Algorithm 1, we show that equations (53) and  (55) yield unique solutions in the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  7  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' If there exist lo > 0, ko > 0, for all l ≥ lo and  k ≥ ko,  rank([IxT xT ,IxT uT ,IxT dT ]) = n(n + 1)  2  + nm+ nz,  (56a)  rank(Ixx) = n(n + 1)  2  ,  (56b)  then, (53) and (55) solve unique solution, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Proof: First, we show that (53) solves unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' This  is aiming to show that  ΦpΩ = 0  (57)  has only the trivial solution Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Now, we prove Ω = 0  by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Assume Ω = [XT  v ,Y T  v ,ZT  v ]T ∈ R  n(n+1)  2  +nm+nz  is a nonzero solution of (57), where Xv ∈ R  n(n+1)  2  , Yv ∈ Rmn,  Zv ∈ Rnz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Then, Xv,Yv,Zv uniquely determine matrices X,Y,Z  by Xv = ˆX, Yv = vec(Y) and Zv = vec(Z), respectively, where  X is a symmetrical matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Define  Inx = [  � t+T  t  xT ⊗ xTdτ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=',  � t+kT  t+(k−1)T xT ⊗ xTdτ]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (58)  Integrating (48) from t to t + T gives  ΦpΩ = Inxvec(E)+ 2IxT uT vec(G)+ 2IxTdT vec(F)= 0, (59)  where  E = ATX + XA− KT  T RY −YT RKT,  (60a)  G = BTX − RY,  (60b)  F = DTX − γ2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (60c)  Since E is a symmetrical matrix, one has Inxvec(E) = IxT xT ˆE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Using this in (59) yields  ΦpΩ = [IxT xT ,2IxT uT ,2IxT dT ]  \uf8ee  \uf8f0  ˆE  vec(G)  vec(F)  \uf8f9  \uf8fb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (61)  Under (56a), we know that [Ixx,Ixu,Ixd] has full column rank,  and thus (61) has only the solution ˆE = 0, vec(G) = 0 and  vec(F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' That is,  ATX + XA− KT  T RY −Y TRKT = 0,  (62a)  BTX = RY,  (62b)  DTX = γ2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (62c)  Since A − BKT is Hurwitz, substituting (62b) and (62c) into  (62a) gives X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' This implies that Y = 0 and Z = 0 due to  R > 0 and γ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' In summary, we have Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' However, this  conflicts with the assumption that Ω is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Therefore, it  concludes that under (56a), (55) solves unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Second, from the integral RL work [8], we conclude that  when (56b) holds for (51), Qi+1 in (51) can be uniquely  determined by (55) with collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  □  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' SIMULATION  We show three simulation experiments, a first one of the  data-driven Algorithm 2 to show its performance, a com-  parison simulation with the bilevel IRL method in [36] to  show the reduction of iteration steps of Algorithm 2, and  a second comparison simulation with the RL-based optimal  tracking control method in [6] to show the improvement of  control performance with the cost function weights correction  of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Simulation result of Algorithm 2  The system dynamics information of the target (1) and the  learner (7) for simulation is  A =  �  −1  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='7  �  ,B =  �  0  3  �  ,D =  �  1  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (63)  For the target agent (1), the actual target cost function objec-  tive (2) consists of QT = diag{8,12},RT = 2I1,γT = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The  disturbance is dT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='003rand(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The expert’s LT, KT and PT  are  KT = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='9869,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='5779],LT = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='4162,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='1472],  PT =  � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='7459  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='3246  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='3246  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='3853  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (64)  For the learner, the behaviour strategy to generate data  is Kb = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='2129,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='2812], and the disturbance is d =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='003rand(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' To start Algorithm 2, the initial weights for cost  function, initial disturbance gain L0, and integral time period  T are given by  Q0 = diag{1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='5},R = I1,γ = 40,L0 = [0, 0],T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='008s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (65)  The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' 1(a) captures the iteration process from the initial  spot to the spot on ∥Ki+1−KT∥ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The final values of Ki,  Qi, Li, and Pi are K∗, Q∗, L∗, and P∗, respectively, as follows  K∗ =  � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='9827  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='5839 �  ,  Q∗ =  �  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='2796  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6670  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6670  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='0151  �  ,  L∗ = 10−3 ×  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='4021  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='4131  �  ,  P∗ =  � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6441  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6622  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6622  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='1968  �  ,  (66)  where K∗ closely approximates the target KT in (64) with  ∥K∗ − KT∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='0073, while Q∗, L∗ and P∗ are not equal to  QT,LT and PT in (64), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' This is the multiple-  solution phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' 1(b), the learner’s state x can  mimic the trajectories of the target xT very well under the  learned K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Therefore, the proposed Algorithm 2 can learn an  appropriate cost function and optimal policy for the learner to  mimic the target trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Comparison simulation case 1  This subsection shows the simulation results of the bilevel  IRL method in [36] that iterates in two-loop to show the  reduction of computational complexity in terms of iteration  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The same expert-learner system, initial parameters for  8  0  10  20  30  40  50  0  1  2  ||Ki − KT||  0  10  20  30  40  50  9  10  11  12  ||Qi − QT||  0  10  20  30  40  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='44095  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='441  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='44105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='4411  ||Li − LT||  0  10  20  30  40  50  Update steps i  (a)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6  ||P i − PT||  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='5  2  Time(s)  (b)  20  10  0  10  20  x1  x2  xT1  xT2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Convergence and imitation performance using Algorithm 2  Algorithm 2 in (63)-(65) are used for this comparison method  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' 2 also captures the iteration process from the initial  spot to the spot on ∥Ki+1 −KT∥ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='01 as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' 1 to show the  difference in iteration steps of the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The inner-  loop iteration figures are omitted since they are too many to  put here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' 2, the final values of K j,Qj,Lj, and P j of  the outer-loop iterations are  K∗ =  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='9822  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='5691  �  ,  Q∗ =  � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='3186  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6974  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6974  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='0506  �  ,  L∗ = 10−3 ×  � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='4053  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='4130 �  ,  P∗ =  � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6486  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6607  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6607  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='1897  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  (67)  where K∗ approximates the target KT in (64) with ∥K∗−KT∥ =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Table I shows that the total iteration steps of the method  is 3370, including 587 outer-loop updates (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' 2) and  2783 inner-loop updates, while Algorithm 2 iterates 51 steps  in total (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The time of the learning process of  Algorithm 2 is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='08s, while that of the comparison method 1 is  169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='936s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' It is proportional to the amount of utilized collected  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Algorithm 2 uses 510 groups of data, and comparison  method 1 uses 21242 groups of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Therefore, Algorithm 2  costs much fewer data and time than the bilevel comparison  method 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  0  100  200  300  400  500  600  0  1  2  ||Kj − KT||  0  100  200  300  400  500  600  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='5  9  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='5  10  ||Qj − QT||  0  100  200  300  400  500  600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='441  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='4411  ||Lj − LT||  0  100  200  300  400  500  600  Outer-loop update steps j  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6  ||P j − PT||  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Convergence of K j,Qj,Lj, and Pj using the comparison method 1  TABLE I  ITERATION STEPS AND LEARNING TIME OF ALGORITHM 2 AND THE  COMPARISON METHOD 1  Methods  Total updates  Learning time  Algorithm 2  51  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='08s  Comparison method 1  3370  169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='936s  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Comparison simulation case 2  In this subsection, the typical RL-based optimal tracking  control method [6] for linear disturbed systems, which com-  putes optimal control policy given cost function weights, is  simulated to show the advantage of Algorithm 2 in control  performance by computing both optimal control policy and  cost function weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  The same system and cost weights shown in (63)-(65) and  discount factor α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='9 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The obtained optimal control  law is u∗ = −[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='1760 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='9139 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='0044 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='7639][xT,rT]T, and the  corresponding imitation performance in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' 3 is not as good  as that of Algorithm 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' By evenly sampling the  trajectory data, the imitation performance of the two methods  is quantified by the error index defined as follows  Te = 1  n  n  ∑  i=1  �  1  a  a  ∑  k=1  |xi(kT)− xTi(kT)|2  where n = 2, a = 250, and T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='008s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' As shown in Table II,  Te of Algorithm 2 is much smaller than that of the comparison  method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The reason is that Algorithm 2 can correct the  given cost function weights when it is inappropriate, but the  comparison method 2 cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Algorithm 2 thus obtains much  better imitation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='  TABLE II  IMITATION INDEX OF ALGORITHM 2 AND THE COMPARISON METHOD 2  Methods  Error index Te  Algorithm 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='0162  Comparison method 2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='4461  9  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content='8  2  20  10  0  10  20  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' Imitation performance of learner’s x to target xT using the comparison  method 2  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' CONCLUSION  This paper proposes a novel data-driven off-policy IRL  approach to determine both cost function and optimal con-  trol policy to stabilize a learner agent suffering from non-  cooperative disturbances by mimicking a target agent’s tra-  jectories using data of both agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The proposed approach  does not need any system dynamics and guarantees stability,  Nash optimality, and imitation performance with single-loop  iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
+page_content=' The rigorous theoretical proofs and simulation ex-  periments verify its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9A0T4oBgHgl3EQfC_9-/content/2301.01997v1.pdf'}
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diff --git a/PdE2T4oBgHgl3EQfrgiA/content/tmp_files/2301.04050v1.pdf.txt b/PdE2T4oBgHgl3EQfrgiA/content/tmp_files/2301.04050v1.pdf.txt
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+arXiv:2301.04050v1  [cs.RO]  10 Jan 2023
+1
+Design, Modeling and Control of a Quadruped Robot SPIDAR:
+Spherically Vectorable and Distributed Rotors Assisted
+Air-Ground Amphibious Quadruped Robot
+Moju Zhao1, Tomoki Anzai2, Takuzumi Nishio2
+Abstract—Multimodal locomotion capability is an emerging
+topic in robotics field, and various novel mobile robots have
+been developed to enable the maneuvering in both terrestrial
+and aerial domains. Among these hybrid robots, several state-of-
+the-art bipedal robots enable the complex walking motion which
+is interlaced with flying. These robots are also desired to have the
+manipulation ability; however, it is difficult for the current forms
+to keep stability with the joint motion in midair due to the cen-
+tralized rotor arrangement. Therefore, in this work, we develop
+a novel air-ground amphibious quadruped robot called SPIDAR
+which is assisted by spherically vectorable rotors distributed in
+each link to enable both walking motion and transformable flight.
+First, we present a unique mechanical design for quadruped robot
+that enables terrestrial and aerial locomotion. We then reveal the
+modeling method for this hybrid robot platform, and further
+develop an integrated control strategy for both walking and
+flying with joint motion. Finally, we demonstrate the feasibility of
+the proposed hybrid quadruped robot by performing a seamless
+motion that involves static walking and subsequent flight. To
+the best of our knowledge, this work is the first to achieve a
+quadruped robot with multimodal locomotion capability, which
+also shows the potential of manipulation in multiple domains.
+Index Terms—Legged Robots; Aerial Systems: Mechanics and
+Control; Motion Control
+I. INTRODUCTION
+During the last decade, robots with multimodal locomotion
+capability have undergone intensive development, and demon-
+strated the versatile maneuvering in multiple domains (i.e.,
+terrestrial, aerial, and aquatic) [1]–[5] which can benefit the
+performance in various situations, such as disaster response
+and industrial surveillance. Among various forms for mul-
+timodal locomotion, the legged type has the advantage in
+unstructured terrains, and several bipedal models have been
+developed to demonstrate the promising walking that is inter-
+laced with flying motion [6], [7]. The legged robots are desired
+to provide not only the advanced locomotion capability, but
+also the manipulation ability by the limb end-effector. Then,
+multilegged (multilimbed) is considered effective from the
+aspect of both the stability in the terrestrial locomotion and
+the freedom in manipulation. In spite of the achievement
+of a simple arm motion in midair by the flying humanoid
+robot in [7], the centralized rotor arrangement in most of the
+existing robot platforms can hardly handle the large change of
+Center-of-Gravity (CoG) caused by the joint motion in midair.
+Besides, the external force at the limb end during the aerial
+manipulation is also difficult to compensate by the centralized
+1 Department of Mechanical Engineering, The University of Tokyo, 2
+Department of Mechano-Infomatics, The University of Tokyo, 7-3-1 Hongo,
+Bunkyo-ku, Tokyo 113-8656, Japan. chou@hnl.t.u-tokyo.ac.jp
+Digital Object Identifier (DOI): see top of this page.
+(A)
+(B)
+Fig. 1.
+Air-ground amphibious quadruped robot SPIDAR: SPherIcally
+vectorable and Distributed rotors assisted Air-ground amphibious quadruped
+Robot. (A) walk on the ground like an usual quadruped robot, but with the
+assistance of thrust force; (B) fly and transform in midair.
+rotors due to the large moment arm. Hence, a distributed rotor
+arrangement is necessary for aerial manipulation. Then, we
+propose a novel quadruped robot platform in this work, which
+can both walk and fly by using the spherically vectorable rotors
+distributed in each link unit as illustrated in Fig. 1.
+One of the efficient type of air-ground hybrid robot plat-
+forms equips an ordinary multirotor for flying, and then de-
+ploys rolling cage or wheels to perform terrestrial locomotion
+[2], [8], [9]. Although the rolling mechanism can achieve the
+stable terrestrial locomotion without any complex control, it is
+relatively difficult for this type to handle the unstructured ter-
+rains (e.g., very few foothold). Then, legged model is proposed
+in several researches [6], [7], [10]–[12] to solve this issue by
+the walking motion. Most of these robots are bipedal model
+which attaches the multirotor unit in their torso for aerial
+maneuvering. Compared with the bipedal model, it is relatively
+easier to achieve the walking stability by the multilegged
+model because of the larger support polygon. Besides, the legs
+can be considered as the limbs for manipulation in midair, and
+thus more limbs can provide more end-effectors for complex
+manipulation task. Therefore, we choose the quadruped model
+that can offer both the stable terrestrial locomotion and the
+potential of aerial manipulation.
+For most of the quadruped robot, the form is bio-inspired
+and specialized for terrestrial locomotion [13]–[15]. However,
+in our work, the robot is also desired to perform manipulation
+task in multiple domains. Several ape-like quadruped robots
+show the manipulation ability by the hands attached on the leg
+ends [16], [17]. The point symmetry is the crucial difference of
+these robots from the skeleton design for common quadruped
+robots. In our work, we also adopt the point symmetry for
+our hybrid quadruped robot to enable the omni-directional
+maneuvering and manipulation in both terrestrial and aerial
+
+2
+domains. Regarding the rotor arrangement, it is difficult for
+rotors centralized in the robot torso like [6] to handle the
+change of CoG caused by the joint motion. Besides, the
+external force acted at the end-effectors can also induce a
+large rotational load for the centralized rotors due to the
+large moment arm. To ensure a sufficient control margin for
+the stable joint motion in midair, [11] proposes a distributed
+rotor arrangement that deploys the thrust units at each limb
+end. However, this rotor arrangement deprives the robot of
+manipulation ability. Therefore, a fully-distributed rotor design
+proposed by [18] is applied in this work. In this design,
+spherically vectorable rotor apparatus is embedded in each link
+and thus can generate individual three-dimensional thrust force
+for the promising maneuvering and manipulation in midair as
+presented in [19].
+For the stable flying motion, the whole platform should be
+lightweight, which leads a relatively compact and weak joint
+actuator for legs. Hence, the thrust force is also required to
+assist the walking motion by reducing the load from gravity.
+For model with centralized rotor arrangement, both the model-
+based [6] and the policy-based [20] methods are developed to
+obtain the assistive thrust. However, these methods are not
+suitable for the model with rotors distributed in all links.
+Then, [21] proposes an assistive thrust control method for a
+bipedal robot with the vectorable rotor attached at each foot,
+whereas [19] presents a control method for a fully-distributed
+rotor model in the aerial domains. Based on these methods, a
+comprehensive investigation of the modeling and control for
+the multilegged model is performed in this work to handle
+multiple types of torques and forces (i.e., the joint torque, the
+contact force on each limb end, the gravity of each link, and
+the thrust force from each vectorable rotor).
+The main contributions of this work can be summarized as
+follows:
+1) We propose a unique mechanical design for air-ground
+quadruped robot with the spherically vectorable rotors
+distributed in all links.
+2) We present a modeling and control methods for this
+multilegged platform for hybrid locomotion in both
+terrestrial and aerial domains.
+3) We achieve the seamless and stable motion that involves
+walking, flying and joint motion in midair by the proto-
+type of quadruped robot.
+The remainder of this paper is organized as follows. The
+mechanical design for this unique quadruped robot is intro-
+duced in Sec. II. The modeling of our robot is presented in
+Sec. III, followed by the integrated control method for hybrid
+locomotion in Sec. IV. We then show the experimental results
+in Sec. V before concluding in Sec. VI.
+II. DESIGN
+In this section, we present the mechanical design for the
+quadruped robot that is capable of terrestrial/aerial hybrid
+locomotion. The key of the whole structure is the spherically
+vectorable rotor embedded in each link unit, along with the
+unique quadruped shape that differs from the common bio-
+inspired type.
+Fig. 2. Mechanical design for the air-ground hybrid quadruped robot.
+(A) skeleton model for common bio-inspired mammal-type quadruped robot.
+(B) proposed point-symmetric skeleton model for hybrid quadruped model.
+(C1)/(C2) two-DoF joint module for each limb, where the yaw axis qi yaw
+comes first followed by the pitch axis qi pitch. (D) spherically vectorable
+rotor apparatus with two vectoring angles (φ, θ), and a combined thrust force
+λi from the counter rotating dual rotors. There is a small offset between two
+vectoring axes because two rods cannot intersect with each other.
+A. Skeleton Model
+As shown in Fig. 2(A), the common skeleton for quadruped
+robot is mammal-type, which puts a priority on the forward
+motion. Thus, the model is plane symmetric, and each leg has
+three DoF (two in the hip, and one in knee). However, our
+robot is desired to enable not only the terrestrial/aerial hybrid
+locomotion, but also the manipulation in midair. Therefore, the
+omni-directional movement is a critical feature for the skeleton
+design. Then, a point symmetric structure is introduced as
+depicted in Fig. 2(B). This is similar to the sprawling-type
+quadruped design proposed by [22], which can provide s wider
+supporting polygon and also a lower CoG than the mammal-
+type. According to this design concept, each limb consists
+of two links that have the same length, and is connected to
+the center torso with a joint module that has two Degree-of-
+Freedom (DoF). For this joint module, the yaw axis qi yaw
+comes first, which is followed by the pitch axis qi pitch
+to allow a larger swinging range for walking as shown in
+Fig. 2(C1)/(C2). The crucial difference from the ordinary
+sprawling-type is that we also introduce an identical two-
+DoF joint module to connect neighboring links, which can
+provide a four-DoF manipulation capability by each limb end
+without the help of the torso motion. Eventually, this robot
+is composed of 8 links with 16 joints for walking and flying.
+Given that the lightweight design is significantly important for
+the flight performance, we deploy a compact servo motor to
+individually actuate each joint at the expense of the torque
+power. Nevertheless, the shortage of the joint torque can be
+compensated by the rotor thrust in our robot.
+B. Spherically Vectorable Rotor
+Rotors embedded in links are used to achieve flight with
+arbitrary joint motion in midair. Besides, it is also necessary
+
+(B)
+(A)
+qi+1_yaw
+i+1_pitch
+yaw
+qi_pitch
+(D)
+C1 top view
+(C2) side view3
+to use the rotor thrust to assist leg lifts for walking, because
+the joint actuator is weak due to the lightweight design as
+mentioned in Sec. II-A. Therefore, the rotor is required to
+point arbitrary direction to handle the change in link ori-
+entation. In other words, it is required to generate a three-
+dimensional thrust force by each rotor module to interact
+with not only the gravity and also the external force (i.e.,
+the contact force on each foot). Then a spherically vectorable
+apparatus proposed by [18] is equipped in each link as depicted
+in Fig. 2(D). To achieve the spherical vectoring around the
+link unit, two rotation axes is necessary. We first introduce
+a roll axis φi around the link rod. Then, we need the second
+orthogonal vectoring axis. If we use a single rotor, the collision
+between the propeller and the link rod will be inevitable
+while performing the second vectoring. Therefore, we apply
+the counter-rotating dual-rotor module to avoid the collision
+as shown in Fig. 2(D). In addition, this dual-rotor can also
+counteract the drag moment and gyroscopic moment. Then,
+we define the pitch axis across dual rotors as the second
+vectoring axis θi. Each vectoring axis is also actuated by an
+individual compact servo motor. Regarding the thrust force,
+since we assume that a pair of rotors rotate with the same
+speed, it is possible to introduced a combined thrust λi for
+each spherically vectorable rotor module. Eventually, there are
+three control input (two vectoring angles φi, θi, and one thrust
+λi) for each vectorable rotor module, and totally 8 sets are used
+to control the whole quadruped model.
+III. MODELING
+In this section, we describe the modeling for this robot
+which can be divided into two parts: the thrust model and
+the whole dynamics model.
+A. Spherically Vectorable Thrust
+Based on the kinematic model depicted in Fig. 3, the three-
+dimensional force fi generated by the i-th rotor module can
+contact
+contact
+contact
+free
+Υ
+Œ•>Ú
+“••
+Œ‰•
+Ε
+Ε>Ú
+Ε>Û
+Ε>Ü
+“•>Ú•
+<(Ü=
+<%K)=
+<.Ü=
+Fig. 3.
+Dynamics model of the proposed quadruped robot. The entire
+dynamics involves the joint torque τj, the contact force at each limb end fci,
+the gravity of each link mig, and the thrust force from each vectorable rotor
+fi. {Li} and {Fi} denote the frame for the i-th link and rotor respectively,
+whereas {CoG} is the CoG frame for the whole model. For the free leg
+during walking, the contact force fci at the limb end disappears.
+be written as:
+fi = λiui,
+(1)
+ui = CoGRFi(q, φi, θi)
+�0
+0
+1�T ,
+(2)
+where CoGRFi denotes a rotation matrix of the rotor frame
+{Fi} w.r.t. the frame {CoG}. For this robot, we define the
+frame {CoG} to have an origin at the CoG point as depicted in
+Fig. 3, and a xyz coordinate that is identical to the baselink at
+the center torso. ui denotes the unit vector for the spherically
+vectorable mechanism that is effected by two vectoring angles
+φi and θi. Besides, this vector also depends on the joint angles
+q ∈ RNJ , NJ is the number of joints.
+Then the total wrench in the frame {CoG} can be given by
+�
+fλ
+τλ
+�
+=
+
+
+Nr
+�
+i=1
+fi
+Nr
+�
+i=1
+pi × fi
+
+
+= Qλ,
+(3)
+Q =
+�
+u1
+u2
+· · ·
+uNr
+p1 × u1
+p2 × u2
+· · ·
+pNr × uNr
+�
+,
+(4)
+λ =
+�λ1
+λ2
+· · ·
+λNr
+�T ,
+where pi is the position of the frame {Fi} origin from the
+frame {CoG} that is influenced by the joint angles q and the
+first vectoring angle φi for the i-th rotor. Nr is the number of
+rotors.
+B. Dynamics of Multilinked Model
+The whole dynamic model can be written as follows:
+˙PΣ =Rcfλ − mΣg +
+Nc
+�
+i=1
+fci,
+(5)
+˙LΣ =τλ +
+Nc
+�
+i=1
+pci × RT
+c fci,
+(6)
+MJ(q)¨q + c(q, ˙q) =τq +
+Nc
+�
+i=1
+JT
+cifci
++
+Nr
+�
+i=1
+JT
+rifi +
+Ns
+�
+i=1
+JT
+simsig.
+(7)
+(5) and (6) denote the centroidal dynamics for the whole
+multibody model. PΣ and LΣ are the entire linear and
+rotational momentum described in the inertial frame {W}
+and the frame {CoG}, respectively. These momentum are
+both affected by the joint angles, vectoring angles, and their
+velocities (i.e., q, ˙q, φ, ˙φ, θ, ˙θ). Rc is the orientatin of the
+frame {CoG} w.r.t. the frame {W}, and is identical to Rb
+that is the orientatin of the baselink. fλ and τλ corresponds
+to the total wrench described in (3). fci is the contact force at
+the i-th limb end (foot) w.r.t. the frame {W}, whereas pci is
+the position of this contact point from the frame {CoG} which
+is also influenced by the joint angles q. Nc is the number of
+contact points (i.e., standing legs). mΣ is the total mass, and
+g is a three-dimensional vector expressing gravity.
+
+4
+(7) corresponds to the joint motion. MJ(q) denotes the
+inertial matrix, whereas c(q, ˙q) is the term related to the
+centrifugal and Coriolis forces in joint motion. J∗i ∈ R3×NJ
+is the Jacobian matrix for the frame of the i-th contact point
+(∗ → c), the i-th rotor (∗ → r), and the i-th segment’s CoG
+(∗ → s), respectively. τq ∈ RNJ is the vector of joint torque
+and fi is the vectoring thrust force corresponding to (1).
+(5) ∼ (7) are highly complex due to the joint motion. Then
+the realtime feedback control based on these nonlinear equa-
+tions is significantly difficult for an onboard computational
+resource. Therefore, for the joint motion, a crucial assumption
+is introduced in our work to simplify the dynamics, i.e., all
+the joints are actuated slowly by individual servo motors.
+This is also called the quasi-static assumption that allows
+˙q ≈ 0; ¨q ≈ 0 during the joint motion.
+Under this assumption, the original dynamic model can be
+approximated as follows:
+mΣ¨rc(q) = Rcfλ − mΣg +
+Nc
+�
+i=1
+fci,
+(8)
+IΣ(q) ˙ω + ω × IΣ(q)ω = τλ +
+Nc
+�
+i=1
+pci × fci,
+(9)
+0 = τq +
+Nc
+�
+i=1
+JT
+cifci +
+Nr
+�
+i=1
+JT
+rifi +
+Ns
+�
+i=1
+JT
+simsig,
+(10)
+where rc is the position of the frame {CoG} w.r.t the frame
+{W}, which can be calculated using the forward-kinematics
+from the baselink states with the joint angles q. ωc is the
+angular velocity of the frame {CoG} w.r.t the frame of
+{CoG}, and is identical to ωb that is the angular velocity
+of the baselink. IΣ(q) is the total inertia tensor that is also
+influenced by the joint angles q.
+(8) and (9) show the property of a single rigid body, whereas
+(10) indicates the equilibrium between the forces and torques
+for the joint motion. Given that we apply the quasi-static
+assumption for joint motion, only the slow terrestrial motion,
+such as the static walk gait, is allowed.
+IV. CONTROL
+In this section, we first describe a unified control framework
+as depicted in Fig. 4, and then present the modification for the
+aerial and terrestrial locomotion, respectively.
+Control
+Allocation
+Centroidal
+Motion 
+Control
+Robot
+Dynamics
+Model
+Approximation
+4Ö×
+˜Ö×
+6˜Ö×
+Ó‰×
+�×
+Ø× Â× Å×
+˜Õ
+4Õ
+6˜Õ
+ÓÕ
+—
+—×
+IŠ +Š
+Vectorable Rotor Control
+4Ö ÓÖ
+˜Ö
+6˜Ö
+—
+Joint
+PD Control
+Îä×
+Fig. 4.
+Unified control framework for terrestrial/aerial locomotion.
+“Model approximation” presented in Sec. III-B is followed by the vectorable
+rotor control based on the centroidal motion. The joint control is performed
+independently.
+A. Centroidal Motion Control
+For the approximated dynamics of (8) and (9), the position
+feedback control based on an ordinary PID control is given by
+f d
+λ = mΣRT
+c (Kf,per + Kf,i
+�
+er + Kf,d ˙er)
++ RT
+c (mΣg −
+Nc
+�
+i=1
+fci),
+(11)
+where er = rd
+c − rc, and Kf,∗ are the PID gain diagonal
+matrices.
+The attitude control follows the SO(3) control method
+proposed by [23]:
+τ d
+λ = IΣ(Kτ,peR + Kτ,i
+�
+eR + Kτ,deω)
++ ωc × IΣωc −
+Nc
+�
+i=1
+pci × RT
+c fci,
+(12)
+eR = 1
+2
+�
+RT
+c Rd
+c − RdT
+c Rc
+�∨ ,
+(13)
+eω = RT
+c Rd
+cωd
+c − ωc,
+(14)
+where [⋆]∨ is the inverse of a skew map.
+Then, the desired wrench w.r.t the frame {CoG} can be
+summarized as follows:
+wd =
+�
+f d
+λ
+τ d
+λ
+�T .
+(15)
+B. Control Allocation
+The goal of vectorable rotor control is to obtain the control
+input (the desired thrust λd and the desired vectoring angles
+φd, θd) from the desired wrench wd. Meanwhile, it is also
+important to suppress the rotor output and the joint load from
+the aspect of the energy consumption. Then, an optimization
+problem should be design to obtain the desired control input.
+Since the vectoring angles φ and θ demand the trigono-
+metric function, nonlinear constraints would appear in the
+optimization problem and thus lead a complex computation.
+To decrease the computational load during the realtime control
+loop, we introduce an alternative three-dimensional forces f
+′
+i
+that is the vectorable thrust w.r.t, the related link frame {Li}:
+f
+′
+i = LiRFi(φi, θi)
+�0
+0
+λi
+�T. The definition of the frames
+of {Li} and {Fi} can be found in Fig. 3. Then the above
+optimization problem can be modified as follows:
+min
+f ′
+i ,τq,fci
+w1
+Nr
+�
+i=1
+∥f
+′
+i∥2 + w2∥τq∥2,
+(16)
+s.t.
+wd =
+Nr
+�
+i=1
+Qif
+′
+i ,
+Qi =
+�E3×3
+[pi×]
+�
+CoGRLi,
+(17)
+τq = −
+Nc
+�
+i=1
+JT
+cifci −
+Nr
+�
+i=1
+JT
+rifi −
+Ns
+�
+i=1
+JT
+simsig,
+(18)
+0 < λi < ¯λ,
+(19)
+− ¯τq < τqi < ¯τq,
+(20)
+0 < fci(2),
+(21)
+
+5
+where w1 and w2 in (16) are the weights for the cost of rotor
+thrust and joint torque, respectively. (17) is the modified form
+of wrench allocation from (3) by using the alternative variable
+f
+′
+i . pi is defined in (3), whereas CoGRLi is the orientation of
+the frame {Li} w.r.t. the frame {CoG}. E3×3 is a 3 × 3
+identity matrix and [·×] denotes the skew symmetric matrix
+of a three dimensional vector. (18) denotes the equilibrium
+between the joint torque τq, the contact force fci, the thrust
+force fi, and the segment gravity msig to satisfy the joint
+quasi-static assumption. (19) and (20) denote the bounds for
+the rotor thrust and joint torque, respectively. The contact force
+fci is also considered as the searching variable, and the z
+element fci(2) should be always non-negative as shown in
+(21).
+Given that all constraints (17) ∼ (21) are linear, an ordinary
+algorithm for quadratic problem can be applied. Once the
+optimized thrust force ˜f
+′
+i is calculated, the true control input
+for the spherically vector rotor apparatus can be obtained as
+follows:
+λi = ∥f
+′
+i∥,
+(22)
+φi = tan−1(−f
+′
+i(1)
+f
+′
+i(2) ),
+(23)
+θi = tan−1(
+f
+′
+i (0)
+−f
+′
+i(1)sin(φi) + f
+′
+i(2)cos(φi)),
+(24)
+where f
+′
+i (0), f
+′
+i(1), and f
+′
+i(2) are the x, y, and z element of
+the vector.
+As a unique mechanical feature of the spherically vectorable
+apparatus depicted in Fig. 2(D), the result of vectoring angles
+φ and θ from (23) and (24) will deviate the position pi in
+(17) because of the small offset between two vectoring axes
+as depicted in Fig. 2(D). Then, the results of (22) ∼ (24) will
+no longer satisfy the constraint (17) because Qi has changed.
+To solve this problem, we apply the iteration process that is
+based on the gradient of a residual term ǫ := wd − Q(θ, φ)λ,
+and finally we can obtain the convergent values of φd, θd and
+λd. The detail can be found in [19].
+C. Joint Control
+The proposed optimization problem of (16) can also pro-
+vides the joint torque that however only satisfies the quasi-
+static assumption for joint motion. In addition, the measure-
+ment bias and noisy from the joint encoders along with the
+slight deformation of the link and joint structure can also
+induce the model error. To handle this model error, it is
+necessary to apply a feed-back control to track the desired
+position for joints. Therefore, a simple PD control for joint
+position is introduced for each joint:
+τ d
+i = kj,p(qd
+i − qi) − kj,d ˙qi,
+(25)
+where qd
+i is the desired joint angle from the walking gait or
+the aerial transformation planning. kj,p and kj,d are the P and
+D control gains. It is also notable that we also used the same
+PD control for the rotor vectoring angles φi and θi.
+D. Aerial Locomotion
+The control mode for aerial locomotion follows the flow
+shown in Fig. 4 but without the contact force fci. Then the
+constraint of (21) and the first term �Nc
+i=1 JT
+cifci at the right
+side of (18) can be omitted. The joint control is executed
+independently to follow the trajectory given by other task
+planing.
+E. Terrestrial Locomotion
+1) Torso altitude control: The terrestrial locomotion is
+totally based on the quasi-static joint motion. Therefore the
+centroidal motion should be also assumed to be static, which
+results in a desired wrench only handling gravity (wd =
+�0
+0
+−mΣg
+0
+0
+0�
+) for (17). Despite of the joint
+position control proposed in (25), a small error regarding
+the torso (i.e., baselink) pose, particularly along the altitude
+direction, would still remain mainly due to the influence of
+gravity. Therefore, we apply a feedback control using the rotor
+thrust for the torso altitude. Instead of the PID position control
+for the centroidal motion as proposed in (11), a truncated
+feedback control for the torso altitude is introduced as follows:
+f d
+z = kb(zd
+b − zb),
+(26)
+where kb is the P gain, and zb is the torso altitude. We assume
+this altitude control is for the “floating” baselink even in the
+terrestrial locomotion mode. Therefore, instead of considering
+f d
+z in (17) and (18), we introduce another independent control
+allocation to obtain the additional thrust force as follows:
+∆wd =
+Nr
+2
+�
+i=1
+Q2i∆f
+′
+2i,
+(27)
+where ∆wd =
+�
+0
+0
+f d
+z
+0
+0
+0
+�T. It is notable that we
+only choose the rotors in the inner link of each leg to suppress
+the influence on the joint quasi-static motion as presented in
+(18). Then ∆f
+′
+2i can be given by
+∆f
+′ = ˜Q#∆wd,
+(28)
+˜Q =
+�Q0
+Q2
+· · ·
+QNr
+�
+,
+∆f
+′ =
+�
+∆f
+′
+0
+∆f
+′
+2
+· · ·
+∆f
+′
+Nr
+�T ,
+where ˜Q# is the psuedo-inverse matrix of ˜Q. Finally, f
+′
+2i →
+f
+′
+2i + ∆f
+′
+2i is performed before substituting it into (22)∼(24).
+2) Static walking gait: In this work, we only focus on the
+static walking gait. Hence only one leg is allowed to lift during
+walking. As the update of the foot step for the lifting leg,
+we analytically solve the inverse-kinematics for the related
+three joint angles: qi yaw, qi pitch, and qi+1 pitch as depicted
+in Fig. 2, which can be uniquely determined. Regarding the
+gait for linear movement, we design a creeping gait that lifts
+the front-left, front-right, rear-right, and rear-left legs in order
+for one gait cycle, and also solely moves the torso in standing
+mode just after the two front legs have moved to the new
+position. To enable the repetition of the gait cycle, the stride
+length of all feet is set equal to the moving distance of torso.
+We further assume the robot only walks on a flat floor, and
+thus the height of feet should be always zero. Then, we first
+
+6
+set an intermediate target position right above the new foot
+step with a small height offset. Thus qi yaw and qi+1 pitch are
+identical to the final target, whereas qi pitch is smaller than
+the final value. Once the lifting leg moves to this intermediate
+pose, the robot starts lowering the leg to reach the new foot
+step only by changing qi pitch. Given that there is no tactile
+sensor on the foot, we introduce a threshold ∆qc for the joint
+angle error of qi pitch to detect touchdown. That is, if qd
+i pitch−
+qi pitch < ∆qc, then switch the lifting leg to the standing
+mode, and thus the number of the contact force fci changes
+from three to four.
+V. EXPERIMENT
+A. Robot Platform
+In this work, we developed a prototype of SIDAR as shown
+in Fig. 5, and the basic specification is summarized in Tab. I.
+Given the lightweight design, we employed CFRP material for
+link rod where cables can pass through. For the joint module,
+we used the Aluminum sheet to connect links, whereas the
+joint servo was Dynamixel XH430-V350R of which the torque
+was enhanced by pulley made from PLA. The range of joint
+angle was [−90◦ 90◦]. For the vectorable rotor module, a
+pair of counter-rotating plastic propellers were enclosed by
+ducts with the aim of safety and increase of thrust, whereas
+Dynamixel XL430-W250T was used for the rotor vectoring.
+Batteries are distributed in each link unit in parallel as shown
+in Fig. 5(G) which can provide a flight duration up to 9 min
+and a longer walk duration up to 20 min. A hemisphere foot
+with anti-slip tape was equipped to ensure the stable point
+contact during the terrestrial locomotion.
+On the center torso as shown in Fig. 5(A), NVIDIA Jetson
+TX2 and an original MCU board called Spinal were deployed
+to perform the realtime control framework as presented in
+Fig. 4. For each link unit, there was a distributed MCU
+board called Neuron that served as relay node between Spinal
+(C1)
+(A)
+(B)
+(D)
+(C2)
+(G)
+(F)
+(A)
+(B)
+(C1)
+(C2)
+(D)
+(E)
+(E)
+(F)
+(G)
+àÜ
+öÜ
+1.1m
+MÜ4w_ê
+MÜ>54w_ê
+MÜ4ngraf
+MÜ>54ngraf
+CAN
+Fig. 5. Prototype of SPIDAR: (A) center torso that employed an original
+red MCU called Spinal and a high level processor (Nvidia Jetson TX2); (B)
+spherically vectorable dual-rotor module; (C1)(C2) two-DoF joint module
+for the “hip” and the “knee”, respectively; (D) single leg (limb) that had the
+maximum length of 1.1 m; (E) small relay board called “Neuron” for each
+link unit that was connected with “Spinal” via CAN; (F) hemisphere foot
+with anti-slip tape; (G) distributed battery attached at each link unit.
+TABLE I
+PROTOTYPE SPECIFICATIONS
+1. Main Feature
+3. Vectorable Rotor
+Attribute
+Value
+Attribute
+Value
+total mass
+15.2 kg
+rotor KV
+1550
+max size (dia.)
+2.7 m
+propeller diameter
+5 inch
+max flight time
+9 min
+max thrust (¯λ)
+42 N
+max walk time
+20 min
+pulley ratio
+1:1.5
+max vectoring torque
+1.5 Nm
+2. Link and Joint
+max vectoring speed
+4.2 rad/s
+Attribute
+Value
+link length
+0.54 m
+4. Lipo Battery
+joint pulley ratio
+1:2
+Attribute
+Value
+max joint speed
+0.34 rad/s
+capacity
+6S 3Ah
+max torque (¯τq)
+6.5 Nm
+amount
+8
+and each actuator. Neurons and Spinal were connected by
+CAN cable. The detail of the onboard communication can be
+found in [18]. Besides, an external motion capture system was
+applied in our experiment to obtain the state of the baselink
+(i.e., rb, ˙rb, Rb, and ωb), which were used to calculate the
+state of centroidal motion based on forward-kinematics.
+B. Basic Experimental Evaluation
+1) Aerial transformation: A unique feature of SPIDAR is
+the aerial maneuvering with joint motion (i.e., aerial transfor-
+mation). To validate the stability during flight, simple transfor-
+mation as shown in Fig. 6 was performed. All limbs changed
+their joints with the same trajectories as plotted in Fig. 7(C).
+(A)
+(B)
+Fig. 6. Stable joint motion in midair: (A) extended pose that has diameter
+of 2.6 m; (B) standing pose, implying the feasibility to takeoff directly from
+the terrestrial mode.
+32
+34
+36
+38
+40
+42
+44 [s]
+(B)
+(A)
+(D)
+0.05
+-0.05
+0
+Š Aåã
+Š Aåä
+Š Aåå
+Š Aåâßß Š AãÜçÖÛ Š AìÔê
+[m]
+1.5
+-1.5
+-1.0
+-0.5
+0.5
+1.0
+0
+[;]
+6
+4
+2
+0
+-2
+-4
+[Nm]
+32
+34
+36
+38
+40
+42
+44 [s]
+(C)
+80
+60
+40
+20
+0
+-20
+[;]
+Š M54ìÔê
+Š M54ãÜçÖÛ
+Š M64ìÔê
+Š M64ãÜçÖÛ
+Š ì54ìÔê
+Š ì54ãÜçÖÛ
+Š ì64ìÔê
+Š ì64ãÜçÖÛ
+Fig. 7. Plots related to Fig. 6 : (A) positional errors of {CoG}; (B) rotational
+errors of {CoG} described in XYZ Euler angles; (C) joint trajectories for
+leg1 (q1 yaw, q1 pitch, q2 yaw, and q2 pitch), other legs followed the same
+joint trajectories; (D) torques for those joints.
+
+MM
+MwM
+M装配件7
+(A)
+(B)
+Fig. 8. lifting a single leg from standing mode: (A) standing mode where
+all feet have contact with the ground; (B) lifting single leg and keeping the
+raised pose with the assistance of rotor thrust.
+For the control gains in (11) and (12), we set Kf,p, Kf,i, Kf,d,
+Kτ,p, Kτ,i, and Kτ,d as D(3.6, 3.6, 2.8), D(0.03, 0.03, 1.2),
+D(4, 4, 2.8), D(15, 15, 10), D(0.3, 0.3, 0.1), and D(5, 5, 5),
+where D(∗, ∗, ∗) ∈ R3×3 is a diagonal matrix. For the
+optimization problem of (16), we omitted the second term
+(i.e., w2 = 0) to put a priority on the minimization of the
+thrust force. Fig. 7(A) and (B) plotted the positional and
+rotational errors during the flight and transformation, and the
+RMS of those errors were [0.014, 0.023, 0.038] m and [0.81,
+0.69, 0.92]◦. The altitude error erz indicated a relatively large
+deviation during the joint motion, which was caused by the
+violation of the quasi-static assumption. Nevertheless, this
+deviation rapidly decreased once the joint motion finished.
+Fig. 7(C) and (D) showed that all joint were well controlled by
+the PD control as presented in (25). Eventually, these results
+demonstrated the stability of both the baselink pose and the
+joint motion in aerial locomotion.
+2) Leg lifting: The key to achieve walking by legged robot
+is the stability while lifting the leg. Then, we evaluated the
+proposed control method by performing a long-term single
+leg lifting as shown in Fig. 8. The cost weights in (16) were
+set as w1 = 1, w2 = 1, and the bound for joint torque ¯τq
+was decreased to 1.5 Nm to ensure sufficient margin for joint
+control. Besides, the gain kb in (26) was set as 25. Leg1
+was lifted by changing q1 pitch from −16 ◦ to −28 ◦, and the
+lifting motion lasted around 30 s as shown in Fig. 9(A). Other
+joints were kept constant in the whole motion as shown in
+Fig. 9(A) and (C), and their torques were within the bounds as
+depicted in Fig. 9(B) and (D). These results demonstrated the
+stability of joint motion against the influence of thrust force.
+Besides the stability of the baselink pose can be confirmed
+in Fig. 9(E) and (F), where both the positional and rotational
+errors converged to the sufficiently small value (i.e., 0.01 m
+and 0.5 ◦). Fig. 9(G) showed the large increase of the thrust
+forces in the lifting leg, whereas Fig. 9(H) showed small
+changes in other standing legs. In addition, these plots also
+confirmed the stable transition between standing mode and
+leg lifting mode. In particular, the shift back to the standing
+model around 40 s demonstrated the smooth touchdown, which
+indicates the promising terrestrial locomotion.
+C. Seamless Terrestrial/Aerial Hybrid Locomotion
+We further evaluated the feasibility of seamless locomotion
+transition as shown in Fig. 10. Fig. 11(A) and (C) demon-
+strated the baselink pose trajectory during walking with five
+15
+20
+25
+30
+35
+40
+[s]
+45
+15
+20
+25
+30
+35
+40
+[s]
+45
+15
+20
+10
+5
+10
+8
+6
+4
+2
+[N]
+[N]
+(F)
+(E)
+(G)
+(H)
+0.02
+-0.02
+0
+[m]
+-0.01
+0.01
+1.5
+-1.5
+-1.0
+-0.5
+0.5
+1.0
+0
+[;]
+Š Aåã Š Aåä Š Aåå
+Š Aåâßß Š AãÜçÖÛ Š AìÔê
+Š ã5 Š ã6
+Š ã7 Š ã8 Š ã9 Š ã:
+(A)
+(B)
+(C)
+(D)
+2
+1
+0
+-1
+-2
+6
+5
+[Nm]
+4
+2
+1
+0
+-1
+3
+-30
+-25
+-20
+-15
+[;]
+80
+[;]
+86
+84
+82
+Š M54ãÜçÖÛ Š M74ãÜçÖÛ Š M94ãÜçÖÛ
+Š ì54ãÜçÖÛ Š ì74ãÜçÖÛ Š ì94ãÜçÖÛ
+Š M64ãÜçÖÛ Š M84ãÜçÖÛ Š M:4ãÜçÖÛ
+Š ì64ãÜçÖÛ Š ì84ãÜçÖÛ Š ì:4ãÜçÖÛ
+Fig. 9. Plots related to Fig. 8: (A) trajectories for hip pitch joints. q7 pitch
+was omitted due to the symmetric pose of leg4 related to leg2; (B) torques
+of joints in (A); (C) trajectories for knee pitch joints; (D) torques of joints in
+(C); (E) positional errors of baselink ; (F) rotational errors of baselink; (G)
+thrust forces in leg1; (H) thrust forces in other legs.
+gait cycles. We observed that the translational drift along the
+walking direction (x axis) and the orthogonal direction (y axis)
+finally grew to 0.18 m and 0.10 m, whereas the rotational drift
+along the yaw axis also increased to 9 ◦. These drifts can be
+attributed to the feed-froward gait planing where the target
+baselink pose was updated based on the last target values
+�
+ 
+!
+"
+0.2m
+0.2m
+0.2m
+5 gait 
+cycles
+gait cycle
+Fig. 10. Seamless Terrestrial/Aerial Hybrid Locomotion: 1⃝ ∼ 3⃝ shows
+the representative phases (moving the front-left leg, the torso, and rear-left
+leg) in one creeping gait cycle. After five gait cycles, robot switched to the
+aerial locomotion directly from the terrestrial pose as shown in 4⃝.
+
+82prMN8
+gait 
+cycle1
+gait 
+cycle2
+gait 
+cycle3
+gait 
+cycle4
+gait 
+cycle5
+stand
+sprawl
+Š NKHH Š LEP?D Š U=S
+--- NKHH×, LEP?D×, U=S×
+-6
+4
+2
+0
+-2
+-4
+-8
+-10
+takeoff
+land
+(B)
+(A)
+(C)
+(D)
+1.2
+0.8
+1.0
+[;]
+-6
+0
+-2
+-4
+-8
+-10
+[;]
+0.6
+0.4
+0.2
+0
+-0.2
+0.8
+0.6
+0.4
+0.2
+0
+-0.2
+[m]
+[m]
+Š Në Š Nì Š Ní
+--- Në× --- Në× --- Në×
+--- U=S×
+20
+40
+60
+80
+100
+120 [s]
+120
+125
+130
+135
+[s]
+Fig. 11. Plots related Fig. 10.(A)/(B) trajectories of baselink position during
+the terrestrial locomotion and the aerial locomotion, respectively; (C)/(D)
+trajectories of baselink orientation during the terrestrial locomotion and the
+aerial locomotion, respectively.
+but not the actual values. Nevertheless, these drifts can be
+considered relatively small compared to the total displacement,
+and are possible to be suppressed by adding a feed-back loop
+in planning as a future work. Furthermore, the deviations
+regarding the z, roll, and pitch axes were sufficiently small,
+which demonstrated the efficiency of the proposed control
+method presented in Sec. IV.
+As shown in Fig. 11(B) and (D), the transition to the
+aerial locomotion was smooth and stable, and the stability in
+midair was also confirmed, Thus, these results demonstrated
+the feasibility of the mechanical design, modeling and control
+methods for the terrestrial/aerial hybrid quadruped platform.
+VI. CONCLUSION
+In this paper, we presented the achievement of the terres-
+trial/aerial hybrid locomotion by the quadruped robot SPIDAR
+that were equipped with the vectorable rotors distributed in all
+links. We first proposed the mechanical design for this unique
+quadruped platform, and then developed the modeling and
+control methods to enable static walking and transformable
+flight. The feasibility of the above methods were verified by
+the experiment of seamless terrestrial/aerial hybrid locomotion
+with the prototype of SPIDAR.
+A crucial issue remained in this work is the oscillation
+and deviation of the baselink pose and joint angles during
+walking. To improve the stability, the rotor thrust should be
+directly used in the joint position control to replace the current
+simple PD control. Furthermore, the gait planning should be
+also robust against the drift by adding a feed-back loop as
+discussed in Sec. V-C. Last but not least, the dynamic walking
+and the aerial manipulation will be investigated to enhance the
+versatility of this robot in both maneuvering and manipulation.
+REFERENCES
+[1] Koji Kawasaki, Moju Zhao, Kei Okada, and Masayuki Inaba. MUWA:
+Multi-field universal wheel for air-land vehicle with quad variable-pitch
+propellers. In 2013 IEEE/RSJ International Conference on Intelligent
+Robots and Systems, pp. 1880–1885, 2013.
+[2] Arash Kalantari and Matthew Spenko.
+Modeling and performance
+assessment of the HyTAQ, a hybrid terrestrial/aerial quadrotor. IEEE
+Transactions on Robotics, Vol. 30, No. 5, pp. 1278–1285, 2014.
+[3] Hiroya Yamada, et al.
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+ACM-R5. In the 36th International Symposium on Robotics (ISR), 2005.
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+E
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf,len=597
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='04050v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='RO]  10 Jan 2023  1  Design, Modeling and Control of a Quadruped Robot SPIDAR:  Spherically Vectorable and Distributed Rotors Assisted  Air-Ground Amphibious Quadruped Robot  Moju Zhao1, Tomoki Anzai2, Takuzumi Nishio2  Abstract—Multimodal locomotion capability is an emerging  topic in robotics field, and various novel mobile robots have  been developed to enable the maneuvering in both terrestrial  and aerial domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Among these hybrid robots, several state-of-  the-art bipedal robots enable the complex walking motion which  is interlaced with flying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' These robots are also desired to have the  manipulation ability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' however, it is difficult for the current forms  to keep stability with the joint motion in midair due to the cen-  tralized rotor arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Therefore, in this work, we develop  a novel air-ground amphibious quadruped robot called SPIDAR  which is assisted by spherically vectorable rotors distributed in  each link to enable both walking motion and transformable flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  First, we present a unique mechanical design for quadruped robot  that enables terrestrial and aerial locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' We then reveal the  modeling method for this hybrid robot platform, and further  develop an integrated control strategy for both walking and  flying with joint motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Finally, we demonstrate the feasibility of  the proposed hybrid quadruped robot by performing a seamless  motion that involves static walking and subsequent flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' To  the best of our knowledge, this work is the first to achieve a  quadruped robot with multimodal locomotion capability, which  also shows the potential of manipulation in multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Index Terms—Legged Robots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Aerial Systems: Mechanics and  Control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Motion Control  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' INTRODUCTION  During the last decade, robots with multimodal locomotion  capability have undergone intensive development, and demon-  strated the versatile maneuvering in multiple domains (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=',  terrestrial, aerial, and aquatic) [1]–[5] which can benefit the  performance in various situations, such as disaster response  and industrial surveillance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Among various forms for mul-  timodal locomotion, the legged type has the advantage in  unstructured terrains, and several bipedal models have been  developed to demonstrate the promising walking that is inter-  laced with flying motion [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The legged robots are desired  to provide not only the advanced locomotion capability, but  also the manipulation ability by the limb end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then,  multilegged (multilimbed) is considered effective from the  aspect of both the stability in the terrestrial locomotion and  the freedom in manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' In spite of the achievement  of a simple arm motion in midair by the flying humanoid  robot in [7], the centralized rotor arrangement in most of the  existing robot platforms can hardly handle the large change of  Center-of-Gravity (CoG) caused by the joint motion in midair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Besides, the external force at the limb end during the aerial  manipulation is also difficult to compensate by the centralized  1 Department of Mechanical Engineering, The University of Tokyo, 2  Department of Mechano-Infomatics, The University of Tokyo, 7-3-1 Hongo,  Bunkyo-ku, Tokyo 113-8656, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' chou@hnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='jp  Digital Object Identifier (DOI): see top of this page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  (A)  (B)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Air-ground amphibious quadruped robot SPIDAR: SPherIcally  vectorable and Distributed rotors assisted Air-ground amphibious quadruped  Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (A) walk on the ground like an usual quadruped robot, but with the  assistance of thrust force;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (B) fly and transform in midair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  rotors due to the large moment arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Hence, a distributed rotor  arrangement is necessary for aerial manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then, we  propose a novel quadruped robot platform in this work, which  can both walk and fly by using the spherically vectorable rotors  distributed in each link unit as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  One of the efficient type of air-ground hybrid robot plat-  forms equips an ordinary multirotor for flying, and then de-  ploys rolling cage or wheels to perform terrestrial locomotion  [2], [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Although the rolling mechanism can achieve the  stable terrestrial locomotion without any complex control, it is  relatively difficult for this type to handle the unstructured ter-  rains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=', very few foothold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then, legged model is proposed  in several researches [6], [7], [10]–[12] to solve this issue by  the walking motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Most of these robots are bipedal model  which attaches the multirotor unit in their torso for aerial  maneuvering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Compared with the bipedal model, it is relatively  easier to achieve the walking stability by the multilegged  model because of the larger support polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Besides, the legs  can be considered as the limbs for manipulation in midair, and  thus more limbs can provide more end-effectors for complex  manipulation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Therefore, we choose the quadruped model  that can offer both the stable terrestrial locomotion and the  potential of aerial manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  For most of the quadruped robot, the form is bio-inspired  and specialized for terrestrial locomotion [13]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' However,  in our work, the robot is also desired to perform manipulation  task in multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Several ape-like quadruped robots  show the manipulation ability by the hands attached on the leg  ends [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The point symmetry is the crucial difference of  these robots from the skeleton design for common quadruped  robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' In our work, we also adopt the point symmetry for  our hybrid quadruped robot to enable the omni-directional  maneuvering and manipulation in both terrestrial and aerial  2  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Regarding the rotor arrangement, it is difficult for  rotors centralized in the robot torso like [6] to handle the  change of CoG caused by the joint motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Besides, the  external force acted at the end-effectors can also induce a  large rotational load for the centralized rotors due to the  large moment arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' To ensure a sufficient control margin for  the stable joint motion in midair, [11] proposes a distributed  rotor arrangement that deploys the thrust units at each limb  end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' However, this rotor arrangement deprives the robot of  manipulation ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Therefore, a fully-distributed rotor design  proposed by [18] is applied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' In this design,  spherically vectorable rotor apparatus is embedded in each link  and thus can generate individual three-dimensional thrust force  for the promising maneuvering and manipulation in midair as  presented in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  For the stable flying motion, the whole platform should be  lightweight, which leads a relatively compact and weak joint  actuator for legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Hence, the thrust force is also required to  assist the walking motion by reducing the load from gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  For model with centralized rotor arrangement, both the model-  based [6] and the policy-based [20] methods are developed to  obtain the assistive thrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' However, these methods are not  suitable for the model with rotors distributed in all links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Then, [21] proposes an assistive thrust control method for a  bipedal robot with the vectorable rotor attached at each foot,  whereas [19] presents a control method for a fully-distributed  rotor model in the aerial domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Based on these methods, a  comprehensive investigation of the modeling and control for  the multilegged model is performed in this work to handle  multiple types of torques and forces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=', the joint torque, the  contact force on each limb end, the gravity of each link, and  the thrust force from each vectorable rotor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  The main contributions of this work can be summarized as  follows:  1) We propose a unique mechanical design for air-ground  quadruped robot with the spherically vectorable rotors  distributed in all links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  2) We present a modeling and control methods for this  multilegged platform for hybrid locomotion in both  terrestrial and aerial domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  3) We achieve the seamless and stable motion that involves  walking, flying and joint motion in midair by the proto-  type of quadruped robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The  mechanical design for this unique quadruped robot is intro-  duced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The modeling of our robot is presented in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' III, followed by the integrated control method for hybrid  locomotion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' We then show the experimental results  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' V before concluding in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' DESIGN  In this section, we present the mechanical design for the  quadruped robot that is capable of terrestrial/aerial hybrid  locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The key of the whole structure is the spherically  vectorable rotor embedded in each link unit, along with the  unique quadruped shape that differs from the common bio-  inspired type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Mechanical design for the air-ground hybrid quadruped robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  (A) skeleton model for common bio-inspired mammal-type quadruped robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  (B) proposed point-symmetric skeleton model for hybrid quadruped model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  (C1)/(C2) two-DoF joint module for each limb, where the yaw axis qi yaw  comes first followed by the pitch axis qi pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (D) spherically vectorable  rotor apparatus with two vectoring angles (φ, θ), and a combined thrust force  λi from the counter rotating dual rotors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' There is a small offset between two  vectoring axes because two rods cannot intersect with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Skeleton Model  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 2(A), the common skeleton for quadruped  robot is mammal-type, which puts a priority on the forward  motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Thus, the model is plane symmetric, and each leg has  three DoF (two in the hip, and one in knee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' However, our  robot is desired to enable not only the terrestrial/aerial hybrid  locomotion, but also the manipulation in midair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Therefore, the  omni-directional movement is a critical feature for the skeleton  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then, a point symmetric structure is introduced as  depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 2(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' This is similar to the sprawling-type  quadruped design proposed by [22], which can provide s wider  supporting polygon and also a lower CoG than the mammal-  type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' According to this design concept, each limb consists  of two links that have the same length, and is connected to  the center torso with a joint module that has two Degree-of-  Freedom (DoF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' For this joint module, the yaw axis qi yaw  comes first, which is followed by the pitch axis qi pitch  to allow a larger swinging range for walking as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 2(C1)/(C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The crucial difference from the ordinary  sprawling-type is that we also introduce an identical two-  DoF joint module to connect neighboring links, which can  provide a four-DoF manipulation capability by each limb end  without the help of the torso motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Eventually, this robot  is composed of 8 links with 16 joints for walking and flying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Given that the lightweight design is significantly important for  the flight performance, we deploy a compact servo motor to  individually actuate each joint at the expense of the torque  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Nevertheless, the shortage of the joint torque can be  compensated by the rotor thrust in our robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Spherically Vectorable Rotor  Rotors embedded in links are used to achieve flight with  arbitrary joint motion in midair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Besides, it is also necessary  (B)  (A)  qi+1_yaw  i+1_pitch  yaw  qi_pitch  (D)  C1 top view  (C2) side view3  to use the rotor thrust to assist leg lifts for walking, because  the joint actuator is weak due to the lightweight design as  mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' II-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Therefore, the rotor is required to  point arbitrary direction to handle the change in link ori-  entation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' In other words, it is required to generate a three-  dimensional thrust force by each rotor module to interact  with not only the gravity and also the external force (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=',  the contact force on each foot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then a spherically vectorable  apparatus proposed by [18] is equipped in each link as depicted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' To achieve the spherical vectoring around the  link unit, two rotation axes is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' We first introduce  a roll axis φi around the link rod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then, we need the second  orthogonal vectoring axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' If we use a single rotor, the collision  between the propeller and the link rod will be inevitable  while performing the second vectoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Therefore, we apply  the counter-rotating dual-rotor module to avoid the collision  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' In addition, this dual-rotor can also  counteract the drag moment and gyroscopic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then,  we define the pitch axis across dual rotors as the second  vectoring axis θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Each vectoring axis is also actuated by an  individual compact servo motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Regarding the thrust force,  since we assume that a pair of rotors rotate with the same  speed, it is possible to introduced a combined thrust λi for  each spherically vectorable rotor module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Eventually, there are  three control input (two vectoring angles φi, θi, and one thrust  λi) for each vectorable rotor module, and totally 8 sets are used  to control the whole quadruped model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' MODELING  In this section, we describe the modeling for this robot  which can be divided into two parts: the thrust model and  the whole dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Spherically Vectorable Thrust  Based on the kinematic model depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 3, the three-  dimensional force fi generated by the i-th rotor module can  contact  contact  contact  free  Œ•  Œ•>Ú  “••  Œ‰•  Ε  Ε>Ú  Ε>Û  Ε>Ü  “•>Ú•  <(Ü=  <%K)=  <.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='Ü=  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Dynamics model of the proposed quadruped robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The entire  dynamics involves the joint torque τj, the contact force at each limb end fci,  the gravity of each link mig, and the thrust force from each vectorable rotor  fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' {Li} and {Fi} denote the frame for the i-th link and rotor respectively,  whereas {CoG} is the CoG frame for the whole model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' For the free leg  during walking, the contact force fci at the limb end disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  be written as:  fi = λiui,  (1)  ui = CoGRFi(q, φi, θi)  �0  0  1�T ,  (2)  where CoGRFi denotes a rotation matrix of the rotor frame  {Fi} w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' the frame {CoG}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' For this robot, we define the  frame {CoG} to have an origin at the CoG point as depicted in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 3, and a xyz coordinate that is identical to the baselink at  the center torso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' ui denotes the unit vector for the spherically  vectorable mechanism that is effected by two vectoring angles  φi and θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Besides, this vector also depends on the joint angles  q ∈ RNJ , NJ is the number of joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Then the total wrench in the frame {CoG} can be given by  �  fλ  τλ  �  =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  Nr  �  i=1  fi  Nr  �  i=1  pi × fi  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  = Qλ,  (3)  Q =  �  u1  u2  · ·  uNr  p1 × u1  p2 × u2  · ·  pNr × uNr  �  ,  (4)  λ =  �λ1  λ2  · ·  λNr  �T ,  where pi is the position of the frame {Fi} origin from the  frame {CoG} that is influenced by the joint angles q and the  first vectoring angle φi for the i-th rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Nr is the number of  rotors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Dynamics of Multilinked Model  The whole dynamic model can be written as follows:  ˙PΣ =Rcfλ − mΣg +  Nc  �  i=1  fci,  (5)  ˙LΣ =τλ +  Nc  �  i=1  pci × RT  c fci,  (6)  MJ(q)¨q + c(q, ˙q) =τq +  Nc  �  i=1  JT  cifci  +  Nr  �  i=1  JT  rifi +  Ns  �  i=1  JT  simsig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  (7)  (5) and (6) denote the centroidal dynamics for the whole  multibody model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' PΣ and LΣ are the entire linear and  rotational momentum described in the inertial frame {W}  and the frame {CoG}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' These momentum are  both affected by the joint angles, vectoring angles, and their  velocities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=', q, ˙q, φ, ˙φ, θ, ˙θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Rc is the orientatin of the  frame {CoG} w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' the frame {W}, and is identical to Rb  that is the orientatin of the baselink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' fλ and τλ corresponds  to the total wrench described in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' fci is the contact force at  the i-th limb end (foot) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' the frame {W}, whereas pci is  the position of this contact point from the frame {CoG} which  is also influenced by the joint angles q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Nc is the number of  contact points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=', standing legs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' mΣ is the total mass, and  g is a three-dimensional vector expressing gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  4  (7) corresponds to the joint motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' MJ(q) denotes the  inertial matrix, whereas c(q, ˙q) is the term related to the  centrifugal and Coriolis forces in joint motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' J∗i ∈ R3×NJ  is the Jacobian matrix for the frame of the i-th contact point  (∗ → c), the i-th rotor (∗ → r), and the i-th segment’s CoG  (∗ → s), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' τq ∈ RNJ is the vector of joint torque  and fi is the vectoring thrust force corresponding to (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  (5) ∼ (7) are highly complex due to the joint motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then  the realtime feedback control based on these nonlinear equa-  tions is significantly difficult for an onboard computational  resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Therefore, for the joint motion, a crucial assumption  is introduced in our work to simplify the dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=', all  the joints are actuated slowly by individual servo motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  This is also called the quasi-static assumption that allows  ˙q ≈ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' ¨q ≈ 0 during the joint motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Under this assumption, the original dynamic model can be  approximated as follows:  mΣ¨rc(q) = Rcfλ − mΣg +  Nc  �  i=1  fci,  (8)  IΣ(q) ˙ω + ω × IΣ(q)ω = τλ +  Nc  �  i=1  pci × fci,  (9)  0 = τq +  Nc  �  i=1  JT  cifci +  Nr  �  i=1  JT  rifi +  Ns  �  i=1  JT  simsig,  (10)  where rc is the position of the frame {CoG} w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='t the frame  {W}, which can be calculated using the forward-kinematics  from the baselink states with the joint angles q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' ωc is the  angular velocity of the frame {CoG} w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='t the frame of  {CoG}, and is identical to ωb that is the angular velocity  of the baselink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' IΣ(q) is the total inertia tensor that is also  influenced by the joint angles q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  (8) and (9) show the property of a single rigid body, whereas  (10) indicates the equilibrium between the forces and torques  for the joint motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Given that we apply the quasi-static  assumption for joint motion, only the slow terrestrial motion,  such as the static walk gait, is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' CONTROL  In this section, we first describe a unified control framework  as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 4, and then present the modification for the  aerial and terrestrial locomotion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Control  Allocation  Centroidal  Motion  Control  Robot  Dynamics  Model  Approximation  4Ö×  ˜Ö×  6˜Ö×  Ó‰×  �×  Ø× Â× Å×  ˜Õ  4Õ  6˜Õ  ÓÕ  —  —×  IŠ +Š  Vectorable Rotor Control  4Ö ÓÖ  ˜Ö  6˜Ö  —  Joint  PD Control  Îä×  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Unified control framework for terrestrial/aerial locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  “Model approximation” presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' III-B is followed by the vectorable  rotor control based on the centroidal motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The joint control is performed  independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Centroidal Motion Control  For the approximated dynamics of (8) and (9), the position  feedback control based on an ordinary PID control is given by  f d  λ = mΣRT  c (Kf,per + Kf,i  �  er + Kf,d ˙er)  + RT  c (mΣg −  Nc  �  i=1  fci),  (11)  where er = rd  c − rc, and Kf,∗ are the PID gain diagonal  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  The attitude control follows the SO(3) control method  proposed by [23]:  τ d  λ = IΣ(Kτ,peR + Kτ,i  �  eR + Kτ,deω)  + ωc × IΣωc −  Nc  �  i=1  pci × RT  c fci,  (12)  eR = 1  2  �  RT  c Rd  c − RdT  c Rc  �∨ ,  (13)  eω = RT  c Rd  cωd  c − ωc,  (14)  where [⋆]∨ is the inverse of a skew map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Then, the desired wrench w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='t the frame {CoG} can be  summarized as follows:  wd =  �  f d  λ  τ d  λ  �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  (15)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Control Allocation  The goal of vectorable rotor control is to obtain the control  input (the desired thrust λd and the desired vectoring angles  φd, θd) from the desired wrench wd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Meanwhile, it is also  important to suppress the rotor output and the joint load from  the aspect of the energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then, an optimization  problem should be design to obtain the desired control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Since the vectoring angles φ and θ demand the trigono-  metric function, nonlinear constraints would appear in the  optimization problem and thus lead a complex computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  To decrease the computational load during the realtime control  loop, we introduce an alternative three-dimensional forces f  ′  i  that is the vectorable thrust w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='t, the related link frame {Li}:  f  ′  i = LiRFi(φi, θi)  �0  0  λi  �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The definition of the frames  of {Li} and {Fi} can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then the above  optimization problem can be modified as follows:  min  f ′  i ,τq,fci  w1  Nr  �  i=1  ∥f  ′  i∥2 + w2∥τq∥2,  (16)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  wd =  Nr  �  i=1  Qif  ′  i ,  Qi =  �E3×3  [pi×]  �  CoGRLi,  (17)  τq = −  Nc  �  i=1  JT  cifci −  Nr  �  i=1  JT  rifi −  Ns  �  i=1  JT  simsig,  (18)  0 < λi < ¯λ,  (19)  − ¯τq < τqi < ¯τq,  (20)  0 < fci(2),  (21)  5  where w1 and w2 in (16) are the weights for the cost of rotor  thrust and joint torque, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (17) is the modified form  of wrench allocation from (3) by using the alternative variable  f  ′  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' pi is defined in (3), whereas CoGRLi is the orientation of  the frame {Li} w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' the frame {CoG}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' E3×3 is a 3 × 3  identity matrix and [·×] denotes the skew symmetric matrix  of a three dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (18) denotes the equilibrium  between the joint torque τq, the contact force fci, the thrust  force fi, and the segment gravity msig to satisfy the joint  quasi-static assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (19) and (20) denote the bounds for  the rotor thrust and joint torque, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The contact force  fci is also considered as the searching variable, and the z  element fci(2) should be always non-negative as shown in  (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Given that all constraints (17) ∼ (21) are linear, an ordinary  algorithm for quadratic problem can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Once the  optimized thrust force ˜f  ′  i is calculated, the true control input  for the spherically vector rotor apparatus can be obtained as  follows:  λi = ∥f  ′  i∥,  (22)  φi = tan−1(−f  ′  i(1)  f  ′  i(2) ),  (23)  θi = tan−1(  f  ′  i (0)  −f  ′  i(1)sin(φi) + f  ′  i(2)cos(φi)),  (24)  where f  ′  i (0), f  ′  i(1), and f  ′  i(2) are the x, y, and z element of  the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  As a unique mechanical feature of the spherically vectorable  apparatus depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 2(D), the result of vectoring angles  φ and θ from (23) and (24) will deviate the position pi in  (17) because of the small offset between two vectoring axes  as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then, the results of (22) ∼ (24) will  no longer satisfy the constraint (17) because Qi has changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  To solve this problem, we apply the iteration process that is  based on the gradient of a residual term ǫ := wd − Q(θ, φ)λ,  and finally we can obtain the convergent values of φd, θd and  λd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The detail can be found in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Joint Control  The proposed optimization problem of (16) can also pro-  vides the joint torque that however only satisfies the quasi-  static assumption for joint motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' In addition, the measure-  ment bias and noisy from the joint encoders along with the  slight deformation of the link and joint structure can also  induce the model error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' To handle this model error, it is  necessary to apply a feed-back control to track the desired  position for joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Therefore, a simple PD control for joint  position is introduced for each joint:  τ d  i = kj,p(qd  i − qi) − kj,d ˙qi,  (25)  where qd  i is the desired joint angle from the walking gait or  the aerial transformation planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' kj,p and kj,d are the P and  D control gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' It is also notable that we also used the same  PD control for the rotor vectoring angles φi and θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Aerial Locomotion  The control mode for aerial locomotion follows the flow  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 4 but without the contact force fci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then the  constraint of (21) and the first term �Nc  i=1 JT  cifci at the right  side of (18) can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The joint control is executed  independently to follow the trajectory given by other task  planing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Terrestrial Locomotion  1) Torso altitude control: The terrestrial locomotion is  totally based on the quasi-static joint motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Therefore the  centroidal motion should be also assumed to be static, which  results in a desired wrench only handling gravity (wd =  �0  0  −mΣg  0  0  0�  ) for (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Despite of the joint  position control proposed in (25), a small error regarding  the torso (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=', baselink) pose, particularly along the altitude  direction, would still remain mainly due to the influence of  gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Therefore, we apply a feedback control using the rotor  thrust for the torso altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Instead of the PID position control  for the centroidal motion as proposed in (11), a truncated  feedback control for the torso altitude is introduced as follows:  f d  z = kb(zd  b − zb),  (26)  where kb is the P gain, and zb is the torso altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' We assume  this altitude control is for the “floating” baselink even in the  terrestrial locomotion mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Therefore, instead of considering  f d  z in (17) and (18), we introduce another independent control  allocation to obtain the additional thrust force as follows:  ∆wd =  Nr  2  �  i=1  Q2i∆f  ′  2i,  (27)  where ∆wd =  �  0  0  f d  z  0  0  0  �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' It is notable that we  only choose the rotors in the inner link of each leg to suppress  the influence on the joint quasi-static motion as presented in  (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then ∆f  ′  2i can be given by  ∆f  ′ = ˜Q#∆wd,  (28)  ˜Q =  �Q0  Q2  · ·  QNr  �  ,  ∆f  ′ =  �  ∆f  ′  0  ∆f  ′  2  · ·  ∆f  ′  Nr  �T ,  where ˜Q# is the psuedo-inverse matrix of ˜Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Finally, f  ′  2i →  f  ′  2i + ∆f  ′  2i is performed before substituting it into (22)∼(24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  2) Static walking gait: In this work, we only focus on the  static walking gait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Hence only one leg is allowed to lift during  walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' As the update of the foot step for the lifting leg,  we analytically solve the inverse-kinematics for the related  three joint angles: qi yaw, qi pitch, and qi+1 pitch as depicted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 2, which can be uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Regarding the  gait for linear movement, we design a creeping gait that lifts  the front-left, front-right, rear-right, and rear-left legs in order  for one gait cycle, and also solely moves the torso in standing  mode just after the two front legs have moved to the new  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' To enable the repetition of the gait cycle, the stride  length of all feet is set equal to the moving distance of torso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  We further assume the robot only walks on a flat floor, and  thus the height of feet should be always zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then, we first  6  set an intermediate target position right above the new foot  step with a small height offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Thus qi yaw and qi+1 pitch are  identical to the final target, whereas qi pitch is smaller than  the final value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Once the lifting leg moves to this intermediate  pose, the robot starts lowering the leg to reach the new foot  step only by changing qi pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Given that there is no tactile  sensor on the foot, we introduce a threshold ∆qc for the joint  angle error of qi pitch to detect touchdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' That is, if qd  i pitch−  qi pitch < ∆qc, then switch the lifting leg to the standing  mode, and thus the number of the contact force fci changes  from three to four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' EXPERIMENT  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Robot Platform  In this work, we developed a prototype of SIDAR as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 5, and the basic specification is summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Given the lightweight design, we employed CFRP material for  link rod where cables can pass through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' For the joint module,  we used the Aluminum sheet to connect links, whereas the  joint servo was Dynamixel XH430-V350R of which the torque  was enhanced by pulley made from PLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The range of joint  angle was [−90◦ 90◦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' For the vectorable rotor module, a  pair of counter-rotating plastic propellers were enclosed by  ducts with the aim of safety and increase of thrust, whereas  Dynamixel XL430-W250T was used for the rotor vectoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Batteries are distributed in each link unit in parallel as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 5(G) which can provide a flight duration up to 9 min  and a longer walk duration up to 20 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' A hemisphere foot  with anti-slip tape was equipped to ensure the stable point  contact during the terrestrial locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  On the center torso as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 5(A), NVIDIA Jetson  TX2 and an original MCU board called Spinal were deployed  to perform the realtime control framework as presented in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' For each link unit, there was a distributed MCU  board called Neuron that served as relay node between Spinal  (C1)  (A)  (B)  (D)  (C2)  (G)  (F)  (A)  (B)  (C1)  (C2)  (D)  (E)  (E)  (F)  (G)  àÜ  öÜ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='1m  MÜ4w_ê  MÜ>54w_ê  MÜ4ngraf  MÜ>54ngraf  CAN  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Prototype of SPIDAR: (A) center torso that employed an original  red MCU called Spinal and a high level processor (Nvidia Jetson TX2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (B)  spherically vectorable dual-rotor module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (C1)(C2) two-DoF joint module  for the “hip” and the “knee”, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (D) single leg (limb) that had the  maximum length of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='1 m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (E) small relay board called “Neuron” for each  link unit that was connected with “Spinal” via CAN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (F) hemisphere foot  with anti-slip tape;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (G) distributed battery attached at each link unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  TABLE I  PROTOTYPE SPECIFICATIONS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Main Feature  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Vectorable Rotor  Attribute  Value  Attribute  Value  total mass  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='2 kg  rotor KV  1550  max size (dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='7 m  propeller diameter  5 inch  max flight time  9 min  max thrust (¯λ)  42 N  max walk time  20 min  pulley ratio  1:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5  max vectoring torque  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5 Nm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Link and Joint  max vectoring speed  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='2 rad/s  Attribute  Value  link length  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='54 m  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Lipo Battery  joint pulley ratio  1:2  Attribute  Value  max joint speed  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='34 rad/s  capacity  6S 3Ah  max torque (¯τq)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5 Nm  amount  8  and each actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Neurons and Spinal were connected by  CAN cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The detail of the onboard communication can be  found in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Besides, an external motion capture system was  applied in our experiment to obtain the state of the baselink  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=', rb, ˙rb, Rb, and ωb), which were used to calculate the  state of centroidal motion based on forward-kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Basic Experimental Evaluation  1) Aerial transformation: A unique feature of SPIDAR is  the aerial maneuvering with joint motion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=', aerial transfor-  mation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' To validate the stability during flight, simple transfor-  mation as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 6 was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' All limbs changed  their joints with the same trajectories as plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 7(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  (A)  (B)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Stable joint motion in midair: (A) extended pose that has diameter  of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='6 m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (B) standing pose, implying the feasibility to takeoff directly from  the terrestrial mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  32  34  36  38  40  42  44 [s]  (B)  (A)  (D)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='05  0  Š Aåã  Š Aåä  Š Aåå  Š Aåâßß Š AãÜçÖÛ Š AìÔê  [m]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='0  0  [;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=']  6  4  2  0  2  4  [Nm]  32  34  36  38  40  42  44 [s]  (C)  80  60  40  20  0  20  [;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=']  Š M54ìÔê  Š M54ãÜçÖÛ  Š M64ìÔê  Š M64ãÜçÖÛ  Š ì54ìÔê  Š ì54ãÜçÖÛ  Š ì64ìÔê  Š ì64ãÜçÖÛ  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Plots related to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 6 : (A) positional errors of {CoG};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (B) rotational  errors of {CoG} described in XYZ Euler angles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (C) joint trajectories for  leg1 (q1 yaw, q1 pitch, q2 yaw, and q2 pitch), other legs followed the same  joint trajectories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (D) torques for those joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  MM  MwM  M装配件7  (A)  (B)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' lifting a single leg from standing mode: (A) standing mode where  all feet have contact with the ground;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (B) lifting single leg and keeping the  raised pose with the assistance of rotor thrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  For the control gains in (11) and (12), we set Kf,p, Kf,i, Kf,d,  Kτ,p, Kτ,i, and Kτ,d as D(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='8), D(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='03, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='2),  D(4, 4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='8), D(15, 15, 10), D(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='1), and D(5, 5, 5),  where D(∗, ∗, ∗) ∈ R3×3 is a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' For the  optimization problem of (16), we omitted the second term  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=', w2 = 0) to put a priority on the minimization of the  thrust force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 7(A) and (B) plotted the positional and  rotational errors during the flight and transformation, and the  RMS of those errors were [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='014, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='023, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='038] m and [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='81,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='69, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='92]◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The altitude error erz indicated a relatively large  deviation during the joint motion, which was caused by the  violation of the quasi-static assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Nevertheless, this  deviation rapidly decreased once the joint motion finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 7(C) and (D) showed that all joint were well controlled by  the PD control as presented in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Eventually, these results  demonstrated the stability of both the baselink pose and the  joint motion in aerial locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  2) Leg lifting: The key to achieve walking by legged robot  is the stability while lifting the leg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Then, we evaluated the  proposed control method by performing a long-term single  leg lifting as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The cost weights in (16) were  set as w1 = 1, w2 = 1, and the bound for joint torque ¯τq  was decreased to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5 Nm to ensure sufficient margin for joint  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Besides, the gain kb in (26) was set as 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Leg1  was lifted by changing q1 pitch from −16 ◦ to −28 ◦, and the  lifting motion lasted around 30 s as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 9(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Other  joints were kept constant in the whole motion as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 9(A) and (C), and their torques were within the bounds as  depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 9(B) and (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' These results demonstrated the  stability of joint motion against the influence of thrust force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  Besides the stability of the baselink pose can be confirmed  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 9(E) and (F), where both the positional and rotational  errors converged to the sufficiently small value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='01 m  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5 ◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 9(G) showed the large increase of the thrust  forces in the lifting leg, whereas Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 9(H) showed small  changes in other standing legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' In addition, these plots also  confirmed the stable transition between standing mode and  leg lifting mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' In particular, the shift back to the standing  model around 40 s demonstrated the smooth touchdown, which  indicates the promising terrestrial locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Seamless Terrestrial/Aerial Hybrid Locomotion  We further evaluated the feasibility of seamless locomotion  transition as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 11(A) and (C) demon-  strated the baselink pose trajectory during walking with five  15  20  25  30  35  40  [s]  45  15  20  25  30  35  40  [s]  45  15  20  10  5  10  8  6  4  2  [N]  [N]  (F)  (E)  (G)  (H)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='02  0  [m]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='0  0  [;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=']  Š Aåã Š Aåä Š Aåå  Š Aåâßß Š AãÜçÖÛ Š AìÔê  Š ã5 Š ã6  Š ã7 Š ã8 Š ã9 Š ã:  (A)  (B)  (C)  (D)  2  1  0  1  2  6  5  [Nm]  4  2  1  0  1  3  30  25  20  15  [;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=']  80  [;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=']  86  84  82  Š M54ãÜçÖÛ Š M74ãÜçÖÛ Š M94ãÜçÖÛ  Š ì54ãÜçÖÛ Š ì74ãÜçÖÛ Š ì94ãÜçÖÛ  Š M64ãÜçÖÛ Š M84ãÜçÖÛ Š M:4ãÜçÖÛ  Š ì64ãÜçÖÛ Š ì84ãÜçÖÛ Š ì:4ãÜçÖÛ  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Plots related to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 8: (A) trajectories for hip pitch joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' q7 pitch  was omitted due to the symmetric pose of leg4 related to leg2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (B) torques  of joints in (A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (C) trajectories for knee pitch joints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (D) torques of joints in  (C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (E) positional errors of baselink ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (F) rotational errors of baselink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (G)  thrust forces in leg1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (H) thrust forces in other legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  gait cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' We observed that the translational drift along the  walking direction (x axis) and the orthogonal direction (y axis)  finally grew to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='18 m and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='10 m, whereas the rotational drift  along the yaw axis also increased to 9 ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' These drifts can be  attributed to the feed-froward gait planing where the target  baselink pose was updated based on the last target values  �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  "  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='2m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='2m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='2m  5 gait  cycles  gait cycle  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Seamless Terrestrial/Aerial Hybrid Locomotion: 1⃝ ∼ 3⃝ shows  the representative phases (moving the front-left leg, the torso, and rear-left  leg) in one creeping gait cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' After five gait cycles, robot switched to the  aerial locomotion directly from the terrestrial pose as shown in 4⃝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  82prMN8  gait  cycle1  gait  cycle2  gait  cycle3  gait  cycle4  gait  cycle5  stand  sprawl  Š NKHH Š LEP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='D Š U=S  --- NKHH×, LEP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='D×, U=S×  6  4  2  0  2  4  8  10  takeoff  land  (B)  (A)  (C)  (D)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='0  [;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=']  6  0  2  4  8  10  [;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=']  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='2  [m]  [m]  Š Në Š Nì Š Ní  --- Në× --- Në× --- Në×  --- U=S×  20  40  60  80  100  120 [s]  120  125  130  135  [s]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Plots related Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (A)/(B) trajectories of baselink position during  the terrestrial locomotion and the aerial locomotion, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' (C)/(D)  trajectories of baselink orientation during the terrestrial locomotion and the  aerial locomotion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  but not the actual values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Nevertheless, these drifts can be  considered relatively small compared to the total displacement,  and are possible to be suppressed by adding a feed-back loop  in planning as a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Furthermore, the deviations  regarding the z, roll, and pitch axes were sufficiently small,  which demonstrated the efficiency of the proposed control  method presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 11(B) and (D), the transition to the  aerial locomotion was smooth and stable, and the stability in  midair was also confirmed, Thus, these results demonstrated  the feasibility of the mechanical design, modeling and control  methods for the terrestrial/aerial hybrid quadruped platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' CONCLUSION  In this paper, we presented the achievement of the terres-  trial/aerial hybrid locomotion by the quadruped robot SPIDAR  that were equipped with the vectorable rotors distributed in all  links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' We first proposed the mechanical design for this unique  quadruped platform, and then developed the modeling and  control methods to enable static walking and transformable  flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The feasibility of the above methods were verified by  the experiment of seamless terrestrial/aerial hybrid locomotion  with the prototype of SPIDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  A crucial issue remained in this work is the oscillation  and deviation of the baselink pose and joint angles during  walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' To improve the stability, the rotor thrust should be  directly used in the joint position control to replace the current  simple PD control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Furthermore, the gait planning should be  also robust against the drift by adding a feed-back loop as  discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' V-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Last but not least, the dynamic walking  and the aerial manipulation will be investigated to enhance the  versatility of this robot in both maneuvering and manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
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+page_content=' 172–178, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
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+page_content='  Design, modeling, and control of an aerial robot  DRAGON: A dual-rotor-embedded multilink robot with the ability  of multi-degree-of-freedom aerial transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' IEEE Robotics and  Automation Letters, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
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+page_content=' 1176–1183, April 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  [19] Moju Zhao, Kei Okada, and Masayuki Inaba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Versatile articulated aerial  robot DRAGON: Aerial manipulation and grasping by vectorable thrust  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' The International Journal of Robotics Research, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  [20] Fan Shi, Tomoki Anzai, Yuta Kojio, Kei Okada, and Masayuki In-  aba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Learning agile hybrid whole-body motor skills for thruster-aided  humanoid robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  In 2022 IEEE/RSJ International Conference on  Intelligent Robots and Systems (IROS), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
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+page_content=' Three-dimensional posture optimization for biped  robot stepping over large ditch based on a ducted-fan propulsion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
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+page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Leok, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' McClamroch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' Geometric tracking control  of a quadrotor UAV on SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' In 49th IEEE Conference on Decision  and Control (CDC), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content=' 5420–5425, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
+page_content='  E' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE2T4oBgHgl3EQfrgiA/content/2301.04050v1.pdf'}
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+CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+1
+Explicitly Solvable Continuous-time Inference for
+Partially Observed Markov Processes
+Daniel Chen, Alexander G. Strang, Andrew W. Eckford Senior Member, IEEE, Peter J. Thomas
+Abstract—Many natural and engineered systems can be mod-
+eled as discrete state Markov processes. Often, only a subset
+of states are directly observable. Inferring the conditional prob-
+ability that a system occupies a particular hidden state, given
+the partial observation, is a problem with broad application.
+In this paper, we introduce a continuous-time formulation of
+the sum-product algorithm, which is a well-known discrete-time
+method for finding the hidden states’ conditional probabilities,
+given a set of finite, discrete-time observations. From our new
+formulation, we can explicitly solve for the conditional probability
+of occupying any state, given the transition rates and observations
+within a finite time window. We apply our algorithm to a
+realistic model of the cystic fibrosis transmembrane conductance
+regulator (CFTR) protein for exact inference of the conditional
+occupancy probability, given a finite time series of partial
+observations.
+I. INTRODUCTION
+Markov processes—dynamic processes whose future be-
+havior depends only on their present state—approximate a
+wide variety of natural and engineered systems. Despite rapid
+advances in high-throughput data acquisition and data pro-
+cessing, many systems of interest contain important degrees
+of freedom that cannot be directly observed. Inferring the
+conditional probability that such a partially observed Markov
+process occupies specific hidden states, given the available ob-
+servations, is a ubiquitous problem in science and engineering.
+Examples appear in robotics [1], ecology [2], neuroscience [3],
+and algorithmic text analysis [4].
+We are motivated by biological examples in the present
+paper. Ion channels in excitable membranes, such as the
+sodium (Na+) and potassium (K+) channels described in
+Hodgkin and Huxley’s quantitative model for action potential
+generation [5], [6], [7], provide an early example. Discrete
+state Markov models based on Hodgkin and Huxley’s K+
+channel contain five states, only one of which conducts an
+ionic current; the other states are “silent” and cannot be distin-
+guished by direct electrophysiological observation. Similarly,
+This work was supported in part by National Institutes of Health BRAIN
+Initiative grant R01 NS118606 and a National Science Foundation grant
+DMS-2052109 to PJT, as well as research support from the Oberlin College
+Libraries, and an NSERC Discovery grant to AWE.
+D. Chen and P. J. Thomas are with the Department of Mathematics, Applied
+Mathematics, and Statistics; Department of Electrical, Control and Systems
+Engineering; Department of Computer and Data Science; Department of
+Biology, Case Western Reserve University, Cleveland, OH 44106 USA (e-
+mail: txc461/pjthomas@case.edu).
+A. G. Strang is with Department of Statistics, University of Chicago,
+Chicago, IL 60637 USA (email: alexstrang@uchicago.edu).
+A. W. Eckford is with Department of Electrical Engineering and Computer
+Science, York University, Toronto, ON M3J 1P3 Canada (e-mail: aeck-
+ford@yorku.ca).
+the Na+ channel has eight states: seven with zero conductance
+and one with nonzero conductance. Colquhoun and Hawkes
+introduced maximum likelihood methods for inferring the rate
+constants of a partially observed Markov process representing
+the nicotinic Acetylcholine receptor [8], [9], [10], [11], but
+did not address the question of inferring microscopic state
+occupancy from observable conductance time series. More
+recently, research into the molecular biology of cystic fibrosis
+(CF) has focused on the CF transmembrane conductance
+regulator (CFTR), which can be modeled as a 7-state system
+with two conducting states and five nonconducting states
+[12] (detailed below in Section V). Beyond these biological
+examples, problems of inferring or estimating hidden states
+from incomplete observations are widely studied in the signal
+processing literature [13].
+The literature contains several approaches to approximating
+the behavior of hidden states of partially observed Markov
+processes. Sampling provides one common technique for
+approximate inference [14]. As an example, recent work by
+Fang et al. demonstrated an efficient algorithm for simulating
+stochastic reaction networks with multiple separated time
+scales using particle filters [15]. In general, Markov Chain
+Monte Carlo is a widely employed sampling technique used
+to infer hidden states [16], [17], that has also been applied to
+ion channels [18]. For partially observed Bayesian networks
+operating in discrete time, message passing algorithms on
+factor graphs provide an efficient and exact inference method
+[19]. The factor-graph formalism is highly flexible. Algorithms
+based on the message passing concept have been extended to
+applications in localization [20], compressed sensing [21], and
+decision fusion [22]. Factor graphs are not limited to models
+with a discrete number of variables. For example, Gaussian
+message passing in linear Gaussian models (e.g. Kalman
+filtering and smoothing) has been developed for continuous-
+time models with discrete-time observations [23], [24], [25].
+However, the state reconstruction problem for continuous-time
+finite-state hidden Markov models has not been addressed in
+the literature, to be best of our knowledge.
+In this work, we extend the message-passing algorithm in
+order to analytically interpolate state-occupancy probabilities
+of a continuous-time system, given a discretely sampled time
+series. That is, we show how to infer the time-dependent
+conditional probabilities of latent states for continuous-time
+discrete-state homogeneous Markov processes given a set of
+partial observations over a finite time window. We derive
+an equivalent formulation of the sum-product algorithm in
+continuous time that allows one to find an explicit ana-
+lytic solution for the state occupancy probability. Having
+arXiv:2301.00843v1  [eess.SP]  2 Jan 2023
+
+CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+2
+explicit solutions lowers the computational cost and, unlike
+sampling-based approximate methods, does not sacrifice ac-
+curacy. Furthermore, the continuous-time formalism leads to
+elegant simplification of the analytic solutions. For certain
+systems—like the three-state systems shown in Figure 2—the
+conditional probability obeys a second-order inhomogeneous
+linear ordinary differential equation. Finally, we demonstrate
+the practical functionality of the algorithm with the 7-state
+model for CFTR using simulated data.
+The paper is organized as follows. Section II reviews the
+message-passing algorithm for Markov processes with binary
+observations. Section III displays the main result of the paper:
+a continuous-time formulation of the sum-product algorithm.
+We present the derivation of the conditional probability using
+the message-passing approach in continuous time, and the
+corresponding sum-product algorithm. Section IV discusses
+the implications of the continuous-time formulation further.
+We give examples of small systems to display how analytic
+solutions may be found. In Section V we demonstrate the value
+of the algorithm for a larger, realistic system.
+II. THEORY OF THE SUM-PRODUCT ALGORITHM
+The sum-product algorithm can be used generally for in-
+ference on probabilistic models that can be written as factor
+graphs [19]. There are many variants of the sum-product
+algorithm, each suitable for accomplishing a different task.
+For our purposes, we will focus on the forward/backward
+algorithm for inference on hidden Markov models.
+We consider a continuous-time, discrete-state homogeneous
+Markov process on a finite state space Ω. Given a discrete,
+uniformly spaced sampling interval, the continuous-time pro-
+cess induces a discrete-time Markov process specified by some
+column-stochastic transition matrix P that is invariant in time.
+Let S ⊂ Ω be a subset of states, and let St ∈ Ω be the state
+of the system at time t. We assume an observer can only
+see whether St is in S or not. Accordingly, let Yt = m(St)
+represent the observable where m(s) : Ω → {0, 1} is the
+indicator function for the set S. The goal of the algorithm
+is to infer the conditional probability of being at a particular
+state, i ∈ Ω, given the binary observation.
+The algorithm involves three vector-valued quantities: a for-
+ward message αt, a backward message βt, and the observation
+message χt. One can interpret the forward message as the
+probability of arriving at a certain state from time 0 to time
+t and the backward message as the likelihood of occupying
+a certain state at time t conditioned on ending up in a given
+target state, or a given target set of states, at the end of the
+measurement T. On the other hand, the observation message
+χt is an indicator function of the possible states given the
+observation. For instance, if at time t, the observation of the
+total system were Yt = 1, then χt would be a vector with 1’s
+on the states in the observation set S and 0 otherwise. Using
+superscripts to denote vector indices, i.e. v(i) denotes the i-
+th element of vector v, we present pseudo-code for the sum-
+product algorithm in Algorithm 1. Note that, the normalization
+constant Z = �
+k α(k)
+t
+β(k)
+t
+in Line 10 of Algorithm 1 is time
+invariant [26].
+Algorithm 1 Forward/Backward Algorithm
+Input: The transition matrix P ∈ Rn×n, the observation
+message χt for t ∈ {1, . . . , T}.
+Output: The inferred probability pt for t ∈ {1, . . . , T}.
+1: Initialize the forward message α0 = π, the stationary
+distribution of P
+2: Initialize the backward message β(i)
+T
+= 1, for all i
+3: for t from 2, . . . , T do
+4:
+αt = diag(χt)Pαt−1
+5: end for
+6: for s from T − 1, . . . , 1 do
+7:
+βs = P ⊺diag(χs+1)βs+1
+8: end for
+9: for t from 1, . . . , T do
+10:
+pt =
+αt⊙βt
+�
+k α(k)
+t
+β(k)
+t
+11: end for
+Fig. 1 illustrates the sum-product algorithm’s application to
+a time series of discretely sampled observations. Consider a
+three-state-chain with symmetric transition rates as shown in
+the top-left of Figure 1. If the system takes states 1 or 2,
+“0” will be observed, and “1” will be observed otherwise.
+Using the sum-product algorithm, we can find the probability
+of the system occupying each state given the discrete-time
+observations. As shown in the bottom row, as the sampling
+time step decreases, the conditional probabilities appear to
+converge to a smooth curve within each interval with a fixed
+observation (either “0” or “1”). Intuitively, there should exist
+a continuous, perhaps piecewise differentiable, representation
+of the conditional probability of a partially observed process.
+We formalize this intuition below.
+III. CONTINUOUS-TIME MESSAGE PASSING
+In this section, we present the main result, namely the
+derivation of the continuous-time message passing algorithm.
+In this new formalism, messages are passed in the form
+of linear differential equations on possible states given the
+observable system. In order to guarantee the existence of the
+continuous sum-product algorithm, we assume the following
+conditions.
+Assumptions:
+A1 The continuous-time process {S(t); t ∈ [0, T]} takes
+values in a finite state space Ω = {1, 2, . . . , N}.
+A2 S(t) has the Markov property, and has exponentially
+distributed waiting times parameterized by a rate matrix
+W, with wji specifying the transition rate from state i to
+state j. Note that W is constant in the interval [0, T].
+A3 There is a distinguished subset S ⊂ Ω such that the
+observable process Y(t) satisfies
+Y(t) =
+�
+1
+if S(t) ∈ S,
+0
+otherwise .
+Further details appear in Appendix A.
+Under these assumptions, we obtain a continuous-time ver-
+sion of the sum-product algorithm by executing the following
+steps (made rigorous in the proof of Theorem 1 below). Write
+
+CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+3
+Fig. 1: Illustrating convergence of conditional state occupancy probabilities to a differentiable function for a three-state model.
+(Top left) The state diagram. (Top middle) True simulated states. (Top right) Binary observation derived from true states.
+(Bottom row) Inference of hidden states via the sum-product algorithm with time steps 2.5 sec (Bottom Left), 1.0 sec (Bottom
+Middle) and 0.5 sec (Bottom Right).
+out the matrix multiplication of the discrete-time algorithm
+element-wise. Focus on one sojourn where the observation
+doesn’t change. Within that time interval, take the limit as the
+time step goes to zero to derive the continuous-time dynamics
+of the conditional probabilities. Extend the solution to the
+full time interval via appropriate boundary conditions at the
+transition between each sojourn. The main result is stated in
+the theorem below.
+Theorem 1. Suppose processes S(t) and the associated
+process Y(t) satisfies assumptions A1, A2, and A3 above.
+Then, given a realization of the process Y(t), the conditional
+probability p(t) = Pr[S(t)|Y(t)] exists, is piecewise smooth
+(C∞), and is C∞ on all intervals where Y(t) is constant. In
+particular, p(t) takes the form
+p(t) =
+ρ(t)
+�
+k ρ(k)(t) =
+α(t) ⊙ β(t)
+�
+k α(k)(t) ⊙ β(k)(t)
+(1)
+where ⊙ denotes the element-wise product, and the quantities
+α(t) and β(t) are functions of time that follow the linear
+ordinary differential equations
+dα(i)(t)
+dt
+=
+�
+k∈S\{i}
+wkiα(k)(t) −
+�
+l̸=i
+wilα(i)(t),
+(2)
+dβ(j)(t)
+dt
+= −
+�
+k∈S\{j}
+wjkβ(k)(t) +
+�
+l̸=j
+wjlβ(j)(t).
+(3)
+Proof. Without loss of generality, focus on the case where
+Y(0) = Y(T) = 0, and Y(t) = 1 for 0 < t < T. We use
+q(t) to denote a quantity, q, evolving in continuous time on
+the interval [0, T], and qt to denote the same process sampled
+at discrete times.
+Let the time interval [0, T] be discretized with a step size
+∆t = T/n for some integer n ≫ 1. At each time step,
+S(t) is sampled. Then, the sum-product algorithm can be used
+to solve for pt. Writing the matrix multiplication out yields
+the following set of equations for the forward and backward
+messages in discrete time:
+α(i)
+t+∆t =
+�
+k
+Pr[st+∆t = i|st = k]α(k)
+t
+χ(i)
+t ,
+(4)
+β(i)
+t
+=
+�
+k
+Pr[st+∆t = k|st = i]β(k)
+t+∆tχ(k)
+t+∆t.
+(5)
+We neglect states not in S because only the probability
+conditioned on the observations is of interest. Then, for any
+state i ∈ S, we argue that the corresponding forward message,
+α(i)(t), and backward message, β(i)(t) in continuous time can
+be written as solutions of systems of differential equations,
+upon taking limits as ∆t → 0. For notational simplicity, for
+
+true states
+observation
+3
+W2
+2
+W/2
+W23
+0
+0
+50
+100
+0
+50
+100
+sample every 2.5 seconds
+sample every 1 seconds
+sample every 0.5 seconds
+1
+0.8
+0.8
+0.8
+0.6
+0.6
+0.6
+0.4
+0.4
+0.4
+0.2
+0.2
+0.2
+0
+0
+0
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+4
+t > τ we let P(i,j)
+t,τ
+= Pr[S(t) = i|S(τ) = j].
+dα(i)(t)
+dt
+= lim
+∆t→0
+α(i)(t + ∆t) − α(i)(t)
+∆t
+(6)
+= lim
+∆t→0
+1
+∆t
+� �
+k
+P(i,k)
+t+∆t,tα(k)(t)χ(i)(t) − α(i)(t)
+�
+(7)
+= lim
+∆t→0
+1
+∆t
+�
+�
+k∈S\{i}
+(wki∆t + o(∆t))α(k)(t) . . .
+−
+�
+l̸=i
+(wil∆t + o(t))α(i)(t)
+�
+(8)
+=
+�
+k∈S\{i}
+wkiα(k)(t) −
+�
+l̸=i
+wilα(i)(t)
+(9)
+dβ(i)(t)
+dt
+= lim
+∆t→0
+β(i)(t + ∆t) − β(i)(t)
+∆t
+(10)
+= lim
+∆t→0
+1
+∆t
+�
+β(i)(t + ∆t) . . .
+−
+�
+k
+P(k,i)
+t+∆t,tβ(k)(t + ∆t)χ(k)(t + ∆t)
+�
+(11)
+= lim
+∆t→0
+1
+∆t
+�
+−
+�
+k∈S\{i}
+(wik∆t + o(∆t))β(k)(t + ∆t)
++
+�
+l̸=i
+(wil∆t + o(t))β(i)(t + ∆t)
+�
+(12)
+= −
+�
+k∈S\{i}
+wikβ(k)(t) +
+�
+l̸=i
+wilβ(i)(t)
+(13)
+where lim∆t→0 o(∆t)/∆t = 0. Readers could refer to Ap-
+pendix A or consult existing literature such as [27] for the
+relationship between transition probabilities and transition
+rates of a Markov jump process.
+The conditional probability can be found by solving the
+differential equations, taking the component-wise product of
+α(t) and β(t) for every t ∈ [0, T], and normalizing so as to
+obtain a valid probability distribution.
+The differential equation formulation is only applicable for
+the time intervals where the observation Y(t) is constant.
+When the observable Y(t) changes (when the systems transi-
+tions from a state in S to a state not in S), the probability with
+respect to time might not be differentiable; in some cases, it
+is not even continuous. Therefore, we must specify boundary
+conditions to connect the probabilities from one sojourn to
+the next. The observable may change either by the system
+entering S or else leaving S. Suppose a transition occurred
+within time (t∗ −∆t, t∗] such that, for t < t∗ −∆t, S(t) ∈ S,
+and S(t) ̸∈ S for t ≥ t∗. Call this event E. Then, we obtain
+the following transition rule for the forward message.
+Pr[S(t∗) = j|E]
+=
+�
+i∈S Pj,i
+t∗,t∗−∆tPr[S(t∗ − ∆t) = i]
+�
+k̸∈S
+�
+i∈S Pk,i
+t∗,t∗−∆tPr[S(t∗ − ∆t) = i]
+(14)
+=
+�
+i∈S(wji∆t + o(∆t))Pr[S(t∗ − ∆t) = i]
+�
+k̸∈S
+�
+i∈S(wki∆t + o(∆t))Pr[S(t∗ − ∆t) = i]
+(15)
+→
+�
+i∈S wjiPr[S(t−
+∗ ) = i]
+�
+k̸∈S
+�
+i∈S wkiPr[S(t−
+∗ ) = i]
+(16)
+as ∆t → 0. Here Pr[S(t−
+∗ ) = i] is the probability of occupying
+state i the instant before the transition, which can be found by
+solving the differential equations introduced above.
+We handle the boundary conditions at state transitions for
+the backward message similarly. Define E as above and let s∗
+be a particular goal state. Then:
+Pr[S(T) = s∗|S(t∗ − ∆t) = j, E]
+=
+�
+i̸∈S
+Pr[S(T) = s∗, S(t∗) = i|S(t∗ − ∆t) = j, E]
+=
+�
+i̸∈S Pr[S(T) = s∗|S(t∗) = i]Pi,j
+t∗,t∗−∆t
+�
+k∈S
+�
+i̸∈S Ps∗,i
+T,t∗Pi,k
+t∗,t∗−∆t
+(17)
+=
+�
+i̸∈S Pr[S(T) = s∗|S(t∗) = i](wij∆t + o(∆t))
+�
+k∈S
+�
+i̸∈S Ps∗,i
+T,t∗(wik∆t + o(∆t))
+(18)
+=
+�
+i̸∈S Pr[S(T) = s∗|S(t∗) = i](wij + o(∆t)
+∆t )
+�
+k∈S
+�
+i̸∈S Ps∗,i
+T,t∗(wik + o(∆t)
+∆t )
+(19)
+→
+�
+i̸∈S wijPr[S(T) = s∗|S(t∗) = i]
+�
+k∈S
+�
+i̸∈S wikPr[S(T) = s∗|S(t∗) = i]
+(20)
+as ∆t → 0. Here, Pr[S(T) = s∗|S(t∗) = i] is a hitting
+probability associated with the backward message at time
+t∗, which can be found by solving the backward-message
+differential equation. These boundary conditions, together with
+the differential equations (2)-(3), give the continuous-time
+evolution of the conditional probability for any finite-length
+observations.
+Note that the equations (2)-(3) extend to the case where
+Y(0) = Y(T) = 1 and Y(t) = 0 for 0 < t < T by viewing
+S ← Ω \ S. So, given a time series observation Y(t) where
+observation (the value of Y(t)) changes at time 0 < t1 < t2 <
+. . . tm, we can solve for the analytic solution at any interval
+with consistent observation (ti, ti+1] using equation (2) and
+(3). Then, use the result to compute the initial condition for the
+next interval — namely, (ti+1, ti+2] for the forward message
+and (ti−1, ti] for the backward message — as specified in (16)
+and (20). Thus, the statement of Theorem 1 holds.
+A general expression for the conditional probability can be
+obtained, but it is not of great utility in most systems. Yet,
+there are certain special cases that yield elegant solutions; we
+introduce several examples in Section IV.
+The continuous-time sum-product algorithm follows directly
+from the derivation above, and is outlined in Algorithm 2.
+
+CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+5
+The discrete algorithm passes information through matrix
+multiplication of a truncated transition matrix; the continuous-
+time algorithm does the same by solving a system of linear
+differential equations using the truncated rate matrix. Since the
+final conditional probability is only piecewise differentiable,
+the boundary condition must be applied whenever a transition
+in or out of the observable set S occurs.
+Algorithm 2 Continuous-time Forward/Backward Algorithm
+Input: The rate transition matrix W ∈ Rn×n, the observed
+process Y(t).
+Output: The inferred probability p(t) for t ∈ [0, T].
+1: Let [t1, t2, . . . , tm] be a list of times where transitions
+occur
+2: τ ← 0
+3: α∗ ← π, the stationary distribution
+4: for j from 1, . . . , m do
+5:
+αj ← solution to the forward message differential
+equation (Equation 2) from τ to tj with initial condition
+α∗
+6:
+τ ← tj
+7:
+α∗ ← distribution specified according to Equation 16
+8: end for
+9: τ ← T
+10: β∗ ← the uniform distribution
+11: for j from m, . . . , 1 do
+12:
+βj ← solution to the backward message differential
+equation (Equation 3) from τ to tj with initial condition
+β∗
+13:
+τ ← tj
+14:
+β∗ ← distribution specified according to Equation 20
+15: end for
+16: α(t), β(t) ← concatenation αj’s and βj’s
+17: Compute ρ(t) = α(t)⊙β(t), the component-wise product
+between α(t) and β(t) pointwise with respect to t
+18: Compute the conditional probability p(t) =
+ρ(t)
+�
+k ρ(k)(t)
+From a practical perspective, having the ability to solve
+for the conditional probabilities exactly through differential
+equations drastically lowers the computational cost of the
+forward/backward algorithm. Traditionally, the discrete-time
+algorithm propagates the forward and backward messages
+through matrix operations at each time step. For long time-
+series and/or high-dimensional systems, this is computation-
+ally prohibitive. Through our continuous-time formalism, we
+solve the differential equations analytically, which is an op-
+eration that is independent of the length of the time-series,
+to find the forward or backward message at any time point.
+This difference effectively reduces the asymptotic scaling
+from O(∆t−1) to O(1), with the later scaling only in the
+number of observable transitions. In scenarios where finding
+the appropriate boundary condition would require an iterative
+procedure of solving the forward and backward messages
+multiple times, our continuous time approach should be much
+more efficient than the traditional discrete-time method. We
+discuss the performance of the continuous-time message-
+Fig. 2: State diagrams of two systems for which the
+continuous-time message passing algorithm exhibit analytic
+simplifications. (Left) 3-state chain with symmetric rates
+w12 = w21. (Right) Irreversible 3-state loop. States marked
+in red return Y(t) = 1 and blue return Y(t) = 0.
+passing algorithm further in Section V.
+IV. ANALYTIC SOLUTION
+Theorem 1 in the previous section shows that the conditional
+probability is always available analytically upon normalizing
+the component-wise product of the forward and backward
+messages,
+i.e. p(t) = ρ(t)/Z where Z = �
+k ρ(k)(t).
+As
+in the discrete-time case, the normalizing term Z is time-
+invariant in the continuous-time case as well. See Appendix
+B. The conditional probability may therefore be expressed in
+a particularly elegant form in certain cases, namely as the
+solution of a linear nonhomogeneous second-order differential
+equation. We begin this section by considering two examples.
+Following the examples, we consider extensions to higher
+dimensions.
+A. Symmetric 3-State Chain
+Consider the three-state chain depicted in the left panel of
+Figure 2, where states 1 and 2 are hidden. Assume that the per-
+capita transition rates within the hidden block are symmetric,
+i.e. w12 = w21 > 0, and assume w13 = w31 = 0. The rates
+w23 > 0 and w32 > 0 may be arbitrary. These assumptions
+result in the following rate matrix:
+W =
+�
+�
+−w21
+w12
+0
+w21
+−(w12 + w32)
+w23
+0
+w32
+−w23
+�
+� .
+(21)
+In this case, let S = {3}, the singleton set of state 3. When
+Y(t) = 1 the inference problem is trivial since the system
+takes state 3 with probability one. Thus, we emphasize the
+intervals when Y(t) = 0.
+First, consider the forward message given by the following
+system of differential equations
+dα
+dt =
+�−w21
+w12
+w21
+−(w12 + w32)
+�
+α .
+(22)
+Note that the matrix defining the system of equations corre-
+sponds to the upper left block of W. To simplify notation, let
+w21 = w12 = a and w32 = b. Then, the submatrix reduces to
+the following form:
+dα
+dt =
+�−a
+a
+a
+−a − b
+�
+α .
+(23)
+
+W
+13
+W
+W
+21
+32
+2
+W
+32
+W
+W
+12
+23
+W
+21
+2CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+6
+This is a linear system of differential equations that can
+be solved exactly. The sub-matrix is real-symmetric, so is
+diagonalizable. Therefore, the solution will be of the form:
+α(t) = Aeλ1tv1 + Beλ2tv2.
+(24)
+where λi is an eigenvalue of the rate submatrix and vi is the
+corresponding eigenvector. The eigenvalues and vectors are:
+λ1/2 = a(−1 ±
+�
+1 + γ2) − γ,
+(25)
+v1/2 =
+�
+γ ±
+�
+1 + γ2
+1
+�
+,
+(26)
+where γ = b/2a. Constants A and B are found through
+the initial condition given below. In the previous section, the
+initial forward message was set to the equilibrium distribution.
+However, the chain structure leaves no ambiguity in the state
+occupied when the observation changes from 1 to 0. So, the
+initial forward message is the delta distribution on state 2:
+α(0) =
+�0
+1
+�
+.
+(27)
+After some time T, the system re-enters the visible state,
+namely, state 3 again. By the same reasoning, we also have an
+unambiguous boundary condition for the backward message:
+β(T) =
+�0
+1
+�
+.
+(28)
+Also, by the symmetry in the rates, the backward message
+evolves according to the same equations as the forward mes-
+sage, but backward in time. Thus, β(t) = α(T − t),
+β(t) = Aeλ1(T −t)v1 + Beλ2(T −t)v2
+(29)
+where λi’s, vi’s, constants A and B all remain the same due
+to symmetry.
+The
+conditional
+probability
+is
+proportional
+to
+the
+component-wise product ρ = α ⊙ β, which yields:
+ρ(t) = A
+�
+eλ1(T −t)+λ2t + eλ1t+λ2(T −t)�
++ B,
+(30)
+where
+A = AB(v1 ⊙ v2) =
+�−AB
+AB�⊺,
+(31)
+B = A2eλ1T (v1 ⊙ v1) + B2eλ2T (v2 ⊙ v2).
+(32)
+To recover the conditional probability, we must normalize such
+that the component sum evaluates to 1. In this case, since the
+submatrix in Equation 22 is symmetric, the eigenvectors of the
+submatrix are orthogonal to each other, which means the sum
+of the components of v1⊙v2, or the inner-product between v1
+and v2, evaluates to 0. Then, the component-wise sum of A is
+zero, so the normalizing constant equals the component-wise
+sum of B.
+Write p(t) to represent the conditional probability. We write
+Z = �
+k B(k) to represent the normalizing constant. The final
+equation describing the conditional probability can thus be
+rewritten in the following form:
+p(t) = 1
+Z
+�
+Ae(λ1−λ2)teλ2T + Ae(λ2−λ1)teλ1T + B
+�
+. (33)
+The components of p(t) may also be expressed as the solutions
+of a second-order ordinary differential equation, cf. (51)-(52).
+B. Irreversible 3-State Loop
+Light-gated Channelrhodopsin-2 (ChR2) receptors can be
+modeled with a 3-state chain where each vertex has out-degree
+1 and forms a cycle, as depicted in Figure 2, right panel [28],
+[29]. Let state 1 be open, and states 2 and 3 be closed. That
+is, S = {1}. The open/closed status of the channel is observed
+through voltage recordings: high conductance indicates that the
+channel is in the open state, and low conductance implies the
+closed state. The rate matrix can be written as the following:
+W =
+�
+�
+−w21
+0
+w13
+w21
+−w32
+0
+0
+w32
+−w13
+�
+�
+(34)
+where wji is the transition rate from state i to state j.
+Conditioning on the channel being closed, the message-passing
+algorithm takes the lower right 2 × 2 block as the transition
+matrix.
+When transitioning from S to a state not in S, the system
+must enter state 2 first and exit through state 3. Thus the
+boundary conditions for the forward and backward message
+are
+α(0) =
+�1
+0
+�
+,
+β(T) =
+�0
+1
+�
+(35)
+where the first component corresponds to state 2 and second
+state 3. Suppose w32 ̸= w13 and let γ =
+w32
+w13−w32 . Then, the
+solution of the message-passing differential equations with the
+initial condition enforced satisfies:
+α(t) = e−w32t
+�1
+γ
+�
+− γe−w31t
+�0
+1
+�
+,
+(36)
+β(t) = e−w32(T −t)
+�
+1
+0
+�
++ 1
+γ e−w13(T −t)
+�
+−γ
+1
+�
+.
+(37)
+To get the conditional probability, first take the component-
+wise product between the forward and backward message at
+time t.
+ρ(t) = e−w32t−w13(T −t)
+�
+−1
+1
+�
++
+� e−w32T
+−e−w13T
+�
+(38)
+As in the previous example, the normalizing constant is
+invariant in time. In this case, Z = e−w32T − e−w13T . Thus,
+the time evolution of the conditional probability can be written
+p(t) = ρ(t)/Z.
+(39)
+Next, we consider a case in which the submatrix is not diag-
+onalizable. We set w = w32 = w13, so that the corresponding
+submatrix:
+U =
+�−w
+0
+w
+−w
+�
+(40)
+admits the following Jordan normal form:
+U =
+�0
+w
+1
+0
+� �−w
+1
+0
+−w
+� � 0
+1
+1
+w
+0
+�
+.
+(41)
+Using the same initial condition as Equation (35) to solve the
+system of differential equations yields the following forward
+message.
+α(t) =
+� e−wt
+te−wt
+�
+(42)
+
+CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+7
+We solve for the backward message by undergoing a similar
+procedure and yield
+β(t) =
+�(T − t)e−w(T −t)
+e−w(T −t)
+�
+.
+(43)
+Upon taking the component-wise product, we observe that the
+evolution of conditional probability is independent of time:
+ρ(t) = e−wT
+�
+T − t
+t
+�
+,
+Z =
+1
+Te−wT .
+(44)
+Since both the time entering and exiting the hidden states are
+fixed, we can see that the probability flows linearly from one
+state to the other at equal rates.
+Time-invariant normalization is a fundamental property of
+the sum-product algorithm [26]. In Appendix B we give an
+elementary demonstration of this property for continuous-time
+systems with time-homogeneous transition rates, as illustrated
+by the two cases presented above.
+C. Generalization
+We show that under certain circumstances, the conditional
+probability follows a second-order nonhomogeneous linear
+ordinary differential equation.
+Corollary 1. For 3-state systems where the truncated subma-
+trix U has distinct eigenvalues, the conditional probability can
+be written as the solution of a nonhomogeneous second-order
+linear differential equation with constant coefficients.
+Proof. Let (λi, vi) be eigenpairs for U and let (λi, wi) be
+eigenpairs for U ⊺. Then the forward and backward messages
+are given by the expressions:
+α(t) = Aeλ1tv1 + Beλ2tv2,
+(45)
+β(t) = Ceλ1(T −t)w1 + Deλ2(T −t)w2.
+(46)
+Taking the component-wise product yields the form:
+ρ(t) = A
+�
+eλ1t+λ2(T −t)�
++ B
+�
+eλ1(T −t)+λ2t�
++ C,
+(47)
+= A
+�
+e(λ1−λ2)teλ2T �
++ B
+�
+e(λ2−λ1)teλ1T �
++ C,
+(48)
+where
+A = AD(v1 ⊙ w2),
+B = BC(v2 ⊙ w1),
+(49)
+C = ACeλ1T (v1 ⊙ w1) + BDeλ2T (v2 ⊙ w2).
+(50)
+Since the left and right eigenvectors corresponding to dif-
+ferent eigenvalues are orthogonal, all time-dependent terms
+cancel when normalizing. Thus the normalization constant
+Z = �
+k C(k). Upon differentiating p(t) = ρ(t)/Z twice with
+respect to time, we arrive at the second-order linear differential
+equation that describes the time evolution of the condition
+probability:
+d2p
+dt2 = (λ1 − λ2)2
+Z
+�
+Ae(λ1−λ2)teλ2T + Be(λ2−λ1)teλ1T �
+(51)
+= (λ1 − λ2)2
+Z
+(p − C).
+(52)
+This second order equation comes equipped with two bound-
+ary conditions set by fixing α(0) and β(T).
+Fig. 3: State diagram for CFTR. When the protein enters one of
+the red states, the ion channels will open and conduct a current.
+On the other hand, when CFTR is in one of the blue states,
+then the ion channels are closed and conducts no current. For
+this system S = {4, 5}.
+Corollary 2. For a system of arbitrary dimension, for which
+the truncated submatrix U has exactly two distinct eigenvalues
+and is diagonalizable, the conditional probability can be
+written as a second-order linear differential equation of the
+same form as Equation 52.
+Proof. Let λ1 and λ2 be the two distinct eigenvalues. Let vi be
+the sum of all right eigenvectors corresponding to eigenvalue
+λi, and let wi be the sum of all left eigenvectors corresponding
+to eigenvalue value λi. Then by diagonalizability, the proof for
+this corollary follows the proof of Corollary 1.
+This result extends Corollary 1 to higher dimensional sys-
+tems, under special conditions. These two corollaries match
+intuition: we are concerned with the state occupancy prob-
+ability conditioned on both the entrance and exit times of
+an observable state, so the probability flow obeys a second-
+order differential equation, which also requires specifying two
+boundary conditions. We note, however, that if the submatrix
+has three or more distinct eigenvalues, we do not expect a
+similar result to hold. Moreover, generic matrices typically
+have as many distinct eigenvalues as their dimension, so we
+do not expect this scenario to occur outside of special cases
+when the rate submatrix is highly regular. In addition, it would
+be difficult to check if a given rate submatrix satisfies these
+conditions through numerical solvers, as equality of repeated
+eigenvalues might be obscured by floating point arithmetic.
+Therefore, as much as the second-order differential equations
+interpretation is intuitively appealing, it is likely not a generic
+phenomenon that we should expect for arbitrary rate matrices.
+V. BIOLOGICAL EXAMPLE: CFTR
+Here, we show that our algorithm applies to a higher
+dimensional system, namely the 7-state model of the cystic
+fibrosis transmembrane conductance regulator (CFTR) protein.
+CF is a common life-threatening genetic disorder. CFTR is
+an important protein that regulates the opening and closing
+of ion channels. Loss of CFTR function causes pancreatic
+insufficiency as well as airway infection due to excessive
+mucus, which in turn can cause a variety of complications
+such as impaired innate immunity and respiratory failure [30].
+Mathematically, the behavior of CFTR can be captured
+with a 7-state hidden Markov process. Figure 3 illustrates
+
+2
+3
+4
+5
+7
+6CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+8
+Fig. 4: The inference result of the continuous-time algorithm (top) and the discrete-time algorithm (middle) for the true state
+occupancy simulated from the Gillespie algorithm (bottom). Sampling time step ∆t = 10−4.
+the state diagram [12]. The ion channel opens (conducts an
+ionic current) when CFTR is in state four or five (marked in
+red). When in the nonconducting states (marked in blue) the
+ion channel is closed. Transmembrane conductance recordings
+report whether CFTR is in a conducting or a nonconducting
+state, but do not directly indicate which of the possible states
+is occupied. Using the numbering shown in Figure 3, the rate
+matrix is defined as follows:
+W =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−9.9
+5.0
+0
+0
+0
+0
+1.7
+9.9
+−12.7
+5.8
+0
+0
+0
+0
+0
+7.7
+−10.7
+10.0
+0
+0
+0
+0
+0
+4.9
+−17.1
+0
+0
+0
+0
+0
+0
+7.1
+−3.0
+7.0
+0
+0
+0
+0
+0
+3.0
+−13.0
+12.8
+0
+0
+0
+0
+0
+6.0
+−14.5
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(53)
+We used Gillespie’s exact stochastic simulation algorithm
+[31], [32], [33] to generate sample traces of the ion channel
+states and recordings. The simulation produces discrete state,
+continuous time trajectories. We introduced a finite sampling
+time step to discretize the simulated data along the time axis,
+consistent with data obtained through experimental recordings.
+Then, we place the generated data into discretized time bins
+where the size of each bin (one can think of this as the sam-
+pling time step) is a parameter that can be altered. We solved
+the differential equations giving the forward and backward
+messages exactly via function handles in Matlab. Figure 4
+compares the traces produced by the classical discrete-time
+algorithm, our continuous-time algorithm, and the true states,
+and shows excellent agreement among the respective curves.
+With a sufficiently small time step (∆t = 10−4), the contin-
+uous and discrete-time algorithms show no visible discrepancy,
+as expected. Figure 5 shows that the maximum discrepancy
+Fig. 5: First-order convergence of the message passing al-
+gorithms as a function of sampling time step. The mean
+maximum difference was plotted along with one standard
+deviation about the mean as the error bar for an ensemble
+of forty trajectories.
+(ℓ∞ norm) between the conditional probabilities generated by
+the continuous time and discrete time algorithms decreases
+linearly with the sampling time step ∆t. The figure shows
+the results from an ensemble of forty independent repeated
+trials for each sample size. Again, we emphasize that while
+the discrete time algorithm requires iterating over all samples,
+which scales O(∆t−1), the continuous time algorithm outputs
+a description of the conditional probability — that is, a
+function that returns a value at an exact time point queried
+— in O(1) time, scaling only with the number of transitions.
+
+inferred probability of states: continuous
+1234567
+0.9
+0.8
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+×104
+0.7
+inferred probability of states: discrete
+1234567
+0.6
+0.5
+0.4
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+×104
+0.3
+true states
+0.2 
+134567
+0.1
+0.5
+1.5
+2
+2.5
+3.5
+4.5
+0
+3
+4
+5
+time
+×10410~2
+maximum difference 
+10~3
+10~5
+10~6
+10-7
+10-7
+10-6
+10-5
+10-4
+10-3
+size of sampling intervalCHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+9
+VI. DISCUSSION & CONCLUSIONS
+This paper presents an algorithm for continuous-time infer-
+ence on partially observable Markov processes with discrete
+state spaces. We show that the well-known sum-product al-
+gorithm can be extended to the continuous-time domain via
+two sets of differential equations and pointwise normalization.
+In the continuous time setting, we were able to solve the
+trajectory of conditional probabilities exactly given a finite
+time-series. Moveover, we find that the dynamics of the state
+occupancy probabilities can be reduced to second-order dif-
+ferential equations under special circumstances. These results
+are valuable not only for their mathematical interest, but
+also because they have the potential to reduce the inference
+problem to solving systems of linear differential equations,
+with a potentially significant reduction in computational com-
+plexity for long time series. Numerically, the continuous-time
+algorithm is consistent with the discrete-time algorithm in the
+limit of small time step, but executes in approximately constant
+time rather than linearly in the number of time steps.
+Our formalism extends naturally to non-binary observations.
+Briefly, let S1, S2, . . . , Sm be a partition over the sample space
+Ω, and suppose the observable process Y is given as:
+Y(t) = i if S(t) ∈ Si.
+(54)
+For a sojourn with observation i, we can apply the forward
+and backward message-passing scheme as introduced for the
+binary case, viewing the observation as either in Si or Ω\Si. At
+the boundaries, the same technique (as in the proof of Theorem
+1) can be used for finding the update rule by noting the
+possible transitions from Si to Sj, for all j ̸= i. This approach
+can finally be extended to an arbitrary collection of subsets
+of Ω where the elements are not necessarily disjoint. One
+may accomplish this extension by expanding all unions and
+intersections as disjoint sets possibly with the same observable.
+We defer detailed investigations in this direction to future
+work.
+Extending the continuous-time message passing to the case
+with inhomogeneous transition rates remains an open problem.
+We expect a derivation similar to the proof of the main theorem
+would be applicable, possibly with smoothness constraints
+on the transition rates. While the normalization constant will
+remain invariant in time, we do not expect a result such as
+Corollary 1 to hold beyond constant transition rates.
+Finally, we note the relationship between conditional prob-
+ability and a second-order differential equation is intuitively
+satisfying: the entry and exit times act as two boundary
+conditions that fix the endpoints of the evolution, whereas
+the unconstrained forward evolution equation, a first-order
+differential equation, requires only the starting condition. It
+is an interesting question for future work to investigate under
+which assumptions a similar result as Corollary 1 would hold
+for systems with more than two distinct eigenvalues.
+APPENDIX A
+NOTATION AND PRELIMINARIES
+For completeness, we define a continuous-time discrete-
+space Markov process with exponential waiting times.
+Definition 1 (Markov Process). Let {S(t); t ∈ [0, T]} be
+a discrete-space, continuous-time stochastic process where
+for each t, S(t) is a random variable with state space
+Ω = {1, 2, . . . , N}, N < ∞. Then, the process S(t) has the
+Markov Property if for any s1, s2, . . . , sm ∈ Ω, 0 < τ1 <
+τ2 < · · · < τm−1 < t,
+Pr[S(t) = sm|S(τ1) = s1, S(τ2) = s2, . . . , S(τm−1) = sm−1]
+= Pr[S(t) = sm|S(τm−1) = sm−1].
+(55)
+Furthermore, the process has exponential waiting times if for
+any i, j ∈ Ω, i ̸= j, there is a constant rate 0 ≤ wji < ∞
+such that
+�
+Pr[S(t + ∆t) = i|S(t) = i] = 1 − �
+k̸=i wki∆t + o(∆t)
+Pr[S(t + ∆t) = j|S(t) = i] = wji∆t + o(∆t)
+Pr[S(t + ∆t) = j, S(t + 2∆t) = k|S(t) = i] = o(∆t),
+∀k
+for sufficiently small ∆t.
+The following table lists notation used in the paper.
+Symbol
+Meaning
+St
+discrete-time process with (integer) time index t
+S(t)
+continuous-time process with time index t
+Ω
+sample space of a process at fixed time
+S
+states that give observable “1”
+Y(t)
+the observed process (indicating S(t) ∈ S)
+P
+transition matrix of a Markov chain
+W
+transition rate matrix of Markov process
+αt; α(t)
+forward message
+βt; β(t)
+backward message
+χt; χ(t)
+observation message
+ρt; ρ(t)
+unnormalized conditional state-occupancy probability
+Z(t)
+normalizing constant for ρ(t)
+pt; p(t)
+conditional state-occupancy probability
+∆t
+sampling time step
+APPENDIX B
+INVARIANT NORMALIZATION IN CONTINUOUS TIME
+Time-invariant invariant normalization is a fundamental
+property of the sum-product algorithm [26]. In the continuous
+time case, we observed that this property can be easily con-
+firmed when the submatrices are diagonalizable, because the
+left and right eigenvectors corresponding to different eigenval-
+ues are orthogonal. When the submatrix has nontrivial Jordan
+blocks, the time-dependent term cancels in less obvious ways.
+Here we show that time-independent normalization holds in
+general, using only elementary calculus without invoking the
+machinery of factor-graphs.
+Recall
+that
+the
+normalizing
+constant
+is
+Z(t)
+=
+�
+i α(i)(t)β(i)(t). Using Equation 2 and 3, we can obtain the
+
+CHEN et. al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES
+10
+following series of expressions.
+dZ(t)
+dt
+= d
+dt
+�
+i∈S
+α(i)(t)β(i)(t)
+(56)
+=
+�
+i∈S
+dα(i)(t)
+dt
+β(i)(t) + α(i)(t)dβ(i)(t)
+dt
+(57)
+=
+�
+i∈S
+�
+� �
+k∈S\{i}
+wkiα(k) −
+�
+l̸=i
+α(i)
+�
+� β(i) . . .
++ α(i)
+�
+�−
+�
+k∈S\{i}
+wikβ(k) +
+�
+l̸=i
+β(i)
+�
+�
+(58)
+=
+�
+i∈S
+�
+k∈S\{i}
+wkiα(k)β(i) − wikα(i)β(k)
+(59)
+= 0
+(60)
+Since the rate of change of Z(t) is zero, the normalizing
+constant is invariant of time.
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+
diff --git a/QdAyT4oBgHgl3EQf7vpN/content/tmp_files/load_file.txt b/QdAyT4oBgHgl3EQf7vpN/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf,len=720
+page_content='CHEN et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES  1  Explicitly Solvable Continuous-time Inference for  Partially Observed Markov Processes  Daniel Chen, Alexander G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Strang, Andrew W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Eckford Senior Member, IEEE, Peter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Thomas  Abstract—Many natural and engineered systems can be mod-  eled as discrete state Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Often, only a subset  of states are directly observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Inferring the conditional prob-  ability that a system occupies a particular hidden state, given  the partial observation, is a problem with broad application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  In this paper, we introduce a continuous-time formulation of  the sum-product algorithm, which is a well-known discrete-time  method for finding the hidden states’ conditional probabilities,  given a set of finite, discrete-time observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' From our new  formulation, we can explicitly solve for the conditional probability  of occupying any state, given the transition rates and observations  within a finite time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' We apply our algorithm to a  realistic model of the cystic fibrosis transmembrane conductance  regulator (CFTR) protein for exact inference of the conditional  occupancy probability, given a finite time series of partial  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' INTRODUCTION  Markov processes—dynamic processes whose future be-  havior depends only on their present state—approximate a  wide variety of natural and engineered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Despite rapid  advances in high-throughput data acquisition and data pro-  cessing, many systems of interest contain important degrees  of freedom that cannot be directly observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Inferring the  conditional probability that such a partially observed Markov  process occupies specific hidden states, given the available ob-  servations, is a ubiquitous problem in science and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Examples appear in robotics [1], ecology [2], neuroscience [3],  and algorithmic text analysis [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  We are motivated by biological examples in the present  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Ion channels in excitable membranes, such as the  sodium (Na+) and potassium (K+) channels described in  Hodgkin and Huxley’s quantitative model for action potential  generation [5], [6], [7], provide an early example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Discrete  state Markov models based on Hodgkin and Huxley’s K+  channel contain five states, only one of which conducts an  ionic current;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' the other states are “silent” and cannot be distin-  guished by direct electrophysiological observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Similarly,  This work was supported in part by National Institutes of Health BRAIN  Initiative grant R01 NS118606 and a National Science Foundation grant  DMS-2052109 to PJT, as well as research support from the Oberlin College  Libraries, and an NSERC Discovery grant to AWE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Chen and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Thomas are with the Department of Mathematics, Applied  Mathematics, and Statistics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Department of Electrical, Control and Systems  Engineering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Department of Computer and Data Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Department of  Biology, Case Western Reserve University, Cleveland, OH 44106 USA (e-  mail: txc461/pjthomas@case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Strang is with Department of Statistics, University of Chicago,  Chicago, IL 60637 USA (email: alexstrang@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Eckford is with Department of Electrical Engineering and Computer  Science, York University, Toronto, ON M3J 1P3 Canada (e-mail: aeck-  ford@yorku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  the Na+ channel has eight states: seven with zero conductance  and one with nonzero conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Colquhoun and Hawkes  introduced maximum likelihood methods for inferring the rate  constants of a partially observed Markov process representing  the nicotinic Acetylcholine receptor [8], [9], [10], [11], but  did not address the question of inferring microscopic state  occupancy from observable conductance time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' More  recently, research into the molecular biology of cystic fibrosis  (CF) has focused on the CF transmembrane conductance  regulator (CFTR), which can be modeled as a 7-state system  with two conducting states and five nonconducting states  [12] (detailed below in Section V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Beyond these biological  examples, problems of inferring or estimating hidden states  from incomplete observations are widely studied in the signal  processing literature [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  The literature contains several approaches to approximating  the behavior of hidden states of partially observed Markov  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Sampling provides one common technique for  approximate inference [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' As an example, recent work by  Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' demonstrated an efficient algorithm for simulating  stochastic reaction networks with multiple separated time  scales using particle filters [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' In general, Markov Chain  Monte Carlo is a widely employed sampling technique used  to infer hidden states [16], [17], that has also been applied to  ion channels [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' For partially observed Bayesian networks  operating in discrete time, message passing algorithms on  factor graphs provide an efficient and exact inference method  [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The factor-graph formalism is highly flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Algorithms  based on the message passing concept have been extended to  applications in localization [20], compressed sensing [21], and  decision fusion [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Factor graphs are not limited to models  with a discrete number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' For example, Gaussian  message passing in linear Gaussian models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Kalman  filtering and smoothing) has been developed for continuous-  time models with discrete-time observations [23], [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  However, the state reconstruction problem for continuous-time  finite-state hidden Markov models has not been addressed in  the literature, to be best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  In this work, we extend the message-passing algorithm in  order to analytically interpolate state-occupancy probabilities  of a continuous-time system, given a discretely sampled time  series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' That is, we show how to infer the time-dependent  conditional probabilities of latent states for continuous-time  discrete-state homogeneous Markov processes given a set of  partial observations over a finite time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' We derive  an equivalent formulation of the sum-product algorithm in  continuous time that allows one to find an explicit ana-  lytic solution for the state occupancy probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Having  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='00843v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='SP]  2 Jan 2023  CHEN et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES  2  explicit solutions lowers the computational cost and, unlike  sampling-based approximate methods, does not sacrifice ac-  curacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Furthermore, the continuous-time formalism leads to  elegant simplification of the analytic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' For certain  systems—like the three-state systems shown in Figure 2—the  conditional probability obeys a second-order inhomogeneous  linear ordinary differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Finally, we demonstrate  the practical functionality of the algorithm with the 7-state  model for CFTR using simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Section II reviews the  message-passing algorithm for Markov processes with binary  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Section III displays the main result of the paper:  a continuous-time formulation of the sum-product algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  We present the derivation of the conditional probability using  the message-passing approach in continuous time, and the  corresponding sum-product algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Section IV discusses  the implications of the continuous-time formulation further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  We give examples of small systems to display how analytic  solutions may be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' In Section V we demonstrate the value  of the algorithm for a larger, realistic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' THEORY OF THE SUM-PRODUCT ALGORITHM  The sum-product algorithm can be used generally for in-  ference on probabilistic models that can be written as factor  graphs [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' There are many variants of the sum-product  algorithm, each suitable for accomplishing a different task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  For our purposes, we will focus on the forward/backward  algorithm for inference on hidden Markov models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  We consider a continuous-time, discrete-state homogeneous  Markov process on a finite state space Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Given a discrete,  uniformly spaced sampling interval, the continuous-time pro-  cess induces a discrete-time Markov process specified by some  column-stochastic transition matrix P that is invariant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Let S ⊂ Ω be a subset of states, and let St ∈ Ω be the state  of the system at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' We assume an observer can only  see whether St is in S or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Accordingly, let Yt = m(St)  represent the observable where m(s) : Ω → {0, 1} is the  indicator function for the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The goal of the algorithm  is to infer the conditional probability of being at a particular  state, i ∈ Ω, given the binary observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  The algorithm involves three vector-valued quantities: a for-  ward message αt, a backward message βt, and the observation  message χt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' One can interpret the forward message as the  probability of arriving at a certain state from time 0 to time  t and the backward message as the likelihood of occupying  a certain state at time t conditioned on ending up in a given  target state, or a given target set of states, at the end of the  measurement T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' On the other hand, the observation message  χt is an indicator function of the possible states given the  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' For instance, if at time t, the observation of the  total system were Yt = 1, then χt would be a vector with 1’s  on the states in the observation set S and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Using  superscripts to denote vector indices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' v(i) denotes the i-  th element of vector v, we present pseudo-code for the sum-  product algorithm in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Note that, the normalization  constant Z = �  k α(k)  t  β(k)  t  in Line 10 of Algorithm 1 is time  invariant [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Algorithm 1 Forward/Backward Algorithm  Input: The transition matrix P ∈ Rn×n, the observation  message χt for t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' , T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Output: The inferred probability pt for t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' , T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  1: Initialize the forward message α0 = π, the stationary  distribution of P  2: Initialize the backward message β(i)  T  = 1, for all i  3: for t from 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' , T do  4:  αt = diag(χt)Pαt−1  5: end for  6: for s from T − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' , 1 do  7:  βs = P ⊺diag(χs+1)βs+1  8: end for  9: for t from 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' , T do  10:  pt =  αt⊙βt  �  k α(k)  t  β(k)  t  11: end for  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' 1 illustrates the sum-product algorithm’s application to  a time series of discretely sampled observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Consider a  three-state-chain with symmetric transition rates as shown in  the top-left of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' If the system takes states 1 or 2,  “0” will be observed, and “1” will be observed otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Using the sum-product algorithm, we can find the probability  of the system occupying each state given the discrete-time  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' As shown in the bottom row, as the sampling  time step decreases, the conditional probabilities appear to  converge to a smooth curve within each interval with a fixed  observation (either “0” or “1”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Intuitively, there should exist  a continuous, perhaps piecewise differentiable, representation  of the conditional probability of a partially observed process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  We formalize this intuition below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' CONTINUOUS-TIME MESSAGE PASSING  In this section, we present the main result, namely the  derivation of the continuous-time message passing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  In this new formalism, messages are passed in the form  of linear differential equations on possible states given the  observable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' In order to guarantee the existence of the  continuous sum-product algorithm, we assume the following  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Assumptions:  A1 The continuous-time process {S(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' t ∈ [0, T]} takes  values in a finite state space Ω = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  A2 S(t) has the Markov property, and has exponentially  distributed waiting times parameterized by a rate matrix  W, with wji specifying the transition rate from state i to  state j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Note that W is constant in the interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  A3 There is a distinguished subset S ⊂ Ω such that the  observable process Y(t) satisfies  Y(t) =  �  1  if S(t) ∈ S,  0  otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Further details appear in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Under these assumptions, we obtain a continuous-time ver-  sion of the sum-product algorithm by executing the following  steps (made rigorous in the proof of Theorem 1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Write  CHEN et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' 1: Illustrating convergence of conditional state occupancy probabilities to a differentiable function for a three-state model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (Top left) The state diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' (Top middle) True simulated states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' (Top right) Binary observation derived from true states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (Bottom row) Inference of hidden states via the sum-product algorithm with time steps 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5 sec (Bottom Left), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='0 sec (Bottom  Middle) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5 sec (Bottom Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  out the matrix multiplication of the discrete-time algorithm  element-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Focus on one sojourn where the observation  doesn’t change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Within that time interval, take the limit as the  time step goes to zero to derive the continuous-time dynamics  of the conditional probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Extend the solution to the  full time interval via appropriate boundary conditions at the  transition between each sojourn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The main result is stated in  the theorem below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Suppose processes S(t) and the associated  process Y(t) satisfies assumptions A1, A2, and A3 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Then, given a realization of the process Y(t), the conditional  probability p(t) = Pr[S(t)|Y(t)] exists, is piecewise smooth  (C∞), and is C∞ on all intervals where Y(t) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' In  particular, p(t) takes the form  p(t) =  ρ(t)  �  k ρ(k)(t) =  α(t) ⊙ β(t)  �  k α(k)(t) ⊙ β(k)(t)  (1)  where ⊙ denotes the element-wise product, and the quantities  α(t) and β(t) are functions of time that follow the linear  ordinary differential equations  dα(i)(t)  dt  =  �  k∈S\\{i}  wkiα(k)(t) −  �  l̸=i  wilα(i)(t),  (2)  dβ(j)(t)  dt  = −  �  k∈S\\{j}  wjkβ(k)(t) +  �  l̸=j  wjlβ(j)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Without loss of generality, focus on the case where  Y(0) = Y(T) = 0, and Y(t) = 1 for 0 < t < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' We use  q(t) to denote a quantity, q, evolving in continuous time on  the interval [0, T], and qt to denote the same process sampled  at discrete times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Let the time interval [0, T] be discretized with a step size  ∆t = T/n for some integer n ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' At each time step,  S(t) is sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Then, the sum-product algorithm can be used  to solve for pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Writing the matrix multiplication out yields  the following set of equations for the forward and backward  messages in discrete time:  α(i)  t+∆t =  �  k  Pr[st+∆t = i|st = k]α(k)  t  χ(i)  t ,  (4)  β(i)  t  =  �  k  Pr[st+∆t = k|st = i]β(k)  t+∆tχ(k)  t+∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (5)  We neglect states not in S because only the probability  conditioned on the observations is of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Then, for any  state i ∈ S, we argue that the corresponding forward message,  α(i)(t), and backward message, β(i)(t) in continuous time can  be written as solutions of systems of differential equations,  upon taking limits as ∆t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' For notational simplicity, for  true states  observation  3  W2  2  W/2  W23  0  0  50  100  0  50  100  sample every 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5 seconds  sample every 1 seconds  sample every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5 seconds  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='2  0  0  0  20  40  60  80  100  20  40  60  80  100  20  40  60  80  100CHEN et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES  4  t > τ we let P(i,j)  t,τ  = Pr[S(t) = i|S(τ) = j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  dα(i)(t)  dt  = lim  ∆t→0  α(i)(t + ∆t) − α(i)(t)  ∆t  (6)  = lim  ∆t→0  1  ∆t  � �  k  P(i,k)  t+∆t,tα(k)(t)χ(i)(t) − α(i)(t)  �  (7)  = lim  ∆t→0  1  ∆t  �  �  k∈S\\{i}  (wki∆t + o(∆t))α(k)(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  −  �  l̸=i  (wil∆t + o(t))α(i)(t)  �  (8)  =  �  k∈S\\{i}  wkiα(k)(t) −  �  l̸=i  wilα(i)(t)  (9)  dβ(i)(t)  dt  = lim  ∆t→0  β(i)(t + ∆t) − β(i)(t)  ∆t  (10)  = lim  ∆t→0  1  ∆t  �  β(i)(t + ∆t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  −  �  k  P(k,i)  t+∆t,tβ(k)(t + ∆t)χ(k)(t + ∆t)  �  (11)  = lim  ∆t→0  1  ∆t  �  −  �  k∈S\\{i}  (wik∆t + o(∆t))β(k)(t + ∆t)  +  �  l̸=i  (wil∆t + o(t))β(i)(t + ∆t)  �  (12)  = −  �  k∈S\\{i}  wikβ(k)(t) +  �  l̸=i  wilβ(i)(t)  (13)  where lim∆t→0 o(∆t)/∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Readers could refer to Ap-  pendix A or consult existing literature such as [27] for the  relationship between transition probabilities and transition  rates of a Markov jump process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  The conditional probability can be found by solving the  differential equations, taking the component-wise product of  α(t) and β(t) for every t ∈ [0, T], and normalizing so as to  obtain a valid probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  The differential equation formulation is only applicable for  the time intervals where the observation Y(t) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  When the observable Y(t) changes (when the systems transi-  tions from a state in S to a state not in S), the probability with  respect to time might not be differentiable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' in some cases, it  is not even continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Therefore, we must specify boundary  conditions to connect the probabilities from one sojourn to  the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The observable may change either by the system  entering S or else leaving S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Suppose a transition occurred  within time (t∗ −∆t, t∗] such that, for t < t∗ −∆t, S(t) ∈ S,  and S(t) ̸∈ S for t ≥ t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Call this event E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Then, we obtain  the following transition rule for the forward message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Pr[S(t∗) = j|E]  =  �  i∈S Pj,i  t∗,t∗−∆tPr[S(t∗ − ∆t) = i]  �  k̸∈S  �  i∈S Pk,i  t∗,t∗−∆tPr[S(t∗ − ∆t) = i]  (14)  =  �  i∈S(wji∆t + o(∆t))Pr[S(t∗ − ∆t) = i]  �  k̸∈S  �  i∈S(wki∆t + o(∆t))Pr[S(t∗ − ∆t) = i]  (15)  →  �  i∈S wjiPr[S(t−  ∗ ) = i]  �  k̸∈S  �  i∈S wkiPr[S(t−  ∗ ) = i]  (16)  as ∆t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Here Pr[S(t−  ∗ ) = i] is the probability of occupying  state i the instant before the transition, which can be found by  solving the differential equations introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  We handle the boundary conditions at state transitions for  the backward message similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Define E as above and let s∗  be a particular goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Then:  Pr[S(T) = s∗|S(t∗ − ∆t) = j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' E]  =  �  i̸∈S  Pr[S(T) = s∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' S(t∗) = i|S(t∗ − ∆t) = j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' E]  =  �  i̸∈S Pr[S(T) = s∗|S(t∗) = i]Pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='j  t∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='t∗−∆t  �  k∈S  �  i̸∈S Ps∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='i  T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='t∗Pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='k  t∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='t∗−∆t  (17)  =  �  i̸∈S Pr[S(T) = s∗|S(t∗) = i](wij∆t + o(∆t))  �  k∈S  �  i̸∈S Ps∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='i  T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='t∗(wik∆t + o(∆t))  (18)  =  �  i̸∈S Pr[S(T) = s∗|S(t∗) = i](wij + o(∆t)  ∆t )  �  k∈S  �  i̸∈S Ps∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='i  T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='t∗(wik + o(∆t)  ∆t )  (19)  →  �  i̸∈S wijPr[S(T) = s∗|S(t∗) = i]  �  k∈S  �  i̸∈S wikPr[S(T) = s∗|S(t∗) = i]  (20)  as ∆t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Here, Pr[S(T) = s∗|S(t∗) = i] is a hitting  probability associated with the backward message at time  t∗, which can be found by solving the backward-message  differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' These boundary conditions, together with  the differential equations (2)-(3), give the continuous-time  evolution of the conditional probability for any finite-length  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Note that the equations (2)-(3) extend to the case where  Y(0) = Y(T) = 1 and Y(t) = 0 for 0 < t < T by viewing  S ← Ω \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' So, given a time series observation Y(t) where  observation (the value of Y(t)) changes at time 0 < t1 < t2 <  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' tm, we can solve for the analytic solution at any interval  with consistent observation (ti, ti+1] using equation (2) and  (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Then, use the result to compute the initial condition for the  next interval — namely, (ti+1, ti+2] for the forward message  and (ti−1, ti] for the backward message — as specified in (16)  and (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Thus, the statement of Theorem 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  A general expression for the conditional probability can be  obtained, but it is not of great utility in most systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Yet,  there are certain special cases that yield elegant solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' we  introduce several examples in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  The continuous-time sum-product algorithm follows directly  from the derivation above, and is outlined in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  CHEN et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES  5  The discrete algorithm passes information through matrix  multiplication of a truncated transition matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' the continuous-  time algorithm does the same by solving a system of linear  differential equations using the truncated rate matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Since the  final conditional probability is only piecewise differentiable,  the boundary condition must be applied whenever a transition  in or out of the observable set S occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Algorithm 2 Continuous-time Forward/Backward Algorithm  Input: The rate transition matrix W ∈ Rn×n, the observed  process Y(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Output: The inferred probability p(t) for t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  1: Let [t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' , tm] be a list of times where transitions  occur  2: τ ← 0  3: α∗ ← π, the stationary distribution  4: for j from 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' , m do  5:  αj ← solution to the forward message differential  equation (Equation 2) from τ to tj with initial condition  α∗  6:  τ ← tj  7:  α∗ ← distribution specified according to Equation 16  8: end for  9: τ ← T  10: β∗ ← the uniform distribution  11: for j from m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' 1 do  12:  βj ← solution to the backward message differential  equation (Equation 3) from τ to tj with initial condition  β∗  13:  τ ← tj  14:  β∗ ← distribution specified according to Equation 20  15: end for  16: α(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' β(t) ← concatenation αj’s and βj’s  17: Compute ρ(t) = α(t)⊙β(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' the component-wise product  between α(t) and β(t) pointwise with respect to t  18: Compute the conditional probability p(t) =  ρ(t)  �  k ρ(k)(t)  From a practical perspective,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' having the ability to solve  for the conditional probabilities exactly through differential  equations drastically lowers the computational cost of the  forward/backward algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Traditionally, the discrete-time  algorithm propagates the forward and backward messages  through matrix operations at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' For long time-  series and/or high-dimensional systems, this is computation-  ally prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Through our continuous-time formalism, we  solve the differential equations analytically, which is an op-  eration that is independent of the length of the time-series,  to find the forward or backward message at any time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  This difference effectively reduces the asymptotic scaling  from O(∆t−1) to O(1), with the later scaling only in the  number of observable transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' In scenarios where finding  the appropriate boundary condition would require an iterative  procedure of solving the forward and backward messages  multiple times, our continuous time approach should be much  more efficient than the traditional discrete-time method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' We  discuss the performance of the continuous-time message-  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' 2: State diagrams of two systems for which the  continuous-time message passing algorithm exhibit analytic  simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' (Left) 3-state chain with symmetric rates  w12 = w21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' (Right) Irreversible 3-state loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' States marked  in red return Y(t) = 1 and blue return Y(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  passing algorithm further in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' ANALYTIC SOLUTION  Theorem 1 in the previous section shows that the conditional  probability is always available analytically upon normalizing  the component-wise product of the forward and backward  messages,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' p(t) = ρ(t)/Z where Z = �  k ρ(k)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  As  in the discrete-time case, the normalizing term Z is time-  invariant in the continuous-time case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' See Appendix  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The conditional probability may therefore be expressed in  a particularly elegant form in certain cases, namely as the  solution of a linear nonhomogeneous second-order differential  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' We begin this section by considering two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Following the examples, we consider extensions to higher  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Symmetric 3-State Chain  Consider the three-state chain depicted in the left panel of  Figure 2, where states 1 and 2 are hidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Assume that the per-  capita transition rates within the hidden block are symmetric,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' w12 = w21 > 0, and assume w13 = w31 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The rates  w23 > 0 and w32 > 0 may be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' These assumptions  result in the following rate matrix:  W =  �  �  −w21  w12  0  w21  −(w12 + w32)  w23  0  w32  −w23  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (21)  In this case, let S = {3}, the singleton set of state 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' When  Y(t) = 1 the inference problem is trivial since the system  takes state 3 with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Thus, we emphasize the  intervals when Y(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  First, consider the forward message given by the following  system of differential equations  dα  dt =  �−w21  w12  w21  −(w12 + w32)  �  α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (22)  Note that the matrix defining the system of equations corre-  sponds to the upper left block of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' To simplify notation, let  w21 = w12 = a and w32 = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Then, the submatrix reduces to  the following form:  dα  dt =  �−a  a  a  −a − b  �  α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (23)  W  13  W  W  21  32  2  W  32  W  W  12  23  W  21  2CHEN et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES  6  This is a linear system of differential equations that can  be solved exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The sub-matrix is real-symmetric, so is  diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Therefore, the solution will be of the form:  α(t) = Aeλ1tv1 + Beλ2tv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (24)  where λi is an eigenvalue of the rate submatrix and vi is the  corresponding eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The eigenvalues and vectors are:  λ1/2 = a(−1 ±  �  1 + γ2) − γ,  (25)  v1/2 =  �  γ ±  �  1 + γ2  1  �  ,  (26)  where γ = b/2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Constants A and B are found through  the initial condition given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' In the previous section, the  initial forward message was set to the equilibrium distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  However, the chain structure leaves no ambiguity in the state  occupied when the observation changes from 1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' So, the  initial forward message is the delta distribution on state 2:  α(0) =  �0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (27)  After some time T, the system re-enters the visible state,  namely, state 3 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' By the same reasoning, we also have an  unambiguous boundary condition for the backward message:  β(T) =  �0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (28)  Also, by the symmetry in the rates, the backward message  evolves according to the same equations as the forward mes-  sage, but backward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Thus, β(t) = α(T − t),  β(t) = Aeλ1(T −t)v1 + Beλ2(T −t)v2  (29)  where λi’s, vi’s, constants A and B all remain the same due  to symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  The  conditional  probability  is  proportional  to  the  component-wise product ρ = α ⊙ β, which yields:  ρ(t) = A  �  eλ1(T −t)+λ2t + eλ1t+λ2(T −t)�  + B,  (30)  where  A = AB(v1 ⊙ v2) =  �−AB  AB�⊺,  (31)  B = A2eλ1T (v1 ⊙ v1) + B2eλ2T (v2 ⊙ v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (32)  To recover the conditional probability, we must normalize such  that the component sum evaluates to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' In this case, since the  submatrix in Equation 22 is symmetric, the eigenvectors of the  submatrix are orthogonal to each other, which means the sum  of the components of v1⊙v2, or the inner-product between v1  and v2, evaluates to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Then, the component-wise sum of A is  zero, so the normalizing constant equals the component-wise  sum of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Write p(t) to represent the conditional probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' We write  Z = �  k B(k) to represent the normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The final  equation describing the conditional probability can thus be  rewritten in the following form:  p(t) = 1  Z  �  Ae(λ1−λ2)teλ2T + Ae(λ2−λ1)teλ1T + B  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' (33)  The components of p(t) may also be expressed as the solutions  of a second-order ordinary differential equation, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' (51)-(52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Irreversible 3-State Loop  Light-gated Channelrhodopsin-2 (ChR2) receptors can be  modeled with a 3-state chain where each vertex has out-degree  1 and forms a cycle, as depicted in Figure 2, right panel [28],  [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Let state 1 be open, and states 2 and 3 be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' That  is, S = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The open/closed status of the channel is observed  through voltage recordings: high conductance indicates that the  channel is in the open state, and low conductance implies the  closed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The rate matrix can be written as the following:  W =  �  �  −w21  0  w13  w21  −w32  0  0  w32  −w13  �  �  (34)  where wji is the transition rate from state i to state j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Conditioning on the channel being closed, the message-passing  algorithm takes the lower right 2 × 2 block as the transition  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  When transitioning from S to a state not in S, the system  must enter state 2 first and exit through state 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Thus the  boundary conditions for the forward and backward message  are  α(0) =  �1  0  �  ,  β(T) =  �0  1  �  (35)  where the first component corresponds to state 2 and second  state 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Suppose w32 ̸= w13 and let γ =  w32  w13−w32 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Then, the  solution of the message-passing differential equations with the  initial condition enforced satisfies:  α(t) = e−w32t  �1  γ  �  − γe−w31t  �0  1  �  ,  (36)  β(t) = e−w32(T −t)  �  1  0  �  + 1  γ e−w13(T −t)  �  −γ  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (37)  To get the conditional probability, first take the component-  wise product between the forward and backward message at  time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  ρ(t) = e−w32t−w13(T −t)  �  −1  1  �  +  � e−w32T  −e−w13T  �  (38)  As in the previous example, the normalizing constant is  invariant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' In this case, Z = e−w32T − e−w13T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Thus,  the time evolution of the conditional probability can be written  p(t) = ρ(t)/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (39)  Next, we consider a case in which the submatrix is not diag-  onalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' We set w = w32 = w13, so that the corresponding  submatrix:  U =  �−w  0  w  −w  �  (40)  admits the following Jordan normal form:  U =  �0  w  1  0  � �−w  1  0  −w  � � 0  1  1  w  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (41)  Using the same initial condition as Equation (35) to solve the  system of differential equations yields the following forward  message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  α(t) =  � e−wt  te−wt  �  (42)  CHEN et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES  7  We solve for the backward message by undergoing a similar  procedure and yield  β(t) =  �(T − t)e−w(T −t)  e−w(T −t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (43)  Upon taking the component-wise product, we observe that the  evolution of conditional probability is independent of time:  ρ(t) = e−wT  �  T − t  t  �  ,  Z =  1  Te−wT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (44)  Since both the time entering and exiting the hidden states are  fixed, we can see that the probability flows linearly from one  state to the other at equal rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Time-invariant normalization is a fundamental property of  the sum-product algorithm [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' In Appendix B we give an  elementary demonstration of this property for continuous-time  systems with time-homogeneous transition rates, as illustrated  by the two cases presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Generalization  We show that under certain circumstances, the conditional  probability follows a second-order nonhomogeneous linear  ordinary differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' For 3-state systems where the truncated subma-  trix U has distinct eigenvalues, the conditional probability can  be written as the solution of a nonhomogeneous second-order  linear differential equation with constant coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Let (λi, vi) be eigenpairs for U and let (λi, wi) be  eigenpairs for U ⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Then the forward and backward messages  are given by the expressions:  α(t) = Aeλ1tv1 + Beλ2tv2,  (45)  β(t) = Ceλ1(T −t)w1 + Deλ2(T −t)w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (46)  Taking the component-wise product yields the form:  ρ(t) = A  �  eλ1t+λ2(T −t)�  + B  �  eλ1(T −t)+λ2t�  + C,  (47)  = A  �  e(λ1−λ2)teλ2T �  + B  �  e(λ2−λ1)teλ1T �  + C,  (48)  where  A = AD(v1 ⊙ w2),  B = BC(v2 ⊙ w1),  (49)  C = ACeλ1T (v1 ⊙ w1) + BDeλ2T (v2 ⊙ w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (50)  Since the left and right eigenvectors corresponding to dif-  ferent eigenvalues are orthogonal, all time-dependent terms  cancel when normalizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Thus the normalization constant  Z = �  k C(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Upon differentiating p(t) = ρ(t)/Z twice with  respect to time, we arrive at the second-order linear differential  equation that describes the time evolution of the condition  probability:  d2p  dt2 = (λ1 − λ2)2  Z  �  Ae(λ1−λ2)teλ2T + Be(λ2−λ1)teλ1T �  (51)  = (λ1 − λ2)2  Z  (p − C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (52)  This second order equation comes equipped with two bound-  ary conditions set by fixing α(0) and β(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' 3: State diagram for CFTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' When the protein enters one of  the red states, the ion channels will open and conduct a current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  On the other hand, when CFTR is in one of the blue states,  then the ion channels are closed and conducts no current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' For  this system S = {4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' For a system of arbitrary dimension, for which  the truncated submatrix U has exactly two distinct eigenvalues  and is diagonalizable, the conditional probability can be  written as a second-order linear differential equation of the  same form as Equation 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Let λ1 and λ2 be the two distinct eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Let vi be  the sum of all right eigenvectors corresponding to eigenvalue  λi, and let wi be the sum of all left eigenvectors corresponding  to eigenvalue value λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Then by diagonalizability, the proof for  this corollary follows the proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  This result extends Corollary 1 to higher dimensional sys-  tems, under special conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' These two corollaries match  intuition: we are concerned with the state occupancy prob-  ability conditioned on both the entrance and exit times of  an observable state, so the probability flow obeys a second-  order differential equation, which also requires specifying two  boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' We note, however, that if the submatrix  has three or more distinct eigenvalues, we do not expect a  similar result to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Moreover, generic matrices typically  have as many distinct eigenvalues as their dimension, so we  do not expect this scenario to occur outside of special cases  when the rate submatrix is highly regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' In addition, it would  be difficult to check if a given rate submatrix satisfies these  conditions through numerical solvers, as equality of repeated  eigenvalues might be obscured by floating point arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Therefore, as much as the second-order differential equations  interpretation is intuitively appealing, it is likely not a generic  phenomenon that we should expect for arbitrary rate matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' BIOLOGICAL EXAMPLE: CFTR  Here, we show that our algorithm applies to a higher  dimensional system, namely the 7-state model of the cystic  fibrosis transmembrane conductance regulator (CFTR) protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  CF is a common life-threatening genetic disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' CFTR is  an important protein that regulates the opening and closing  of ion channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Loss of CFTR function causes pancreatic  insufficiency as well as airway infection due to excessive  mucus, which in turn can cause a variety of complications  such as impaired innate immunity and respiratory failure [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Mathematically, the behavior of CFTR can be captured  with a 7-state hidden Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Figure 3 illustrates  2  3  4  5  7  6CHEN et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES  8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' 4: The inference result of the continuous-time algorithm (top) and the discrete-time algorithm (middle) for the true state  occupancy simulated from the Gillespie algorithm (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Sampling time step ∆t = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  the state diagram [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The ion channel opens (conducts an  ionic current) when CFTR is in state four or five (marked in  red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' When in the nonconducting states (marked in blue) the  ion channel is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Transmembrane conductance recordings  report whether CFTR is in a conducting or a nonconducting  state, but do not directly indicate which of the possible states  is occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Using the numbering shown in Figure 3, the rate  matrix is defined as follows:  W =  �  �  �  �  �  �  �  �  �  −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='0  0  0  0  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='7  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='9  −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='8  0  0  0  0  0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='7  −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='7  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='0  0  0  0  0  0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='9  −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='1  0  0  0  0  0  0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='1  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='0  0  0  0  0  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='0  −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='8  0  0  0  0  0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='0  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  �  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (53)  We used Gillespie’s exact stochastic simulation algorithm  [31], [32], [33] to generate sample traces of the ion channel  states and recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The simulation produces discrete state,  continuous time trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' We introduced a finite sampling  time step to discretize the simulated data along the time axis,  consistent with data obtained through experimental recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Then, we place the generated data into discretized time bins  where the size of each bin (one can think of this as the sam-  pling time step) is a parameter that can be altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' We solved  the differential equations giving the forward and backward  messages exactly via function handles in Matlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Figure 4  compares the traces produced by the classical discrete-time  algorithm, our continuous-time algorithm, and the true states,  and shows excellent agreement among the respective curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  With a sufficiently small time step (∆t = 10−4), the contin-  uous and discrete-time algorithms show no visible discrepancy,  as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Figure 5 shows that the maximum discrepancy  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' 5: First-order convergence of the message passing al-  gorithms as a function of sampling time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The mean  maximum difference was plotted along with one standard  deviation about the mean as the error bar for an ensemble  of forty trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (ℓ∞ norm) between the conditional probabilities generated by  the continuous time and discrete time algorithms decreases  linearly with the sampling time step ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' The figure shows  the results from an ensemble of forty independent repeated  trials for each sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Again, we emphasize that while  the discrete time algorithm requires iterating over all samples,  which scales O(∆t−1), the continuous time algorithm outputs  a description of the conditional probability — that is, a  function that returns a value at an exact time point queried  — in O(1) time, scaling only with the number of transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  inferred probability of states: continuous  1234567  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  5  ×104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='7  inferred probability of states: discrete  1234567  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  5  ×104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='3  true states  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='2  134567  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='5  0  3  4  5  time  ×10410~2  maximum difference  10~3  10~5  10~6  10-7  10-7  10-6  10-5  10-4  10-3  size of sampling intervalCHEN et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES  9  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' DISCUSSION & CONCLUSIONS  This paper presents an algorithm for continuous-time infer-  ence on partially observable Markov processes with discrete  state spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' We show that the well-known sum-product al-  gorithm can be extended to the continuous-time domain via  two sets of differential equations and pointwise normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  In the continuous time setting, we were able to solve the  trajectory of conditional probabilities exactly given a finite  time-series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Moveover, we find that the dynamics of the state  occupancy probabilities can be reduced to second-order dif-  ferential equations under special circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' These results  are valuable not only for their mathematical interest, but  also because they have the potential to reduce the inference  problem to solving systems of linear differential equations,  with a potentially significant reduction in computational com-  plexity for long time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Numerically, the continuous-time  algorithm is consistent with the discrete-time algorithm in the  limit of small time step, but executes in approximately constant  time rather than linearly in the number of time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Our formalism extends naturally to non-binary observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Briefly, let S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' , Sm be a partition over the sample space  Ω, and suppose the observable process Y is given as:  Y(t) = i if S(t) ∈ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (54)  For a sojourn with observation i, we can apply the forward  and backward message-passing scheme as introduced for the  binary case, viewing the observation as either in Si or Ω\\Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' At  the boundaries, the same technique (as in the proof of Theorem  1) can be used for finding the update rule by noting the  possible transitions from Si to Sj, for all j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' This approach  can finally be extended to an arbitrary collection of subsets  of Ω where the elements are not necessarily disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' One  may accomplish this extension by expanding all unions and  intersections as disjoint sets possibly with the same observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  We defer detailed investigations in this direction to future  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Extending the continuous-time message passing to the case  with inhomogeneous transition rates remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  We expect a derivation similar to the proof of the main theorem  would be applicable, possibly with smoothness constraints  on the transition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' While the normalization constant will  remain invariant in time, we do not expect a result such as  Corollary 1 to hold beyond constant transition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Finally, we note the relationship between conditional prob-  ability and a second-order differential equation is intuitively  satisfying: the entry and exit times act as two boundary  conditions that fix the endpoints of the evolution, whereas  the unconstrained forward evolution equation, a first-order  differential equation, requires only the starting condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' It  is an interesting question for future work to investigate under  which assumptions a similar result as Corollary 1 would hold  for systems with more than two distinct eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  APPENDIX A  NOTATION AND PRELIMINARIES  For completeness, we define a continuous-time discrete-  space Markov process with exponential waiting times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Definition 1 (Markov Process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Let {S(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' t ∈ [0, T]} be  a discrete-space, continuous-time stochastic process where  for each t, S(t) is a random variable with state space  Ω = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' , N}, N < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Then, the process S(t) has the  Markov Property if for any s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' , sm ∈ Ω, 0 < τ1 <  τ2 < · · · < τm−1 < t,  Pr[S(t) = sm|S(τ1) = s1, S(τ2) = s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' , S(τm−1) = sm−1]  = Pr[S(t) = sm|S(τm−1) = sm−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  (55)  Furthermore, the process has exponential waiting times if for  any i, j ∈ Ω, i ̸= j, there is a constant rate 0 ≤ wji < ∞  such that  �  Pr[S(t + ∆t) = i|S(t) = i] = 1 − �  k̸=i wki∆t + o(∆t)  Pr[S(t + ∆t) = j|S(t) = i] = wji∆t + o(∆t)  Pr[S(t + ∆t) = j, S(t + 2∆t) = k|S(t) = i] = o(∆t),  ∀k  for sufficiently small ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  The following table lists notation used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Symbol  Meaning  St  discrete-time process with (integer) time index t  S(t)  continuous-time process with time index t  Ω  sample space of a process at fixed time  S  states that give observable “1”  Y(t)  the observed process (indicating S(t) ∈ S)  P  transition matrix of a Markov chain  W  transition rate matrix of Markov process  αt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' α(t)  forward message  βt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' β(t)  backward message  χt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' χ(t)  observation message  ρt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' ρ(t)  unnormalized conditional state-occupancy probability  Z(t)  normalizing constant for ρ(t)  pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' p(t)  conditional state-occupancy probability  ∆t  sampling time step  APPENDIX B  INVARIANT NORMALIZATION IN CONTINUOUS TIME  Time-invariant invariant normalization is a fundamental  property of the sum-product algorithm [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' In the continuous  time case, we observed that this property can be easily con-  firmed when the submatrices are diagonalizable, because the  left and right eigenvectors corresponding to different eigenval-  ues are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' When the submatrix has nontrivial Jordan  blocks, the time-dependent term cancels in less obvious ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Here we show that time-independent normalization holds in  general, using only elementary calculus without invoking the  machinery of factor-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  Recall  that  the  normalizing  constant  is  Z(t)  =  �  i α(i)(t)β(i)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' Using Equation 2 and 3, we can obtain the  CHEN et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' al: EXPLICITLY SOLVABLE CONTINUOUS-TIME INFERENCE FOR PARTIALLY OBSERVED MARKOV PROCESSES  10  following series of expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  dZ(t)  dt  = d  dt  �  i∈S  α(i)(t)β(i)(t)  (56)  =  �  i∈S  dα(i)(t)  dt  β(i)(t) + α(i)(t)dβ(i)(t)  dt  (57)  =  �  i∈S  �  � �  k∈S\\{i}  wkiα(k) −  �  l̸=i  α(i)  �  � β(i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
+page_content='  + α(i)  �  �−  �  k∈S\\{i}  wikβ(k) +  �  l̸=i  β(i)  �  �  (58)  =  �  i∈S  �  k∈S\\{i}  wkiα(k)β(i) − wikα(i)β(k)  (59)  = 0  (60)  Since the rate of change of Z(t) is zero, the normalizing  constant is invariant of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQf7vpN/content/2301.00843v1.pdf'}
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diff --git a/SNE0T4oBgHgl3EQf1wKp/content/tmp_files/2301.02704v1.pdf.txt b/SNE0T4oBgHgl3EQf1wKp/content/tmp_files/2301.02704v1.pdf.txt
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+Revista Mexicana de Física 3 040909 (2022) 1–4
+September 2022
+Rivet and the analysis preservation in heavy-ion collisions experiments
+Antonio Carlos Oliveira da Silva (for the ALICE Collaboration)
+University of Tennessee, Knoxville, 1408 Circle Drive, Knoxville TN 37996-1200
+Received 3 July 2022; accepted 15 September 2022
+The comparison of experimental data and theoretical predictions is important for our understanding of the mechanisms for interactions and
+particle production in hadron collisions, both at the Large Hadron Collider and at the Relativistic Heavy-Ion Collider experiments. Several
+tools were ideated to help with that. Rivet (Robust Independent Validation of Experiment and Theory) is a framework that facilitates the
+comparison between measurements from high-energy physics experiments and Monte Carlo event generators able to produce outputs using
+the HepMC package. Rivet contains a repository with analysis algorithms developed by experiments, providing analysis documentation and
+preservation.
+The recent developments for the implementation of centrality and multiplicity classes in Rivet are presented in this contribution.
+Keywords:
+1
+Introduction
+Currently, the data and analysis preservation in high-energy
+physics experiments is becoming a common concern. Previ-
+ous experiments and collaborations are losing the power of
+reproducing their measurements since the data are not prop-
+erly kept in a accessible way.
+The old code, which con-
+tains crucial and detailed information like detector accep-
+tance, particle and event selections, and corrections, is no
+longer maintained and it is very difficult, if possible, to be
+run again. Comparisons of previous measurements with new
+models is, therefore, very challenging.
+Robust Independent Validation of Experiment and The-
+ory (Rivet) [1] is a framework that aims to facilitate the com-
+parison between data and Monte Carlo (MC) event genera-
+tors.
+2
+Rivet framework
+Rivet analyses are written in C++ and it currently contains
+more than 1000 analyses from several high-energy physics
+collaborations. The data, when available, are downloaded di-
+rectly from HepData [2]. Any model that is incorporated in
+an event generator able to produce output that complies with
+HepMC framework [3] can be used by Rivet for the compar-
+ison with data. The integration of Rivet with HepMC and
+HepData is pictured in the scheme presented in Fig. 1.
+The references for the event generators in Fig.
+1 can
+be found in [4–12]. Not all of them provide output using
+HepMC standards.
+In principle, an article presenting a measurement should
+present enough information to make the measurement able
+to be reproduced by another experiment or theoretician inter-
+ested in comparing the data with a model. However, some
+subtle details about detector acceptances, particle selections,
+trigger conditions, etc, could be missing or not clearly de-
+scribed. This can be the case even in internal notes in large
+collaborations.
+FIGURE 1. Schematic diagram showing how Rivet is integrated
+with HepMC and HepData.
+Rivet aims at preserving all the analysis details. Further-
+more, it should reproduce the methods used in a measurement
+as close as possible. One of the pillars of the Rivet philoso-
+phy is that we should treat Monte Carlo simulations in the
+same way as data.
+In order to assure maximum fidelity to the original ac-
+ceptances, selections and methods used in a measurement,
+whenever possible, the rivet analysis should be implemented
+by the collaboration that published the article containing the
+measurement. Currently, ALICE has an official internal pro-
+cedure for Rivet analysis approvals.
+3
+Centrality and multiplicity determination
+Measurements in heavy-ion collisions are commonly differ-
+ential in centrality intervals to study different physical phe-
+nomena. Therefore, centrality determination is a crucial fea-
+ture Rivet has to provide in order to reproduce measurements
+in heavy-ion collisions.
+Centrality determination in ALICE is commonly pro-
+vided by the V0 detector [13], which consists of two ar-
+rays, V0-A and V0-C covering the pseudorapidity ranges 2.8
+< η < 5.1 and 3.7 < η < 1.7 respectively.
+Figure 2 presents the distribution of the total energy de-
+arXiv:2301.02704v1  [physics.data-an]  6 Jan 2023
+
+AMPT
+FXR
+HIJING
+JETSCAPE
+PYTHIA
+Monte Carlo Mode
+JEWEL
+EPOS
+PHSI
+smash
+HepMC
+HEpData
+Rivet2
+ANTONIO CARLOS OLIVEIRA DA SILVA
+posited in the V0 scintillators (amplitude). The most cen-
+tral collisions are associated with those event with highest
+V0 amplitude. The details of the centrality determination are
+described in [14].
+FIGURE 2. Distribution of the total amplitude in the V0 scintilla-
+tors in black points. The data are fitted using a Negative Binomial
+Distribution (NBD) using parameters from the Glauber model.
+b
+b
+b
+b
+b b
+b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b
+b b b b b b b b b b
+b b b b
+b
+b b b b b b b b b b b b b b
+b b b b b b
+b b b b b b
+b b
+b b b b
+b b
+b b b b b b b b b b b b b b b b b b b b b b b b
+b b b b b
+b b b b b b b b b b b b b b b b b b b b b b b b b b b b
+b
+b b b b
+b
+b
+b
+b b b b
+b b b b b b b b
+b b
+b
+b b b b
+b b
+b b b b b b b b b b
+b
+b b b b b b b
+b b b b b b b b b b b b b b b
+b
+b b b
+b
+b b b b b
+b b b
+b b b
+b b
+b
+b b b
+b b b
+b b
+b
+b
+b b b b b b b b b b b
+b
+b b
+b b b b b
+b b
+b b
+b b
+b
+b
+b b
+b b b
+b
+b b b b b
+b
+b
+b
+b b
+b b
+b b
+b
+b
+b
+b b
+b
+b
+b b
+b
+b b
+b
+b
+b
+b
+b b b
+b
+b b b b
+b
+Data
+calibration ALICE PbPb2760GeV
+0
+5000
+1.0·104
+1.5·104
+2.0·104
+2.5·104
+3.0·104
+3.5·104
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+1
+VZERO amplitude (a.u.) or ch. particle multiplicity
+Events
+FIGURE 3. V0 amplitude in arbitrary units (black markers) mea-
+sured by ALICE and the charged particle multiplicity in the V0 ac-
+ceptance calculated with Rivet from Pb–Pb collisions at 2.76 TeV
+events simulated with PYTHIA 8 Angantyr (red line).
+The centrality determination in Rivet uses the multiplic-
+ity of charged particles in the acceptance of the V0. Since
+the multiplicity of particles in heavy-ion collision events in
+Monte Carlo event generators is model dependent, it is neces-
+sary to create a calibration file that depends on collisions sys-
+tem, energy and event generator. Figure 3 shows the V0 am-
+plitude distribution measured by ALICE and the multiplicity
+of charged particles in Pb–Pb collisions at √sNN = 2.76 TeV
+generated with PYTHIA 8 Angantyr [6].
+Rivet divides the charged particle multiplicity presented
+in figure 3 (red line) in centrality percentiles.
+So the
+most central events are associated to the highest multiplic-
+ity events. When running the Rivet analysis that requires
+centrality determination, this calibration has to be provided.
+A similar strategy is used for multiplicity determination in
+pp and p–Pb collisions. Currently, ALICE is developing the
+possibility to characterize the event using the self-normalized
+charged particle multiplicity distribution. This development
+is presented in sections 4 and 5.
+4
+Self-normalized multiplicity
+Forward-rapidity multiplicity classes can be defined in
+ALICE using the V0 detector. Figure 4 shows the distribution
+of the V0M amplitude, which is proportional to the number
+of charged particles passing through the V0A and V0C de-
+tectors, scaled by its average value ⟨V0M⟩ [15].
+FIGURE 4. Distribution of the V0M amplitude scaled by its aver-
+age value ⟨V0M⟩ used to determined forward-rapidity multiplicity
+classes in pp collisions at √s = 13 TeV.
+The Silicon Pixel Detector (SPD) [16, 17] is the closest
+detector to the interaction point in ALICE. The SPD provides
+mid-rapidity multiplicity classes determination using the re-
+constructed tracklets, which are track segments that connects
+hits in the two SPD layers pointing to the primary vertex. The
+self-normalized estimator is obtained with the distribution of
+SPD tracklets NSPD tracklets in -2 < η < 2 scaled by the av-
+erage of its value ⟨NSPD tracklets⟩. The self-normalized SPD
+tracklets distribution is presented in Fig. 5.
+Supl. Rev. Mex. Fis. 3 040909
+
+Yield (a.u.)
+10-3
+10-3
++
+Data
+Glauberfit
+80-90%
+70-80%
+60-70%
+10-4
+10-4
+500
+1000
+10-5
+50-60%
+40-50%
+0-40%
+20-30%
+10-20%
+10-6
+5-10%
+-5%
+8
+0
+4000
+8000
+12000
+16000
+20000
+VZERO Amplitude (a.u.)
+ALI-PUB-89941103
+ALICE
+Forward Multiplicity Classes
+Min. Bias data (%)
+102
+Normalised counts
+pp, Vs = 13 TeV
+0-1
+1-5
+5-10
+10-15
+10
+15-20
+20-30
+30-40
+40-50
+50-70
+70-100
+High-Mult data (%)
+10-
+0-0.01
+0.01-0.1
+10-3
+10
+10
+5
+0
+2
+4
+6
+8
+10
+VOM/VOM)
+ALI-PUB-499489A LATEXTEMPLATE FOR THE RMF, RMF-E, SRMF
+3
+FIGURE 5. Distribution of the number of SPD tracklets scaled by its
+average value ⟨NSPD tracklets⟩ used to determine midrapidity mul-
+tiplicity classes in pp collisions at √s = 13 TeV.
+The event characterization using self-normalized multi-
+plicity estimators in Rivet is under development in ALICE.
+Similar to what was discussed in the previous section, instead
+of using the V0M amplitude, Rivet uses the charged particle
+multiplicity in the V0 acceptance. In particular, the forward-
+rapidity self-normalized estimator uses the multiplicity of
+charged particles in the V0 acceptance scaled by its average
+value. Figure 6 shows the V0M/⟨V0M⟩ distribution calcu-
+lated with Rivet using pp collisions at 13 TeV generated with
+PYTHIA 8 Monash 2013 tune [4,18].
+Rivet pp13TeV
+0
+2
+4
+6
+8
+10
+10−5
+10−4
+10−3
+10−2
+10−1
+1
+V0M/⟨V0M⟩
+Normalized counts
+FIGURE 6. Self-normalized multiplicity distribution of charged par-
+ticles in the acceptance of the V0 detector in pp collisions at √s =
+13 TeV generated with PYTHIA 8 Monash 2013.
+Similarly to what is done for the V0M, the self-
+normalized multiplicity estimator at mid-rapidity uses the
+number of charged particles in the acceptance of the SPD.
+Figure 7 shows the distribution of the charged particles in the
+SPD acceptance scaled by its average value in pp collisions
+at 13 TeV generated with PYTHIA 8 Monash 2013.
+Rivet pp13TeV
+0
+2
+4
+6
+8
+10
+10−4
+10−3
+10−2
+10−1
+1
+NSPD tracklets
+SPD tracklets
+SPD tracklets/⟨NSPD tracklets
+SPD tracklets
+SPD tracklets⟩
+Normalized counts
+FIGURE 7. Self-normalized multiplicity distribution of charged par-
+ticles in the acceptance of the SPD detector in pp collisions at
+√s = 13 TeV generated with PYTHIA 8 Monash 2013.
+5
+Results using the self-normalized multiplic-
+ity estimators
+The self-normalized multiplicity estimators framework in
+Rivet is currently work in progress and being tested using ar-
+ticles published by ALICE that use such estimators. The first
+measurement used to test the V0M/⟨V0M⟩ estimator was the
+transverse momentum (pT) of jets in different multiplicity in-
+tervals in pp collisions at √s = 13 TeV [19].
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+Data
+Rivet [cent=GEN]
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+1/Nevent d2N/dpTd
+event d2N/dpTd
+event d2N/dpTdηηη (GeV/ccc)))−1
+−1
+−1
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+0.5
+0.6
+0.7
+0.8
+0.91
+1.1
+1.2
+1.3
+1.4
+jet pT
+pT
+pT (GeV/ccc)))
+MC/Data
+FIGURE 8. Charged-particle jet transverse momentum distribution
+in pp collisions at √s = 5.02 TeV for the 0-1% multiplicity
+class corresponding to the self-normalized V0M-based multiplic-
+ity estimator. Jets were reconstructed using jet resolution parame-
+ter R = 0.2. Data (black markers) are compared with PYTHIA 8
+Monash 2013 (red line).
+Supl. Rev. Mex. Fis. 3 040909
+
+103
+ALICE
+Central Multiplicity Classes
+Min. Bias data (%)
+102
+pp, Vs = 13 TeV
+Normalised counts
+0-1
+1-5
+5-10
+10-15
+10
+15-20
+20-30
+30-40
+40-50
+50-70
+70-100
+10
+10
+3
+10
+10
+5
+0
+2
+4
+6
+8
+10
+KN
+SPD Tracklet4
+ANTONIO CARLOS OLIVEIRA DA SILVA
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+Data
+Rivet SPD pp5TeV [cent=GEN]
+0
+5
+10
+15
+20
+25
+dN/dη
+dN/dη
+dN/dη
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+b
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+0.5
+0.6
+0.7
+0.8
+0.91
+1.1
+1.2
+1.3
+1.4
+ηηη
+MC/Data
+FIGURE 9. Charged particle pseudorapidity distribution in pp colli-
+sions at √s = 5.02 TeV for the 0-1% multiplicity class correspond-
+ing to the self-normalized SPD-based multiplicity estimator. Data
+(black markers) are compared with PYTHIA 8 Monash 2013 (red
+line).
+The transverse momentum of jets reconstructed with
+FastJet anti-kT algorithm [20] and resolution parameter R =
+0.2 in the multiplicity class 0-1% in pp collisions at √s =
+13 TeV is presented in figure 8. The measurement was com-
+pared with PYTHIA 8 Monash 2013 using Rivet and the self-
+normalized V0M multiplicity framework. The agreement of
+the model to data is a positive indication that the framework
+can reproduce the multiplicity determination method used by
+ALICE. Other multiplicity classes presented a similar perfor-
+mance.
+The self-normalized estimator using the SPD was also
+tested using the measurements in [15].
+Figure 9 presents
+the charged particle pseudorapidity distribution in pp colli-
+sions at √s = 5.02 TeV. The ALICE data are compared
+with PYTHIA 8 Monash 2013 using Rivet and the SPD self-
+normalized framework. The results fairly reproduce the com-
+parisons to MC presented in the cited article.
+6
+Summary
+Rivet is a valuable tool for analysis preservation and compar-
+ison of data to Monte Carlo event generators. The develop-
+ment of additional tools to facilitate the implementation of
+Rivet analyses is an important contribution to the framework
+and can be of benefit both for the experiment and the theory
+side. The self-normalized multiplicity estimators are provid-
+ing consistent results between MC curves in Rivet and those
+provided by experiments. The final goal is to make this mul-
+tiplicity framework available soon in the Rivet official frame-
+work.
+1. A. Buckley, et al., Rivet user manual (2010).
+2. E. Maguire, L. Heinrich, and G. Watt, HEPData: a repository
+for high energy physics data, Journal of Physics: Conference
+Series 898 (2017) 102006,
+10.1088/1742-6596/898/
+10/102006
+3. M. Dobbs and J. B. Hansen, The HepMC C++ Monte Carlo
+Event Record for High Energy Physics,
+Tech. rep., CERN,
+Geneva (2000), URL https://cds.cern.ch/record/
+684090, Revised version number 1 submitted on 2001-02-27
+09:54:32.
+4. T. Sjöstrand, S. Mrenna, and P. Z. Skands, A Brief Introduc-
+tion to PYTHIA 8.1, Comput. Phys. Commun. 178 (2008) 852,
+10.1016/j.cpc.2008.01.036
+5. Z.-W. Lin, et al., Multiphase transport model for relativistic
+heavy ion collisions, Physical Review C 72 (2005), 10.1103/
+physrevc.72.064901
+6. C. Bierlich, et al., The Angantyr model for Heavy-Ion Col-
+lisions in PYTHIA8,
+JHEP 10 (2018) 134,
+10.1007/
+JHEP10(2018)134
+7. J. H. Putschke, et al., The JETSCAPE framework (2019), 10.
+48550/ARXIV.1903.07706,
+URL https://arxiv.
+org/abs/1903.07706.
+8. J. Weil, et al., Particle production and equilibrium properties
+within a new hadron transport approach for heavy-ion colli-
+sions, Physical Review C 94 (2016), 10.1103/physrevc.
+94.054905
+9. T. Pierog, et al.,
+EPOS LHC: Test of collective hadroniza-
+tion with data measured at the CERN Large Hadron Collider,
+Phys. Rev. C 92 (2015) 034906, 10.1103/PhysRevC.92.
+034906
+10. K. Zapp, et al., A Monte Carlo model for ‘jet quenching’, The
+European Physical Journal C 60 (2009), 10.1140/epjc/
+s10052-009-0941-2
+11. X.-N. Wang and M. Gyulassy, hijing: A Monte Carlo model
+for multiple jet production in pp, pA, and AA collisions, Phys.
+Rev. D 44 (1991) 3501, 10.1103/PhysRevD.44.3501
+12. W. Cassing and E. Bratkovskaya,
+Parton–hadron–string dy-
+namics: An off-shell transport approach for relativistic ener-
+gies,
+Nuclear Physics A 831 (2009) 215,
+10.1016/j.
+nuclphysa.2009.09.007
+13. T. A. Collaboration, Performance of the ALICE VZERO sys-
+tem, Journal of Instrumentation 8 (2013) P10016, 10.1088/
+1748-0221/8/10/p10016
+14. B. Abelev, J. Adam, and D. Adamov,
+Centrality determina-
+tion of Pb-Pb collisions at √sNN = 2.76 TeV with ALICE 88
+(2013), 10.1103/physrevc.88.044909
+15. J. Adam et al.,
+Pseudorapidity distributions of charged par-
+ticles as a function of mid- and forward rapidity multiplic-
+Supl. Rev. Mex. Fis. 3 040909
+
+A LATEXTEMPLATE FOR THE RMF, RMF-E, SRMF
+5
+ities in pp collisions at √s = 5.02, 7 and 13 TeV (2021),
+10.1140/epjc/s10052-021-09349-5
+16. ALICE Inner Tracking System (ITS): Technical Design Report,
+Technical design report. ALICE (CERN, Geneva, 1999), URL
+http://cds.cern.ch/record/391175.
+17. R. Santoro, et al., The ALICE Silicon Pixel Detector: readiness
+for the first proton beam, Journal of Instrumentation 4 (2009)
+P03023, 10.1088/1748-0221/4/03/p03023
+18. P. Skands, S. Carrazza, and J. Rojo,
+Tuning PYTHIA 8.1:
+the Monash 2013 tune, The European Physical Journal C 74
+(2014), 10.1140/epjc/s10052-014-3024-y
+19. ALICE Collaboration,
+Multiplicity dependence of charged-
+particle jet production in pp collisions at √s = 13 TeV
+(2022), 10.48550/ARXIV.2202.01548, URL https:
+//arxiv.org/abs/2202.01548.
+20. M. Cacciari, G. P. Salam, and G. Soyez,
+FastJet user man-
+ual, The European Physical Journal C 72 (2012), 10.1140/
+epjc/s10052-012-1896-2
+Supl. Rev. Mex. Fis. 3 040909
+
diff --git a/SNE0T4oBgHgl3EQf1wKp/content/tmp_files/load_file.txt b/SNE0T4oBgHgl3EQf1wKp/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f26d0321c78467b3157c20cc2f01da015740effe
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@@ -0,0 +1,382 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf,len=381
+page_content='Revista Mexicana de Física 3 040909 (2022) 1–4  September 2022  Rivet and the analysis preservation in heavy-ion collisions experiments  Antonio Carlos Oliveira da Silva (for the ALICE Collaboration)  University of Tennessee, Knoxville, 1408 Circle Drive, Knoxville TN 37996-1200  Received 3 July 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' accepted 15 September 2022  The comparison of experimental data and theoretical predictions is important for our understanding of the mechanisms for interactions and  particle production in hadron collisions, both at the Large Hadron Collider and at the Relativistic Heavy-Ion Collider experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Several  tools were ideated to help with that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rivet (Robust Independent Validation of Experiment and Theory) is a framework that facilitates the  comparison between measurements from high-energy physics experiments and Monte Carlo event generators able to produce outputs using  the HepMC package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rivet contains a repository with analysis algorithms developed by experiments, providing analysis documentation and  preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The recent developments for the implementation of centrality and multiplicity classes in Rivet are presented in this contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Keywords:  1  Introduction  Currently, the data and analysis preservation in high-energy  physics experiments is becoming a common concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Previ-  ous experiments and collaborations are losing the power of  reproducing their measurements since the data are not prop-  erly kept in a accessible way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The old code, which con-  tains crucial and detailed information like detector accep-  tance, particle and event selections, and corrections, is no  longer maintained and it is very difficult, if possible, to be  run again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Comparisons of previous measurements with new  models is, therefore, very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Robust Independent Validation of Experiment and The-  ory (Rivet) [1] is a framework that aims to facilitate the com-  parison between data and Monte Carlo (MC) event genera-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  2  Rivet framework  Rivet analyses are written in C++ and it currently contains  more than 1000 analyses from several high-energy physics  collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The data, when available, are downloaded di-  rectly from HepData [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Any model that is incorporated in  an event generator able to produce output that complies with  HepMC framework [3] can be used by Rivet for the compar-  ison with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The integration of Rivet with HepMC and  HepData is pictured in the scheme presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The references for the event generators in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  1 can  be found in [4–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Not all of them provide output using  HepMC standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  In principle, an article presenting a measurement should  present enough information to make the measurement able  to be reproduced by another experiment or theoretician inter-  ested in comparing the data with a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' However, some  subtle details about detector acceptances, particle selections,  trigger conditions, etc, could be missing or not clearly de-  scribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' This can be the case even in internal notes in large  collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Schematic diagram showing how Rivet is integrated  with HepMC and HepData.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Rivet aims at preserving all the analysis details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Further-  more, it should reproduce the methods used in a measurement  as close as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' One of the pillars of the Rivet philoso-  phy is that we should treat Monte Carlo simulations in the  same way as data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  In order to assure maximum fidelity to the original ac-  ceptances, selections and methods used in a measurement,  whenever possible, the rivet analysis should be implemented  by the collaboration that published the article containing the  measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Currently, ALICE has an official internal pro-  cedure for Rivet analysis approvals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  3  Centrality and multiplicity determination  Measurements in heavy-ion collisions are commonly differ-  ential in centrality intervals to study different physical phe-  nomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Therefore, centrality determination is a crucial fea-  ture Rivet has to provide in order to reproduce measurements  in heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Centrality determination in ALICE is commonly pro-  vided by the V0 detector [13], which consists of two ar-  rays, V0-A and V0-C covering the pseudorapidity ranges 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='8  < η < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='7 < η < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Figure 2 presents the distribution of the total energy de-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='02704v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='data-an]  6 Jan 2023  AMPT  FXR  HIJING  JETSCAPE  PYTHIA  Monte Carlo Mode  JEWEL  EPOS  PHSI  smash  HepMC  HEpData  Rivet2  ANTONIO CARLOS OLIVEIRA DA SILVA  posited in the V0 scintillators (amplitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The most cen-  tral collisions are associated with those event with highest  V0 amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The details of the centrality determination are  described in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Distribution of the total amplitude in the V0 scintilla-  tors in black points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
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+page_content='0·104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5·104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='0·104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5·104  10−6  10−5  10−4  10−3  10−2  10−1  1  VZERO amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=') or ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' particle multiplicity  Events  FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' V0 amplitude in arbitrary units (black markers) mea-  sured by ALICE and the charged particle multiplicity in the V0 ac-  ceptance calculated with Rivet from Pb–Pb collisions at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='76 TeV  events simulated with PYTHIA 8 Angantyr (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The centrality determination in Rivet uses the multiplic-  ity of charged particles in the acceptance of the V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Since  the multiplicity of particles in heavy-ion collision events in  Monte Carlo event generators is model dependent, it is neces-  sary to create a calibration file that depends on collisions sys-  tem, energy and event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Figure 3 shows the V0 am-  plitude distribution measured by ALICE and the multiplicity  of charged particles in Pb–Pb collisions at √sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='76 TeV  generated with PYTHIA 8 Angantyr [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Rivet divides the charged particle multiplicity presented  in figure 3 (red line) in centrality percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  So the  most central events are associated to the highest multiplic-  ity events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' When running the Rivet analysis that requires  centrality determination, this calibration has to be provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  A similar strategy is used for multiplicity determination in  pp and p–Pb collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Currently, ALICE is developing the  possibility to characterize the event using the self-normalized  charged particle multiplicity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' This development  is presented in sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  4  Self-normalized multiplicity  Forward-rapidity multiplicity classes can be defined in  ALICE using the V0 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Figure 4 shows the distribution  of the V0M amplitude, which is proportional to the number  of charged particles passing through the V0A and V0C de-  tectors, scaled by its average value ⟨V0M⟩ [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  FIGURE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Distribution of the V0M amplitude scaled by its aver-  age value ⟨V0M⟩ used to determined forward-rapidity multiplicity  classes in pp collisions at √s = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The Silicon Pixel Detector (SPD) [16, 17] is the closest  detector to the interaction point in ALICE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The SPD provides  mid-rapidity multiplicity classes determination using the re-  constructed tracklets, which are track segments that connects  hits in the two SPD layers pointing to the primary vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The  self-normalized estimator is obtained with the distribution of  SPD tracklets NSPD tracklets in -2 < η < 2 scaled by the av-  erage of its value ⟨NSPD tracklets⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The self-normalized SPD  tracklets distribution is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Supl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' 3 040909  Yield (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=')  10-3  10-3  +  Data  Glauberfit  80-90%  70-80%  60-70%  10-4  10-4  500  1000  10-5  50-60%  40-50%  0-40%  20-30%  10-20%  10-6  5-10%  5%  8  0  4000  8000  12000  16000  20000  VZERO Amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=')  ALI-PUB-89941103  ALICE  Forward Multiplicity Classes  Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Bias data (%)  102  Normalised counts  pp, Vs = 13 TeV  0-1  1-5  5-10  10-15  10  15-20  20-30  30-40  40-50  50-70  70-100  High-Mult data (%)  10-  0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='01-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1  10-3  10  10  5  0  2  4  6  8  10  VOM/VOM)  ALI-PUB-499489A LATEXTEMPLATE FOR THE RMF, RMF-E, SRMF  3  FIGURE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Distribution of the number of SPD tracklets scaled by its  average value ⟨NSPD tracklets⟩ used to determine midrapidity mul-  tiplicity classes in pp collisions at √s = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The event characterization using self-normalized multi-  plicity estimators in Rivet is under development in ALICE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Similar to what was discussed in the previous section, instead  of using the V0M amplitude, Rivet uses the charged particle  multiplicity in the V0 acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' In particular, the forward-  rapidity self-normalized estimator uses the multiplicity of  charged particles in the V0 acceptance scaled by its average  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Figure 6 shows the V0M/⟨V0M⟩ distribution calcu-  lated with Rivet using pp collisions at 13 TeV generated with  PYTHIA 8 Monash 2013 tune [4,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Rivet pp13TeV  0  2  4  6  8  10  10−5  10−4  10−3  10−2  10−1  1  V0M/⟨V0M⟩  Normalized counts  FIGURE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Self-normalized multiplicity distribution of charged par-  ticles in the acceptance of the V0 detector in pp collisions at √s =  13 TeV generated with PYTHIA 8 Monash 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Similarly to what is done for the V0M, the self-  normalized multiplicity estimator at mid-rapidity uses the  number of charged particles in the acceptance of the SPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Figure 7 shows the distribution of the charged particles in the  SPD acceptance scaled by its average value in pp collisions  at 13 TeV generated with PYTHIA 8 Monash 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Rivet pp13TeV  0  2  4  6  8  10  10−4  10−3  10−2  10−1  1  NSPD tracklets  SPD tracklets  SPD tracklets/⟨NSPD tracklets  SPD tracklets  SPD tracklets⟩  Normalized counts  FIGURE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Self-normalized multiplicity distribution of charged par-  ticles in the acceptance of the SPD detector in pp collisions at  √s = 13 TeV generated with PYTHIA 8 Monash 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  5  Results using the self-normalized multiplic-  ity estimators  The self-normalized multiplicity estimators framework in  Rivet is currently work in progress and being tested using ar-  ticles published by ALICE that use such estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The first  measurement used to test the V0M/⟨V0M⟩ estimator was the  transverse momentum (pT) of jets in different multiplicity in-  tervals in pp collisions at √s = 13 TeV [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  Data  Rivet [cent=GEN]  10−6  10−5  10−4  10−3  10−2  10−1  1/Nevent d2N/dpTd  event d2N/dpTd  event d2N/dpTdηηη (GeV/ccc)))−1  −1  −1  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  10  20  30  40  50  60  70  80  90  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='4  jet pT  pT  pT (GeV/ccc)))  MC/Data  FIGURE 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Charged-particle jet transverse momentum distribution  in pp collisions at √s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='02 TeV for the 0-1% multiplicity  class corresponding to the self-normalized V0M-based multiplic-  ity estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Jets were reconstructed using jet resolution parame-  ter R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Data (black markers) are compared with PYTHIA 8  Monash 2013 (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Supl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' 3 040909  103  ALICE  Central Multiplicity Classes  Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Bias data (%)  102  pp, Vs = 13 TeV  Normalised counts  0-1  1-5  5-10  10-15  10  15-20  20-30  30-40  40-50  50-70  70-100  10  10  3  10  10  5  0  2  4  6  8  10  KN  SPD Tracklet4  ANTONIO CARLOS OLIVEIRA DA SILVA  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  Data  Rivet SPD pp5TeV [cent=GEN]  0  5  10  15  20  25  dN/dη  dN/dη  dN/dη  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='4  ηηη  MC/Data  FIGURE 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Charged particle pseudorapidity distribution in pp colli-  sions at √s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='02 TeV for the 0-1% multiplicity class correspond-  ing to the self-normalized SPD-based multiplicity estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Data  (black markers) are compared with PYTHIA 8 Monash 2013 (red  line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The transverse momentum of jets reconstructed with  FastJet anti-kT algorithm [20] and resolution parameter R =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='2 in the multiplicity class 0-1% in pp collisions at √s =  13 TeV is presented in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The measurement was com-  pared with PYTHIA 8 Monash 2013 using Rivet and the self-  normalized V0M multiplicity framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The agreement of  the model to data is a positive indication that the framework  can reproduce the multiplicity determination method used by  ALICE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Other multiplicity classes presented a similar perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  The self-normalized estimator using the SPD was also  tested using the measurements in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  Figure 9 presents  the charged particle pseudorapidity distribution in pp colli-  sions at √s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='02 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The ALICE data are compared  with PYTHIA 8 Monash 2013 using Rivet and the SPD self-  normalized framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The results fairly reproduce the com-  parisons to MC presented in the cited article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  6  Summary  Rivet is a valuable tool for analysis preservation and compar-  ison of data to Monte Carlo event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The develop-  ment of additional tools to facilitate the implementation of  Rivet analyses is an important contribution to the framework  and can be of benefit both for the experiment and the theory  side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The self-normalized multiplicity estimators are provid-  ing consistent results between MC curves in Rivet and those  provided by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' The final goal is to make this mul-  tiplicity framework available soon in the Rivet official frame-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
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+page_content=' 3 040909  A LATEXTEMPLATE FOR THE RMF, RMF-E, SRMF  5  ities in pp collisions at √s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='02, 7 and 13 TeV (2021),  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1140/epjc/s10052-021-09349-5  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' ALICE Inner Tracking System (ITS): Technical Design Report,  Technical design report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' ALICE (CERN, Geneva, 1999), URL  http://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
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+page_content=', The ALICE Silicon Pixel Detector: readiness  for the first proton beam, Journal of Instrumentation 4 (2009)  P03023, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1088/1748-0221/4/03/p03023  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
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+page_content=' Carrazza, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
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+page_content='1:  the Monash 2013 tune, The European Physical Journal C 74  (2014), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1140/epjc/s10052-014-3024-y  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' ALICE Collaboration,  Multiplicity dependence of charged-  particle jet production in pp collisions at √s = 13 TeV  (2022), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='01548, URL https:  //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='org/abs/2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='01548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
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+page_content=' Cacciari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Salam, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Soyez,  FastJet user man-  ual, The European Physical Journal C 72 (2012), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content='1140/  epjc/s10052-012-1896-2  Supl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
+page_content=' 3 040909' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQf1wKp/content/2301.02704v1.pdf'}
diff --git a/TNAzT4oBgHgl3EQfXfzv/content/tmp_files/load_file.txt b/TNAzT4oBgHgl3EQfXfzv/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf,len=895
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='01321v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='CR]  3 Jan 2023  Cheesecloth: Zero-Knowledge Proofs of Real-World Vulnerabilities  Santiago Cuéllar∗  Galois, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Bill Harris  Galois, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  James Parker  Galois, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Stuart Pernsteiner  Galois, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Eran Tromer  Columbia University  Abstract  Currently, when a security analyst discovers a vulnerabil-  ity in critical software system, they must navigate a fraught  dilemma: immediately disclosing the vulnerability to the  public could harm the system’s users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' whereas disclosing the  vulnerability only to the software’s vendor lets the vendor  disregard or deprioritize the security risk, to the detriment of  unwittingly-affected users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  A compelling recent line of work aims to resolve this by  using Zero Knowledge (ZK) protocols that let analysts prove  that they know a vulnerability in a program, without reveal-  ing the details of the vulnerability or the inputs that exploit  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In principle, this could be achieved by generic ZK tech-  niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In practice, ZK vulnerability proofs to date have been  restricted in scope and expressibility, due to challenges re-  lated to generating proof statements that model real-world  software at scale and to directly formulating violated proper-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  This paper presents  CHEESECLOTH, a  novel proof-  statement compiler, which proves practical vulnerabilities in  ZK by soundly-but-aggressively preprocessing programs on  public inputs, selectively revealing information about exe-  cuted control segments, and formalizing information leak-  age using a novel storage-labeling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' CHEESECLOTH’s  practicality is demonstrated by generating ZK proofs of  well-known vulnerabilities in (previous versions of) critical  software, including the Heartbleed information leakage in  OpenSSL and a memory vulnerability in the FFmpeg graph-  ics framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  1  Introduction  Ideally, programs that process sensitive information would  always execute safely and securely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' With this ideal remain-  ing difficult to achieve for the foreseeable future, it is criti-  cal that that when programs are found to be vulnerable, the  program’s affected users are alerted quickly and safely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This  ∗Authors listed alphabetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  requirement presents a challenge: convincingly disclosing a  vulnerability appears to require sharing the vulnerability’s  details (such as an exploit that triggers it), thereby placing  users at greater risk in the short term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  A promising approach to disclosing vulnerabilities con-  vincingly yet safely is to leverage Zero-Knowledge (ZK)  proofs: protocols in which one party—designated as the  prover—convinces another party—designated as the veri-  fier—of the validity of a claim without revealing any addi-  tional information about the claim’s evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Such a use of ZK proofs has arguably been a conceptual  possibility ever since the initial fundamental results establish-  ing that they exist for all problems in NP [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' It has become  more realistic with improvements to underlying ZK proto-  cols and with the emergence of schemes for encoding knowl-  edge of executions of programs written in convenient lan-  guages (starting with [10,24] and discussed further below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  In order to prove vulnerabilities in ZK about practical soft-  ware, several open problems remaing to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' First,  proof frameworks must scale to compile proofs of vulner-  abilities that require considerably more steps of execution  and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' TinyRAM [10–12] is sufficiently flexible to val-  idate the executions of applications, but it is expensive, in  part due to the fact that it simulates every instruction in the  modeled CPU’s ISA in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' TinyRAM’s performance  is surpassed by those of Pantry [17] and Buffet [52], but both  frameworks require loops to be unrolled to a public bound:  publicly revealing these bounds leaks information about the  underlying vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  A second open problem is to efficiently compile state-  ments from an understandable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' One immediate ap-  proach is to execute a program under a dynamic safety moni-  tor for well-understood safety properties, such as those those  implemented in Valgrind [41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' however, directly encoding  the additional monitoring would induce prohibitively large  overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Approaches for verifying low-level exploits in  ZK [27] rely on being able to efficiently compile directly-  understandable properties into statements of control-location  reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  1  To address these problems, we present CHEESECLOTH, an  optimizing ZK proof-statement generator that efficently en-  codes vulnerabilities in practical software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The contributions  behind CHEESECLOTH’s design include:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Optimizations of ZK statements that verify the executions  of programs, taking advantage of program structure but  without revealing additional information about the exe-  cution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Specifically, Public-PC segments construct execu-  tion traces from segments with public program counters,  thus enabling aggressive constant folding, without leak-  ing information about the overall execution trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Simi-  larly, instructions which are publicly-determined to be ex-  ecuted infrequently are sparsely supported (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' can’t be  executed at every step), making the statement smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Novel, efficient ZK encodings of memory errors preva-  lent in practical software, specifically out-of-bound ac-  cess, use-after-free, free-after-free, and uninitialized ac-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Previous related work focused primarily on proving  knowledge of a valid execution without proving existence  of a vulnerability [10,11] or encoded proofs of vulnerabil-  ity using a less efficient memory model [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' A novel, efficient encoding of statements that a program  always leaks data (when given an exploit as a secret in-  put).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Our scheme enables proofs of program properties  that are related to, but critically distinct from, existing pro-  gram monitors and type systems that prove that a program  may leak data [20,39], optionally in ZK [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  We implemented these optimizations and encodings in  CHEESECLOTH, a full compilation toolchain for encoding  vulnerabilities of real-world programs into efficient ZK  proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The toolchain extends previous approaches based  on TinyRAM, and includes a full definition of a novel  TinyRAM extension (named MicroRAM) and a compiler to  MicroRAM from the LLVM intermediate language,enabling  proofs of vulnerabilities in programs provided in C, C++, or  Rust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  We evaluated our implementation by proving in ZK the  existence of three vulnerabilities in practical systems soft-  ware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Specifically, we proved that previous versions of the  GRIT and FFmpeg [2] graphics processing libraries con-  tained buffer-overflow vulnerabilites, and that the OpenSSL  cryptograhy toolkit [3] was vulnerable to the notorious Heart-  bleed vulnerability [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' CHEESECLOTH takes the software  C/C++ source code and a flag denoting a vulnerabilities  class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' it combines these with an emulation of the runtime  environment (operating system and libraries), and applies  the aforementioned techniques, to derive a statement directly  provable in ZK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The ZK proof can then be given, as a wit-  ness, the concrete exploit used to demonstrate the original at-  tack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' CHEESECLOTH contains implementations of powerful  program analyses that, when combined with manual program  partitioning in some cases, dramatically increase the scale of  programs that it can process, compared to a more naive com-  piler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The remainder of this paper is organized as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 2  reviews the background that this work builds upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 3  presents the implementation details of our CHEESECLOTH  compilation pipeline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 4 covers the critical and aggres-  sive optimizations we make to verify the ZK execution of a  program;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 5 describes our ZK encodings to efficiently de-  tect memory and information leakage vulnerabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 6  describes our practical experience using CHEESECLOTH to  prove vulnerabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 7 compares our approach to re-  lated work and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 8 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  2  Background  In this section, we review prior work on which our contribu-  tion builds upon, specifically Zero-Knowledge (ZK) proofs of  program executions (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1), information leakage by pro-  grams (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2), and partial program evaluation (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1  Zero-Knowledge Proofs  Zero-knowledge proofs enable a prover party to prove to  a verifier party that the prover knows the correctness of a  computational statement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', that a given Boolean circuit  is satisfiable), without revealing information about their evi-  dence for the claim (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', the witness that satisfied the circuit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  There exist ZK protocols for proving knowledge of solutions  to all problems in NP [26], and in recent years, numerous ef-  ficient protocols have been developed and implemented for  ZK proofs of general statements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', [8, 10, 12, 14–16, 22,  24,28,29,31,32,38,43,51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Some of these works specifically address statements about  correct execution of programs running on a general-purpose  architecture that include Random Access Memory (RAM),  where the program is expressed in low-level machine code  or a high-level language [9,10,12,14,16,17,22,27,29,31,32,  38,51,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Our compiler uses a hybrid of step-by-step CPU emula-  tion, similar to TinyRAM [10–12], a MIPS-like CPU that  can simulate programs in C and similar low-level languages  that access RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The TinyRAM encoder, given a public  TinyRAM program and bound on the number of steps of ex-  ecution to simulate and a private program input, generates  an R1CS that is satisfied by encodings of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The con-  straint system consists of (1) a family of constraint systems  that validate computations purely over registers in each step  and (2) a novel memory-checking sub-circuit that verifies the  correctness of RAM operations using a permutation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  This CPU-unrolling technique is excellent for supporting lan-  guage features such as data-dependent loops, control-flow  and self-modifying code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The technique can also naturally  leverage existing tools such as compilers front-ends and li-  braries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  2  We combine TinyRAM-style emulation with direct com-  pilation of program blocks into circuit gates [17, 24, 52]  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The compiler’s output is a circuit whose satisfia-  bility is equivalent to the existence of a vulnerability in the  source program, and whose structure does not reveal the vul-  nerability or how it may be triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  In our evaluation, the underlying ZK protocol is the  Mac’n’Cheese [9] protocol for proving circuit satisfiability,  as implemented by the Swanky [23] library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This is an inter-  active protocol, where the prover and verifier engage in mul-  tiple rounds of communication to evaluate the circuit, at the  end of which the verifier learns that the circuit accepted the  secret witness provided by the prover (and nothing else).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2  Information Flow  One core contribution of our work is a practical scheme for  proving in zero knowledge that a program leaks data, which  we have applied to prove that previous versions of OpenSSL  leak private data, as triggered by the Heartbleed vulnerabil-  ity (described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The scheme’s design  requires a formal treatment of information flow: specifically,  a treatment sufficiently formal that we could generate logical  circuits that would be satisfied only by witnesses to leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  In the interest of space and clarity, we will omit a definition  of information flow and leakage for a full programming lan-  guage, but we will describe ours in sfufficient datail to com-  municate the key challenges and approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  A labeling L is a subset of a program’s input variables I  designated as the private inputs, and a subset of its output  variables O denoted as public outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Program P satisfies  noninterference with respect to L if each pair of inputs that  are only distinct at private inputs result in values that are the  same at all public outputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' P leaks with respect to L if, with  respect to L, it does not satisfy noninterference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' It follows  from the above definition that a leak is witnessed by a pair  of executions that differ only at L-labeled inputs and produce  distinct L-labeled outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Noninterference has a precise but accessible formal defi-  nition that can capture the flow requirements of some criti-  cal software [20], but its shortcomings in practice are well  known [39, 40, 45]: the complete information flow spec-  ifications of practical programs often are not noninterfer-  ence properties, intuitively because programs that take sen-  sitive inputs typically do need to reveal some partial infor-  mation about them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' and even when desired flow properties  are noninterference properties, proving that a program sat-  isfies the property in general can involve careful reasoning  about unbounded data and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' A rich body of prior  work [13, 18, 19, 25] has considered generalizations of non-  interference involving equivalences over observable events,  along with rich programming languages and type systems  and attempt to prove their satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' However, noninterfer-  ence properties still constitute aspects of a program’s com-  plete information flow requirements that unfortunately are  both critical and are violated in practice (Heartbleed being  a prominent example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This pattern justisfies the current  work’s primary focus on proving noninterference violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1  Labeled Programs and Executions  In their most general form, information flow and leakage are  defined over pairs of executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Practical program moni-  tors [20,49] and type systems [39] prove facts about all exe-  cution pairs, by labeling the program’s data and control struc-  tures with metadata which is tracked through the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  These approaches can be carried out by a programmer or au-  tomated analysis that directly annotates the program or exe-  cution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' However, the requisite guarantees are different in our  proof-of-vulnerability context compared to their usual appli-  cations, as seen next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  At a high level, the guarantees provided by dynamic infor-  mation flow monitors are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' A labeling of a program  execution over n steps is an assignment from each program  variable and step 0 ≤ i < n to a sensitivity label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' A label-  ing over-approximates information flow if, from any two ex-  ecutions starting from states that only differ at high inputs,  the program produces results that differ only at high-labeled  timestamped storage cells (static analyses and type systems  lift this property to be defined over all pairs of executions  that differ only at sensitive inputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Such over-approximation allows for “false positives” in  identifying information flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' For example, in a context  where only input x is sensitive and the return value is pub-  lic, the following function always_true does not leak any  information about its input secret because it returns true  for each input value:  bool always_true(bool secret) {  if (secret) return secret;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  else return !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='secret;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' }  However, many natural taint analyses would label the re-  turned values as sensitive because it is computed from  secret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Over-approximation of potential leaks is often still valu-  able for aiding programmers to ensure that their program  does not leak: falsely determining that a secure program may  leak may constitute a nuisance, and may need be mitigated  to ensure practicality, but can to some degree to tolerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  However, in our setting of proving a leak in ZK, it is un-  acceptable for the verifier to learn only that a program may  leak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The whole point is to prove that it does leak (given the  purported exploit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We will thus create a labeling which is an  under-approximation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', when the labels say so, a leakage  is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' It will then remain to empirically show that label-  ing indeed detects leakage for the vulnerabilities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3  Partial Evaluation  In many practical contexts, a program may receive different  subsets of its input at different times after it has been written  and compiled: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', after being installed, a configuration file  may be included that remains the same over all executions  on distinct inputs subsequently received from a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' A  natural objective is, given a program and a subset of its in-  puts that can be fixed, to generate a specialized program that  processes the remaining inputs with improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Stated more precisely, for program P(X,Y) with input vari-  ables X and Y, a partial evaluation of P on an assignment  A : X → Words from X to data values Words is a program PA  such that P(A,B) is the same as PA(B) for each assignment  B : Y → Words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Partially evaluating programs in a practical language  brings several complexities [35];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' the underlying technique  amounts to: (1) evaluate the program under a symbolic state,  in which registers and memory addresses may be mapped  either to memory addresses or terms defined over symbolic  variables that denoted unknown values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' (2) using computed  symbolic states that describe all possible states at each con-  trol point, simplify the control structure at each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Vari-  ations of this technique may be viewed as aggressive exten-  sions and generalizations of the constant propagation analy-  sis and constant folding transformation implemented in con-  ventional optmizing compilers [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  3  CHEESECLOTH Implementation  CHEESECLOTH produces ZK proofs of real world vulnera-  bilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' It takes as input a public LLVM program (typically  compiled from C, C++, or Rust) and, when run as the prover,  a secret exploit that triggers a vulnerability in the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  CHEESECLOTH outputs a ZK circuit that verifies the execu-  tion trace of the program and checks whether or not a vulner-  ability occurred during that execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The pipeline enables a  prover to demonstrate to a verifier that there is a vulnerability  in a program while keeping the vulnerability and triggering  exploit secret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  CHEESECLOTH produces ZK circuits in multiple stan-  dard representations including Rank-1 Constraints Systems  (R1CS) [10] and SIEVE IR [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Because the circuits are seri-  alized in standardized formats, CHEESECLOTH is agnostic to  the ZK protocol applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' When run as the prover, CHEESE-  CLOTH outputs the accompanying witness for the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  CHEESECLOTH can be extended to check different proper-  ties about a program’s execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Users can selectively en-  able which extensions to run by providing different input  flags to the compilation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' These extensions are how  the memory and information leakage vulnerability detection  checks described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 5 are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This section  covers the baseline design of the CHEESECLOTH compila-  tion pipeline which includes (1) the MicroRAM assembly  language, (2) the MicroRAM Compiler, and (3) the Witness  Checker Generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 4 describes optimizations for this  design that enable it to scale to real world vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1  MicroRAM  The MicroRAM assembly language is critical to CHEESE-  CLOTH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' It is the core IR language that CHEESECLOTH op-  erates on and is the language that the MicroRAM Compiler  compiles LLVM programs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The Witness Checker Genera-  tor produces ZK circuits that verify program executions ac-  cording to MicroRAM’s architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  MicroRAM is heavily inspired by TinyRAM [10, 11],  which is a practical and efficient assembly language with  a simple transition function that is ideal for ZK execution  verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We describe MicroRAM and its architecture be-  low, and we precisely describe how its design diverges from  TinyRAM in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  MicroRAM is a random-access machine designed to effi-  ciently detect vulnerabilities in program executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' It is a  reduced instruction set computer (RISC) with a Harvard ar-  chitecture and byte-addressable random-access memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  MicroRAM instructions are relatively simple and include  4 boolean operations, 8 arithmetic operations for signed and  unsigned integers, 2 shift operations, 5 compare operations,2  move operations, 3 jump instructions, 2 operations for read-  ing and writing to memory, and 1 answer operation that re-  turns and halts the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Floating-point and vector arith-  metic are not directly supported in the MicroRAM machine  and must be implemented in software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Instructions take two  registers and one operand (either a register or an immediate)  as arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' As an example, instruction xor ri rj 255  writes to register ri the exclusive-orof register rj and the im-  mediate 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' CHEESECLOTH extensions like those described  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 5 can introduce additional instructions as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The state of the MicroRAM machine consists of the pro-  gram counter (pc), k 64-bit registers, a memory of 264 64-bit  words, a flag indicating whether or not the execution so far is  valid (inv_flag), and a flag tracking whether a vulnerability  has occurred (vuln_flag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' CHEESECLOTH extensions can  extend the state of the MicroRAM machine as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  To demonstrate the existence of a vulnerability in a pro-  gram, a prover must present a secret input that results in a  valid execution trace that triggers a vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Formally,  given a MicroRAM program, P, and an initial memory, m0,  P(m0) demonstrates a vulnerability in T steps if inv_flag  is false and vuln_flag is true in the final MicroRAM  state of the program’s execution trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' inv_flag is set to  false if any of the checks validating the program’s execu-  tion fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The extensions implementing the vulnerability de-  tection checks set vuln_flag to true if they observe a vul-  nerability during the program’s execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1  Beyond TinyRAM  As mentioned above, our MicroRAM machine is inspired by  TinyRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Here we report on how MicroRAM’s design de-  parts from the TinyRAM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  MicroRAM’s memory model is byte-addressable while  TinyRAM is word-addressable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Byte-addressable memory  is necessary to support functionality like string manipula-  tions and packed structs, without adding subroutines to ac-  cess bytes within full words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  TinyRAM receives input via input tapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In MicroRAM,  input is passed directly in memory, which saves many cy-  cles that TinyRAM spends copying input to memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' A Mi-  croRAM program can request non-deterministic advice in  several ways, however the prover does not have to commit  to the advice ahead of time on a tape;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' instead they provide  the advice upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This approach is better suited to  support backends that exploit parallelism or streaming, and  it results in smaller circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  TinyRAM uses a 1-bit condition flag for branching while  MicroRAM does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This is advantageous since Micro-  RAM targets a variety of backends including non-boolean  arithmetic circuits where the flag is more expensive than a  regular register1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In addition, the semantics without a flag  are much simpler so the compiler, interpreter, and circuit  generator are simpler as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We found that even when  targeting boolean circuits, the benefits of having a condi-  tion flag are outweighed by the extra complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  We have not yet explored using a von Neumann architec-  ture [12] for MicroRAM because, despite the asymptotic  benefits, the instruction fetching circuit is not yet a limit-  ing factor in our ZK statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2  MicroRAM Compiler  The MicroRAM Compiler is implemented as a LLVM back-  end that takes LLVM IR programs as input and produces Mi-  croRAM assembly as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We currently support C, C++,  and Rust programs by compiling them to LLVM IR with the  Clang and rustc compiler frontends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Support for other lan-  guages such as C#, Haskell, or Scala can be added in the  future by connecting their appropriate LLVM frontends and  writing the appropriate standard libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Our compiler backend supports a large subset of the  LLVM IR language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The compiler supports all boolean and  arithmetic operations for integers of different sizes, bitwise  operations, all non-concurrent memory operations including  pointer arithmetic with getelementptr, conversion opera-  tions, function calls, variable arguments, comparisons, and  phi nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Complex operations like floating-point opera-  tions are implemented in software via a LLVM compiler  1If full words fit in a field element, then the flag is the same size as a  register, but requires special circuitry and has more restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Exceptions, and all exception handling instructions, are  not supported;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' but we we can still tolerate programs with ex-  ceptions as long as the prover is disclose that the execution of  interest, which triggers a vulnerability,does not throw any ex-  ceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This is since the MicroRAM Compiler translates all  exception handling instructions to traps that mark the trace as  invalid by setting the inv_flag flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' By inserting traps, the  MicroRAM Compiler can process programs with any num-  ber of unsupported features, as long as the prover is will-  ing to reveal that those features are not involved in the vul-  nerable execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' With this simple trick, users can compile  real-world programs without having to manually remove un-  supported features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' When enabling traps, provers must take  care not to reveal too much information about the underlying  vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3 presents a more detailed discussion  about the security implications of how proof statements can  reveal information about their witnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1  Standard library  MicroRAM supports a significant portion of the C standard  library and POSIX system calls, using Picolibc [4]: a library  that offers the C standard library APIs and was originally de-  signed for small embedded systems with limited RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Picol-  ibc supports multiple widely deployed target architectures,  including ARM, RISC-V, and x86-64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  We implemented MicroRAM as a target architecture for  Picolibc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This enables the MicroRAM compiler to support  most of the C standard library and POSIX system calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' It  is also convenient as it allows prover to publicly customize  the behavior of system calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' For example, in our case study  of OpenSSL, the victim server receives the malicious re-  quest from the attacker over the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We customized the  behavior of read when compiled natively to intercept and  record all data received over the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' When compiled  for MicroRAM, read returns the previously recorded exploit  request, which is loaded from secret memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We also cus-  tomize the implementation of malloc and free to efficiently  detect memory vulnerabilities (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2  Generating advice  As we will see in later sections, CHEESECLOTH requires non-  deterministic advice to efficiently generate a ZK circuit that  verifies the consistency of memory in an execution (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3)  and the presence of a vulnerability (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' To aid the  prover in producing that advice, the MicroRAM compiler  runs two interpreter passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The first pass executes the pro-  gram without any advice and records the necessary advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The second execution runs the nondeterministic semantics  and records the trace, which is passed to the Witness Checker  Generator to produce the witness for the prover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3  Security  MicroRAM produces zero-knowledge proofs which ensure  that no additional information is revealed about the wit-  ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' However, the proof statement itself can reveal informa-  tion about the secret input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' For example, in MicroRAM and  TinyRAM the circuit reveals a time bound T on the execution  length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In Pantry/Buffet, the circuit discloses an upper bound  Ti on every loop (and recursive function) in the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In  vRAM [53], every instruction run during the execution is re-  vealed to the verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We argue that a formalization of this  information leakage is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Interesting and important  future work will be to define a formal framework to analyse  how secure these encodings are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='4  Preprocessing public inputs  One opportunity for aggressive optimization is to publicly  evaluate logic that is determined by the program’s public in-  puts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Many practical programs collect inputs from multiple  sources, some of which are not secret (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', irrelevant to the  vulnerability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' If the prover and verifier agree when defining  a proof statement that only some inputs are sensitive secrets  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', data packets received from a network connection) while  others are not (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', straightforward configuration options),  then the resulting proof statement could be immediately op-  timized by generating the proof statement and partially eval-  uating the resulting circuit on its input wires corresponding  to non-sensitive inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  CHEESECLOTH supports such cases with a compiler pass  that determines the largest program prefix in which no op-  eration depends on secret inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The MicroRAM compiler  then separates the public prefix from the remaining program  suffix and compiles them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' When the interpreter is  executed by both the prover and verifier, it executes the pre-  fix and defines a public snapshot of the resulting state, includ-  ing both registers and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' When executed by the prover,  the interpreter then executes the remaining suffix using both  the snapshot and their private input generate to generate the  statement’s witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In practice, this simple optimization has  significant impact, reducing the number of execution steps in  OpenSSL’s ZK proof statement from 25M to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3M (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The compiler optimization implements a relatively re-  stricted form of partial evaluation and constant folding  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Our initial experience indicates that further ex-  tensions could improve CHEESECLOTH’s performance dras-  tically: a key technical challenge is that while programs may  perform much processing of public data over the course of  the entire execution, the processing is often interleaved with  computation over sensitive inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Evaluating each of the in-  terleaved phases of public computation is sound in principle,  but can only be automated by ensuring that regions of storage  used by public and secret phases are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Such automa-  tion could potentially be achieved by applying points-to and  shape analyses [46–48], including separation logic [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3  Witness Checker Generator  The Witness Checker Generator takes as input a MicroRAM  program and generates a ZK circuit, serialized in standard-  ized formats including R1CS and SIEVE IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' It also accepts  nondeterministic advice as input and outputs the secret wit-  ness to the circuit when run as the prover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The Witness Checker Generator builds arithmetic circuits  for the prime field 2128−159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' As an optimization, it automat-  ically constant folds gates that are independent of secret in-  puts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' To scale to large circuits and avoid running out of mem-  ory, it streams the circuit serialization to a file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This stream-  ing is independent of secret witnesses, so the same circuit is  generated for the prover and verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The nondeterministic advice the Witness Checker Genera-  tor accepts provides a description of a program’s execution  together with the advice necessary to run it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Concretely, the  advice for an execution of T steps contains the initial pro-  gram memory, the T MicroRAM states making up the execu-  tion trace, and a mapping from step number to additional ad-  vice given at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This additional advice includes mem-  ory ports for what is read or written to memory and stutters  that indicate the execution should pause for the current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The Witness Checker Generator produces a ZK circuit that  verifies that the witness describes a valid execution trace for  the program and that a vulnerability occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The circuit is  split into four key pieces: (1) the transition function circuit,  (2) the memory consistency circuit, (3) a state transition net-  work, and (4) public-pc segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We describe the first two  here, which follow a similar structure to the circuit construc-  tion for TinyRAM [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The other two are described later in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Transition function circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The transition function circuit  checks a single step of execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' These checks are chained  together to validate the entire execution trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 1 shows  pseudocode for the transition function circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' It takes as  input the circuit’s wire representation of the current Micro-  RAM state, the next state, and any additional advice needed  for the current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The circuit then fetches the instruction  to execute based on the program counter and pulls out the  instruction’s argument values by indexing into machine reg-  isters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' It calculates the expected result of the step by multi-  plexing over the instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Finally, the circuit ensures that  the calculated expected state matches the next state provided  as advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Memory consistency circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The memory consistency cir-  cuit is similar to TinyRAM’s except addresses are byte-  addressable instead of word-addressable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Each step may  have a corresponding memory port advice that states the ad-  dress and what was read or written to memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The transi-  tion function circuit verifies that the execution trace matches  the memory port advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' All of the memory ports are sorted  by address and step number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The memory consistency cir-  6  1  fn transition_func(circuit, current_st, next_st) {  2  let expected_st = current_st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='clone();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  3  let instr = fetch_instr(circuit, current_st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='pc);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  4  let arg1 = index(circuit, current_st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='regs, instr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='op1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  5  let arg2 = index(circuit, current_st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='regs, instr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='op2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  6  7  let result = circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='mux(instr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='opcode == XOR,  8  xor(circuit, arg1, arg2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=');' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  9  expected_st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='pc = circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='mux(  10  is_jump(circuit, instr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='opcode),  11  result,  12  circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='add(current_st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='pc, 1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  13  write_index(circuit, expected_st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='regs, instr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='dest,  14  result);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  15  16  circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='assert(expected_st == next_st);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' }  Figure 1: Pseudocode for the transition function circuit that  validates a single MicroRAM step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  cuit linearly scans the memory ports to ensure that all reads  and writes to a given address are consistent with the previ-  ous memory operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' For example, a read should return  the same value that was previously written to an address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Fi-  nally, the memory consistency circuit checks that the sorted  memory ports are a permutation of the memory ports used by  the transition function circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1 describes how these  checks are enhanced to efficiently detect memory vulnerabil-  ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  4  Optimizations  This section describes two of CHEESECLOTH’s key optimiza-  tions: constructing executions with public program coun-  ters (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1) and tuning steps based on instruction sparsity  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='4 contains an empirical evaluation of the  optimizations’ effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1  Public-PC segments  The MicroRAM machine is design to minimize the size of  the circuit that checks the transition function circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' How-  ever, even with MicroRAM’s small instruction set, the tran-  sition function circuit is still large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This is due to the fact  that every transition function must support every operation  in every step of execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' What if we could remove all the  unused functionality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This is the approach of vRAM [53],  where the circuit is tuned to check the instruction that is exe-  cuted at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The resulting circuit is much smaller, but  unfortunately the trace of executed instructions is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The values in memory and registers would still be kept secret,  however a verifier could easily discover where the vulnerable  code is in the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In this section, we present public-pc  segments which generate much smaller circuits without re-  vealing the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  A public-pc segment is a sequence of transition circuits  with a hardcoded and public program counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Using con-  stant folding, all the instruction fetches of the public-pc seg-  ments are known and the unused functionality of every step  can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' For example, with a public program counter,  the fetch_instr and subsequent mux operations over the in-  struction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 1 can be constant folded away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We gener-  ate public-pc segments for all straightline code segments in  a program, and we implemented a compiler pass that uses a  naive control-flow analysis to estimate how many times each  segment will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The analysis takes a global bound spec-  ifying how many times to unroll loops and estimates how  many times a function will be called by counting the number  of call sites for that function in the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  To preserve the security of the trace, the cycle counter of  segments is kept private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In addition, we introduce a state  routing network so that the end state of a segment could be  the initial state for any other segment in the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Just like  the memory routing network, the routing information for the  state routing network is given by the prover and kept secret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  As a further optimization, we avoid using the state rout-  ing network when possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' For example, when a public-pc  segment branches to two statically known locations, we di-  rectly connect the end state of that segment to the segments  representating those two locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  It is possible that the bound for unrolling loops is not large  enough to support certain executions, so the pipeline would  not generate enough public-pc segments for a section of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  As backup, the pipeline also produces private-pc segments  which are just like public-pc segments except the program  counter is not revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Private-pc segments look similar to a  much smaller TinyRAM circuit with the difference that their  start and end states come from the state routing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The  circuits for these segments are significantly larger, but can ex-  ecute any part of the program at any point during execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2  Sparsity  With the naive CPU unrolling described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3, ev-  ery transition function must contain a memory port, which  causes the memory consistency network to grow at a rate of  O(T logT), where T is the number of steps executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Unfor-  tunately, most of those gates are wasted by execution steps  that do not access memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' CHEESECLOTH mitigates this ex-  cess by removing some of the unused memory ports, thereby  reducing the size of the memory consistency circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The key observation for this optimization is that memory  operations are rarely contiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Even when a program per-  forms a memory-intensive operation, other instructions are  often interleaved between memory instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' For exam-  ple, when adding the values in a buffer, it takes some steps  to increment the pointer and add the values between memory  reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This enables us to share one memory port among s  contiguous steps, shrinking the memory consistency network  7  by a factor of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  We define the memory sparsity, s, as the number of steps  that share a single memory port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' CHEESECLOTH chooses s  based on a static analysis of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The analysis deter-  mines the minimum distance between two memory opera-  tions in any possible execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Across statically-unknown  jumps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' calling a function from a pointer dereference),  the analysis naively considers all the instructions the control  flow can possibly jump to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This memory sparsity number s is  then used by the MicroRAM Compiler and Witness Checker  Generator to generate the optimized circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Given a memory sparsity s, the Witness Checker Generator  will group s consecutive steps and create a single memory  port for all of these steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' A multiplexer connects the single  memory port to the entire group and sends the result, using  nondeterministic advice, to the right step (if any).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  If s is larger than the actual sparsity displayed by a trace,  then (if unlucky) multiple memory accesses can fall into  the same group of steps, which has a single memory port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  CHEESECLOTH handles this situation by inserting stutter  instructions that delay memory operations until they are  pushed into the next group with separate memory ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In-  serting stutter instructions can be expensive, but reducing the  size of the memory consistency circuit is more beneficial  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In future work, we will explore swapping program  instructions to reduce stutter instructions and determine the  optimal s parameter for most programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  5  Encoding Vulnerabilities  This section describes how CHEESECLOTH encodes two  prevalent and critical classes of software vulnerabilities:  memory unsafety (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1) and data leakage (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1  Memory unsafety  We now describe how CHEESECLOTH efficiently models  memory and represents memory vulnerabilties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In CHEESE-  CLOTH, memory is an array of 264 bytes with half reserved  for the heap and the rest for global variables and the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Our approach is to keep track of valid memory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' allo-  cated arrays) and report a vulnerability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', set bug_flag)  when the program accesses non-valid memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' At the start  of the execution, the only valid memory is where the global  variables are stored and, during execution,malloc makes allo-  cated regions valid and free makes them invalid again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' With  this technique we can catch the following memory errors:  Uninitialized access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' All uninitialized memory is invalid,  so any use triggers a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Use-after-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' When a region is freed it becomes invalid,  so any use triggers a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Free-after-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The implementation of free starts by read-  ing a word from the region to be freed, if the region is not  1  void* malloc(size_t size) {  2  // Get pointer from advice  3  char* addr = __cc_malloc(size);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  4  /* Compute and validate the size of  5  the allocation provided by the  6  prover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' */  7  size_t region_size =  8  1ull << ((addr >> 58) & 63);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  9  /* The allocated region must have  10  space for `size` bytes, plus  11  an additional word for metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  12  /  13  __cc_valid_if(  14  region_size >= size + sizeof(uintptr_t),  15  "allocated region size is too small");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  16  /* `region_size` is always a power of  17  two and is at least the word size,  18  so the address must be a multiple  19  of the word size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' */  20  __cc_valid_if(addr % region_size == 0,  21  "allocated address is misaligned for"  22  "its region size");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  23  /* Write 1 (allocated) to the metadata  24  field, and poison it to prevent  25  tampering, invalidating the trace  26  if the metadata word is already  27  poisoned (this happens if the  28  prover tries to return the same  29  region for two separate  30  allocations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' */  31  uintptr_t* metadata = (uintptr_t*)  32  (addr + region_size - sizeof(uintptr_t));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  33  __cc_write_and_poison(metadata, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  34  35  // further computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  36  return (void*)addr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' }  Figure 2: Implementation of non-deterministic malloc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  valid it triggers a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Out-of-bound access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' If the program accesses an address  out of bounds, that new location might (see below) not be  valid and this triggers a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  It is clear that a normal execution with such bound check-  ing might miss out-of-bound access bugs, when the ac-  cess happens to fall on another valid region, and free-after-  free/use-after-free bugs, if an intermediate malloc makes the  region valid before the bug is triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' However, we only  need to show that the bug exists in one execution, so we im-  plement a malloc guided by nondeterministicadvice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' this lets  the prover choose the allocation layout to ensure the bug is  triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  While the techniques described here are specific to heap  memory bugs, the same ideas can be applied to the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1  Encoding dynamic memory allocation  An implementation of malloc with nondeterminism poses  its own challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' If left unchecked, the prover could man-  ufacture an execution that triggers a false bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' For example  the prover could malloc overlapping regions such that if one  8  is freed and the other one is accessed a false bug is triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Thus, our implementation of malloc and free (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 2) focuses  on verifying that the nondeterminisitc choices are legal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' If  foul play is detected, the execution is flagged as invalid with  inv_flag and will not be accepted by the verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  To ensure that malloc never returns overlapping regions,  we predetermine aligned non-overlapping regions of differ-  ent sizes for malloc to choose from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Concretely, we divide  memory into 26 pools of size 258, then subdivide pool i into  regions of size 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' malloc rounds up the requested size to the  next power of two, then returns the start of an unallocated  region of that size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' For example, malloc(15) must return a  region in the 4th pool and be 16-byte aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In fact, we can  easily verify that malloc has allocated a correct region just  by looking at the pointer returned: the first 6 bits determine  the pool and the rest the alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Finally, malloc must not return the same pointer twice  without it being freed in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' To do so, we add to each  region one word reserved for metadata that is marked and  made invalid when the region is allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' If the region was  allocated again, the invalid metadata would be made invalid  again which makes the trace invalid by setting inv_flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Furthermore, an implementation of malloc/free that tracks  the validity of all memory locations would be quite inef-  ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Luckily, the prover knows exactly where the bug  will happen and thus the malloc/free implementation only  needs to track the status of that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' At the beginning  of the execution, the prover commits to a secret location  stored in the global variable __cc_memory_error_address  and then malloc/free only track the validity of that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  In particular, if an allocated/freed region does not contain  __cc_memory_error_address then malloc/free do not check  for errors, and run in constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2  Data leakage  A straightforward approach to proving leakage would be to  directly encode the definition of noninterference in the ZK  circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This could be accomplished by verifying two pro-  gram executions where only sensitive inputs are distinct but  public outputs are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' However, suchan approach would  result in a statement of twice the size required for validating  a single execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Instead, we might hope to prove a leak  using a single execution in which storage is annotated with  labels (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' However, such systems tranditionally have  only been designed to prove that a program may leak infor-  mation, which is unacceptable for definitively proving a leak  without providing a violating execution directly (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Specifying leakage To identify sensitive sources and sinks,  the instructions source and sink are added to the Mi-  croRAM instruction set, and are directly wrapped by user-  level functions taintSource and taintSink, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  source annotates that a given byte of data carries sensitive  data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' sink annotates that a given byte is output to a channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Instantiating the general definitions of information flow and  leakage (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2) for this extended ISA, a MicroRAM pro-  gram leaks if it has two executions whose inputs only differ  at addresses given to source, but result in different values  at an address given in calls to sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Leakage is established  by the prover and verifier collaborating to extend the subject  program to call the taintSource and taintSink to anno-  tate sensitive sources and sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Proving leakage To soundly and precisely prove leakage, we  propose a novel labeling system that tracks what program  storage may and must hold secret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' There are four labels, de-  noted and partially ordered as  ⊥ ⊑ ℓ0,ℓ1 ⊑ ⊤  with a least-upper bound (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', join) operation denoted ⊔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' La-  bels ℓ0 and ℓ1 annotates data that must belong to one of two  principals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' ⊤ denotes that the data’s sensitivity is unknown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  ⊥ denotes data that must not be influenced by a principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  With this labeling scheme, leakage of ℓ-labeled data written  to a ℓc-labeled sink must occur when ℓ ̸= ⊤ ∧ℓ ̸⊑ ℓc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  MicroRAM state is extended so that every register and  byte of memory is associated with a label, similar to previ-  ous leakage monitors [20, 42, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Two additional labels  model effects of instructions other than register arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The control context label γ is maintained to be ⊥ if the pro-  gram execution has not branched on secret data, and ⊤ other-  wise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' similarly, the storage context label σ is maintained to  be ⊥ if the program has not stored to a secret address, and ⊤  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Each assignment x:=e sets the label of x to L(e) ⊔ γ ⊔ σ  (where L(e) is the label of e, defined below);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' thus, if the pro-  gram has branched on secret data or written to a secret ad-  dress, the label of x is set to ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' If e is an arithmetic/logical  operation f(y), then L(e) is ⊥ when L(y) is ⊥ and ⊤ other-  wise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' L(e) = L(y) if f is a bijection: our current implemen-  tation conservatively only labels single-register expressions  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', copy sources) as L(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' If e is a load *p, then L(e) is  L(*p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Conditional branches update γ and memory stores up-  date σ according to the labels’ descriptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' we omit formal  descriptions here, due to space constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Plenty of natural programs leak but cannot be proved to  do so by this labeling system, potentially because a leakage  happens after branching or storing to a ⊤-labeled value, or  because a secret value is propagated over an operation not  recognized as a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Such cases restrict the situations in  which the labeling scheme can be applied to prove leakage,  but do not threaten its validity when it claims that a given  program leaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' They might be addressed in future work that  refines instruction interpretations using valid logical axioms  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', the fact that for each value x, x + 0 = x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Such threats  and mitigations are dual to threats to precision, and possi-  ble refinements when proving that a program execution may  leak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  9  6  Evaluation  We evaluate CHEESECLOTH with three case studies that  demonstrate ZK proofs of real world software vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The vulnerabilities scale by code size and execution trace  length to showcase the capabilities of CHEESECLOTH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We  also benchmark the optimizations (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 4) to evaluate their  effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 1 presents the results of using CHEESECLOTH to pro-  duce ZK proofs for our case studies which include GRIT,  FFMPEG, and OpenSSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' For each case study, we report  the size of the program in terms of the number of Micro-  RAM instructions, the number of execution steps required  to demonstrate the vulnerability, and the number of multipli-  cation gates in the resulting ZK circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We prove satisfia-  bility of the ZK circuit using the Mac’n’Cheese [9] interac-  tive ZK protocol, as implemented by the Swanky [23] library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  We record the protocol running time and communication cost  between the prover and verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' All measurements were per-  formed on a 128 core Intel Xeon E7-8867 CPU with 2 TB of  RAM running Debian 11, although our implementation typi-  cally uses considerably less memory (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1  Memory unsafety in GRIT  The GBA Raster Image Transmogrifier (GRIT) [34] converts  bitmap image files to a graphics format that is readable by  the Game Boy Advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' A bitmap image includes headers, a  palette array indicating the colors in the image, and the pixels  for the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' For 24bpp images, GRIT’s parser assumes the  palette size is zero and allocates a buffer without space for  the palette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' When populating the buffer, it checks the image  header for the numberof palette entries without checking that  this matches the assumed palette size that was used during  allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' As a result, a malformed 24bpp image can write  an arbitrary amount of data (up to the length of the file) past  the allocated buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  To demonstrate this memory error, we construct a 24bpp  exploit image with 0x3000 bytes of pixel data and 12 bytes of  palette data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' On Linux, the 12 byte overflow overwrites heap  metadata and triggers an assertion failure in the memory allo-  cator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' When run through CHEESECLOTH, we generate a ZK  proof that a memory error is triggered within six thousand  steps of GRIT’s execution without revealing the triggering  image or where the error occurred in the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2  Memory unsafety in FFmpeg  FFmpeg is a tool for recording, converting, and streaming au-  dio and video [2], and is used in popular software projects  such as Chrome, Firefox, iTunes, VLC, and YouTube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' FFm-  peg is written in C and has been plagued by vulnerabilities  that compromise memory safety, enabling attackers to exe-  cute code and share local files over the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Versions of  FFmpeg prior to v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2 contained a vulnerability [1] caused  by the memory error in the function gif_copy_img_rect,  which copies the frame of a GIF file between buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Previ-  ous versions of gif_copy_img_rect insecurely calculated  a pointer to the end of a memory buffer by directly using  the input image’s height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This calculation allowed an attacker  to provide a carefully crafted GIF which causes FFmpeg to  write to memory outside of an array’s bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  To prove memory unsafety of FFmpeg in ZK, we manually  crafted a GIF image that exploits the described memory vul-  nerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We passed this image and a program module that  invokes FFmpeg’s video decoder to CHEESECLOTH, which  generated a proof of a out-of-bound access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The only facts re-  vealed about the exploitative GIF are what are implied by the  fact that it trigger an out-of-bound access within 76K steps  of execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Preprocessing FFmpeg on public inputs There was po-  tential to aggressively optimize FFmpeg’s proof state-  ment, which was ultimately achieved by applying CHEESE-  CLOTH’s constant folding transformation pass after manual  program partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The need for partitioning arose due to  the interleaving of public and secret computation in the GIF  modules, which executes by: (1) demultiplexing a given se-  cret GIF file into a sequence of data packets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' (2) initializing  the state of the decoder, using public configuration settings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  (3) executing the codec that contains the vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Although phase (2) computes entirely over public data, it  would not be optimized by CHEESECLOTH’s constant fold-  ing pass because the pass halts upon detecting computation  that uses secret data, and thus would not optimize any pro-  gram segment after phase (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' To address this issue, we man-  ually partitioned the program by phase, applied CHEESE-  CLOTH’s constant folding pass to each, and linked the result-  ing optimized MicroRAM code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In general, our case study  of FFmpeg motivates the further study and design of more  aggressive constant folding passes, which might apply more  sophisticated static program analyses (Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3  Leakage in OpenSSL  OpenSSL [3] is a widely deployed open-source crypto-  graphic library that contains implementations of the SSL and  TLS protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' OpenSSL versions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1f contained  a devastating vulnerability dubbed Heartbleed, discovered in  2014 [5], that could be exploited by a remote attacker to  completely leak information stored over the protocol’s exe-  cution, including other clients’ sensitive information and pri-  vate keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Comprehensive descriptions of SSL and OpenSSL are be-  yond the scope of this paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' for the purposes of our work,  it suffices to note that SSL parties support multiple requests,  including both requests to store data from the another party  and requests to reply to a heartbeat signal: a signal sent only  10  Program  Code size (K instrs)  Execution steps (K)  Mult gates (M)  Protocol time  Protocol memory  GRIT  3  5  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='7  3m 40s  845 MB  FFmpeg  24  79  672.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='7  1h 22m  19 GB  OpenSSL  340  1,300  17,049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='5  36h 45m  460 GB  Table 1: Results for generating and running a ZK proof of software vulnerability for each case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  1  void process_heartbeat(SSLRequest *req) {  2  unsigned int len = parse_heartbeat_len(req);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  3  unsigned char *heartbeat = get_heartbeat(req);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  4  unsigned char *response = malloc(len);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  5  memcpy(response, heartbeat, len);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  6  write(response, len);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' }  Figure 3: Pseudocode depicting the Heartbleed vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  to obtain a response to ensure that the other party is still re-  sponsive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The heartbeat request and response is critical to the oper-  ation of the Heartbleed vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' A well-formed request  consists of a data buffer d and a length field n < |d|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' A cor-  rect response to such a request returns the first n bytes con-  tained in d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' However, a party could potentially transmit an  ill-formed request, in which n > |d|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The correct response to  such an ill-formed request is to reject it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The implementation of OpenSSL (illustrated by the pseu-  docode function process_heartbeat in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 3) crucially  failed to implement this aspect of the protocol and instead  returned the n bytes of memory contiguous with the input  buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' process_heartbeat takes a heartbeat request from  a client and echos the provided heartbeat string back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' It does  so by first parsing the length of the heartbeat string from  the client’s request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The function then gets a pointer to the  heartbeat string in the request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Next, it allocates a response  buffer and copies len bytes from the heartbeat string into  the response buffer, which is subsequently sent back to the  client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Since process_heartbeat does not check the pro-  vided heartbeat length against the actual length of the heart-  beat string, if the claimed length is larger than the actual  length of the provided heartbeat string, memory beyond the  client’s request is sent back to the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This is practically  exploitable, and has been demonstrated to reveal sensitive in-  memory data such as cryptographic keys and passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Using CHEESECLOTH, we proved in ZK that OpenSSL  version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1f leaks arbitrary user information in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3M steps  of execution, propagating data purely over register copies,  loads, and stores;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' while the statement reveals a bound on the  amount of computation required to perform the leak and in-  formation about the types of instructions used to perform the  leak (described below), it gives no direct indication of what  validation is missing in the function for processing heartbeat  requests, or that heartbeat requests are involved in the leak at  all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We describe the statement proved, along with technical  challenges and solutions, in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  1  int login_handler(  2  SSLConn *c, char *password, int len) {  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  4  label l = getLabel(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  5  for (size_t i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' i < len;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' i++)  6  taintSource(password + i, l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  7  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' }  8  int ssl3_write(  9  SSLConn *c, char *buf, int len) {  10  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  11  label l = getLabel(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  12  for (size_t i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' i < len;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' i++)  13  taintSink(buf + i, l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  14  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' }  Figure  4:  Versions  of  the  OpenSSL  functions  login_handler  and ssl3_write  that we  augmented  with operations that specify information sources and sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Passwords are tainted with the label of the current connec-  tion, and leaks are detected if data written to the network has  a label from a different connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Specifying OpenSSL’s leakage A primary challenge of our  work was to provide a scheme for identifying sensitive  sources and sinks such that:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' A verifier with only an understanding of the data that a  subject program handles should be able to inspect the  modified program and definitively conclude that it cor-  rectly defines information sources and sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Any modifications to the program to enable the defini-  tion of sources and sinks between which information is  leaked must not reveal additional information about the  leak’s triggering input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Our mechanism for defining sensitivity of sources and  sinks consists of the designated functions taintSource and  taintSink (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We found that such a library served  well for specifying information flow in in OpenSSL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' psue-  docode of C functions modified in the OpenSSL codebase  to label sources and sinks are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The function  login_handler, given an SSL connection c and a buffer  password presumed to contain len bytes of sensitive infor-  mation to be transmitted over c, labels len addresses be-  ginning with password with the label of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The function  ssl3_write, given an SSL connection c and buffer buf pre-  sumed to output len bytes, denotes sinks at the output chan-  nel with the label of c for len addresses beginning with buf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  The modifications to login_handler and ssl3_write  11  Program  Mult gates without  public-pc segments (M)  Mult gates without  sparsity (M)  GRIT  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3 (37%)  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='9 (4%)  FFmpeg  716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='9 (6%)  709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='9 (5%)  Table 2: The number of multiplication gates in the circuit  with the different optimizations disabled, as well as the per-  centage increase in size over the baseline numbers from  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  illustrate the utility of first-order labels that can be opera-  tionally collected and set, as opposed to operations that set  addresses as only high sources or low sinks, even in a setting  in which the information belonging to only one principal is of  interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' By using first-order labels, we we able to write small  specification functions that only unified the labels between a  network connection and a given buffer, and then succinctly  modified the original program logic in contexts that readily  provided a connection and related buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Proving OpenSSL’s leakage Once OpenSSL has been suit-  ably modified to call the taintSource and taintSink func-  tions, its leakage can be proved by generating a statement  whose solution corresponds to an execution of a server run-  ning OpenSSL that leaks sensitive data from one connec-  tion to another connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We have generated such a state-  ment where the server first responds to a public login request  where the password is marked as sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The server then  handles a secret malicious heartbeat request that returns the  password from the previous request’s connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Using CHEESECLOTH we prove OpenSSL’s leakage by  validating the previously described execution which is de-  rived from one of its originally disclosed exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The leak-  age is detected through the source and sink annotations ac-  cording to our proposed must-leak labeling scheme (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2)  and the verifier only learns an upper bound on the length of  the malicious request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We found that the labeling scheme en-  abled leakage to be proved much more efficiently, reducing  the overall circuit size by 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='6% over the two trace approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  CHEESECLOTH proved the vulnerability of OpenSSL in ap-  proximately 37 hours, using 460 GB of protocol communica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='4  Optimizations  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 2 contains the improvements yielded by our key opti-  mizations (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' We ran the GRIT and FFmpeg case stud-  ies with each optimization disabled and report on the num-  ber of multiplication gates in the resulting ZK circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In  addition, we provide the percentage improvement over the  baseline numbers from Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The public-pc optimization  reduced gate size by 37% in the shorter GRIT execution and  6% for FFmpeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' While this is an improvement, these results  indicate there is still room for improvement in our analysis  that determines the number of public segments to generate  for longer executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The sparsity optimization with s = 2  offers modest improvements of 4%–5% in gate size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  7  Related Work  Recent work has provided the first, exciting steps toward  proofs of vulnerability in ZK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' BubbleRAM [29] is an ef-  ficient framework for proving vulnerability, leverage novel  protocols for converting between computations in arithmetic  and Boolean fields, efficiently handling both read-only and  read-write memory, and the Stack protocol [30] for prov-  ing satisfaction of circuits with explicit disjunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Al-  though our current statement compiler partially overlaps with  BubbleRAM because it implements an older scheme for mod-  eling RAM computations [10], most of our paper’s key con-  tributions, namely simplifying unrolled computations using  partial evaluation and the novel scheme for generating state-  ments of application leakage, are largely independent of the  contributions of [29,30], and we believe that the approaches  could be composed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In particular, Stack was evaluated on  code snippets representative of a practical CVE of up to 50  LoC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' due to its efficient support of disjunctions, it could scale  to prove that one of many more such snippets is vulnera-  ble, but it likely strongly benefit from CHEESECLOTH’s pro-  gram optimizations if any particular code segment increased  in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Reverie [27] is a framework for proving exploits in mi-  croprocessor code, consisting of a circuit generator that com-  piles a given program to an arithmetic circuit and an instan-  tiation [36] of the “MPC in the Head” protocol [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The  compiler generates statements from exploits that have been  formalized as executing a designated instruction that signals  an error condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=', violations of reachability properties,  formalized directly in the program’s control flow);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' the eval-  uation of Reverie demonstrates that it can be used to prove  Capture the Flag (CTF) exploits that require up to 51K cy-  cles on an MSP430 microprocessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The core contributions  of Reverie are largely complementary to those of CHEESE-  CLOTH, which could potentially be adapted to efficiently  compile vulnerability statements about programs in interme-  diate languages to control reachability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Recent work on static program analysis in ZK [21] has  presented techniques for proving over-approximations of all  program executions without revealing further details of the  program, and instantiates the framework on an abstract do-  main for information flow based on taint tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The static  analysis itself is designed to prove that a program may leak  information: thus, it cannot yield results that directly imply  that a program must leak, although in many cases it could  provide evidence that could strongly inform an analysts be-  lieft that a program may in fact leak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Our MicroRAM machine is inspired by TinyRAM [10] but  departs from their design in sevaral important ways discussed  12  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' There are also some key differences in scope and  capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' TinyRAM is designed to express correctness  of any nondeterministic computations while MicroRAM fo-  cuses on vulnerable programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' For example, SNARKs for C  [10] approach cannot encode proofs of memory-safety vul-  nerability in ZK directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Instead, they encode knowledge  of the existence of a complete, concrete vulnerability trace,  which includes copies of exact values in all local variables  and the values in memory at each point in the trace and the  bug must be evident in the execution’s return value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Our ap-  proach encodes memory vulnerabilities directly, resulting in  a significantly more succinct witnesses to vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In  particular we can disregard the trace after the bug is found  and we don’t rely on the programs return value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Furthermore, TinyRAM approach does not scale to proofs  of vulnerabilities in practical programs and has only been  evaluated on programs with less than 1,200 low-level instruc-  tions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' In contrast, the optimizations proposed in this  work enables us to support programs with more than 340,000  lines of low-level code (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Beyond scalability, Micro-  RAM supports a much broader subset of the C language, in-  cluding most of the standard C library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Pantry and Buffet [17, 52] represent computation as arith-  metic constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' a solution to the constraints is a valid trace  of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' After implementing the memory consis-  tency approach of TinyRAM, they report results orders of  magnitude better than TinyRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Buffet supports all features  in the C language, with the exceptions of goto statements  and function pointers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' To translate computation into a con-  straint system, Pantry and Buffet must unroll loops to pub-  licly revealed bound (although the original work does not  explicitly discuss encoding recursive functions, we hypoth-  esize that they would be encoded similarly, using bounded  function inlining).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' The constraint system must include every  branch of conditionals and every iteration of every loop (mul-  tiplicatively with nested loops) which could lead to blowups  in the constraint system, howeverthe authors suggest that this  would only happen in degenerated cases and would not be  common in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' A variant Pantry/Buffet that uses zero-  knowledge techniques to keep the state private with the same  efficiency benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' When presenting our approach, we com-  pare facts about private inputs that it reveals to those revealed  from public loop bounds (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  vRAM [53] has achieved further efficiency with a inge-  nious universal preprocessing that allows the parties to use  a smaller circuit tailored to verifying the specific program  on the chosen inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Unfortunately, such tailored circuits  can reveal significant information about the input provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Our public-pc optimization (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='1) attempts to balance the  gains of a tailored circuit and the privacy requirements of the  prover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  8  Conclusion  Due to a sustainted successes in the development of ZK  protocols, recent techniques have reached the cusp of prov-  ing knowledge of realistic vulnerabilities and proving sub-  tle exploits in low-level code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' This paper describes how  a host of core techniques from compiler design—namely,  conservative instruction profiling and under-approximating  information-flow tainting—can be implemented in an opti-  mizing proof-statement generator to produce proofs of vul-  nerability in commodity software that can be triggered only  be using a considerable amount of time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Our practical experience has produced a zero-knowledge  proof of memory unsafety in FFmpeg and a proof of leakakge  in OpenSSL that directly used the Heartbleed exploit as a wit-  ness and demonstrates that zero knowledge proofs of vulner-  ability in critical application software are now practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Availability and Ethical Considerations  We are in process of open sourcing the implementation  of CHEESECLOTH for publication and artifact evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  CHEESECLOTH aids in responsible disclosure by produc-  ing zero-knowledge proofs of the existence of vulnerabili-  ties while keeping the vulnerabilities and exploits secret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' All  vulnerabilities used in our evaluation have been previously  disclosed publicly, and fixes are widely deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Thus, the  work presented in this paper does not constitute an unethical  disclosure of potentially harmful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Acknowledgments  This material is based upon work supported by the Defense  Advanced Research Projects Agency (DARPA) under Con-  tract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' HR001120C0085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Any opinions, findings and con-  clusions or recommendations expressed in this material are  those of the author(s) and do not necessarily reflect the  views of the Defense Advanced Research Projects Agency  (DARPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Approved for Public Release, Distribution Unlim-  ited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  References  [1] CVE-2013-0864.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  https://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='org/  cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='name=CVE-2013-0864.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Accessed: 2022-10-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  [2] FFmpeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' https://ffmpeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Accessed: 2022-09-  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  [3] OpenSSL:  Cryptography  and  SSL/TLS  toolkit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  https://openssl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Accessed: 2022-09-05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  13  [4] Picolibc: C libraries for smaller embedded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  https://keithp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='com/picolibc/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Accessed: 2022-  10-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  [5] The Heartbleed Bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  https://heartbleed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Accessed: 2022-09-05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  [6] zkInterface:  SIEVE  intermediate  representation  (IR) proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  https://hackmd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='io/@danib31/  BkP9HBp2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content=' Accessed: 2022-10-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  [7] Alfred V Aho, Monica S Lam, Ravi Sethi, and Jeffrey D  Ullman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Compilers: principles, techniques, & tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
+page_content='  Pearson Education India, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf'}
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diff --git a/TtE4T4oBgHgl3EQfLwyC/content/tmp_files/2301.04941v1.pdf.txt b/TtE4T4oBgHgl3EQfLwyC/content/tmp_files/2301.04941v1.pdf.txt
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+arXiv:2301.04941v1  [math.RT]  12 Jan 2023
+RIGID INTEGRAL REPRESENTATIONS OF QUIVERS REVISITED
+WILLIAM CRAWLEY-BOEVEY
+Abstract. In earlier work, the author classified rigid representations of a quiver by
+means of finitely generated free modules over a principal ideal ring.
+We show that
+the classification of exceptional pointwise free lattices can be extended to arbitrary
+commutative rings and that the classification of rigid lattices can be extended to reduced
+commutative rings.
+1. Introduction
+In [4] we studied rigid representations of a quiver over a principal ideal domain. Here
+we generalize the results to more general commutative base rings.
+Let R be a commutative ring (always non-zero). Given an R-module X and a homo-
+morphism R → S we write XS for the induced S-module S ⊗R X. Recall that a finitely
+generated projective R-module P has constant rank n if Pp is a free Rp-module of rank
+n for each prime ideal p of R, or equivalently if dimK P K = n for all homomorphisms
+R → K with K a field, which we may take to be algebraically closed.
+Let Q = (Q0, Q1, h, t) be a finite quiver and let RQ be the path algebra of Q over R,
+with a trivial path ei for each vertex i ∈ Q0. By an RQ-lattice we mean an RQ-module
+which is finitely generated projective as an R-module. We say that an RQ-module X
+is pointwise free if eiX is a free R-module for all i. We say that an RQ-lattice X has
+rank vector α ∈ NQ0 if eiX has constant rank αi for each i, and that X has pointwise
+constant rank if it has rank vector α for some α. If R → S is a homomorphism, then XS
+is naturally an SQ-module.
+We say that an RQ-module X is rigid if Ext1
+RQ(X, X) = 0, and exceptional if also the
+natural map R → EndRQ(X) is an isomorphism. If K is an algebraically closed field, then
+by definition the possible dimension vectors of exceptional KQ-modules are the real Schur
+roots for Q. We call the possible dimension vectors of rigid KQ-modules rigid dimension
+vectors. By results of [2] or [3] these do not depend on the field K. If X is a rigid
+(respectively exceptional) RQ-lattice of rank vector α, then XK is a rigid (respectively
+exceptional) KQ-module of dimension vector α for any homomorphism R → K with K
+a field (see Lemmas 3.1,3.2), and so α must be a rigid dimension vector (respectively real
+Schur root).
+Theorem A. For any commutative ring R and real Schur root α for Q, there is a unique
+rigid pointwise free RQ-lattice with rank vector α and it is exceptional.
+The existence follows easily from [4], and the uniqueness follows from the next result.
+As mentioned, the case when R is a principal ideal domain is in [4]. The case when R is
+a truncated polynomial ring K[ǫ]/(ǫn) is a special case of [6, Theorem 1.2]. Recall that a
+commutative ring R is reduced if it has no nilpotent elements.
+2020 Mathematics Subject Classification. Primary 16G20; Secondary 16G30,16H20,13C10.
+Key words and phrases. Quiver representations, Rigid representations, Lattices over orders.
+Partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
+SFB-TRR 358/1 2023 – 491392403.
+1
+
+Theorem B. If R is reduced, then any rigid RQ-lattice X of pointwise constant rank
+has a decomposition
+X ∼= (X1 ⊗R P1) ⊕ · · · ⊕ (Xr ⊗R Pr)
+where the Xi are pairwise non-isomorphic exceptional pointwise free RQ-lattices satisfying
+Ext1
+RQ(Xi, Xj) = 0 for all i, j and the Pi are finitely generated projective R-modules of
+constant rank. Moreover this decomposition is unique up to isomorphism and reordering.
+In particular any exceptional RQ-lattice is of the form Y ⊗R P with Y an exceptional
+pointwise free RQ-module and P a projective R-module of constant rank 1.
+Note that in general one doesn’t have uniqueness for rigid pointwise free lattices of a
+given rank vector. For example let Q be the quiver 1 → 2 and R a reduced ring with a
+stably free projective module P which is not free, say P ⊕Rn ∼= Rm. Then (RQe2⊗RP)⊕
+(RQe1 ⊗R Rn) is pointwise free, but not isomorphic to (RQe2 ⊗R Rm−n) ⊕(RQe1 ⊗R Rn).
+2. Commutative rings
+Concerning our definitions and results, note that every finitely generated projective
+R-module has constant rank if and only if Spec R is connected, or equivalently R has no
+idempotents other than 0 and 1, see for example Exercise 20.12 in [5].
+Lemma 2.1. Let R be a commutative ring.
+(i) If X is a finitely generated R-module, then X = 0 if and only if XK = 0 for all
+homomorphisms R → K with K an algebraically closed field.
+(ii) If θ : X → Y is a homomorphism of R-modules and Y is finitely generated, then
+θ is onto if and only if θK is onto for all R → K with K an algebraically closed
+field.
+(iii) If R is reduced and X is a finitely generated R-module, then X is projective of
+constant rank n if and only if dimK XK = n for all homomorphisms R → K with
+K an algebraically closed field.
+(iv) If R is reduced and θ : X → Y is a homomorphism with X, Y finitely generated
+projective of constant rank, then θ is a split monomorphism if and only if θK is a
+monomorphism for all R → K with K an algebraically closed field.
+Proof. (i) Since X is finitely generated, if non-zero it has a map onto some R/m, and
+taking K to be the algebraic closure of this field gives a surjective map XK → R/m⊗RK ∼=
+K.
+(ii) Clear.
+(iii) See Exercise 20.13 in [5].
+(iv) If θK is a monomorphism for all R → K, then Coker θK has constant dimension,
+so Coker θ is projective of constant rank. Then Im θ is a summand of Y , so projective.
+But then 0 → Ker θ → X → Im θ → 0 splits, so is exact after inducing to K. Thus
+Ker θK = 0. Also Ker θ is summand of X, so finitely generated. Thus Ker θ is zero.
+□
+Lemma 2.2. Any commutative ring R can be written as a quotient R ∼= ˜R/I with ˜R an
+integral domain and I is an ideal contained in the Jacobson radical of ˜R.
+Proof. Let A = Z[xr : r ∈ R] be the polynomial ring over Z with indeterminates indexed
+by the elements of R. There is a surjective homomorphism θ : A → R sending each xr
+to r and the set
+S = {a ∈ A : θ(a) is invertible}
+is a multiplicative subset. We set ˜R to be the localization AS, an integral domain. The
+homomorphism θ extends to a surjective homomorphism ˜θ : ˜R → R.
+2
+
+Let x ∈ Ker ˜θ. If y ∈ ˜R, then xy ∈ Ker ˜θ. Write xy = as−1 with a ∈ A and s ∈ S.
+Then θ(a) = 0, so s + a ∈ S. Thus 1 + xy = (s + a)s−1 is invertible in ˜R. Since this is
+true for all y, it follows that x is in the Jacobson radical of ˜R.
+□
+3. Lattices
+Let R be a commutative ring and Q a quiver. Lemma 1 of [4] generalizes immediately
+to this setting.
+Lemma 3.1. If X is an RQ-lattice, then proj. dim X ≤ 1 and
+Ext1
+RQ(X, Y )S ∼= Ext1
+SQ(XS, Y S)
+for any homomorphism R → S and any RQ-module Y . In particular if X is a rigid
+RQ-lattice, then XS is a rigid SQ-lattrice for all homomorphisms R → S. Conversely, if
+X is an RQ-lattice and XK rigid for all homomorphisms R → K with K an algebraically
+closed field, then X is rigid.
+Proof. Letting S be the R-subalgebra of RQ with basis the trivial paths ei and B be the
+free R-submodule of RQ with basis the arrows, the path algebra RQ is isomorphic to the
+tensor algebra of B over S, so there is a standard resolution
+0 → RQ ⊗S B ⊗S RQ → RQ ⊗S RQ → RQ → 0,
+or equivalently
+0 →
+�
+a∈Q1
+RQeh(a) ⊗R et(a)RQ →
+�
+i∈Q0
+RQei ⊗R eiR → RQ → 0.
+Tensoring with the RQ-module X gives
+0 = TorRQ
+1 (RQ, X) →
+�
+a∈Q1
+RQeh(a) ⊗R et(a)X →
+�
+i∈Q0
+RQei ⊗R eiX → X → 0.
+and this is a resolution of X by finitely generated projective RQ-modules. We denote it
+by 0 → P1 → P0 → X → 0. Since X is projective over R, the induced sequence
+0 → P S
+1 → P S
+0 → XS → 0
+is exact, so a resolution of XS by projective SQ-modules. Now if P is a finitely generated
+projective RQ-module, then
+HomRQ(P, Y )S ∼= HomSQ(P S, Y S).
+Thus we get a commutative diagram with exact rows
+HomRQ(P0, Y )S −−−→ HomRQ(P1, Y )S −−−→ Ext1
+RQ(X, Y )S
+−−−→ 0
+�
+�
+HomSQ(P S
+0 , Y S) −−−→ HomSQ(P S
+1 , Y S) −−−→ Ext1
+SQ(XS, Y S) −−−→ 0
+in which the vertical maps are isomorphisms, and hence an isomorphism
+Ext1
+RQ(X, Y )S ∼= Ext1
+SQ(XS, Y S).
+Now Ext1
+RQ(X, X) is a finitely generated R-module, so if Ext1
+RQ(X, X)K = 0 for all
+homomorphisms from R to an algebraically closed field K then Ext1
+RQ(X, X) = 0, that
+is, X is rigid.
+□
+3
+
+Over an algebraically closed field K, the isomorphism classes of representations X of
+Q of dimension vector α correspond to orbits OX of a group GL(α) acting on an affine
+space Rep(Q, α). Moreover X is rigid if and only if OX is open orbit. Since the affine
+space it irreducible, it follows that there is at most one rigid module of dimension α, up
+to isomorphism.
+The general dimension of Hom(X, Y ) and Ext1(X, Y ) for X of dimension α and Y
+of dimension β is denoted hom(α, β) and ext(α, β). In particular this is these are the
+dimensions for X and Y rigid. Based on work of Schofield [11], it is proved in [3] that
+hom(α, β) and ext(α, β) do not depend on the algebraically closed field K.
+Lemma 3.2. If R is reduced, and if X, Y are rigid RQ-lattices of pointwise constant
+rank α, β, then
+HomRQ(X, Y )S ∼= HomSQ(XS, Y S)
+for all homomorphisms R → S. Moreover Ext1
+RQ(X, Y ) and HomRQ(X, Y ) are finitely
+generated projective R-modules of constant rank ext(α, β) and hom(α, β).
+Proof. For any R → K with K an algebraically closed field, Ext1
+RQ(X, Y )K has constant
+dimension ext(α, β), so Ext1
+RQ(X, Y ) is projective over R of this rank by Lemma 2.1(iii).
+Now the projective resolution 0 → P1 → P0 → X → 0 of X gives an exact sequence
+0 → HomRQ(X, Y ) → HomRQ(P0, Y ) → HomRQ(P1, Y ) → Ext1
+RQ(X, Y ) → 0.
+The last three are projective R-modules of constant rank, so this sequence splits. Thus
+the first module is also projective over R. Moreover the sequence remains exact under
+induction using a homomorphism R → S. It follows that
+HomRQ(X, Y )S ∼= HomSQ(XS, Y S).
+Considering the case when S is a field, we see that HomRQ(X, Y ) has constant rank
+hom(α, β).
+□
+The following result is well known.
+Lemma 3.3. The full subquiver of Q on the support of any rigid dimension vector α has
+no oriented cycles.
+Proof. If not, one can easily construct representations of Q of dimension α over an alge-
+braically closed field K on which the trace of the oriented cycle is arbitrary, but the variety
+of representations has a dense orbit, so the trace must be constant. (This argument is
+already used in [4, Theorem 1] for real Schur roots.)
+□
+4. Mutations
+Lemma 4.1. If R is a commutative ring and α is real Schur root for Q, then there exists
+an exceptional pointwise free RQ-lattice of dimension vector α.
+Proof. By [4, Theorem 2], there is a (unique) exceptional ZQ-lattice X0 of rank vector
+α. Now we obtain an RQ-lattice XR = X ⊗Z R, and it is exceptional by Lemmas 3.1
+and 3.2.
+□
+The cited theorem [4, Theorem 2] depends on a braid group action for exceptional
+sequences of lattices, defined using mutations of exceptional pairs.
+The result about
+mutations in turn depends on the construction of mutations of pairs of representations of
+KQ where K is a field, for which [4] cites [10] and [2] cites [7]. We take this opportunity
+to give a direct proof, based on the following generalization of the Happel-Ringel Lemma
+[8, Lemma 4.1].
+4
+
+Lemma 4.2. Let θ : X → Y be a homomorphism between finite-dimensional A-modules
+for a hereditary algebra A and suppose that Ext1
+A(Y, X) = 0. If Y is indecomposable, then
+either θ is an epimorphism or the map X → Im(θ) is a split epimorphism.
+Proof. The map θ gives exact sequences
+ξ : 0 → Im(θ)
+α−→ Y → Coker(θ) → 0
+and
+η : 0 → Ker(θ) → X
+β−→ Im(θ) → 0.
+From Ext1
+A(Coker(θ), η) we get an exact sequence
+· · · → Ext1
+A(Coker(θ), X)
+f−→ Ext1
+A(Coker(θ), Im(θ)) → Ext2
+A(Coker(θ), Ker(θ)) = 0
+so ξ = f(ζ) for some ζ. Thus there is a commutative diagram
+ζ : 0 −−−→
+X
+δ
+−−−→ Z −−−→ Coker(θ) −−−→ 0
+β
+�
+γ
+�
+���
+ξ : 0 −−−→ Im(θ)
+α
+−−−→ Y −−−→ Coker(θ) −−−→ 0
+Now the sequence
+0 → X
+
+δ
+β
+
+
+−−−→ Z ⊕ Im(θ)
+�
+γ
+−α
+�
+−−−−−−→ Y → 0
+is exact, so splits since Ext1
+A(Y, X) = 0. Thus there are maps
+�p
+q�
+: Z ⊕ Im(θ) → X
+and
+�
+r
+s
+�
+: Y → Z ⊕ Im(θ)
+with
+�
+1Z
+0
+0
+1Im(θ)
+�
+=
+�
+δ
+β
+� �p
+q�
++
+�
+r
+s
+� �γ
+−α�
+.
+Thus 1Im(θ) = βq − sα. Now if θ is not an epimorphism, then αs cannot be an auto-
+morphism of Y . Since Y is indecomposable, αs is nilpotent. This implies that βq is
+invertible, and hence that β is a split epimorphism.
+□
+If X is also indecomposable, one recovers the Happel-Ringel Lemma: if θ is non-
+zero and not an epimorphism, then X → Im(θ) must be an isomorphism, so θ is a
+monomorphism. Dually we have:
+Lemma 4.3. Let θ : X → Y be a homomorphism between finite-dimensional A-modules
+for a hereditary algebra A and suppose that Ext1
+A(Y, X) = 0. If X is indecomposable,
+then either θ is a monomorphism or the map Im(θ) → Y is a split monomorphism.
+A sequence of exceptional RQ-lattices (X1, . . . , Xr) is called an exceptional sequence
+provided that HomRQ(Xi, Xj) = Ext1
+RQ(Xi, Xj) = 0 for all i > j. The result about
+mutations is as follows.
+Theorem 4.4. Suppose R is reduced. If (X, Y ) is an exceptional pair of RQ-lattices, then
+there are exceptional pairs (LXY, X) and (Y, RY X) given as follows. If HomRQ(X, Y ) = 0
+then LXY and RY X are given by the universal exact sequences
+0 → Y → LXY → X ⊗R Ext1
+RQ(X, Y ) → 0,
+0 → Y ⊗R D Ext1
+RQ(X, Y ) → RY X → X → 0.
+where D = HomR(−, R). If HomRQ(X, Y ) ̸= 0, then the universal map
+f : X ⊗R HomRQ(X, Y ) → Y
+5
+
+is an epimorphism or a monomorphism, and LXY is its kernel or cokernel, and the
+universal map
+g : X → HomR(HomRQ(X, Y ), Y ) ∼= Y ⊗R D HomRQ(X, Y )
+is an epimorphism or a monomorphism and RY X is its kernel or cokernel.
+Proof. Since R is reduced and the lattices are rigid, the Hom and Ext spaces are projective
+over R of constant rank, independent of R. Moreover using Lemma 2.1 we can, as in [4,
+Lemma 4], reduce to the case when R = K is an algebraically closed field.
+In this setting, the claim follows easily once one knows that if HomKQ(X, Y ) ̸= 0 then
+Ext1
+KQ(X, Y ) = 0, and that f and g are either epimorphisms or monomorphisms.
+Now if θ ∈ HomKQ(X, Y ) is a non-zero map, then by the Happel-Ringel Lemma θ is
+either an epimorphism or a monomorphism. In the first case we have a exact sequence
+· · · → Ext1
+KQ(X, X) → Ext1
+KQ(X, Y ) → Ext2
+KQ(X, Ker(θ)) = 0
+and in the second case we have
+· · · → Ext1
+KQ(Y, Y ) → Ext1
+KQ(X, Y ) → Ext2
+KQ(Coker(θ), Y ) = 0.
+Since Ext1
+KQ(X, X) = Ext1
+KQ(Y, Y ) = 0, in both cases we get Ext1
+KQ(X, Y ) = 0.
+If the map f : X ⊗K HomKQ(X, Y ) → Y is not an epimorphism, then the induced
+map f ′ : X ⊗K HomKQ(X, Y ) → Im(f) is a split epimorphism by Lemma 4.2. Since
+EndKQ(X) = K, the map f can be identified with the universal map from a direct sum
+of copies of X to Y given by a basis of HomKQ(X, Y ), from which it is easy to see
+that f is right minimal [1, §I.2]. Thus by [1, Corollary I.2.3] there is no non-zero direct
+summand of the domain on which f vanishes. Thus f ′ is an isomorphism, and hence f
+is a monomorphism. The argument for g is dual.
+□
+5. Classification of rigids
+We prove the following result. Once Theorem A is proved, it gives Theorem B. The
+proof of the theorem is the same as Theorem 2(ii) of [4].
+Theorem 5.1. Let R be reduced. For each real Schur root α, we choose an exceptional
+pointwise free RQ-lattice X(α) of rank vector α. Any rigid RQ-lattice X of pointwise
+constant rank has a decomposition
+X ∼= (X(α1) ⊗R P1) ⊕ · · · ⊕ (X(αr) ⊗R Pr)
+where the α1, . . . , αr are distinct real Schur roots, ext(αi, αj) = 0 for all i, j and the Pi
+are finitely generated projective R-modules of constant rank. Moreover this decomposition
+is unique up to isomorphism and reordering. In particular any exceptional RQ-lattice of
+rank vector α is of the form X(α) ⊗R P with X exceptional pointwise free and P a
+projective R-module of constant rank 1.
+Proof. By Lemma 3.3, we may assume that Q has no oriented cycles. We fix temporarily
+a homomorphism R → K with K an algebraically closed field. Then XK is rigid, so
+decomposes as direct sum
+XK ∼= Mm1
+1
+⊕ · · · ⊕ Mmr
+r
+.
+with the Mi pairwise non-isomorphic exceptional KQ-modules, Ext1
+KQ(Mi, Mj) = 0 for
+all i, j and all mi > 0. By [8, Corollary 4.2], we can order the Mi so that (M1, . . . , Mr) is
+an exceptional sequence, so Hom(Mi, Mj) = 0 for i > j. Let Mi have dimension vector αi.
+We have Hom(X(αi), X(αj)) = 0 for i > j since dim Hom(Mi, Mj) = hom(αi, αj).
+6
+
+Now Pr = HomRQ(X(αr), X) is a finitely generated projective R-module of constant
+rank. Consider the evaluation map θ : X(αr) ⊗R Pr → X and let C be its cokernel. For
+an arbitrary homomorphism R → K with K an algebraically closed field (no longer the
+one fixed above), using Lemma 3.2, we can identify θK with the evaluation map
+X(αr)K ⊗K HomKQ(X(αr)K, XK) → X
+but we know that
+XK ∼= (X(α1)K)m1 ⊕ · · · ⊕ (X(αr)K)mr
+since both sides are rigid KQ-modules of the same dimension vector. Thus θK is the
+inclusion of (X(αr)K)mr as a direct summand of XK. Thus by Lemma 2.1, θ is a split
+monomorphism of R-modules, so C is projective over R. Also
+CK ∼= Coker(θK) ∼= (X(α1)K)m1 ⊕ · · · ⊕ (X(αr−1)K)mr−1
+so Ext1
+KQ(CK, CK) = 0 and Ext1
+KQ(CK, X(αr)K) = 0. Thus C is rigid and Ext1(C, Xr) =
+0. Thus X ∼= X(αr) ⊗R Pr ⊕C. Now the result follows by induction (on r or on the total
+rank of X).
+For uniqueness, note that if we have another decomposition
+X = (X(β1) ⊗ P ′
+1) ⊕ · · · ⊕ (X(βs) ⊗ P ′
+s)
+with the βi being distinct real Schur roots and the P ′
+i projective R-modules of constant
+rank, then using a fixed homomorphism R → K and the uniqueness of rigid KQ-modules
+of a given dimension vector, we see that s = r and that the βi are a permutation of the
+αi. Thus we reduce to the case that βi = αi. But then
+P ′
+r ∼= HomRQ(X(αr), X) ∼= Pr
+and then the corresponding cokernels of the evaluation maps are isomorphic, so by in-
+duction P ′
+i ∼= Pi for all i.
+The special case of exceptional lattices follows.
+□
+6. Uniqueness for exceptional pointwise frees
+We now prove Theorem A. The existence part was Lemma 4.1. Suppose X is a point-
+wise free lattice for RQ of rank vector α, a real Schur root. By Lemma 2.2 we can write
+R ∼= ˜R/I with ˜R reduced and I contained in the Jacobson radical of ˜R.
+We define a pointwise free ˜RQ-lattice ˜X by giving it as a representation of Q by free
+˜R-modules as follows. For each vertex i we take ei ˜X to be a free ˜R-module of rank αi,
+and we fix an isomorphism
+(ei ˜X)/I(ei ˜X) ∼= (ei ˜X)R ∼= eiX.
+For each arrow a ∈ Q1, multiplication by a induces an R-linear map et(a)X → eh(a)X,
+and we lift this to an ˜R-linear map et(a) ˜X → eh(a) ˜X. This defines ˜X as a representation
+of Q and clearly we have ˜XR ∼= X.
+Let E = Ext1
+˜RQ( ˜X, ˜X), a finitely generated ˜R-module. Now ER ∼= Ext1
+RQ(X, X) = 0
+by Lemma 3.1, so E = IE, so by Nakayama’s Lemma E = 0. Thus ˜X is rigid. (This
+idea of lifting X to a rigid representation ˜X is taken from [6].)
+By Theorem B we have ˜X ∼= X(α)⊗ ˜R ˜P, where X(α) is a chosen exceptional pointwise
+free ˜RQ-lattice of rank vector α and ˜P is a finitely generated projective ˜R-module of
+constant rank 1.
+7
+
+Tensoring with R, we get X ∼= Y ⊗R P where Y = X(α)R is an exceptional pointwise
+free RQ-lattice and P = ˜P R is a finitely generated projective R-module of constant
+rank 1.
+Now since X and Y are pointwise free of rank vector α, we have
+Rαi ∼= eiX ∼= eiY ⊗R P ∼= Rαi ⊗R P ∼= P αi
+for all i. Since α is a real root for Q, it is indivisible, that is, its components are coprime.
+Thus we can find ai, bi ∈ N such that
+1 +
+�
+i
+aiαi =
+�
+i
+biαi.
+Then
+P ⊕ R
+�
+i aiαi ∼= P ⊕
+�
+i
+(Rαi)ai ∼= P ⊕
+�
+i
+(P αi)ai ∼=
+�
+i
+(P αi)bi ∼=
+�
+i
+(Rαi)bi ∼= R
+�
+i biαi.
+Thus P is stably free. Now any stably free projective module of constant rank 1 for a
+commutative ring is free, see for example [9, Theorem 4.11], so P ∼= R. Thus X ∼= Y .
+We have already observed that Y is exceptional, hence so is X.
+References
+[1] M. Auslander, I. Reiten and S. O. Smalø, Representation theory of Artin algebras, Cambridge
+Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge, 1995.
+[2] W. Crawley-Boevey, Exceptional sequences of representations of quivers, in ‘Representations of
+algebras’ (Ottawa, ON, 1992), 117-–124, CMS Conf. Proc., 14, Amer. Math. Soc., Providence, RI,
+1993.
+[3] W. Crawley-Boevey, Subrepresentations of general representations of quivers, Bull. London Math.
+Soc. 28 (1996), 363—366.
+[4] W. Crawley-Boevey, Rigid integral representations of quivers, in ‘Representation theory of
+algebras’ (Cocoyoc, 1994), 155-–163, CMS Conf. Proc., 18, Amer. Math. Soc., Providence, RI,
+1996.
+[5] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Graduate Texts in
+Mathematics, 150. Springer-Verlag, New York, 1995.
+[6] C. Geiß, B. Leclerc and J. Schr¨oer, Rigid modules and Schur roots. Math. Z. 295 (2020), no. 3-4,
+1245–1277.
+[7] A. L. Gorodentsev, Exceptional bundles on surfaces with a moving anticanonical class (Russian),
+Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 740-–757, 895; translation in Math. USSR-Izv. 33
+(1989), 67-–83.
+[8] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399-–443.
+[9] T. Y. Lam, Serre’s problem on projective modules, Springer Monographs in Mathematics.
+Springer-Verlag, Berlin, 2006.
+[10] A. N. Rudakov, Exceptional collections, mutations and helices, in ‘Helices and vector bundles’,
+1-–6, London Math. Soc. Lecture Note Ser., 148, Cambridge Univ. Press, Cambridge, 1990.
+[11] A. Schofield, General representations of quivers, Proc. London Math. Soc. (3) 65 (1992), 46-–64.
+Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, 33501 Bielefeld, Germany
+Email address: wcrawley@math.uni-bielefeld.de
+8
+
diff --git a/TtE4T4oBgHgl3EQfLwyC/content/tmp_files/load_file.txt b/TtE4T4oBgHgl3EQfLwyC/content/tmp_files/load_file.txt
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index 0000000000000000000000000000000000000000..9f88b3d6e1cd225b58846f3c76f34fbc4c5dc5dc
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@@ -0,0 +1,326 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf,len=325
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='04941v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='RT]  12 Jan 2023  RIGID INTEGRAL REPRESENTATIONS OF QUIVERS REVISITED  WILLIAM CRAWLEY-BOEVEY  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' In earlier work, the author classified rigid representations of a quiver by  means of finitely generated free modules over a principal ideal ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  We show that  the classification of exceptional pointwise free lattices can be extended to arbitrary  commutative rings and that the classification of rigid lattices can be extended to reduced  commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Introduction  In [4] we studied rigid representations of a quiver over a principal ideal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Here  we generalize the results to more general commutative base rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Let R be a commutative ring (always non-zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Given an R-module X and a homo-  morphism R → S we write XS for the induced S-module S ⊗R X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Recall that a finitely  generated projective R-module P has constant rank n if Pp is a free Rp-module of rank  n for each prime ideal p of R, or equivalently if dimK P K = n for all homomorphisms  R → K with K a field, which we may take to be algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Let Q = (Q0, Q1, h, t) be a finite quiver and let RQ be the path algebra of Q over R,  with a trivial path ei for each vertex i ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' By an RQ-lattice we mean an RQ-module  which is finitely generated projective as an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We say that an RQ-module X  is pointwise free if eiX is a free R-module for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We say that an RQ-lattice X has  rank vector α ∈ NQ0 if eiX has constant rank αi for each i, and that X has pointwise  constant rank if it has rank vector α for some α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If R → S is a homomorphism, then XS  is naturally an SQ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  We say that an RQ-module X is rigid if Ext1  RQ(X, X) = 0, and exceptional if also the  natural map R → EndRQ(X) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If K is an algebraically closed field, then  by definition the possible dimension vectors of exceptional KQ-modules are the real Schur  roots for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We call the possible dimension vectors of rigid KQ-modules rigid dimension  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' By results of [2] or [3] these do not depend on the field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If X is a rigid  (respectively exceptional) RQ-lattice of rank vector α, then XK is a rigid (respectively  exceptional) KQ-module of dimension vector α for any homomorphism R → K with K  a field (see Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2), and so α must be a rigid dimension vector (respectively real  Schur root).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' For any commutative ring R and real Schur root α for Q, there is a unique  rigid pointwise free RQ-lattice with rank vector α and it is exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  The existence follows easily from [4], and the uniqueness follows from the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  As mentioned, the case when R is a principal ideal domain is in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The case when R is  a truncated polynomial ring K[ǫ]/(ǫn) is a special case of [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Recall that a  commutative ring R is reduced if it has no nilpotent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Primary 16G20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Secondary 16G30,16H20,13C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Quiver representations, Rigid representations, Lattices over orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –  SFB-TRR 358/1 2023 – 491392403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  1  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If R is reduced, then any rigid RQ-lattice X of pointwise constant rank  has a decomposition  X ∼= (X1 ⊗R P1) ⊕ · · · ⊕ (Xr ⊗R Pr)  where the Xi are pairwise non-isomorphic exceptional pointwise free RQ-lattices satisfying  Ext1  RQ(Xi, Xj) = 0 for all i, j and the Pi are finitely generated projective R-modules of  constant rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Moreover this decomposition is unique up to isomorphism and reordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  In particular any exceptional RQ-lattice is of the form Y ⊗R P with Y an exceptional  pointwise free RQ-module and P a projective R-module of constant rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Note that in general one doesn’t have uniqueness for rigid pointwise free lattices of a  given rank vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' For example let Q be the quiver 1 → 2 and R a reduced ring with a  stably free projective module P which is not free, say P ⊕Rn ∼= Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Then (RQe2⊗RP)⊕  (RQe1 ⊗R Rn) is pointwise free, but not isomorphic to (RQe2 ⊗R Rm−n) ⊕(RQe1 ⊗R Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Commutative rings  Concerning our definitions and results, note that every finitely generated projective  R-module has constant rank if and only if Spec R is connected, or equivalently R has no  idempotents other than 0 and 1, see for example Exercise 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='12 in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Let R be a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (i) If X is a finitely generated R-module, then X = 0 if and only if XK = 0 for all  homomorphisms R → K with K an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (ii) If θ : X → Y is a homomorphism of R-modules and Y is finitely generated, then  θ is onto if and only if θK is onto for all R → K with K an algebraically closed  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (iii) If R is reduced and X is a finitely generated R-module, then X is projective of  constant rank n if and only if dimK XK = n for all homomorphisms R → K with  K an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (iv) If R is reduced and θ : X → Y is a homomorphism with X, Y finitely generated  projective of constant rank, then θ is a split monomorphism if and only if θK is a  monomorphism for all R → K with K an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' (i) Since X is finitely generated, if non-zero it has a map onto some R/m, and  taking K to be the algebraic closure of this field gives a surjective map XK → R/m⊗RK ∼=  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (ii) Clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (iii) See Exercise 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='13 in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  (iv) If θK is a monomorphism for all R → K, then Coker θK has constant dimension,  so Coker θ is projective of constant rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Then Im θ is a summand of Y , so projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  But then 0 → Ker θ → X → Im θ → 0 splits, so is exact after inducing to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus  Ker θK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Also Ker θ is summand of X, so finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus Ker θ is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Any commutative ring R can be written as a quotient R ∼= ˜R/I with ˜R an  integral domain and I is an ideal contained in the Jacobson radical of ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Let A = Z[xr : r ∈ R] be the polynomial ring over Z with indeterminates indexed  by the elements of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' There is a surjective homomorphism θ : A → R sending each xr  to r and the set  S = {a ∈ A : θ(a) is invertible}  is a multiplicative subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We set ˜R to be the localization AS, an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The  homomorphism θ extends to a surjective homomorphism ˜θ : ˜R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  2  Let x ∈ Ker ˜θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If y ∈ ˜R, then xy ∈ Ker ˜θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Write xy = as−1 with a ∈ A and s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Then θ(a) = 0, so s + a ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus 1 + xy = (s + a)s−1 is invertible in ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since this is  true for all y, it follows that x is in the Jacobson radical of ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Lattices  Let R be a commutative ring and Q a quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Lemma 1 of [4] generalizes immediately  to this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If X is an RQ-lattice, then proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' dim X ≤ 1 and  Ext1  RQ(X, Y )S ∼= Ext1  SQ(XS, Y S)  for any homomorphism R → S and any RQ-module Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' In particular if X is a rigid  RQ-lattice, then XS is a rigid SQ-lattrice for all homomorphisms R → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Conversely, if  X is an RQ-lattice and XK rigid for all homomorphisms R → K with K an algebraically  closed field, then X is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Letting S be the R-subalgebra of RQ with basis the trivial paths ei and B be the  free R-submodule of RQ with basis the arrows, the path algebra RQ is isomorphic to the  tensor algebra of B over S, so there is a standard resolution  0 → RQ ⊗S B ⊗S RQ → RQ ⊗S RQ → RQ → 0,  or equivalently  0 →  �  a∈Q1  RQeh(a) ⊗R et(a)RQ →  �  i∈Q0  RQei ⊗R eiR → RQ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Tensoring with the RQ-module X gives  0 = TorRQ  1 (RQ, X) →  �  a∈Q1  RQeh(a) ⊗R et(a)X →  �  i∈Q0  RQei ⊗R eiX → X → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  and this is a resolution of X by finitely generated projective RQ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We denote it  by 0 → P1 → P0 → X → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since X is projective over R, the induced sequence  0 → P S  1 → P S  0 → XS → 0  is exact, so a resolution of XS by projective SQ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Now if P is a finitely generated  projective RQ-module, then  HomRQ(P, Y )S ∼= HomSQ(P S, Y S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Thus we get a commutative diagram with exact rows  HomRQ(P0, Y )S −−−→ HomRQ(P1, Y )S −−−→ Ext1  RQ(X, Y )S  −−−→ 0  \uf8e6\uf8e6�  \uf8e6\uf8e6�  HomSQ(P S  0 , Y S) −−−→ HomSQ(P S  1 , Y S) −−−→ Ext1  SQ(XS, Y S) −−−→ 0  in which the vertical maps are isomorphisms, and hence an isomorphism  Ext1  RQ(X, Y )S ∼= Ext1  SQ(XS, Y S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Now Ext1  RQ(X, X) is a finitely generated R-module, so if Ext1  RQ(X, X)K = 0 for all  homomorphisms from R to an algebraically closed field K then Ext1  RQ(X, X) = 0, that  is, X is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  3  Over an algebraically closed field K, the isomorphism classes of representations X of  Q of dimension vector α correspond to orbits OX of a group GL(α) acting on an affine  space Rep(Q, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Moreover X is rigid if and only if OX is open orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since the affine  space it irreducible, it follows that there is at most one rigid module of dimension α, up  to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  The general dimension of Hom(X, Y ) and Ext1(X, Y ) for X of dimension α and Y  of dimension β is denoted hom(α, β) and ext(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' In particular this is these are the  dimensions for X and Y rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Based on work of Schofield [11], it is proved in [3] that  hom(α, β) and ext(α, β) do not depend on the algebraically closed field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If R is reduced, and if X, Y are rigid RQ-lattices of pointwise constant  rank α, β, then  HomRQ(X, Y )S ∼= HomSQ(XS, Y S)  for all homomorphisms R → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Moreover Ext1  RQ(X, Y ) and HomRQ(X, Y ) are finitely  generated projective R-modules of constant rank ext(α, β) and hom(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' For any R → K with K an algebraically closed field, Ext1  RQ(X, Y )K has constant  dimension ext(α, β), so Ext1  RQ(X, Y ) is projective over R of this rank by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Now the projective resolution 0 → P1 → P0 → X → 0 of X gives an exact sequence  0 → HomRQ(X, Y ) → HomRQ(P0, Y ) → HomRQ(P1, Y ) → Ext1  RQ(X, Y ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  The last three are projective R-modules of constant rank, so this sequence splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus  the first module is also projective over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Moreover the sequence remains exact under  induction using a homomorphism R → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' It follows that  HomRQ(X, Y )S ∼= HomSQ(XS, Y S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Considering the case when S is a field, we see that HomRQ(X, Y ) has constant rank  hom(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  The following result is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The full subquiver of Q on the support of any rigid dimension vector α has  no oriented cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If not, one can easily construct representations of Q of dimension α over an alge-  braically closed field K on which the trace of the oriented cycle is arbitrary, but the variety  of representations has a dense orbit, so the trace must be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' (This argument is  already used in [4, Theorem 1] for real Schur roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=')  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Mutations  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If R is a commutative ring and α is real Schur root for Q, then there exists  an exceptional pointwise free RQ-lattice of dimension vector α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' By [4, Theorem 2], there is a (unique) exceptional ZQ-lattice X0 of rank vector  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Now we obtain an RQ-lattice XR = X ⊗Z R, and it is exceptional by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1  and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  The cited theorem [4, Theorem 2] depends on a braid group action for exceptional  sequences of lattices, defined using mutations of exceptional pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  The result about  mutations in turn depends on the construction of mutations of pairs of representations of  KQ where K is a field, for which [4] cites [10] and [2] cites [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We take this opportunity  to give a direct proof, based on the following generalization of the Happel-Ringel Lemma  [8, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  4  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Let θ : X → Y be a homomorphism between finite-dimensional A-modules  for a hereditary algebra A and suppose that Ext1  A(Y, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If Y is indecomposable, then  either θ is an epimorphism or the map X → Im(θ) is a split epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The map θ gives exact sequences  ξ : 0 → Im(θ)  α−→ Y → Coker(θ) → 0  and  η : 0 → Ker(θ) → X  β−→ Im(θ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  From Ext1  A(Coker(θ), η) we get an exact sequence  · · → Ext1  A(Coker(θ), X)  f−→ Ext1  A(Coker(θ), Im(θ)) → Ext2  A(Coker(θ), Ker(θ)) = 0  so ξ = f(ζ) for some ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus there is a commutative diagram  ζ : 0 −−−→  X  δ  −−−→ Z −−−→ Coker(θ) −−−→ 0  β  \uf8e6\uf8e6�  γ  \uf8e6\uf8e6�  ���  ξ : 0 −−−→ Im(θ)  α  −−−→ Y −−−→ Coker(θ) −−−→ 0  Now the sequence  0 → X  \uf8eb  \uf8edδ  β  \uf8f6  \uf8f8  −−−→ Z ⊕ Im(θ)  �  γ  −α  �  −−−−−−→ Y → 0  is exact, so splits since Ext1  A(Y, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus there are maps  �p  q�  : Z ⊕ Im(θ) → X  and  �  r  s  �  : Y → Z ⊕ Im(θ)  with  �  1Z  0  0  1Im(θ)  �  =  �  δ  β  � �p  q�  +  �  r  s  � �γ  −α�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Thus 1Im(θ) = βq − sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Now if θ is not an epimorphism, then αs cannot be an auto-  morphism of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since Y is indecomposable, αs is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' This implies that βq is  invertible, and hence that β is a split epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  If X is also indecomposable, one recovers the Happel-Ringel Lemma: if θ is non-  zero and not an epimorphism, then X → Im(θ) must be an isomorphism, so θ is a  monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Dually we have:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Let θ : X → Y be a homomorphism between finite-dimensional A-modules  for a hereditary algebra A and suppose that Ext1  A(Y, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If X is indecomposable,  then either θ is a monomorphism or the map Im(θ) → Y is a split monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  A sequence of exceptional RQ-lattices (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' , Xr) is called an exceptional sequence  provided that HomRQ(Xi, Xj) = Ext1  RQ(Xi, Xj) = 0 for all i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The result about  mutations is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Suppose R is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If (X, Y ) is an exceptional pair of RQ-lattices, then  there are exceptional pairs (LXY, X) and (Y, RY X) given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If HomRQ(X, Y ) = 0  then LXY and RY X are given by the universal exact sequences  0 → Y → LXY → X ⊗R Ext1  RQ(X, Y ) → 0,  0 → Y ⊗R D Ext1  RQ(X, Y ) → RY X → X → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  where D = HomR(−, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' If HomRQ(X, Y ) ̸= 0, then the universal map  f : X ⊗R HomRQ(X, Y ) → Y  5  is an epimorphism or a monomorphism, and LXY is its kernel or cokernel, and the  universal map  g : X → HomR(HomRQ(X, Y ), Y ) ∼= Y ⊗R D HomRQ(X, Y )  is an epimorphism or a monomorphism and RY X is its kernel or cokernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since R is reduced and the lattices are rigid, the Hom and Ext spaces are projective  over R of constant rank, independent of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Moreover using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1 we can, as in [4,  Lemma 4], reduce to the case when R = K is an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  In this setting, the claim follows easily once one knows that if HomKQ(X, Y ) ̸= 0 then  Ext1  KQ(X, Y ) = 0, and that f and g are either epimorphisms or monomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Now if θ ∈ HomKQ(X, Y ) is a non-zero map, then by the Happel-Ringel Lemma θ is  either an epimorphism or a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' In the first case we have a exact sequence  · · → Ext1  KQ(X, X) → Ext1  KQ(X, Y ) → Ext2  KQ(X, Ker(θ)) = 0  and in the second case we have  · · → Ext1  KQ(Y, Y ) → Ext1  KQ(X, Y ) → Ext2  KQ(Coker(θ), Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Since Ext1  KQ(X, X) = Ext1  KQ(Y, Y ) = 0, in both cases we get Ext1  KQ(X, Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  If the map f : X ⊗K HomKQ(X, Y ) → Y is not an epimorphism, then the induced  map f ′ : X ⊗K HomKQ(X, Y ) → Im(f) is a split epimorphism by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since  EndKQ(X) = K, the map f can be identified with the universal map from a direct sum  of copies of X to Y given by a basis of HomKQ(X, Y ), from which it is easy to see  that f is right minimal [1, §I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus by [1, Corollary I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='3] there is no non-zero direct  summand of the domain on which f vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus f ′ is an isomorphism, and hence f  is a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The argument for g is dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Classification of rigids  We prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Once Theorem A is proved, it gives Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The  proof of the theorem is the same as Theorem 2(ii) of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Let R be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' For each real Schur root α, we choose an exceptional  pointwise free RQ-lattice X(α) of rank vector α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Any rigid RQ-lattice X of pointwise  constant rank has a decomposition  X ∼= (X(α1) ⊗R P1) ⊕ · · · ⊕ (X(αr) ⊗R Pr)  where the α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' , αr are distinct real Schur roots, ext(αi, αj) = 0 for all i, j and the Pi  are finitely generated projective R-modules of constant rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Moreover this decomposition  is unique up to isomorphism and reordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' In particular any exceptional RQ-lattice of  rank vector α is of the form X(α) ⊗R P with X exceptional pointwise free and P a  projective R-module of constant rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='3, we may assume that Q has no oriented cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' We fix temporarily  a homomorphism R → K with K an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Then XK is rigid, so  decomposes as direct sum  XK ∼= Mm1  1  ⊕ · · · ⊕ Mmr  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  with the Mi pairwise non-isomorphic exceptional KQ-modules, Ext1  KQ(Mi, Mj) = 0 for  all i, j and all mi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' By [8, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2], we can order the Mi so that (M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' , Mr) is  an exceptional sequence, so Hom(Mi, Mj) = 0 for i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Let Mi have dimension vector αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  We have Hom(X(αi), X(αj)) = 0 for i > j since dim Hom(Mi, Mj) = hom(αi, αj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  6  Now Pr = HomRQ(X(αr), X) is a finitely generated projective R-module of constant  rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Consider the evaluation map θ : X(αr) ⊗R Pr → X and let C be its cokernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' For  an arbitrary homomorphism R → K with K an algebraically closed field (no longer the  one fixed above), using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2, we can identify θK with the evaluation map  X(αr)K ⊗K HomKQ(X(αr)K, XK) → X  but we know that  XK ∼= (X(α1)K)m1 ⊕ · · · ⊕ (X(αr)K)mr  since both sides are rigid KQ-modules of the same dimension vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus θK is the  inclusion of (X(αr)K)mr as a direct summand of XK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1, θ is a split  monomorphism of R-modules, so C is projective over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Also  CK ∼= Coker(θK) ∼= (X(α1)K)m1 ⊕ · · · ⊕ (X(αr−1)K)mr−1  so Ext1  KQ(CK, CK) = 0 and Ext1  KQ(CK, X(αr)K) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus C is rigid and Ext1(C, Xr) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus X ∼= X(αr) ⊗R Pr ⊕C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Now the result follows by induction (on r or on the total  rank of X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  For uniqueness, note that if we have another decomposition  X = (X(β1) ⊗ P ′  1) ⊕ · · · ⊕ (X(βs) ⊗ P ′  s)  with the βi being distinct real Schur roots and the P ′  i projective R-modules of constant  rank, then using a fixed homomorphism R → K and the uniqueness of rigid KQ-modules  of a given dimension vector, we see that s = r and that the βi are a permutation of the  αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus we reduce to the case that βi = αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' But then  P ′  r ∼= HomRQ(X(αr), X) ∼= Pr  and then the corresponding cokernels of the evaluation maps are isomorphic, so by in-  duction P ′  i ∼= Pi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  The special case of exceptional lattices follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Uniqueness for exceptional pointwise frees  We now prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' The existence part was Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Suppose X is a point-  wise free lattice for RQ of rank vector α, a real Schur root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='2 we can write  R ∼= ˜R/I with ˜R reduced and I contained in the Jacobson radical of ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  We define a pointwise free ˜RQ-lattice ˜X by giving it as a representation of Q by free  ˜R-modules as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' For each vertex i we take ei ˜X to be a free ˜R-module of rank αi,  and we fix an isomorphism  (ei ˜X)/I(ei ˜X) ∼= (ei ˜X)R ∼= eiX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  For each arrow a ∈ Q1, multiplication by a induces an R-linear map et(a)X → eh(a)X,  and we lift this to an ˜R-linear map et(a) ˜X → eh(a) ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' This defines ˜X as a representation  of Q and clearly we have ˜XR ∼= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Let E = Ext1  ˜RQ( ˜X, ˜X), a finitely generated ˜R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Now ER ∼= Ext1  RQ(X, X) = 0  by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='1, so E = IE, so by Nakayama’s Lemma E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus ˜X is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' (This  idea of lifting X to a rigid representation ˜X is taken from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=')  By Theorem B we have ˜X ∼= X(α)⊗ ˜R ˜P, where X(α) is a chosen exceptional pointwise  free ˜RQ-lattice of rank vector α and ˜P is a finitely generated projective ˜R-module of  constant rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  7  Tensoring with R, we get X ∼= Y ⊗R P where Y = X(α)R is an exceptional pointwise  free RQ-lattice and P = ˜P R is a finitely generated projective R-module of constant  rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Now since X and Y are pointwise free of rank vector α, we have  Rαi ∼= eiX ∼= eiY ⊗R P ∼= Rαi ⊗R P ∼= P αi  for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Since α is a real root for Q, it is indivisible, that is, its components are coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Thus we can find ai, bi ∈ N such that  1 +  �  i  aiαi =  �  i  biαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Then  P ⊕ R  �  i aiαi ∼= P ⊕  �  i  (Rαi)ai ∼= P ⊕  �  i  (P αi)ai ∼=  �  i  (P αi)bi ∼=  �  i  (Rαi)bi ∼= R  �  i biαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Thus P is stably free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Now any stably free projective module of constant rank 1 for a  commutative ring is free, see for example [9, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='11], so P ∼= R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Thus X ∼= Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  We have already observed that Y is exceptional, hence so is X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
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+page_content=' Gorodentsev, Exceptional bundles on surfaces with a moving anticanonical class (Russian),  Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
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+page_content=' Nauk SSSR Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
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+page_content=' translation in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' USSR-Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
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+page_content='  [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
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+page_content=' 274 (1982), 399-–443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Lam, Serre’s problem on projective modules, Springer Monographs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Springer-Verlag, Berlin, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Lecture Note Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=', 148, Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Press, Cambridge, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Schofield, General representations of quivers, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content=' (3) 65 (1992), 46-–64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='  Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, 33501 Bielefeld, Germany  Email address: wcrawley@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='uni-bielefeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
+page_content='de  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf'}
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+The spectral reconstruction of inclusive rates
+John Bulava 𝑎,∗
+𝑎Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany
+E-mail: john.bulava@desy.de
+A recently re-discovered variant of the Backus-Gilbert algorithm for spectral reconstruction en-
+ables the controlled determination of smeared spectral densities from lattice field theory correlation
+functions. A particular advantage of this approach is the a priori specification of the kernel with
+which the underlying spectral density is smeared, allowing for variation of its peak position,
+smearing width, and functional form. If the unsmeared spectral density is sufficiently smooth
+in the neighborhood of a particular energy, it can be obtained from an extrapolation to zero
+smearing-kernel width at fixed peak position. A natural application for this approach is scattering
+processes summed over all hadronic final states. As a proof-of-principle test, an inclusive rate
+is computed in the two-dimensional O(3) sigma model from a two-point correlation function of
+conserved currents. The results at finite and zero smearing radius are in good agreement with
+the known analytic form up to energies at which 40-particle states contribute, and are sensitive to
+the 4-particle contribution to the inclusive rate. The straight-forward adaptation to compute the
+𝑅-ratio in lattice QCD from two-point functions of the electromagnetic current is briefly discussed.
+The 39th International Symposium on Lattice Field Theory,
+8th-13th August, 2022,
+Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.04072v1  [hep-lat]  10 Jan 2023
+
+The spectral reconstruction of inclusive rates
+John Bulava
+1.
+Introduction
+Lattice QCD simulations proceed by computing 𝑛-point euclidean correlation functions of
+(quasi-) local interpolating operators. Single-hadron states and finite-volume few-hadron states are
+isolated from correlation functions in the asymptotic large euclidean time limit. However, some
+hadronic phenomena are best studied by other means. As an example, this work considers inclusive
+rates defined as a sum over all hadronic final states produced by an external current. At large
+center-of-mass energies, a finite-volume approach to such a process is impractical since it requires
+the isolation of all individual finite-volume levels with arbitrarily many particles. Such processes
+are a cornerstone of QCD and connect the low-energy hadronic and high-energy perturbative
+regimes [1], serving as a manifestation of ‘quark-hadron duality’ [2] whereby perturbative QCD in
+terms of quarks and gluons becomes increasingly effective at computing inclusive rates summed
+over final states consisting entirely of hadrons.
+For concreteness, consider the QCD part of the process 𝑒+𝑒− → hadrons
+𝜌(𝑠) = 𝑅(𝑠)
+12𝜋2 ,
+𝑅(𝑠) = 𝜎 [𝑒+𝑒− → hadrons] (𝑠)
+4𝜋𝛼em(𝑠)2/(3𝑠)
+,
+(1)
+𝜌𝜇𝜈(𝑘) = 1
+2𝜋
+ˆ
+𝑑4𝑥 𝑒−𝑖𝑘·𝑥⟨Ω| ˆ𝑗em
+𝜇 (𝑥) ˆ𝑗em
+𝜈 (0)|Ω⟩ = (𝑔𝜇𝜈𝑘2 − 𝑘𝜇𝑘𝜈) 𝜌(𝑘2),
+(2)
+where ˆ𝑗em
+𝜇
+is the quark-level electromagnetic current. The desired inclusive rate is given by the
+spectral density 𝜌(𝑠), which is also present in the analogous infinite-volume Euclidean correlator
+𝐶(𝑡) =
+ˆ
+𝑑3𝒙 ⟨Ω| ˆ𝑗em
+𝑧 (𝒙) 𝑒− ˆ𝐻𝑡 ˆ𝑗em
+𝑧 (0)†|Ω⟩ =
+ˆ ∞
+0
+𝑑𝜔 𝜔2𝜌(𝜔2) 𝑒−𝜔𝑡 .
+(3)
+The direct determination of 𝜌(𝑠) in lattice QCD is not straightforward, however. First, the inversion
+of integral equations like Eq. 3 using 𝐶(𝑡) evaluated at a finite number of discrete times with
+statistical errors is notoriously ill-posed.
+Furthermore, the finite volume introduces additional
+complication. Even if the inverse problem were solved successfully and the finite-volume euclidean
+correlator 𝐶𝐿(𝑡) used to determine its spectral density 𝜌𝐿(𝑠), it differs qualitatively from its infinite-
+volume counterpart 𝜌(𝑠). While 𝜌𝐿(𝑠) is a sum over Dirac 𝛿-functions for each finite-volume state,
+𝜌(𝑠) is smooth apart from non-analyticities due to the opening of thresholds. In no way does 𝜌𝐿(𝑠)
+‘approach’ 𝜌(𝑠) as 𝐿 → ∞.
+The bridge between finite and infinite volume is made more effectively using the smeared
+spectral density
+𝜌𝜖 (𝐸) =
+ˆ ∞
+0
+𝑑𝜔 𝛿𝜖 (𝐸 − 𝜔) 𝜌(𝜔) ,
+(4)
+where lim𝜖 →0 𝛿𝜖 (𝑥) = 𝛿(𝑥), 𝛿(𝑥) is the Dirac-delta function, and
+´ ∞
+−∞ 𝑑𝑥 𝛿𝜖 (𝑥) = 1. This solves
+(in principle) both of the difficulties mentioned above: the inverse problem can be made arbitrarily
+mild by increasing in the smearing width 𝜖 and 𝜌𝐿,𝜖 (𝐸) approaches its infinite-volume counterpart
+in a well-defined manner. The goal is now to take the ordered double limit [3]
+𝜌(𝐸) = lim
+𝜖 →0+ lim
+𝐿→∞ 𝜌𝐿,𝜖 (𝐸),
+(5)
+2
+
+The spectral reconstruction of inclusive rates
+John Bulava
+the asymptotic corrections to which are discussed in Sec. 21.
+Although spectral reconstruction has a long history in lattice QCD, particularly at finite temper-
+ature [4], the treatment of the inverse problem in Eq. 3 demands special care. In order to define the
+result of the spectral reconstruction procedure, precise knowledge of the smearing kernel in Eq. 4
+is required. As detailed in Sec. 2, the Backus-Gilbert approach [5, 6] is suitable in this respect2.
+Since the estimator for the smeared spectral density ˆ𝜌𝜖 (𝐸) is simply a linear combination of the
+input correlator data ˆ𝜌𝜖 (𝐸) = �
+𝑡 𝑔𝑡 (𝜖, 𝐸)𝐶(𝑡), the resultant smearing kernel is given by the same
+linear combination of decaying exponentials in Eq. 3
+ˆ𝜌𝜖 (𝐸) =
+ˆ
+𝑑𝜔 ˆ𝛿𝜖 (𝐸, 𝜔) 𝜌(𝜔),
+ˆ𝛿𝜖 (𝐸, 𝜔) =
+∑︁
+𝑡
+𝑔𝑡 (𝜖, 𝐸)𝜔2𝑒−𝜔𝑡.
+(6)
+Explicit knowledge of ˆ𝛿𝜖 (𝐸, 𝜔) is a minimum requirement for a well-defined spectral reconstruction
+procedure.
+Naively the Backus-Gilbert approach provides knowledge of the kernel in Eq. 6 only a posteriori
+for a given choice of coefficients. However, the coefficients themselves can be chosen to approximate
+a particular smearing kernel specified a priori [9]. This important innovation is employed here
+and was applied to lattice field theory for the first time in Ref. [10]. In order to understand this
+reconstruction algorithm, and in particular demonstrate control over the systematic errors, a test in a
+controlled context is warranted. Such a test has been performed in Ref. [11] for the two-dimensional
+O(3) sigma model together with attempts at saturating the ordered double limit of Eq. 5.
+The remainder of this work is organized as follows. The spectral reconstruction method is
+presented in Sec. 2 in the context of the O(3) model test mentioned above. Prospects for adapting
+the method to current correlators in lattice QCD is discussed in Sec. 3 and Sec. 4 concludes.
+2.
+O(3) model test
+This section reviews a spectral reconstruction test which was recently published in Ref. [11].
+It employs the spectral reconstruction procedure of Ref. [10] in the two-dimensional O(3) sigma
+model. Consider the standard lattice discretization
+𝑆[𝜎] = 𝛽
+2
+∑︁
+𝑥∈Λ
+𝑎2 ∑︁
+𝜇
+ˆ𝜕𝜇𝜎(𝑥) · ˆ𝜕𝜇𝜎(𝑥) = 𝛽
+∑︁
+𝑥∈Λ
+∑︁
+𝜇
+[1 − 𝜎(𝑥) · 𝜎(𝑥 + 𝑎 ˆ𝜇)] ,
+(7)
+where 𝜎(𝑥) ∈ R3, |𝜎(𝑥)| = 1, and ˆ𝜕𝜇 𝑓 (𝑥) = 1
+𝑎 [ 𝑓 (𝑥 + 𝑎 ˆ𝜇) − 𝑓 (𝑥)]. This model has a conserved
+current
+𝑗 𝑎
+𝜇(𝑥) = 𝛽𝜖 𝑎𝑏𝑐𝜎𝑏(𝑥) ˆ𝜕𝜇𝜎𝑐(𝑥)
+(8)
+at finite lattice spacing, and possesses a dynamically-generated mass gap 𝑚. The total zero spatial
+momentum euclidean current-current correlation function analogous to Eq. 3 (but without the factor
+1Finite lattice spacing effects must also be removed by taking the continuum limit, which is here performed at fixed
+𝜖 and 𝐸.
+2Another spectral reconstruction algorithm for which the smearing kernel is formally known a posteriori is the
+Chebyshev polynomial approach of Ref. [7]. Ref. [8] compares that approach to the one employed here.
+3
+
+The spectral reconstruction of inclusive rates
+John Bulava
+n=2
+n=4
+10
+20
+30
+40
+50
+μ/m
+0.5
+1.0
+1.5
+2.0
+2.5
+ρ(n)(μ)
+Figure 1:
+Left: exactly known 𝑛 = 2 and 𝑛 = 4 particle contributions to the (continuum, infinite-
+volume) spectral density associated with the conserved current in the two-dimensional O(3) sigma model.
+Contributions from states with more particles are insignificant in the energy range shown here. Right: the
+four smearing kernels 𝛿x
+𝜖 (𝑥), where x = {g, c0, c1, c2}, defined in Eq. 9 plotted against 𝑥 with 𝜖 = 1 .
+of 𝜔2) is computed by numerical simulations using the single-cluster algorithm of Ref. [12]. A
+variety of ensembles are generated, with 𝑚𝐿 ≈ 30 − 60 and 𝑎𝑚 ∈ [0.01, 0.04] to assess finite
+volume effects and take the continuum limit. In the continuum the contributions to the associated
+spectral density 𝜌(𝜔) from each fixed-particle number sector can be computed exactly [13]. Below
+energies 𝐸 < 50𝑚, only the 𝑛 = 2 and 𝑛 = 4 particle contributions are significant and are shown in
+Fig.1.
+In order to demonstrate the a priori specification of the smearing kernel, consider four kernels
+with different profiles as a function of 𝑥 = 𝐸 − 𝜔:
+𝛿g
+𝜖 (𝑥) =
+1
+√
+2𝜋𝜖
+exp
+�
+− 𝑥2
+2𝜖2
+�
+,
+𝛿c0
+𝜖 (𝑥) = 1
+𝜋
+𝜖
+𝑥2 + 𝜖2 ,
+(9)
+𝛿c1
+𝜖 (𝑥) = 2
+𝜋
+𝜖3
+(𝑥2 + 𝜖2)2 ,
+𝛿c2
+𝜖 (𝑥) = 8
+3𝜋
+𝜖5
+(𝑥2 + 𝜖2)3 ,
+(10)
+including the gaussian (denoted ‘g’) and three Cauchy-like kernels denoted ‘c𝑛’, for which 𝑛 = 0, 1, 2
+distinguishes the power of the pole. These kernels are depicted in Fig. 1.
+The method advocated in Ref. [10] to reconstruct the smeared spectral density 𝜌x
+𝜖 (𝐸), where
+x = {g, c0, c1, c2}, is based on two criteria. First, the reconstructed smearing kernel ˆ𝛿x
+𝜖 (𝐸, 𝜔)
+should be close to the desired one 𝛿x
+𝜖 (𝐸 − 𝜔). Second, the coefficients {𝑔𝑡 (𝜖, 𝐸)} in Eq. 6 should
+not induce a large statistical variance on the estimator ˆ𝜌x
+𝜖 (𝐸). These two considerations are encoded
+in the functionals
+𝐴[𝑔] =
+ˆ ∞
+𝐸0
+𝑑𝜔
+�
+𝛿x
+𝜖 (𝐸 − 𝜔) − ˆ𝛿x
+𝜖 (𝐸, 𝜔)
+�2 ,
+(11)
+𝐵[𝑔] = Var[ ˆ𝜌x
+𝜖 (𝐸)] =
+∑︁
+𝑡𝑡′
+𝑔𝑡 (𝜖, 𝐸) 𝑔𝑡′(𝜖, 𝐸) Cov[𝐶(𝑡), 𝐶(𝑡′)]
+(12)
+respectively. The coefficients are then chosen to minimize the combination functional 𝐺𝜆[𝑔] = (1−
+𝜆)𝐴[𝑔]/𝐴[0] + 𝜆𝐵[𝑔], where the ‘trade-off’ parameter 𝜆 is introduced. For small 𝜆, the ‘accuracy’
+functional 𝐴[𝑔] takes preference over the ‘precision’ one 𝐵[𝑔] resulting in small systematic but large
+4
+
+The spectral reconstruction of inclusive rates
+John Bulava
+3
+−
+10
+2
+−
+10
+1
+−
+10
+1
+0.66
+0.67
+0.68
+0.69
+0.7
+0.71
+0.72
+0.73
+(E)
+ερ
+ =  0.20
+c
+λ
+ = 0.75m, gauss, 
+ε
+E = 3.0m, 
+A[q]/A[0]
+(E)
+ερ
+1.5
+−
+1
+−
+0.5
+−
+0
+0.5
+1
+1.5
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+ε
+)/
+ω
+(E-
+ε
+λ
+δ
+ε
+ε
+λ
+δ
+ε
+Figure 2: Left: indicative illustration of the trade-off between statistical and systematic errors for a particular
+choice of 𝐸 and 𝜖 on a single ensemble. Each point corresponds to a different 𝜆 and the horizontal band
+indicates the chosen reconstruction (with 𝜆 = 𝜆c = 0.20) for which statistical errors dominate systematic ones.
+Right: for this same setup and 𝜆 = 𝜆c, the reconstructed kernel ˆ𝛿g
+𝜖 (𝐸, 𝜔), together with the desired kernel
+𝛿g
+𝜖 (𝐸 − 𝜔) shown as a solid line. All reconstructions employ the correlator timeslices 𝑡 = 1𝑎, . . . , 160𝑎.
+statistical errors. By contrast, large 𝜆 results in small statistical errors but a reconstructed smearing
+kernel which does not resemble the desired one. The choice of the parameter 𝜆 is performed
+automatically and results in an approximate balance of these two criteria. The effect of varying
+𝜆 is illustrated in Fig. 2 together with the reconstructed kernel ˆ𝛿g
+𝜖 (𝐸, 𝜔) for a sample setup. The
+trade-off between statistical and systematic errors is familiar to lattice field theorists and the left
+panel of Fig. 2 resembles the identification of a plateau in effective mass plots.
+The procedure described above is performed for a variety of 𝐸 and 𝜖, and for all four smearing
+kernels. Next, finite volume effects must be assessed and the continuum limit taken independently
+for each 𝐸, 𝜖, and kernel. Finite-volume effects are assessed at a single lattice spacing by simulating
+two additional ensembles with doubled spatial and temporal extents, respectively. The differences
+Δ𝐿,𝑇 between the spectral reconstruction on the doubled lattices divided by the statistical error are
+shown in Fig. 3, which show (at most) moderately significant hints for finite-𝐿 effects at energies
+near the two-particle production threshold.
+With the finite-volume effects demonstrably controlled for the (𝐸, 𝜖) values and the kernels in
+question, the continuum limit can be investigated. Cutoff effects for ‘on-shell’ quantities in the two-
+dimensional O(3) sigma model have a long history, due to their apparently linear behaviour which
+is caused by large logarithmic corrections [14]. Unfortunately, the analysis there is incomplete
+for the ‘off-shell’ smeared spectral density considered here. In order to explore the influence of
+logarithmic cutoff effects, the fit forms
+𝑄(𝑎) = 𝑄(0) + 𝐶𝑎2𝛽𝑟,
+𝑟 = 0, 3, 6
+(13)
+are explored to extrapolate the smeared spectral densities to the continuum limit. A comparison
+of the extrapolation forms is shown in Fig. 4. The continuum limits are generally mild and well-
+constrained by the data, although the slope becomes steeper for increasing 𝐸.
+Given the assessment of systematic errors due to spectral reconstruction, finite 𝐿 and 𝑇 effects,
+and finite lattice spacing, it is time to confront the computations of 𝜌x
+𝜖 (𝐸) with the exact spectral
+density 𝜌(𝜔) (comprised of the two-, four-, and six-particle contributions) smeared with the exact
+5
+
+The spectral reconstruction of inclusive rates
+John Bulava
+Figure 3: The difference Δ𝐿,𝑇 between spectral reconstruction on ensembles using 𝐿 and 2𝐿 (top row), and
+using ensembles with 𝑇 and 2𝑇 in the bottom row. In both cases Δ𝐿,𝑇 is divided by the statistical error on
+the smaller ensemble. Perhaps some marginally significant hints for finite-𝐿 effects are observed at small 𝐸
+near the two-particle production threshold.
+Figure 4: The continuum limit for a single 𝐸, 𝜖, and smearing kernel using the ansatz of Eq. 13. The shaded
+band indicates the fit, and the horizontal dotted region the extrapolated value for 𝑟 = 3, which is taken as the
+final result.
+6
+
+The spectral reconstruction of inclusive rates
+John Bulava
+Figure 5: Lattice results for 𝜌x
+𝜖 (𝐸) in the continuum limit (the data points shown in the legend) compared
+against the exact spectral density including two-, four-, and six-particle contributions smeared with the exact
+kernel 𝛿x
+𝜖 (𝐸 − 𝜔). The exact results are shown as lines in the top row, and the bottom row shows the ‘pull’
+between the numerical data and the exact result, divided by the statistical and systematic error combined
+in quadrature. A naive histogram of the differences, which ignores correlations among the data, is shown
+horizontally in the bottom row and approximately resembles the unit gaussian.
+smearing kernel 𝛿x
+𝜖 (𝐸 − 𝜔).
+Such a comparison is performed in Fig. 5 where the numerical
+computations are demonstrably consistent with the exact results, within the quoted errors. These
+errors take into account the statistical errors due to the reconstruction, with any residual systematic
+errors from finite-𝐿,𝑇 effects and the continuum limit added in quadrature.
+At this point the verification of the spectral reconstruction approach of Ref. [10] is complete.
+Smeared spectral densities in the two-dimensional O(3) sigma model have been reconstructed with
+smearing kernels specified a priori, which are consistent with the exact result after the continuum
+limit has been taken. Consider now the 𝜖 → 0 limit in Eq. 5. For this, an important property of the
+unsmeared spectral density evident in Fig. 1 is required, namely that it varies increasingly slowly
+with increasing 𝐸. This circumvents the limitations in reconstructing kernels with a fixed 𝜖 and
+increasing 𝐸 evident in Fig. 5: larger smearing widths are sufficient at larger 𝐸. To this end, the
+smearing width is scaled 𝜖 ∝ (𝐸 −2𝑚). Also, rather than using a single smearing kernel, 𝜌x
+𝜖 (𝐸) for
+all kernels are used to perform constrained extrapolations. For this the small-𝜖 expansion is useful
+𝜌x
+𝜖 (𝐸) ≡
+ˆ ∞
+0
+𝑑𝜔 𝛿x
+𝜖 (𝐸 − 𝜔) 𝜌(𝜔) = 𝜌(𝐸) +
+∞
+∑︁
+𝑘=1
+𝑤x
+𝑘𝑎𝑘(𝐸)𝜖 𝑘 ,
+(14)
+7
+
+The spectral reconstruction of inclusive rates
+John Bulava
+x
+𝑤x
+𝑘, even 𝑘
+𝑤x
+𝑘, odd 𝑘
+𝑤x
+1
+𝑤x
+2
+𝑤x
+3
+𝑤x
+4
+g
+𝑘!
+(−2)𝑘/2(𝑘/2)!
+0
+0
+−1
+0
+3
+c0
+1
+1
+1
+1
+1
+1
+c1
+(1 − 𝑘)
+(1 − 𝑘)
+0
+−1
+−2
+−3
+c2
+1
+3 (𝑘 − 3)(𝑘 − 1)
+1
+3 (𝑘 − 3)(𝑘 − 1)
+0
+−1/3
+0
+1
+Table 1: The kernel-dependent coefficients 𝑤x
+𝑘 appearing in the small-𝜖 expansion of Eq. (14). For the c1
+and c2 kernels, 𝑤c1
+3 and 𝑤c2
+5 (respectively) are the non-zero coefficients with lowest odd order.
+where the contribution at the 𝑘th order in 𝜖 is the product of a kernel-independent factor
+𝑎𝑘(𝐸) =
+������
+������
+(−1)𝑘/2
+𝑘!
+�
+𝑑
+𝑑𝐸
+� 𝑘
+𝜌(𝐸) ,
+𝑘 even
+lim𝜂→0+ (−1) (𝑘−1)/2
+2𝜋
+´ ∞
+−∞ d𝜔 𝜌(𝐸+𝜔)+𝜌(𝐸−𝜔)
+(𝜔+𝑖𝜂)𝑘+1
+,
+𝑘 odd
+.
+(15)
+which depends on the unsmeared spectral density, and a kernel-independent piece 𝑤(x)
+𝑘
+which is
+however independent of 𝜌(𝜔). The 𝑤(x)
+𝑘
+for the kernels used here are given for all orders in Tab. 1.
+The c0 kernel is however not practically useful in such extrapolations due to the O(𝜖) term.
+A representative constrained extrapolation, in which all kernels (apart from c0) are used to fit
+for 𝜌(𝐸) and the 𝑎𝑘(𝐸) up to a certain order, is shown in Fig. 6. A final estimate for 𝜌(𝐸) is chosen
+with a statistical error larger than the variation between different extrapolation orders and ranges.
+Repeating this procedure for all values of 𝐸 yields the final results for the spectral density 𝜌(𝐸)
+shown in Fig. 10. Not only do the numerical results agree with the exact spectral density including
+two-, four-, and six-particle contributions, but differ significantly from the two-particle contribution
+alone, indicating the sensitivity to four-particle states. Furthermore, the largest energy of 𝐸 = 40𝑚
+is statistically consistent with the two-loop perturbative result, demonstrating that 𝜌(𝐸) has been
+computed up to the onset of the perturbative regime.
+3.
+Prospects for QCD
+It is in principle straightforward to adopt the analysis of the O(3) sigma model in Sec. 2 to the
+lattice QCD computation of current spectral densities. However, while it is difficult to compare
+the density of finite-volume states in one and three spatial dimensions, the O(3) model setup with
+𝑚𝐿 ≈ 30 may be difficult to achieve in QCD. Fortunately, the masterfield paradigm [15–17] offers
+the possibility of large lattice volumes by accumulating statistics from widely-separated space-time
+regions rather than widely-separated Markov chain elements.
+Work in this direction has been detailed at this conference in talks by M. Cè and P. Fritzsch.
+This section describes preliminary work toward the spectral reconstruction of the isovector vector
+8
+
+The spectral reconstruction of inclusive rates
+John Bulava
+Figure 6: Left: a sample constrained extrapolation using the known coefficients in Tab. 1 up to and including
+O(𝜖4) terms for a fixed energy 𝐸 = 14𝑚. The relative fit ranges of the different smearing kernels are adjusted
+so that each kernel has an equal amount of support between the two-particle threshold 2𝑚 and 𝐸. Right:
+variation of the extrapolated value of 𝜌(𝐸) for different extrapolation ranges and orders. The final result is
+conservatively taken as the horizontal shaded region.
+Figure 7: Left: a selection of some of values of 𝜖 (given in the legend) used in the 𝜖 → 0 extrapolation,
+together with the exact smeared spectral density shown as solid lines for the gaussian kernel. Right: the
+final extrapolated results for 𝜌(𝐸) together with the exact two-particle contribution to the spectral density
+and the sum of the two-, four-, and six-particle contributions. The two-loop perturbative spectral density is
+also shown.
+current spectral density with the collaborators and setup mentioned in those talks. Using 𝑁f = 2+1
+dynamical flavors of stabilized Wilson fermions [17] at 𝑎 = 0.09 fm, two ensembles were generated
+with (𝐿/𝑎)4 = 964 and 1924. The analysis described below is based on two and five thermalized,
+widely separated configuations on the 𝐿/𝑎 = 96 and 192 ensembles, respectively. Details about the
+construction of the correlators and the estimation of the statistical errors were given by M. Cè. For
+the data presented here, a variant of the bootstrap procedure is employed.
+As suggested in Sec. 1, the prototypical QCD analogue of Sec. 2 is the hadronic component
+to 𝑒+𝑒− → hadrons, which can be obtained by solving the inverse problem of the euclidean
+current-current correlator projected onto zero spatial momentum in Eq. 3. However, including both
+the isoscalar and isovector components of the electromagnetic current requires valence quark-line
+9
+
+The spectral reconstruction of inclusive rates
+John Bulava
+disconnected Wick contractions, incurring additional computational cost and statistical variance.
+Consider then the simpler case of the isovector-vector correlator. Phenomenologically this spectral
+density can be accessed directly from hadronic decays of the tau lepton [18]. A state-of-the-art
+phenomenological determination of the isovector-vector spectral density is performed in Ref. [19].
+The spectral reconstruction approach of Sec. 2 is adopted nearly identically here, apart from
+some key differences.
+First, the basis functions provided by the correlator data in Sec. 2 are
+𝑏𝑡 (𝜔) = e−𝜔𝑡 + e−𝜔(𝑇 −𝑡), but those employed in this analysis from Eq. 3 are 𝑏𝑡 (𝜔) = 𝜔2 e−𝜔𝑡.
+The flexibility of the formalism of Sec. 2 to handle these different basis functions is an advantage
+over the Chebyshev approach of Ref. [7]. Also, for these large lattices the finite temporal extent
+can be demonstrably ignored. For a first test of the approach in QCD, only the gaussian smearing
+kernel from Eq. 9 is considered. All correlator timeslices from 𝑡min = 𝑎 to 𝑡max = 35𝑎 are used in
+the reconstruction, and all arithmetic operations are performed with 400 bits of computer precision
+using the Arb library [20].
+Another innovation for this analysis compared to Sec. 2 is the procedure for choosing the 𝜆 at
+which statistical errors dominate the systematic errors. As suggested by the left panel in Fig. 2, the
+procedure in Sec. 2 which balances the two functionals 𝐴[𝑔]/𝐴[0] and 𝐵[𝑔] from Eq. 11 is perhaps
+over-conservative and somewhat arbitrary. The alternative approach employed here makes use of
+one of the possible constraints introduced in Ref. [11]. By the addition of a lagrange multiplier, it is
+possible to enforce constraints on the reconstructed smearing kernel ˆ𝛿g
+𝜖 (𝐸, 𝜔). Ref. [11] describes
+how to impose the coincidence of the reconstructed and desired kernels at a particular point
+ˆ𝛿g
+𝜖 (𝐸, 𝜔∗) = 𝛿g
+𝜖 (𝐸 − 𝜔∗).
+(16)
+Although Ref. [11] only considers 𝜔 = 𝐸, the generalization to arbitrary 𝜔∗, even outside the
+interval [𝐸0, ∞), is straightforward.
+Using this ‘equal value’ constraint on the reconstructed kernel, it is possible to estimate how
+small 𝐴[𝑔]/𝐴[0] must be for the statistical errors to dominate. An ‘ensemble’ of reconstructions
+are performed with different values of 𝜔∗, in addition to the unconstrained one. The systematic error
+estimate is then obtained from the variation of ˆ𝜌g
+𝜖 (𝐸) among this ensemble at similar 𝐴[𝑔]/𝐴[0].
+The point at which this variation is smaller than the statistical error on the unconstrained result is
+taken as the optimal reconstruction. Of course this procedure depends on the ensemble of constraint
+points {𝜔∗} which are considered. However, it is sensitive the unsmeared spectral density 𝜌(𝜔), in
+contrast to the approach of Ref. [11]. If additional values of 𝜔∗ are added for which 𝜌(𝜔∗) has little
+support, these will likely differ little from the unconstrained case, apart from possible variations in
+ˆ𝛿g
+𝜖 (𝐸, 𝜔) away from 𝜔∗ induced by the constraint at 𝜔∗. An illustration of this procedure is given
+in Fig. 8.
+After applying the procedure discussed above for a variety of 𝜖 and 𝐸 for the gaussian kernel
+on each of the 𝐿 = 9 fm and 18 fm ensembles, finite volume effects can be examined. This is done
+in Fig. 9, using 𝑣1(𝑠) = 2𝜋2𝜌(𝑠) for a variety of energies at two different values of the smearing
+width 𝜖. While there are possibly hints of finite-volume effects at the one-to-few sigma level at both
+𝜖, these effects are generally under control. Additional volumes will however elucidate the situation
+in the future.
+We finally turn to a comparison of the reconstructed isovector vector spectral density with
+experiment [21]. For this a preliminary value of the vector current renormalization factor 𝑍𝑉 is
+10
+
+The spectral reconstruction of inclusive rates
+John Bulava
+3
+−
+10
+2
+−
+10
+1
+−
+10
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1
+0.11
+0.12
+0.13
+A[g]/A[0]
+(E)
+ε
+g
+ρ
+π
+ = 0.5m
+ε
+, 
+π
+E = 2.5m
+none
+π
+2.5m
+π
+2m
+π
+3m
+π
+4m
+π
+5m
+π
+6m
+π
+7m
+π
+8m
+π
+9m
+π
+10m
+π
+11m
+π
+12m
+2
+−
+0
+2
+4
+6
+8
+10
+12
+0.05
+−
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+ε
+ - E)/
+ω
+(
+εδ
+ε
+ε
+g
+δ
+ε
+none
+π
+2.5m
+π
+2m
+π
+3m
+π
+4m
+π
+5m
+π
+6m
+π
+7m
+π
+8m
+π
+9m
+π
+10m
+π
+11m
+π
+12m
+Figure 8: Illustration of the method for choosing the optimal tradeoff parameter 𝜆 described in the text for the
+gaussian reconstruction on the 𝐿 = 18 fm ensemble with 𝜖 = 0.5𝑚 𝜋 and 𝐸 = 2.5𝑚 𝜋. Left: different values of
+𝜆 for the unconstrained reconstruction and reconstructed kernels constrained to agree with 𝛿g
+𝜖 (𝐸 − 𝜔∗) at the
+various values of 𝜔∗ indicated in the legend. The horizontal band indicates the chosen estimate for which the
+statistical error on the unconstrained reconstruction covers the spread given by the ensemble of constraints.
+For comparison, the method for balancing statistical and systematic errors of Sec. 2 (and Ref. [11]) chooses
+the unconstrained point with 𝐴[𝑔]/𝐴[0] ≈ 0.0016. Right: the reconstructed smearing kernel compared
+to the desired gaussian (solid line) for each member of the constraint ensemble near the chosen value of
+𝐴[𝑔]/𝐴[0] indicated by the horizontal band in the left plot. The residual variation between the different
+constraints is evidently smaller than the statistical error on the constrained reconstruction, although perhaps
+additional values of 𝜔∗ near 𝜔∗ − 𝐸 ≈ 2𝜖 should be added in the future.
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+)
+2
+s (GeV
+(s)
+ε
+1,
+g
+v
+π
+ = 0.5m
+ε
+L = 9 fm
+L = 18 fm
+0
+0.5
+1
+1.5
+2
+2.5
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+)
+2
+s (GeV
+(s)
+ε
+1,
+g
+v
+π
+ = 1.0m
+ε
+L = 9 fm
+L = 18 fm 
+Figure 9: Finite volume effects in the reconstructed vector isovector spectral density on the two masterfield
+ensembles described in the text. Gaussian smearing is used for a variety of energies at smearing width
+𝜖 = 0.5𝑚 𝜋, shown on the left, and 𝜖 = 𝑚 𝜋, shown on the right. These effects are generally small apart from
+some mild discrepancies near 𝑠 = 0.4 GeV2 for 𝜖 = 0.5𝑚 𝜋 and 𝑠 = 0.75GeV2 for 𝜖 = 𝑚 𝜋. Additional smaller
+lattice volumes could further examine these potential finite volume effects.
+11
+
+The spectral reconstruction of inclusive rates
+John Bulava
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+0
+0.5
+1
+1.5
+2
+2.5
+3
+)
+2
+s (GeV
+(s)
+ε
+1,
+g
+v
+ = 265 MeV, a = 0.09 fm
+π
+L = 18 fm, m
+π
+ = 0.5m
+ε
+π
+ = 0.75m
+ε
+π
+ = 1.0m
+ε
+π
+ = 1.25m
+ε
+0
+0.5
+1
+1.5
+2
+2.5
+3
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+τ– → V–ντ
+π–π0
+π–3π0, 2π–π+π0, (6π)–
+ωπ–, ηπ–π0, (KK
+–(π))–
+QCD prediction
+parton model
+s   (GeV2)
+v1(s)
+ALEPH
+Figure 10:
+Comparison of the lattice QCD results for the isovector vector spectral density on the larger
+𝐿/𝑎 = 192 master field ensemble discussed in the text (shown on the left), with experimental results for
+hadronic 𝜏-decays on the right. Statistical errors due to the scale setting and the renormalization of the vector
+current are not yet taken into account.
+employed, which was presented at this conference by J. Kuhlmann. The statistical error on 𝑍𝑉
+is ignored in these preliminary results, as is the error on the lattice scale, which is crudely set
+by assuming 𝑚 𝜋 = 265 MeV. The results are summarized in Fig. 10, and broadly resemble the
+experimental plot, with a narrow peak likely due to the 𝜌(770) vector resonance followed by a slow
+rise due to four-particle states. Particularly interesting is the mild indication of this rise in the lattice
+QCD data, which (like in the O(3) model) show the effects of four-pion states. It should be noted
+that the current state-of-the-art for the finite-volume approach to lattice QCD scattering amplitudes
+is the numerical computation of (exclusive rather than inclusive) three-pion scattering amplitudes3.
+4.
+Conclusions
+Alternative techniques are required to compute phenomena arising from many hadronic states.
+The spectral reconstruction of smeared spectral densities from euclidean correlator data not only
+bridges the gap between finite and infinite volume, but also helps to regulate the ill-posed nature
+of the problem. The application discussed here is the computation of inclusive rates summed over
+all final states produced by an external current. In the two-dimensional O(3) model, after taking
+the continuum limit, the algorithm presented in Sec. 2 (first proposed for lattice field theory in
+Ref. [10]) results in smeared spectral densities consistent with known analytic results. Spectral
+reconstruction algorithms based on the Backus-Gilbert approach [5, 6] enable a precise definition
+of the smeared spectral density that has been computed, while the modification of Ref. [9] further
+allows the a priori specification of a desired smearing kernel. The simple linear ansatz on which
+these approaches are based enables the direct expression for the smearing kernel given in Eq. 6.
+Smeared spectral densities are useful not only for inclusive decay rates. An incomplete list
+of recent applications of the Backus-Gilbert approach includes the nucleon hadronic tensor [24],
+3For a review of the current status of computations of three-particle scattering amplitudes using the finite-volume
+approach, see the presentation by F. Romero-López and the recent reviews in Refs. [22, 23].
+12
+
+The spectral reconstruction of inclusive rates
+John Bulava
+the determination of PDFs from Ioffe time data [25], and the photon emissivity of the quark-
+gluon plasma [26]. These applications do not employ the algorithmic variant enabling a priori
+specification of the smearing kernel, but could perhaps benefit from it in the future. This a priori
+specification of the kernel enabled in Refs. [9, 10] is also present in the Chebyshev approach of
+Ref. [7], but the stabilizing effect of the functional 𝐵[𝑔] in Eq. 11 is naively not present. The work
+of Ref. [8] is a first step towards comparing the two approaches.
+The advantages of the a priori approach are leveraged in the two-dimensional O(3) model to
+perform joint constrained 𝜖 → 0 extrapolations with several different kernels. The presence of the
+narrow 𝜌(770) peak in the isovector vector spectral density in QCD discussed in Sec. 3 complicates
+such an extrapolation and more work is required toward an implementation. A similar approach has
+been employed to compute inclusive decay rates in Refs. [27, 28], and taken up by additional groups
+in Refs. [8, 29, 30]. Work towards computing the 𝑅-ratio was reported in this conference [31], as
+well as a similar analyses of the total hadronic tau decay rate [32, 33], albeit with a wider gaussian
+smearing radius than employed here. The interplay between the spatial extent and the smallest
+achievable smearing width requires further study. Furthermore, the a priori approach of Ref. [7]
+led to the direct computation of the Borel transform of a current-current correlator required for
+the Shifman-Vainshtein-Zakharov sum rule in Ref. [34], possibly opening the door for additional
+interaction between lattice QCD and QCD sum rules. Another interesting application is pursued in
+Ref. [35] in which fits to smeared spectral densities are considered as an alternative to ‘standard’
+spectroscopy. Additional applications could appear in the future. The a priori approach in principle
+enables the computation of exclusive scattering amplitudes using Refs. [36, 37], while the formalism
+for inclusive rates was developed already in Refs. [3, 38].
+References
+[1] A. Pich, Precision physics with inclusive QCD processes, Prog. Part. Nucl. Phys. 117 (2021)
+103846, [arXiv:2012.04716].
+[2] E. C. Poggio, H. R. Quinn, and S. Weinberg, Smearing the Quark Model, Phys. Rev. D 13
+(1976) 1958.
+[3] M. T. Hansen, H. B. Meyer, and D. Robaina, From deep inelastic scattering to heavy-flavor
+semileptonic decays: Total rates into multihadron final states from lattice QCD, Phys. Rev. D
+96 (2017), no. 9 094513, [arXiv:1704.08993].
+[4] O. Kaczmarek and H.-T. Shu, Spectral and Transport Properties from Lattice QCD, Lect.
+Notes Phys. 999 (2022) 307–345, [arXiv:2206.14676].
+[5] G. Backus and F. Gilbert, The resolving power of gross earth data, Geophysical Journal
+International 16 (1968), no. 2 169–205.
+[6] G. Backus and F. Gilbert, Uniqueness in the inversion of inaccurate gross earth data,
+Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and
+Engineering Sciences 266 (1970), no. 1173 123–192,
+[http://rsta.royalsocietypublishing.org/content/266/1173/123.full.pdf].
+13
+
+The spectral reconstruction of inclusive rates
+John Bulava
+[7] G. Bailas, S. Hashimoto, and T. Ishikawa, Reconstruction of smeared spectral function from
+Euclidean correlation functions, PTEP 2020 (2020), no. 4 043B07, [arXiv:2001.11779].
+[8] A. Barone, S. Hashimoto, A. Jüttner, T. Kaneko, and R. Kellermann, Inclusive semi-leptonic
+𝐵(𝑠) mesons decay at the physical 𝑏 quark mass, in 39th International Symposium on Lattice
+Field Theory, 11, 2022. arXiv:2211.15623.
+[9] F. Pijpers and M. Thompson, Faster formulations of the optimally localized averages method
+for helioseismic inversions, Astronomy and Astrophysics 262 (08, 1992) L33–L36.
+[10] M. Hansen, A. Lupo, and N. Tantalo, Extraction of spectral densities from lattice correlators,
+Phys. Rev. D 99 (2019), no. 9 094508, [arXiv:1903.06476].
+[11] J. Bulava, M. T. Hansen, M. W. Hansen, A. Patella, and N. Tantalo, Inclusive rates from
+smeared spectral densities in the two-dimensional O(3) non-linear 𝜎-model, JHEP 07
+(2022) 034, [arXiv:2111.12774].
+[12] M. Luscher and U. Wolff, How to Calculate the Elastic Scattering Matrix in Two-dimensional
+Quantum Field Theories by Numerical Simulation, Nucl. Phys. B339 (1990) 222–252.
+[13] J. Balog and M. Niedermaier, Off-shell dynamics of the O(3) NLS model beyond Monte
+Carlo and perturbation theory, Nucl. Phys. B 500 (1997) 421–461, [hep-th/9612039].
+[14] J. Balog, F. Niedermayer, and P. Weisz, The Puzzle of apparent linear lattice artifacts in the
+2d non-linear sigma-model and Symanzik’s solution, Nucl. Phys. B 824 (2010) 563–615,
+[arXiv:0905.1730].
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diff --git a/U9E2T4oBgHgl3EQftwgF/content/tmp_files/load_file.txt b/U9E2T4oBgHgl3EQftwgF/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf,len=683
+page_content='The spectral reconstruction of inclusive rates  John Bulava 𝑎,∗  𝑎Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany  E-mail: john.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='bulava@desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='de  A recently re-discovered variant of the Backus-Gilbert algorithm for spectral reconstruction en-  ables the controlled determination of smeared spectral densities from lattice field theory correlation  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' A particular advantage of this approach is the a priori specification of the kernel with  which the underlying spectral density is smeared, allowing for variation of its peak position,  smearing width, and functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' If the unsmeared spectral density is sufficiently smooth  in the neighborhood of a particular energy, it can be obtained from an extrapolation to zero  smearing-kernel width at fixed peak position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' A natural application for this approach is scattering  processes summed over all hadronic final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' As a proof-of-principle test, an inclusive rate  is computed in the two-dimensional O(3) sigma model from a two-point correlation function of  conserved currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The results at finite and zero smearing radius are in good agreement with  the known analytic form up to energies at which 40-particle states contribute, and are sensitive to  the 4-particle contribution to the inclusive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The straight-forward adaptation to compute the  𝑅-ratio in lattice QCD from two-point functions of the electromagnetic current is briefly discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  The 39th International Symposium on Lattice Field Theory,  8th-13th August, 2022,  Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='04072v1  [hep-lat]  10 Jan 2023  The spectral reconstruction of inclusive rates  John Bulava  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Introduction  Lattice QCD simulations proceed by computing 𝑛-point euclidean correlation functions of  (quasi-) local interpolating operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Single-hadron states and finite-volume few-hadron states are  isolated from correlation functions in the asymptotic large euclidean time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' However, some  hadronic phenomena are best studied by other means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' As an example, this work considers inclusive  rates defined as a sum over all hadronic final states produced by an external current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' At large  center-of-mass energies, a finite-volume approach to such a process is impractical since it requires  the isolation of all individual finite-volume levels with arbitrarily many particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Such processes  are a cornerstone of QCD and connect the low-energy hadronic and high-energy perturbative  regimes [1], serving as a manifestation of ‘quark-hadron duality’ [2] whereby perturbative QCD in  terms of quarks and gluons becomes increasingly effective at computing inclusive rates summed  over final states consisting entirely of hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  For concreteness, consider the QCD part of the process 𝑒+𝑒− → hadrons  𝜌(𝑠) = 𝑅(𝑠)  12𝜋2 ,  𝑅(𝑠) = 𝜎 [𝑒+𝑒− → hadrons] (𝑠)  4𝜋𝛼em(𝑠)2/(3𝑠)  ,  (1)  𝜌𝜇𝜈(𝑘) = 1  2𝜋  ˆ  𝑑4𝑥 𝑒−𝑖𝑘·𝑥⟨Ω| ˆ𝑗em  𝜇 (𝑥) ˆ𝑗em  𝜈 (0)|Ω⟩ = (𝑔𝜇𝜈𝑘2 − 𝑘𝜇𝑘𝜈) 𝜌(𝑘2),  (2)  where ˆ𝑗em  𝜇  is the quark-level electromagnetic current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The desired inclusive rate is given by the  spectral density 𝜌(𝑠), which is also present in the analogous infinite-volume Euclidean correlator  𝐶(𝑡) =  ˆ  𝑑3𝒙 ⟨Ω| ˆ𝑗em  𝑧 (𝒙) 𝑒− ˆ𝐻𝑡 ˆ𝑗em  𝑧 (0)†|Ω⟩ =  ˆ ∞  0  𝑑𝜔 𝜔2𝜌(𝜔2) 𝑒−𝜔𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  (3)  The direct determination of 𝜌(𝑠) in lattice QCD is not straightforward, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' First, the inversion  of integral equations like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 3 using 𝐶(𝑡) evaluated at a finite number of discrete times with  statistical errors is notoriously ill-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Furthermore, the finite volume introduces additional  complication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Even if the inverse problem were solved successfully and the finite-volume euclidean  correlator 𝐶𝐿(𝑡) used to determine its spectral density 𝜌𝐿(𝑠), it differs qualitatively from its infinite-  volume counterpart 𝜌(𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' While 𝜌𝐿(𝑠) is a sum over Dirac 𝛿-functions for each finite-volume state,  𝜌(𝑠) is smooth apart from non-analyticities due to the opening of thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' In no way does 𝜌𝐿(𝑠)  ‘approach’ 𝜌(𝑠) as 𝐿 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  The bridge between finite and infinite volume is made more effectively using the smeared  spectral density  𝜌𝜖 (𝐸) =  ˆ ∞  0  𝑑𝜔 𝛿𝜖 (𝐸 − 𝜔) 𝜌(𝜔) ,  (4)  where lim𝜖 →0 𝛿𝜖 (𝑥) = 𝛿(𝑥), 𝛿(𝑥) is the Dirac-delta function, and  ´ ∞  −∞ 𝑑𝑥 𝛿𝜖 (𝑥) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' This solves  (in principle) both of the difficulties mentioned above: the inverse problem can be made arbitrarily  mild by increasing in the smearing width 𝜖 and 𝜌𝐿,𝜖 (𝐸) approaches its infinite-volume counterpart  in a well-defined manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The goal is now to take the ordered double limit [3]  𝜌(𝐸) = lim  𝜖 →0+ lim  𝐿→∞ 𝜌𝐿,𝜖 (𝐸),  (5)  2  The spectral reconstruction of inclusive rates  John Bulava  the asymptotic corrections to which are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Although spectral reconstruction has a long history in lattice QCD, particularly at finite temper-  ature [4], the treatment of the inverse problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 3 demands special care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' In order to define the  result of the spectral reconstruction procedure, precise knowledge of the smearing kernel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 4  is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' As detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2, the Backus-Gilbert approach [5, 6] is suitable in this respect2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Since the estimator for the smeared spectral density ˆ𝜌𝜖 (𝐸) is simply a linear combination of the  input correlator data ˆ𝜌𝜖 (𝐸) = �  𝑡 𝑔𝑡 (𝜖, 𝐸)𝐶(𝑡), the resultant smearing kernel is given by the same  linear combination of decaying exponentials in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 3  ˆ𝜌𝜖 (𝐸) =  ˆ  𝑑𝜔 ˆ𝛿𝜖 (𝐸, 𝜔) 𝜌(𝜔),  ˆ𝛿𝜖 (𝐸, 𝜔) =  ∑︁  𝑡  𝑔𝑡 (𝜖, 𝐸)𝜔2𝑒−𝜔𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  (6)  Explicit knowledge of ˆ𝛿𝜖 (𝐸, 𝜔) is a minimum requirement for a well-defined spectral reconstruction  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Naively the Backus-Gilbert approach provides knowledge of the kernel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 6 only a posteriori  for a given choice of coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' However, the coefficients themselves can be chosen to approximate  a particular smearing kernel specified a priori [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' This important innovation is employed here  and was applied to lattice field theory for the first time in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' In order to understand this  reconstruction algorithm, and in particular demonstrate control over the systematic errors, a test in a  controlled context is warranted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Such a test has been performed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [11] for the two-dimensional  O(3) sigma model together with attempts at saturating the ordered double limit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  The remainder of this work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The spectral reconstruction method is  presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 in the context of the O(3) model test mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Prospects for adapting  the method to current correlators in lattice QCD is discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 3 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 4 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  O(3) model test  This section reviews a spectral reconstruction test which was recently published in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  It employs the spectral reconstruction procedure of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [10] in the two-dimensional O(3) sigma  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Consider the standard lattice discretization  𝑆[𝜎] = 𝛽  2  ∑︁  𝑥∈Λ  𝑎2 ∑︁  𝜇  ˆ𝜕𝜇𝜎(𝑥) · ˆ𝜕𝜇𝜎(𝑥) = 𝛽  ∑︁  𝑥∈Λ  ∑︁  𝜇  [1 − 𝜎(𝑥) · 𝜎(𝑥 + 𝑎 ˆ𝜇)] ,  (7)  where 𝜎(𝑥) ∈ R3, |𝜎(𝑥)| = 1, and ˆ𝜕𝜇 𝑓 (𝑥) = 1  𝑎 [ 𝑓 (𝑥 + 𝑎 ˆ𝜇) − 𝑓 (𝑥)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' This model has a conserved  current  𝑗 𝑎  𝜇(𝑥) = 𝛽𝜖 𝑎𝑏𝑐𝜎𝑏(𝑥) ˆ𝜕𝜇𝜎𝑐(𝑥)  (8)  at finite lattice spacing, and possesses a dynamically-generated mass gap 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The total zero spatial  momentum euclidean current-current correlation function analogous to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 3 (but without the factor  1Finite lattice spacing effects must also be removed by taking the continuum limit, which is here performed at fixed  𝜖 and 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  2Another spectral reconstruction algorithm for which the smearing kernel is formally known a posteriori is the  Chebyshev polynomial approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [8] compares that approach to the one employed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  3  The spectral reconstruction of inclusive rates  John Bulava  n=2  n=4  10  20  30  40  50  μ/m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  ρ(n)(μ)  Figure 1:  Left: exactly known 𝑛 = 2 and 𝑛 = 4 particle contributions to the (continuum, infinite-  volume) spectral density associated with the conserved current in the two-dimensional O(3) sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Contributions from states with more particles are insignificant in the energy range shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Right: the  four smearing kernels 𝛿x  𝜖 (𝑥), where x = {g, c0, c1, c2}, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 9 plotted against 𝑥 with 𝜖 = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  of 𝜔2) is computed by numerical simulations using the single-cluster algorithm of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' A  variety of ensembles are generated, with 𝑚𝐿 ≈ 30 − 60 and 𝑎𝑚 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='04] to assess finite  volume effects and take the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' In the continuum the contributions to the associated  spectral density 𝜌(𝜔) from each fixed-particle number sector can be computed exactly [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Below  energies 𝐸 < 50𝑚, only the 𝑛 = 2 and 𝑛 = 4 particle contributions are significant and are shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  In order to demonstrate the a priori specification of the smearing kernel, consider four kernels  with different profiles as a function of 𝑥 = 𝐸 − 𝜔:  𝛿g  𝜖 (𝑥) =  1  √  2𝜋𝜖  exp  �  − 𝑥2  2𝜖2  �  ,  𝛿c0  𝜖 (𝑥) = 1  𝜋  𝜖  𝑥2 + 𝜖2 ,  (9)  𝛿c1  𝜖 (𝑥) = 2  𝜋  𝜖3  (𝑥2 + 𝜖2)2 ,  𝛿c2  𝜖 (𝑥) = 8  3𝜋  𝜖5  (𝑥2 + 𝜖2)3 ,  (10)  including the gaussian (denoted ‘g’) and three Cauchy-like kernels denoted ‘c𝑛’, for which 𝑛 = 0, 1, 2  distinguishes the power of the pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' These kernels are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  The method advocated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [10] to reconstruct the smeared spectral density 𝜌x  𝜖 (𝐸), where  x = {g, c0, c1, c2}, is based on two criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' First, the reconstructed smearing kernel ˆ𝛿x  𝜖 (𝐸, 𝜔)  should be close to the desired one 𝛿x  𝜖 (𝐸 − 𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Second, the coefficients {𝑔𝑡 (𝜖, 𝐸)} in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 6 should  not induce a large statistical variance on the estimator ˆ𝜌x  𝜖 (𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' These two considerations are encoded  in the functionals  𝐴[𝑔] =  ˆ ∞  𝐸0  𝑑𝜔  �  𝛿x  𝜖 (𝐸 − 𝜔) − ˆ𝛿x  𝜖 (𝐸, 𝜔)  �2 ,  (11)  𝐵[𝑔] = Var[ ˆ𝜌x  𝜖 (𝐸)] =  ∑︁  𝑡𝑡′  𝑔𝑡 (𝜖, 𝐸) 𝑔𝑡′(𝜖, 𝐸) Cov[𝐶(𝑡), 𝐶(𝑡′)]  (12)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The coefficients are then chosen to minimize the combination functional 𝐺𝜆[𝑔] = (1−  𝜆)𝐴[𝑔]/𝐴[0] + 𝜆𝐵[𝑔], where the ‘trade-off’ parameter 𝜆 is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' For small 𝜆, the ‘accuracy’  functional 𝐴[𝑔] takes preference over the ‘precision’ one 𝐵[𝑔] resulting in small systematic but large  4  The spectral reconstruction of inclusive rates  John Bulava  3  −  10  2  −  10  1  −  10  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='67  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='73  (E)  ερ  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='20  c  λ  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='75m, gauss,  ε  E = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='0m,  A[q]/A[0]  (E)  ερ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  −  1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='4  ε  )/  ω  (E-  ε  λ  δ  ε  ε  λ  δ  ε  Figure 2: Left: indicative illustration of the trade-off between statistical and systematic errors for a particular  choice of 𝐸 and 𝜖 on a single ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Each point corresponds to a different 𝜆 and the horizontal band  indicates the chosen reconstruction (with 𝜆 = 𝜆c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='20) for which statistical errors dominate systematic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Right: for this same setup and 𝜆 = 𝜆c, the reconstructed kernel ˆ𝛿g  𝜖 (𝐸, 𝜔), together with the desired kernel  𝛿g  𝜖 (𝐸 − 𝜔) shown as a solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' All reconstructions employ the correlator timeslices 𝑡 = 1𝑎, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' , 160𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' By contrast, large 𝜆 results in small statistical errors but a reconstructed smearing  kernel which does not resemble the desired one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The choice of the parameter 𝜆 is performed  automatically and results in an approximate balance of these two criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The effect of varying  𝜆 is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 together with the reconstructed kernel ˆ𝛿g  𝜖 (𝐸, 𝜔) for a sample setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The  trade-off between statistical and systematic errors is familiar to lattice field theorists and the left  panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 resembles the identification of a plateau in effective mass plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  The procedure described above is performed for a variety of 𝐸 and 𝜖, and for all four smearing  kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Next, finite volume effects must be assessed and the continuum limit taken independently  for each 𝐸, 𝜖, and kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Finite-volume effects are assessed at a single lattice spacing by simulating  two additional ensembles with doubled spatial and temporal extents, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The differences  Δ𝐿,𝑇 between the spectral reconstruction on the doubled lattices divided by the statistical error are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 3, which show (at most) moderately significant hints for finite-𝐿 effects at energies  near the two-particle production threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  With the finite-volume effects demonstrably controlled for the (𝐸, 𝜖) values and the kernels in  question, the continuum limit can be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Cutoff effects for ‘on-shell’ quantities in the two-  dimensional O(3) sigma model have a long history, due to their apparently linear behaviour which  is caused by large logarithmic corrections [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Unfortunately, the analysis there is incomplete  for the ‘off-shell’ smeared spectral density considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' In order to explore the influence of  logarithmic cutoff effects, the fit forms  𝑄(𝑎) = 𝑄(0) + 𝐶𝑎2𝛽𝑟,  𝑟 = 0, 3, 6  (13)  are explored to extrapolate the smeared spectral densities to the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' A comparison  of the extrapolation forms is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The continuum limits are generally mild and well-  constrained by the data, although the slope becomes steeper for increasing 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Given the assessment of systematic errors due to spectral reconstruction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' finite 𝐿 and 𝑇 effects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  and finite lattice spacing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' it is time to confront the computations of 𝜌x  𝜖 (𝐸) with the exact spectral  density 𝜌(𝜔) (comprised of the two-,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' four-,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' and six-particle contributions) smeared with the exact  5  The spectral reconstruction of inclusive rates  John Bulava  Figure 3: The difference Δ𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='𝑇 between spectral reconstruction on ensembles using 𝐿 and 2𝐿 (top row),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' and  using ensembles with 𝑇 and 2𝑇 in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' In both cases Δ𝐿,𝑇 is divided by the statistical error on  the smaller ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Perhaps some marginally significant hints for finite-𝐿 effects are observed at small 𝐸  near the two-particle production threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Figure 4: The continuum limit for a single 𝐸, 𝜖, and smearing kernel using the ansatz of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The shaded  band indicates the fit, and the horizontal dotted region the extrapolated value for 𝑟 = 3, which is taken as the  final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  6  The spectral reconstruction of inclusive rates  John Bulava  Figure 5: Lattice results for 𝜌x  𝜖 (𝐸) in the continuum limit (the data points shown in the legend) compared  against the exact spectral density including two-, four-, and six-particle contributions smeared with the exact  kernel 𝛿x  𝜖 (𝐸 − 𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The exact results are shown as lines in the top row, and the bottom row shows the ‘pull’  between the numerical data and the exact result, divided by the statistical and systematic error combined  in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' A naive histogram of the differences, which ignores correlations among the data, is shown  horizontally in the bottom row and approximately resembles the unit gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  smearing kernel 𝛿x  𝜖 (𝐸 − 𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Such a comparison is performed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 5 where the numerical  computations are demonstrably consistent with the exact results, within the quoted errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' These  errors take into account the statistical errors due to the reconstruction, with any residual systematic  errors from finite-𝐿,𝑇 effects and the continuum limit added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  At this point the verification of the spectral reconstruction approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [10] is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Smeared spectral densities in the two-dimensional O(3) sigma model have been reconstructed with  smearing kernels specified a priori, which are consistent with the exact result after the continuum  limit has been taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Consider now the 𝜖 → 0 limit in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' For this, an important property of the  unsmeared spectral density evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 1 is required, namely that it varies increasingly slowly  with increasing 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' This circumvents the limitations in reconstructing kernels with a fixed 𝜖 and  increasing 𝐸 evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 5: larger smearing widths are sufficient at larger 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' To this end, the  smearing width is scaled 𝜖 ∝ (𝐸 −2𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Also, rather than using a single smearing kernel, 𝜌x  𝜖 (𝐸) for  all kernels are used to perform constrained extrapolations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' For this the small-𝜖 expansion is useful  𝜌x  𝜖 (𝐸) ≡  ˆ ∞  0  𝑑𝜔 𝛿x  𝜖 (𝐸 − 𝜔) 𝜌(𝜔) = 𝜌(𝐸) +  ∞  ∑︁  𝑘=1  𝑤x  𝑘𝑎𝑘(𝐸)𝜖 𝑘 ,  (14)  7  The spectral reconstruction of inclusive rates  John Bulava  x  𝑤x  𝑘, even 𝑘  𝑤x  𝑘, odd 𝑘  𝑤x  1  𝑤x  2  𝑤x  3  𝑤x  4  g  𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  (−2)𝑘/2(𝑘/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  0  0  −1  0  3  c0  1  1  1  1  1  1  c1  (1 − 𝑘)  (1 − 𝑘)  0  −1  −2  −3  c2  1  3 (𝑘 − 3)(𝑘 − 1)  1  3 (𝑘 − 3)(𝑘 − 1)  0  −1/3  0  1  Table 1: The kernel-dependent coefficients 𝑤x  𝑘 appearing in the small-𝜖 expansion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' For the c1  and c2 kernels, 𝑤c1  3 and 𝑤c2  5 (respectively) are the non-zero coefficients with lowest odd order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  where the contribution at the 𝑘th order in 𝜖 is the product of a kernel-independent factor  𝑎𝑘(𝐸) =  ������  ������  (−1)𝑘/2  𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  �  𝑑  𝑑𝐸  � 𝑘  𝜌(𝐸) ,  𝑘 even  lim𝜂→0+ (−1) (𝑘−1)/2  2𝜋  ´ ∞  −∞ d𝜔 𝜌(𝐸+𝜔)+𝜌(𝐸−𝜔)  (𝜔+𝑖𝜂)𝑘+1  ,  𝑘 odd  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  (15)  which depends on the unsmeared spectral density, and a kernel-independent piece 𝑤(x)  𝑘  which is  however independent of 𝜌(𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The 𝑤(x)  𝑘  for the kernels used here are given for all orders in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  The c0 kernel is however not practically useful in such extrapolations due to the O(𝜖) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  A representative constrained extrapolation, in which all kernels (apart from c0) are used to fit  for 𝜌(𝐸) and the 𝑎𝑘(𝐸) up to a certain order, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' A final estimate for 𝜌(𝐸) is chosen  with a statistical error larger than the variation between different extrapolation orders and ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Repeating this procedure for all values of 𝐸 yields the final results for the spectral density 𝜌(𝐸)  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Not only do the numerical results agree with the exact spectral density including  two-, four-, and six-particle contributions, but differ significantly from the two-particle contribution  alone, indicating the sensitivity to four-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Furthermore, the largest energy of 𝐸 = 40𝑚  is statistically consistent with the two-loop perturbative result, demonstrating that 𝜌(𝐸) has been  computed up to the onset of the perturbative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Prospects for QCD  It is in principle straightforward to adopt the analysis of the O(3) sigma model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 to the  lattice QCD computation of current spectral densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' However, while it is difficult to compare  the density of finite-volume states in one and three spatial dimensions, the O(3) model setup with  𝑚𝐿 ≈ 30 may be difficult to achieve in QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Fortunately, the masterfield paradigm [15–17] offers  the possibility of large lattice volumes by accumulating statistics from widely-separated space-time  regions rather than widely-separated Markov chain elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Work in this direction has been detailed at this conference in talks by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Cè and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Fritzsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  This section describes preliminary work toward the spectral reconstruction of the isovector vector  8  The spectral reconstruction of inclusive rates  John Bulava  Figure 6: Left: a sample constrained extrapolation using the known coefficients in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 1 up to and including  O(𝜖4) terms for a fixed energy 𝐸 = 14𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The relative fit ranges of the different smearing kernels are adjusted  so that each kernel has an equal amount of support between the two-particle threshold 2𝑚 and 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Right:  variation of the extrapolated value of 𝜌(𝐸) for different extrapolation ranges and orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The final result is  conservatively taken as the horizontal shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Figure 7: Left: a selection of some of values of 𝜖 (given in the legend) used in the 𝜖 → 0 extrapolation,  together with the exact smeared spectral density shown as solid lines for the gaussian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Right: the  final extrapolated results for 𝜌(𝐸) together with the exact two-particle contribution to the spectral density  and the sum of the two-, four-, and six-particle contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The two-loop perturbative spectral density is  also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  current spectral density with the collaborators and setup mentioned in those talks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Using 𝑁f = 2+1  dynamical flavors of stabilized Wilson fermions [17] at 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='09 fm, two ensembles were generated  with (𝐿/𝑎)4 = 964 and 1924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The analysis described below is based on two and five thermalized,  widely separated configuations on the 𝐿/𝑎 = 96 and 192 ensembles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Details about the  construction of the correlators and the estimation of the statistical errors were given by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Cè.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' For  the data presented here, a variant of the bootstrap procedure is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  As suggested in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 1, the prototypical QCD analogue of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 is the hadronic component  to 𝑒+𝑒− → hadrons, which can be obtained by solving the inverse problem of the euclidean  current-current correlator projected onto zero spatial momentum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' However, including both  the isoscalar and isovector components of the electromagnetic current requires valence quark-line  9  The spectral reconstruction of inclusive rates  John Bulava  disconnected Wick contractions, incurring additional computational cost and statistical variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Consider then the simpler case of the isovector-vector correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Phenomenologically this spectral  density can be accessed directly from hadronic decays of the tau lepton [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' A state-of-the-art  phenomenological determination of the isovector-vector spectral density is performed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  The spectral reconstruction approach of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 is adopted nearly identically here, apart from  some key differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  First, the basis functions provided by the correlator data in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 are  𝑏𝑡 (𝜔) = e−𝜔𝑡 + e−𝜔(𝑇 −𝑡), but those employed in this analysis from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 3 are 𝑏𝑡 (𝜔) = 𝜔2 e−𝜔𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  The flexibility of the formalism of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 to handle these different basis functions is an advantage  over the Chebyshev approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Also, for these large lattices the finite temporal extent  can be demonstrably ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' For a first test of the approach in QCD, only the gaussian smearing  kernel from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 9 is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' All correlator timeslices from 𝑡min = 𝑎 to 𝑡max = 35𝑎 are used in  the reconstruction, and all arithmetic operations are performed with 400 bits of computer precision  using the Arb library [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Another innovation for this analysis compared to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 is the procedure for choosing the 𝜆 at  which statistical errors dominate the systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' As suggested by the left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2, the  procedure in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 which balances the two functionals 𝐴[𝑔]/𝐴[0] and 𝐵[𝑔] from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 11 is perhaps  over-conservative and somewhat arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The alternative approach employed here makes use of  one of the possible constraints introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' By the addition of a lagrange multiplier, it is  possible to enforce constraints on the reconstructed smearing kernel ˆ𝛿g  𝜖 (𝐸, 𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [11] describes  how to impose the coincidence of the reconstructed and desired kernels at a particular point  ˆ𝛿g  𝜖 (𝐸, 𝜔∗) = 𝛿g  𝜖 (𝐸 − 𝜔∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  (16)  Although Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [11] only considers 𝜔 = 𝐸, the generalization to arbitrary 𝜔∗, even outside the  interval [𝐸0, ∞), is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Using this ‘equal value’ constraint on the reconstructed kernel, it is possible to estimate how  small 𝐴[𝑔]/𝐴[0] must be for the statistical errors to dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' An ‘ensemble’ of reconstructions  are performed with different values of 𝜔∗, in addition to the unconstrained one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The systematic error  estimate is then obtained from the variation of ˆ𝜌g  𝜖 (𝐸) among this ensemble at similar 𝐴[𝑔]/𝐴[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  The point at which this variation is smaller than the statistical error on the unconstrained result is  taken as the optimal reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Of course this procedure depends on the ensemble of constraint  points {𝜔∗} which are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' However, it is sensitive the unsmeared spectral density 𝜌(𝜔), in  contrast to the approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' If additional values of 𝜔∗ are added for which 𝜌(𝜔∗) has little  support, these will likely differ little from the unconstrained case, apart from possible variations in  ˆ𝛿g  𝜖 (𝐸, 𝜔) away from 𝜔∗ induced by the constraint at 𝜔∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' An illustration of this procedure is given  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  After applying the procedure discussed above for a variety of 𝜖 and 𝐸 for the gaussian kernel  on each of the 𝐿 = 9 fm and 18 fm ensembles, finite volume effects can be examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' This is done  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 9, using 𝑣1(𝑠) = 2𝜋2𝜌(𝑠) for a variety of energies at two different values of the smearing  width 𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' While there are possibly hints of finite-volume effects at the one-to-few sigma level at both  𝜖, these effects are generally under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Additional volumes will however elucidate the situation  in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  We finally turn to a comparison of the reconstructed isovector vector spectral density with  experiment [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' For this a preliminary value of the vector current renormalization factor 𝑍𝑉 is  10  The spectral reconstruction of inclusive rates  John Bulava  3  −  10  2  −  10  1  −  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='13  A[g]/A[0]  (E)  ε  g  ρ  π  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5m  ε  ,  π  E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5m  none  π  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5m  π  2m  π  3m  π  4m  π  5m  π  6m  π  7m  π  8m  π  9m  π  10m  π  11m  π  12m  2  −  0  2  4  6  8  10  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='05  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='4  ε  E)/  ω  (  εδ  ε  ε  g  δ  ε  none  π  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5m  π  2m  π  3m  π  4m  π  5m  π  6m  π  7m  π  8m  π  9m  π  10m  π  11m  π  12m  Figure 8: Illustration of the method for choosing the optimal tradeoff parameter 𝜆 described in the text for the  gaussian reconstruction on the 𝐿 = 18 fm ensemble with 𝜖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5𝑚 𝜋 and 𝐸 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5𝑚 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Left: different values of  𝜆 for the unconstrained reconstruction and reconstructed kernels constrained to agree with 𝛿g  𝜖 (𝐸 − 𝜔∗) at the  various values of 𝜔∗ indicated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The horizontal band indicates the chosen estimate for which the  statistical error on the unconstrained reconstruction covers the spread given by the ensemble of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  For comparison, the method for balancing statistical and systematic errors of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 (and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [11]) chooses  the unconstrained point with 𝐴[𝑔]/𝐴[0] ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='0016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Right: the reconstructed smearing kernel compared  to the desired gaussian (solid line) for each member of the constraint ensemble near the chosen value of  𝐴[𝑔]/𝐴[0] indicated by the horizontal band in the left plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The residual variation between the different  constraints is evidently smaller than the statistical error on the constrained reconstruction, although perhaps  additional values of 𝜔∗ near 𝜔∗ − 𝐸 ≈ 2𝜖 should be added in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='8  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='8  )  2  s (GeV  (s)  ε  1,  g  v  π  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5m  ε  L = 9 fm  L = 18 fm  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='9  1  )  2  s (GeV  (s)  ε  1,  g  v  π  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='0m  ε  L = 9 fm  L = 18 fm  Figure 9: Finite volume effects in the reconstructed vector isovector spectral density on the two masterfield  ensembles described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Gaussian smearing is used for a variety of energies at smearing width  𝜖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5𝑚 𝜋, shown on the left, and 𝜖 = 𝑚 𝜋, shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' These effects are generally small apart from  some mild discrepancies near 𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='4 GeV2 for 𝜖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5𝑚 𝜋 and 𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='75GeV2 for 𝜖 = 𝑚 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Additional smaller  lattice volumes could further examine these potential finite volume effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  11  The spectral reconstruction of inclusive rates  John Bulava  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  3  )  2  s (GeV  (s)  ε  1,  g  v  = 265 MeV, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='09 fm  π  L = 18 fm, m  π  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5m  ε  π  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='75m  ε  π  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='0m  ε  π  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='25m  ε  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='5  τ– → V–ντ  π–π0  π–3π0, 2π–π+π0, (6π)–  ωπ–, ηπ–π0, (KK  –(π))–  QCD prediction  parton model  s   (GeV2)  v1(s)  ALEPH  Figure 10:  Comparison of the lattice QCD results for the isovector vector spectral density on the larger  𝐿/𝑎 = 192 master field ensemble discussed in the text (shown on the left), with experimental results for  hadronic 𝜏-decays on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Statistical errors due to the scale setting and the renormalization of the vector  current are not yet taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  employed, which was presented at this conference by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Kuhlmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The statistical error on 𝑍𝑉  is ignored in these preliminary results, as is the error on the lattice scale, which is crudely set  by assuming 𝑚 𝜋 = 265 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The results are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 10, and broadly resemble the  experimental plot, with a narrow peak likely due to the 𝜌(770) vector resonance followed by a slow  rise due to four-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Particularly interesting is the mild indication of this rise in the lattice  QCD data, which (like in the O(3) model) show the effects of four-pion states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' It should be noted  that the current state-of-the-art for the finite-volume approach to lattice QCD scattering amplitudes  is the numerical computation of (exclusive rather than inclusive) three-pion scattering amplitudes3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Conclusions  Alternative techniques are required to compute phenomena arising from many hadronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  The spectral reconstruction of smeared spectral densities from euclidean correlator data not only  bridges the gap between finite and infinite volume, but also helps to regulate the ill-posed nature  of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The application discussed here is the computation of inclusive rates summed over  all final states produced by an external current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' In the two-dimensional O(3) model, after taking  the continuum limit, the algorithm presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 (first proposed for lattice field theory in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [10]) results in smeared spectral densities consistent with known analytic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Spectral  reconstruction algorithms based on the Backus-Gilbert approach [5, 6] enable a precise definition  of the smeared spectral density that has been computed, while the modification of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [9] further  allows the a priori specification of a desired smearing kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The simple linear ansatz on which  these approaches are based enables the direct expression for the smearing kernel given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Smeared spectral densities are useful not only for inclusive decay rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' An incomplete list  of recent applications of the Backus-Gilbert approach includes the nucleon hadronic tensor [24],  3For a review of the current status of computations of three-particle scattering amplitudes using the finite-volume  approach, see the presentation by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Romero-López and the recent reviews in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  12  The spectral reconstruction of inclusive rates  John Bulava  the determination of PDFs from Ioffe time data [25], and the photon emissivity of the quark-  gluon plasma [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' These applications do not employ the algorithmic variant enabling a priori  specification of the smearing kernel, but could perhaps benefit from it in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' This a priori  specification of the kernel enabled in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [9, 10] is also present in the Chebyshev approach of  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [7], but the stabilizing effect of the functional 𝐵[𝑔] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 11 is naively not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The work  of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [8] is a first step towards comparing the two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  The advantages of the a priori approach are leveraged in the two-dimensional O(3) model to  perform joint constrained 𝜖 → 0 extrapolations with several different kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The presence of the  narrow 𝜌(770) peak in the isovector vector spectral density in QCD discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 3 complicates  such an extrapolation and more work is required toward an implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' A similar approach has  been employed to compute inclusive decay rates in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [27, 28], and taken up by additional groups  in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [8, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Work towards computing the 𝑅-ratio was reported in this conference [31], as  well as a similar analyses of the total hadronic tau decay rate [32, 33], albeit with a wider gaussian  smearing radius than employed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The interplay between the spatial extent and the smallest  achievable smearing width requires further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Furthermore, the a priori approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [7]  led to the direct computation of the Borel transform of a current-current correlator required for  the Shifman-Vainshtein-Zakharov sum rule in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [34], possibly opening the door for additional  interaction between lattice QCD and QCD sum rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Another interesting application is pursued in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [35] in which fits to smeared spectral densities are considered as an alternative to ‘standard’  spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Additional applications could appear in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' The a priori approach in principle  enables the computation of exclusive scattering amplitudes using Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [36, 37], while the formalism  for inclusive rates was developed already in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' [3, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
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+page_content=' Meyer, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Robaina, From deep inelastic scattering to heavy-flavor  semileptonic decays: Total rates into multihadron final states from lattice QCD, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
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+page_content=' D  96 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 9 094513, [arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='08993].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  [4] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Kaczmarek and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Shu, Spectral and Transport Properties from Lattice QCD, Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  Notes Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 999 (2022) 307–345, [arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='14676].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Backus and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Gilbert, The resolving power of gross earth data, Geophysical Journal  International 16 (1968), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 2 169–205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Backus and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' Gilbert, Uniqueness in the inversion of inaccurate gross earth data,  Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and  Engineering Sciences 266 (1970), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 1173 123–192,  [http://rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
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+page_content=' D 102  (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content=' 11 114516, [arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='01253].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
+page_content='  16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf'}
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+arXiv:2301.00860v1  [hep-ph]  2 Jan 2023
+Dijet azimuthal decorrelation in e+e− annihilation
+Hana Benslamaa, Yazid Delendaa,∗, Kamel Khelifa-Kerfab
+aLaboratoire de Physique des Rayonnements et de leurs Interactions avec la Mati`ere,
+D´epartement de Physique, Facult´e des Sciences de la Mati`ere,
+Universit´e de Batna-1, Batna 05000, Algeria
+bLaboratoire de Math´ematique et Applications,
+D´epartement de Physique, Facult´e des Sciences et Technologies
+Universit´e Ahmed Zabana de Relizane, Relizane 48000, Algeria
+Abstract
+We examine non-global and clustering logarithms in the distribution of the azimuthal decorrelation between
+two jets in e+e− → dijet events, where the jets are defined with E-scheme recombination in the generalized kt
+algorithm. We calculate at one loop and to all orders the leading global single logarithms in the distribution
+of the said observable.
+We also compute at fixed order up to four loops the non-global and clustering
+logarithms, and numerically resum them to all orders in the large-Nc approximation.
+We compare our
+results at O(αs) and O(α2
+s) with those of the EVENT2 fixed-order Monte Carlo program and find agreement
+of the leading singular behavior of the azimuthal decorrelation distribution. We find that the impact of
+non-global logarithms on the resummed distribution in the anti-kt algorithm is substantial, while it is
+significantly smaller in the kt algorithm. Furthermore, the combined clustering and non-global logarithms
+in the kt algorithm have an even smaller effect on the distribution. Finally, we use the program Gnole
+to calculate the resummed distribution at NLL accuracy, thus achieving state-of-the-art accuracy for the
+resummation of this quantity.
+Keywords:
+QCD, Jets, Resummation
+1. Introduction
+The production of jets in e+e− collisions is a simple and clean environment, yet rich of physics, to test
+QCD and the Standard Model. It will be used in future colliders such as the ILC and FCC-ee in order to
+make precise measurements of QCD-related quantities, which together with detailed theoretical calculations
+will pave the way towards potential discovery of new-physics phenomena.
+At lowest order two correlated jets are produced back-to-back with a relative azimuthal angle equal to
+π. At higher orders the jets manifest a decorrelation of azimuthal angle δφ which is enhanced near the
+back-to-back limit. The quantity δφ, being sensitive to soft/collinear QCD effects, is of great interest in
+the phenomenology of perturbative and non-perturbative QCD dynamics. For instance it has been used
+to study unintegrated parton distribution functions in deep-inelastic e − p scattering (DIS) [1] and small-x
+BFKL effects [2], as well as measurements of the QCD coupling at various scales [3]. Many studies have
+been devoted to the distribution of δφ in various processes, such as dijet production in p − p [4, 5, 6] (and
+even p − pb [7]) collisions and DIS [2, 8]. Experimentally, boson-jet (in pp collisions) [9] and lepton-jet or
+photon-jet (in DIS) [10, 11] decorrelations have been measured.
+Near the back-to-back limit, the distribution of the azimuthal decorrelation is characterized by large
+logarithms preventing the convergence of the perturbative series, and thus need to be resummed to all
+∗Corresponding author.
+Email addresses: hana.benslama@univ-batna.dz (Hana Benslama), yazid.delenda@univ-batna.dz (Yazid Delenda,
+kamel.khelifakerfa@univ-relizane.dz (Kamel Khelifa-Kerfa)
+Preprint submitted to Physics Letters B
+January 4, 2023
+
+orders. Depending on the nature of the algorithm being used to define the jets, the leading logarithms in
+this distribution can be double or single logarithms. For instance, in pt-weighted recombination scheme of
+the kt [12, 13, 14], anti-kt [15] and Cambridge/Aachen algorithms [16, 17], the leading logarithms are double,
+αn
+s L2n, while in E-scheme recombination they are single, αn
+s Ln, with L = ln(1/δφ). In the former scheme,
+the jets recoil against emissions everywhere in the phase space, and in particular soft and collinear emissions
+to these jets, which leads to the double logarithms in δφ. On the other hand, in the latter (E-scheme), the
+jets recoil only against emissions that do not get clustered to them, and hence only away-from-jets emissions
+contribute to δφ, resulting in leading soft wide-angle single logarithmic contributions.
+In addition to this, the classification of the δφ observable, in E-scheme definition, falls in the “non-global”
+category, and as a consequence its distribution receives contributions from single non-global (NGLs) [18, 19]
+and/or clustering (CLs) [20, 21] logarithms. The resummation of these logarithms is not straightforward,
+and is usually performed numerically via Monte Carlo (MC) programs in the planar (large-Nc) limit.
+In this letter, we are interested in the calculation of NGLs and/or CLs for the δφ distribution both in the
+kt and anti-kt algorithms. We compute the coefficients of these logarithms as a function of the jet radius R
+up to O(α3
+s), and at O(α4
+s) in the anti-kt algorithm at small R. We use the fixed-order MC program EVENT2
+[22, 23] in order to compare the leading singular behavior of the δφ distribution with our results at O(αs)
+and O(α2
+s). We also compute the resummed NGLs and CLs at all orders in the large-Nc limit using the MC
+code of refs. [18, 19] as well as the recently-published program Gnole [24, 25] (in the anti-kt algorithm).
+The latter program is also used to compute the resummed differential δφ distribution at next-to-leading
+logarithmic (NLL) accuracy, in which we additionally control all the sub-leading logarithms αn+1
+s
+Ln in the
+exponent of the resummation, and quantify the corresponding scale uncertainties.
+This letter is organized as follows. In the next section we compute at O(αs) the leading-order distribution
+focusing on the logarithmic contribution, and compare with fixed-order MC programs at this order. In
+section 3, we present the calculation of NGLs and CLs at O(α2
+s) and show plots of the coefficients of these
+logarithms as a function of the jet radius and comment on the relative size of these coefficients. We also
+compare at this order the calculated δφ distribution with the output of the program EVENT2, thus confirming
+our results. In section 4 we extend the calculation to O(α3
+s) and (in the anti-kt algorithm and at small R)
+O(α4
+s), and point out the significantly different color structure of NGLs in kt clustering. In section 5 we
+present the all-orders resummation of the NGLs and/or CLs in the large-Nc limit up to LL accuracy for the
+kt algorithm, and NLL accuracy in the anti-kt clustering. Finally we draw our conclusions in section 6.
+2. One-loop calculation and the global form factor
+In this letter we consider the process of dijet production in e+e− annihilation at centre-of-mass energy
+√s. The jets are reconstructed with the kt [14] or anti-kt [15] algorithms, suited for e+e− annihilation, with
+merging and stopping distances dij and di defined by
+dij = min
+�
+E2p
+i , E2p
+j
+� 1 − cos θij
+1 − cos R ,
+di = E2p
+i
+,
+(1)
+where p = +1 for the kt algorithm and p = −1 for the anti-kt algorithm. Here R is the jet radius, Ei
+is the energy of the ith parton in the final state, and θij is the opening angle between partons i and j.
+The algorithm sequentially merges objects i and j whenever dij is the smallest of all merging and stopping
+distances, and if an object i has its stopping distance as the smallest then it gets admitted to the list of
+final inclusive jets. The algorithm keeps recursing until all partons are clustered into jets. In this letter,
+we assume the jet kinematics to be defined with E-scheme recombination, such that the 4-momentum of a
+merged object simply equals the vectorial sum of the momenta of its constituents.
+At the Born level, the two jets are produced back-to-back, and their relative azimuthal angle (with
+respect to the beam axis) is exactly π. The observable we are interested in is the deviation from π of this
+relative azimuthal angle, δφ, when soft gluons are emitted at higher orders. It is straightforward to obtain
+the following expression for δφ in terms of the transverse momenta of the emitted gluons κti and their
+2
+
+azimuthal angles ϕi, with respect to the beam axis
+δφ =
+������
+�
+i/∈jets
+κti
+pt
+sin ϕi
+������
+,
+(2)
+where pt is the jet transverse momentum. The (algebraic) sum is over all emitted gluons that are not clustered
+to any of the two measured (leading) jets. This definition is valid only at single leading logarithmic (LL)
+accuracy, and we shall give the proper definition, valid at NLL accuracy, in section 5. Furthermore, the
+expression of δφ in eq. (2) only applies in E-scheme recombination. Alternative jet recombination schemes
+exist for which the jet kinematics take a different form, e.g. the pt-weighted scheme, and the resummation
+takes an entirely different structure [8].
+At one loop the cumulative cross-section for events with azimuthal decorrelation δϕ less than some ∆,
+normalized to the Born cross-section, reads
+Σ1(∆) = −2 CF
+� αs(kt)
+π
+dkt
+kt
+d cos θ
+sin2 θ
+dφ
+2π ωk
+q¯q Θout(k) Θ (κt| sin ϕ|/pt − ∆) ,
+(3)
+where θ, φ and kt are the polar angle, azimuthal angle and transverse momentum of the emitted soft gluon
+k, with respect to the jet (thrust) axis (the back-to-back outgoing jets are aligned along the z axis), CF
+is the color factor associated with the emission of the gluon off the hard q¯q dipole, and αs is the strong
+coupling with argument kt. The invariant antenna function ωk
+q¯q is given by
+ωk
+q¯q = k2
+t
+2
+pq · p¯q
+(pq · k)(p¯q · k) = 1 ,
+(4)
+with pi denoting the momentum of particle i
+pq =
+√s
+2 (1, 0, 0, 1) ,
+(5a)
+p¯q =
+√s
+2 (1, 0, 0, −1) ,
+(5b)
+k = E (1, cos φ sin θ, sin φ sin θ, cos θ) ,
+(5c)
+where kt = E sin θ. The constraint Θout(k) restricts the emitted gluon to be outside the jets and forbids it
+from being clustered to any of them in order to induce a non-zero azimuthal decorrelation. It depends on
+the jet radius R and is given, in the generalized algorithm, by
+Θout(k) = Θ (cos R − | cos θ|) .
+(6)
+Since the soft emission is restricted to be outside both jets then there are no collinear logarithms as-
+sociated with this observable. This means that the leading logarithms are single, which allows us at LL
+accuracy to simply change Θ(κt| sin ϕ|/pt − ∆) → Θ(2 kt/√s − ∆), since any factor multiplying kt will only
+induce sub-leading logarithms. 1 We can then perform the integration over kt using the one-loop running
+of the coupling (which formally enters the distribution at higher orders), and write the result in terms of
+the evolution parameter t defined by
+t(∆) ≡
+� √s/2 αs(kt)
+π
+dkt
+kt
+Θ
+�
+2 kt/√s − ∆
+�
+= −
+1
+2πβ0
+ln(1 − 2λ) ,
+(7)
+where λ = αs(√s/2)β0 ln(1/∆) and β0 is the one-loop coefficient of the QCD beta function. The angular
+integration is straightforward and we obtain
+Σ1(∆) = −2 CF t(∆)
+� 1
+−1
+dc
+1 − c2 Θ (cos R − |c|) = −2 CF t(∆) ln 1 + cos R
+1 − cos R ,
+(8)
+1Notice that κt and ϕ are different from kt and φ.
+3
+
+with c standing for cos θ. Note that since only emissions in the inter-jet (gap) region are integrated over,
+this result may be cast in terms of the rapidity-gap width ∆η
+∆η ≡ ln 1 + cos R
+1 − cos R .
+(9)
+The all-orders resummed global form factor is simply the exponential of the one-loop distribution. That is
+Σglobal(∆) = exp [−2 CF t(∆) ∆η] .
+(10)
+An identical expression was also arrived at in ref. [26] for jet shapes in e+e− annihilation.
+The leading-order result can be verified by comparing it with the output of the MC program EVENT2 at
+O(αs) [22, 23]. Specifically we compare the differential distribution
+2π
+αs
+dΣ1
+dL = 4 CF ln 1 + cos R
+1 − cos R ,
+(11)
+where L = ln ∆, with the same MC distribution, for the chosen value of R = 0.5. We show in figure 1 a
+plot of the difference between the MC distribution and the expansion of the resummation at O(αs), where,
+as expected, this difference tends to zero in the logarithmically-enhanced region.
+Figure 1: The difference between the leading-order EVENT2 differential distribution 2π/αs dΣ1/dL and the resummed distribu-
+tion expanded at O(αs). The singular behavior of the MC distribution is exactly cancelled by the expanded result.
+3. Two-loops calculation: NGLs and CLs
+When employing the kt or anti-kt clustering algorithms with E-scheme recombination, the resummation
+of the azimuthal decorrelation distribution requires the treatment of NGLs and/or CLs. The corresponding
+cumulative distribution at O(α2
+s) can then be split into three contributions
+Σ2(∆) = 1
+2! [Σ1(∆)]2 + ΣNG
+2
+(∆) + ΣCL
+2 (∆) ,
+(12)
+with ΣCL
+2 (∆) = 0 for anti-kt clustering. Let us first discuss the NGLs contribution ΣNG
+2
+(∆) in both algo-
+rithms, and then compute the CLs contribution ΣCL
+2 (∆) for kt clustering.
+3.1. Calculation of NGLs
+The origin of NGLs at two loops is the emission of a soft gluon k1 inside any of the two outgoing jets
+which itself emits a softer gluon k2 outside the jets without being clustered back to them.
+While this
+configuration results in a non-zero δφ, its virtual correction (specifically when k2 is virtual) gives δφ = 0,
+4
+
+and thus we have a real-virtual mis-cancellation of the soft singularities. We express the contribution of the
+uncancelled virtual correction to the integrated azimuthal decorrelation distribution as follows
+ΣNG
+2
+(∆) = S2(R) t2
+2! ,
+(13a)
+S2(R) = −2 CF CA
+�
+dc1
+1 − c2
+1
+dφ1
+2π
+dc2
+1 − c2
+2
+dφ2
+2π A12
+q¯q ΞNG
+2
+(R) ,
+(13b)
+where CA is the color factor associated with the non-Abelian emission of gluon k2 off k1. The irreducible
+two-loops antenna function A12
+q¯q is given by [27]
+A12
+q¯q = ω1
+q¯q
+�
+ω2
+q1 + ω2
+1¯q − ω2
+q¯q
+�
+=
+1 − c1c2
+1 − c1c2 − s1s2 cos(φ1 − φ2) − 1 ,
+(14)
+with si ≡ sin θi. The function ΞNG
+2
+restricts the angular phase-space of integration and is given, in the
+anti-kt and kt algorithms respectively, by
+ΞNG, akt
+2
+= Θin(k1)Θout(k2) ,
+(15a)
+ΞNG, kt
+2
+= Θin(k1)Θout(k2)Θ(d12 − d2) .
+(15b)
+The step function Θ(d12 − d2) forbids gluon k2 from being clustered back to the jet in the kt algorithm. It
+is given by Θ(cos R − cos θ12), with cos θ12 = c1c2 + s1s2 cos(φ1 − φ2).
+In the anti-kt algorithm the integration is simple and its result can be expressed in the same form as
+that of the rapidity-gap NGLs coefficient found in ref. [19]
+Sakt
+2
+(R) = −CF CA
+�π2
+6 + 2∆η2 − 2∆η ln
+�
+e2∆η − 1
+�
+− Li2
+�
+e−2∆η�
+− Li2
+�
+1 − e2∆η��
+= −CF CA
+�π2
+3 − R4
+8 − R6
+24 − 29 R8
+2560 + O
+�
+R10��
+,
+(16)
+where ∆η(R) is defined in eq. (9). The above formula shows that in the limit R → 0 the two-loops NGLs
+coefficient does not vanish, but rather reaches its maximum value. This feature was observed in ref. [19]
+and was ascribed to the fact that NGLs originate from the edge of jets, since this is the phase-space region
+where gluons k1 and k2 are collinear and thus the amplitude squared (14) is most singular.
+In the kt algorithm, and at small values of R (using small angles), we can write the NGLs coefficient as
+Skt
+2 (R ∼ 0) = −8 CF CA
+� 1
+0
+dθ1
+� ∞
+1
+dθ2
+� 2π
+0
+dφ
+2π
+1
+(θ2
+1 + θ2
+2) sec φ − 2 θ1 θ2
+Θ
+�
+θ2
+1 + θ2
+2 − 2 θ1 θ2 cos φ − 1
+�
+= −2π2
+27 CF CA ,
+(17)
+where we made the following changes of variables: φ = φ1 − φ2 and θi → R θi. Away from the small-R
+limit one may perform the integration numerically to obtain the full-R result for the two-loops coefficient of
+NGLs. We present the results in the following subsection together with the CLs coefficient.
+3.2. CLs with kt clustering
+To compute the CLs we consider the Abelian primary emission of two strongly-ordered gluons directly
+off the hard q¯q dipole, whereby the harder gluon k1 is inside one of the two jets and the softer k2 is outside
+both of them, with the constraint d12 < d2, such that gluon k2 gets clustered into the jet by gluon k1, which
+leads to δφ = 0. However, when k1 is virtual then gluon k2 remains in the gap causing the hard jets to
+decorrelate. In this case we obtain a large single logarithmic contribution to the δφ distribution given by
+ΣCL
+2 (∆) = Ckt
+2 (R) t2
+2! ,
+(18a)
+Ckt
+2 (R) = 4 C2
+F
+�
+dc1
+1 − c2
+1
+dφ1
+2π
+dc2
+1 − c2
+2
+dφ2
+2π w1
+q¯q w2
+q¯q Θ(|c1| − cos R) Θ(cos R − |c2|) Θ(cos θ12 − cos R) .
+(18b)
+5
+
+Note again that Cakt
+2
+(R) = 0, as there are no CLs for anti-kt clustering.
+First, let us consider the small-R limit of this integral. In this case we may write
+Ckt
+2 (R ∼ 0) = 8 C2
+F
+� 1
+0
+dθ1
+θ1
+� ∞
+1
+dθ2
+θ2
+� 2π
+0
+dφ
+2π Θ
+�
+−θ2
+1 − θ2
+2 + 2 θ1 θ2 cos φ + 1
+�
+= 5π2
+27 C2
+F .
+(19)
+Away from this small-R limit we perform the integration numerically. We show in figure 2 a plot of the
+coefficients of NGLs and CLs at O(α2
+s) as a function of the jet radius R. Also shown is the combined
+coefficient of CLs and NGLs in the kt algorithm; Fkt
+2 = Skt
+2 + Ckt
+2 .
+Figure 2: CLs and NGLs coefficients at two loops with kt and anti-kt clustering.
+Notice that NGLs coefficient in the kt clustering algorithm is significantly smaller than that in the anti-
+kt. The CLs coefficient is positive and quite small, with the advantage that it cancels the NGLs coefficient,
+particularly at small values of R. Note that the overall coefficient F2 is less than 1 for values of R smaller
+than about 0.5. For small R values, the anti-kt NGLs coefficient computed here is identical to that found
+for the single hemisphere mass [28] and jet mass with a jet veto [26].
+3.3. Comparison to EVENT2
+We compare our two-loops results with the exact MC distribution at O(α2
+s) obtained with the EVENT2
+program. The latter splits the distribution at NLO into three color contributions; O(C2
+F), O(CF CA) and
+O(CF Tf nf), with Tf = 1/2 being the normalization constant of the generators of the SU(3) group in the
+fundamental representation and nf = 5 is the number of active quark flavors. The expansion of the resummed
+distribution at this order, including running coupling effects, is written, after differentiating with respect to
+L, as
+�2π
+αs
+�2 dΣ2
+dL =
+�
+C2
+F 4
+�
+4 ∆η2 + C2
+�
+− CF CA
+4
+3 (3 S2 + 11∆η) + CF Tf nf
+16
+3 ∆η
+�
+L + O(1) .
+(20)
+At this order this differential distribution is linear in L, but does not capture the O(1) constant, which, in
+the cumulative integrated distribution, is an NLL O(α2
+s L) term. We plot the difference between the MC
+distribution and the expansion (20) for each color contribution separately. All curves should tend towards a
+constant when L grows large and negative. The results are shown in figure 3. Here the O(C2
+F) part contains
+the CLs coefficient in kt clustering, while the O(CF CA) term contains the NGLs coefficient both in kt and
+anti-kt algorithms. The O(CF Tf nf) contribution is algorithm independent at LL accuracy, but the NLL
+O(1) constant does depend on the jet algorithm.
+6
+
+Figure 3: The difference between the NLO EVENT2 differential distribution (2π/αs)2 dΣ2/dL and the resummed distribution
+expanded at O(α2
+s). The leading logarithmic behavior of the MC distribution O(α2
+sL) is cancelled, leaving a constant behavior
+at large values of L.
+4. NGLs and CLs at three loops
+4.1. NGLs at three loops
+At O(α3
+s), the cumulative distribution can be written as follows
+Σ3(∆) = 1
+3! [Σ1(∆)]3 + Σ1(∆) × ΣNG
+2
+(∆) + Σ1(∆) × ΣCL
+2 (∆) + ΣNG
+3
+(∆) + ΣCL
+3 (∆) .
+(21)
+Focusing first on the pure NGLs contribution ΣNG
+3
+(i.e., excluding the cross-talk between the one-loop global
+and two-loops non-global logarithms) in the anti-kt algorithm, we may write it in the form
+ΣNG,akt
+3
+(∆) = Sakt
+3
+(R) t3
+3! ,
+(22a)
+Sakt
+3
+(R) = 2 CF C2
+A
+� � 3
+�
+i=1
+dci
+1 − c2
+i
+dφi
+2π
+�
+�
+A12
+q¯q ¯
+A13
+q¯q Θin(k1)Θout(k2)Θout(k3) − B123
+q¯q Θin(k1)Θin(k2)Θout(k3)
+�
+,
+(22b)
+where we define ¯
+A13
+q¯q = A13
+q¯q/ω1
+q¯q, and the 3-loops irreducible cascade antenna function is given by
+B123
+q¯q = ω1
+q¯q
+�
+A23
+q1 + A23
+1¯q − A23
+q¯q
+�
+.
+(23)
+It is worth mentioning that, for the cascade term, there is a non-negligible contribution from the configuration
+in which gluon k1 is in one jet emitting gluon k2 in the other jet which itself emits the softest gluon k3 in
+the gap between the two jets. At small values of R the integration is quite simple and yields the result
+Sakt
+3
+(R ∼ 0) = CF C2
+A 2 ζ3 .
+(24)
+Notice that this result is twice that found for the hemisphere mass observable [29]. Away from the small-R
+limit the integration can be performed numerically and we shall present the results in the next subsection.
+The pure NGLs contribution in the kt algorithm is given by an identical form to that of the anti-kt (22a)
+ΣNG,kt
+3
+(∆) = Skt
+3 (R) t3
+3! .
+(25)
+We perform for the first time in the literature a calculation of NGLs in the kt algorithm beyond two loops.
+Due to non-linearity of kt clustering we shall find a class of NGLs that have a non-standard color factor,
+namely C2
+F CA at O(α3
+s).
+Such terms usually (in anti-kt clustering) only arise in the cross-talk of the
+expansion of the primary-emission global form factor (10) at O(αs) together with NGLs at O(α2
+s), i.e., the
+term Σ1(∆) × ΣNG
+2
+(∆) in eq. (21). However, in the kt algorithm, we find them as “pure” irreducible NGLs.
+That is, they are part of the term ΣNG
+3
+(∆) in eq. (21).
+7
+
+To proceed, we consider three types of emissions at O(α3
+s), as shown in figure 4: (a) one primary + two
+correlated emissions, (b) ladder emissions, and (c) cascade emissions. In each case we consider all possible
+virtual-correction Feynman diagrams as well as angular configurations of the emitted gluons that affect the
+clustering procedure, and look for a mis-match between the soft divergences of these emissions.
+Figure 4: The three types of emissions to consider for NGLs calculation at O(α3
+s): (a) one primary + two correlated emissions,
+(b) ladder emissions, and (c) cascade emissions.
+Starting first with type (a) contributions, there are three possible permutations of the gluons.
+2
+For the permutation in which k3 is emitted in correlation with k2, and which has a squared amplitude
+4 C2
+F CA ω1
+q¯q A23
+q¯q, we find that the angular phase space of integration that yields a logarithmic contribution
+is given by
+ΞNG,kt
+31
+(R) = Θin(k1)Θin(k2)Θout(k3)Θ(d3 − d13)Θ(d23 − d3)+
++ Θout(k1)Θin(k2)Θout(k3)Θ(d23 − d3)+
+− Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)Θ(d2 − d12) .
+(26)
+To see how one obtains this result let us give one example of angular configurations that result in a logarithmic
+contribution. There are four Feynman diagrams in this case, shown in figure 5. Consider the situation when
+particles k3 and k2 are outside the jet regions, d3j > d3 and d2j > d2, while particle k1 is inside, d1j < d1. In
+this scenario, both diagrams in which k1 is virtual (i.e., diagrams (3) and (4) in figure 5) yield δφ ̸= 0, since
+in both diagrams particle k2 is real and remains in the gap region after applying the clustering. However,
+these two diagrams contribute equally and with opposite signs, so they cancel each other. For the remaining
+two diagrams, (1) and (2), we have a mismatch when particle k2 gets pulled inside the jet by the real particle
+k1 while k3 remains in the gap. This happens when d12 is smaller than d2, and both d13 and d23 are greater
+than d3. While in diagram (1) we have a real unclustered gluon k3 in the gap region (i.e., it forms a jet
+on its own) giving δφ ̸= 0, in diagram (2) the gap is empty and the hard jets are exactly back-to-back
+with δφ = 0. The virtual-correction diagram (2) contributes fully to the cumulative distribution while the
+real-emission diagram (1) cancels this contribution only up to δφ = ∆, leaving uncancelled virtual-correction
+contributions with a negative sign. This is the last term in eq. (26).
+Figure 5: Feynman diagrams corresponding to the squared amplitude 4 C2
+F CA ω1
+q¯q A23
+q¯q.
+2Note that for all permutations the transverse momenta of the three gluons are strongly ordered as follows: kt1 ≫ kt2 ≫ kt3.
+8
+
+Similarly, we obtain for the second and third gluon permutations of type (a) diagrams, with squared
+amplitudes 4 C2
+F CA ω2
+q¯q A13
+q¯q and 4 C2
+F CA ω3
+q¯q A12
+q¯q, respectively, as well as type (b) (ladder-emission) contri-
+butions, with squared amplitude 2 CF C2
+A ¯
+A12
+q¯q A13
+q¯q, the same phase space function. It is given by
+ΞNG,kt
+32
+(R) = Θin(k1)Θin(k2)Θout(k3)Θ(d13 − d3)Θ(d3 − d23)+
++ Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d3 − d23)+
++ Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)Θ(d12 − d2) .
+(27)
+Finally, for type (c) (cascade emission) contributions, corresponding to the squared amplitude 2 CF C2
+A B123
+q¯q ,
+the phase space function reads
+ΞNG,kt
+33
+(R) = − Θin(k1)Θin(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)−
+− Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)Θ(d2 − d12) .
+(28)
+Before performing the integration, we subtract off the part of the phase space that produces the interfer-
+ence term between the one-loop global primary logarithm and the two-loops NGLs, which only comes from
+type (a) emissions. For the first permutation of gluons, this part of phase space is identified by the second
+term in eq. (26), Θout(k1)Θin(k2)Θout(k3)Θ(d23 − d3), where gluon k1 reproduces the one-loop global term
+(8) and the other correlated gluons, k2 and k3, give the two-loops NGLs (eq. (13b) with phase space (15b)).
+However, for the other two gluon permutations of type (a) emissions, the phase space (27) does not simply
+contain such interference terms. Strictly speaking, this means that NGLs do not cleanly factorize from the
+global form factor in the kt algorithm. Nevertheless, we can manually add and subtract the interference
+terms and write the total distribution in the factorizable form (21).
+We can then write the “pure” NGLs coefficient Skt
+3 , given in eq. (25), in the following form
+Skt
+3 (R) = S(a)
+3 (R) + S(b)+(c)
+3
+(R) ,
+(29)
+where we split the result according to the color factor, such that for type (a) emissions we have
+S(a)
+3 (R) = 4 C2
+F CA
+� � 3
+�
+i=1
+dci
+1 − c2
+i
+dφi
+2π
+� �
+ω1
+q¯q A23
+q¯q �Ξ31(R) + ω2
+q¯q A13
+q¯q �Ξ32(R) + ω3
+q¯q A12
+q¯q �Ξ33(R)
+�
+,
+(30)
+with modified phase space that subtracts away the interference terms
+�Ξ31(R) = ΞNG,kt
+31
+(R) − Θout(k1)Θin(k2)Θout(k3)Θ(d23 − d3) ,
+(31a)
+�Ξ32(R) = ΞNG,kt
+32
+(R) − Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3) ,
+(31b)
+�Ξ33(R) = ΞNG,kt
+32
+(R) − Θin(k1)Θout(k2)Θout(k3)Θ(d12 − d2) ,
+(31c)
+and for type (b) and (c) emissions
+S(b)+(c)
+3
+(R) = 2 CF C2
+A
+� � 3
+�
+i=1
+dci
+1 − c2
+i
+dφi
+2π
+� �
+A12
+q¯q ¯
+A13
+q¯q ΞNG,kt
+32
+(R) + B123
+q¯q ΞNG,kt
+33
+(R)
+�
+.
+(32)
+We are now in a position to perform the integrations numerically as a function of the jet radius R. We
+show the results in the next subsection, in which we also compute the clustering logarithmic contribution
+at O(α3
+s).
+4.2. CLs with kt clustering at three loops
+Following the same steps as for NGLs calculation, the phase space clustering function at O(α3
+s) for the
+CLs contribution, which results from the mismatch of soft singularities between real and virtual emissions
+9
+
+of three primary soft gluons, with a squared amplitude 8 C3
+F w1
+q¯q w2
+q¯q w3
+q¯q, is given by
+ΞCL,kt
+3
+(R) = − Θin(k1)Θin(k2)Θout(k3)Θ(d3 − d13)Θ(d3 − d23)−
+− Θout(k1)Θin(k2)Θout(k3)Θ(d3 − d23)−
+− Θin(k1)Θout(k2)Θout(k3)Θ(d3 − d13)−
+− Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)Θ(d2 − d12) .
+(33)
+Note that the above phase-space clustering function is similar (but not exactly identical) to that found for
+the jet mass observable [30]. Extracting the interference terms between the CLs at two loops and the global
+logarithm at one loop, i.e., the term Σ1(∆) × ΣCL
+2 (∆) in eq. (21), we reduce the above phase space function
+to that of the “pure” CLs contribution as
+�ΞCL,kt
+3
+(R) = − Θin(k1)Θin(k2)Θout(k3)Θ(d3 − d13)Θ(d3 − d23)+
++ Θin(k1)Θout(k2)Θout(k3)Θ(d2 − d12)[1 − Θ(d13 − d3)Θ(d23 − d3)] .
+(34)
+Then, the clustering logarithmic contribution to the cumulative cross-section is given by
+ΣCL
+3 (∆) = Ckt
+3 (R) t3
+3! ,
+(35a)
+Ckt
+3 (R) = 8 C3
+F
+� � 3
+�
+i=1
+dci
+1 − c2
+i
+dφi
+2π
+�
+w1
+q¯q w2
+q¯q w3
+q¯q �ΞCL,kt
+3
+(R) .
+(35b)
+We show in figure 6 a plot of the coefficients of NGLs and CLs in the kt and anti-kt algorithms as a
+function of the jet radius R. Shown also is the combined coefficient Fkt
+3 = Skt
+3 + Ckt
+3 for the kt algorithm.
+Figure 6: CLs and NGLs coefficients at three loops with kt and anti-kt clustering.
+As in the previous section, we notice that the NGLs coefficient in the anti-kt algorithm at this order is
+also quite large. Clearly, the application of the kt clustering has reduced the significance of NGLs by almost
+a factor of 30 for values less than R ∼ 0.6. The CLs coefficient is also small such that the overall coefficient
+F3 is smaller in magnitude than the Sakt
+3
+by about a factor of 3 for most values of R.
+5. Four loops and beyond
+5.1. Four-loops NGLs with anti-kt at small R
+The calculation of NGLs with anti-kt clustering proceeds in a similar manner at fourth order, and can
+easily be deduced from previous calculations of NGLs in the literature. In fact, the phase space of integration
+10
+
+is similar to that of the hemisphere mass distribution reported in ref. [29], and thus the cumulative cross-
+section, at this order, may be cast in the following way
+Σakt
+4
+(∆) = 1
+4! [Σ1(∆)]4+ 1
+2! [Σ1(∆)]2 ΣNG,akt
+2
+(∆)+Σ1(∆) ΣNG,akt
+3
+(∆)+ 1
+2!
+�
+ΣNG,akt
+2
+(∆)
+�2
++ΣNG,akt
+4
+(∆) , (36)
+with pure NGLs contribution given by
+ΣNG,akt
+4
+(∆) = Sakt
+4
+(R) t4
+4! ,
+(37a)
+Sakt
+4
+(R) = 2 CF C3
+A
+� � 4
+�
+i=1
+dci
+1 − c2
+i
+dφi
+2π
+� �
+− A12
+q¯q ¯
+A13
+q¯q ¯
+A14
+q¯q Θin(k1)Θout(k2)Θout(k3)Θout(k4)+
++ 3 ¯
+A12
+q¯q B134
+q¯q Θin(k1)Θout(k2)Θin(k3)Θout(k4) + U1234
+q¯q
+Θin(k1)Θin(k2)Θout(k3)Θout(k4)−
+− C1234
+q¯q
+Θin(k1)Θin(k2)Θin(k3)Θout(k4)
+�
+−
+− 2 CF C3
+A
+�
+1 − 2 CF
+CA
+� � � 4
+�
+i=1
+dci
+1 − c2
+i
+dφi
+2π
+�
+A1234
+q¯q
+Θin(k1)Θin(k2)Θout(k3)Θout(k4) ,
+(37b)
+where the irreducible antenna functions read [27]
+C1234
+q¯q
+= w1
+q¯q
+�
+B234
+q1 + B234
+1¯q − B234
+q¯q
+�
+,
+(38a)
+U1234
+q¯q
+= w1
+q¯q
+�
+A23
+q1 ¯
+A24
+q1 + A23
+1¯q ¯
+A24
+1¯q − A23
+q¯q ¯
+A24
+q¯q
+�
+,
+(38b)
+A1234
+q¯q
+= ω1
+q¯q ω2
+q1
+� ¯
+A23
+q¯q − ¯
+A23
+q1
+� � ¯
+A24
+q1 − ¯
+A24
+1¯q
+�
++ ω1
+q¯q ω2
+1¯q
+� ¯
+A23
+q¯q − ¯
+A23
+1¯q
+� � ¯
+A24
+1¯q − ¯
+A24
+q1
+�
+−
+− ω1
+q¯q ω2
+q¯q
+� ¯
+A23
+q1 − ¯
+A23
+q¯q
+� � ¯
+A24
+1¯q − ¯
+A24
+q¯q
+�
++ k3 ↔ k4 .
+(38c)
+At small values of R the integration has been performed in ref. [29], and the corresponding result is
+given by
+Sakt
+4
+(R ∼ 0) = −CF C3
+A ζ4
+�29
+4 −
+�
+1 − 2 CF
+CA
+��
+.
+(39)
+5.2. LL resummation
+In this section we present numerical results for the resummation of NGLs and CLs in the large-Nc
+approximation. For this we use the MC code first developed in refs. [18, 19] with modification of kt clustering
+in terms of distances (1). The results are shown in figure 7 for the resummed cumulative distribution Σ as
+a function of the evolution parameter t (7), for the particular value of the jet radius R = 0.5. Shown in the
+figure are: results for the (primary) global distribution (black curve), obtained by running the MC program
+using anti-kt clustering and allowing only for primary emissions, and the full anti-kt distribution (solid pink
+curve) which additionally includes NGLs at large Nc. We observe the very large impact of NGLs on the
+distribution, reducing the global form factor by a factor of 10 for t = 0.3.
+We also show in the same figure the primary-emission distribution obtained by running the above-
+mentioned program with kt clustering (solid green curve), which includes the global form factor together
+with the resummed CLs, as well as the overall distribution in kt clustering which includes in addition the
+resummed NGLs (solid purple curve). We observe that the distribution in kt clustering is affected by both
+CLs and NGLs, but the combined impact of the two is noticeably small. This means that CLs tend to cancel
+NGLs in kt clustering, as noted with the fixed-order calculations performed above.
+Moreover, we show in figure 7 the analytical results for the resummed distribution in each case (dashed
+lines), which are estimated from the observed pattern of exponentiation
+Σ(∆) = exp [Σ1(∆)] exp
+� ∞
+�
+i=2
+ΣNG
+i
+(∆)
+�
+exp
+� ∞
+�
+i=2
+ΣCL
+i
+(∆)
+�
+.
+(40)
+11
+
+Figure 7: Numerically resummed NGLs and CLs at large Nc.
+It is clear that the truncation of the series at i = 3 in the exponent, though quite close to the numerical
+result, does not give an accurate fit of the MC distribution. This means that higher-order contributions
+cannot be ignored.
+5.3. NLL resummation with anti-kt
+We present here a resummed result for the azimuthal decorrelation distribution with anti-kt clustering
+at NLL accuracy in the large-Nc limit, obtained using the recently-published program Gnole [24, 25]. The
+distribution can be obtained from this program by defining our observable within the code, using the full
+definition rather than the LL approximation (2). Explicitly written the former reads
+δφ =
+������
+sin−1 �
+i/∈jets
+k⊥i
+ptr
+������
+,
+(41)
+where k⊥i is the component of the transverse momentum of the emission i perpendicular to the thrust (or
+leading jet) axis, and ptr is the recoiling jet’s transverse momentum. For simplicity we take the jets to be
+at threshold, i.e., transverse to the beam direction.
+Figure 8: NLL numerical resummation of the δφ distribution at large Nc.
+In figure 8 we present the results for the NLL resummed distribution together with the LL one, both
+obtained with Gnole. We note here that the LL distribution is obtained using the definition of the observable
+(41), while that obtained with the MC code of refs. [18, 19] (figure 7) is essentially equivalent to the transverse
+12
+
+energy distribution (in other words, definition 2 without the sin φ part). This does in fact numerically affect
+the distribution even at LL accuracy. 3
+We observe that the NLL corrections to the distribution are quite important, just like the transverse
+energy distribution shown in ref. [25]. Furthermore, the scale-uncertainty band, obtained by varying the
+renormalization and resummation scales by factors of 2 and 1/2 around their central values (µR = √s
+and Q0 = √s/2, respectively, with √s = MZ), gets significantly reduced in the NLL curve. It is worth
+mentioning that one still needs to account for the matching in order to fully control all sub-leading NLL
+logarithms at the tail of the distribution.
+We note that the actual distribution does not possess a Sudakov peak at low values of δφ, instead it tends
+towards a constant value. This is explained by the fact that the very low values of δφ are not suppressed
+by soft emissions, but are rather enhanced by vectorial cancellation of semi-hard emissions.
+6. Conclusions
+In this letter we have presented both fixed-order and all-orders results for the distribution of the azimuthal
+decorrelation observable for the specific QCD process e+e− annihilation into two jets. The said observable
+is of non-global nature and hence its distribution contains, in addition to the usual global logarithms, non-
+global and/or clustering logarithms. These logarithms are jet-algorithm dependent, and start to appear at
+two loops and are quite delicate to compute.
+For NGLs, we have calculated the full-R expression analytically at two loops and numerically at three
+loops for the anti-kt jet algorithm. At four loops, they have been determined only for small-R values for
+the same said jet algorithm. For the kt clustering algorithm, calculations have been performed up to three
+loops and only numerically.
+Moreover, CLs, which are absent in the anti-kt algorithm, have been computed numerically up to three
+loops. The usual reduction in the significance of NGLs due to kt clustering has been observed confirming
+previous findings. Furthermore, the combined impact of NGLs and CLs on the distribution is observed to
+be very small which has important phenomenological implications in terms of accuracy of the resummed
+distribution. Our results at two loops have been checked against the output of the MC program EVENT2.
+Numerical estimates of the all-orders distribution has been presented both at LL and NLL accuracy.
+The achievement of the latter accuracy has been made possible by the recently-published Gnole code. NLL
+resummation exhibits a better distribution both in terms of accuracy and scale uncertainty. It is thus worth
+investigating the impact of NLL effects with kt clustering. Of similar worthiness is computing NGLs and
+CLs at four loops with full-R dependence.
+Acknowledgements
+This work is supported by PRFU research project B00L02UN050120230003. The authors wish to thank
+the Algerian Ministry of Higher Education and Scientific Research and DGRSDT for financial support.
+The numerical calculations presented here have been performed in the HPC cluster at the University of
+Batna 2 (UB2-HPC).
+We thank Andrea Banfi for clarifications about using Gnole program.
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+14
+
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+page_content=' Yazid Delendaa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Kamel Khelifa-Kerfab  aLaboratoire de Physique des Rayonnements et de leurs Interactions avec la Mati`ere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  D´epartement de Physique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Facult´e des Sciences de la Mati`ere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Universit´e de Batna-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Batna 05000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Algeria  bLaboratoire de Math´ematique et Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  D´epartement de Physique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Facult´e des Sciences et Technologies  Universit´e Ahmed Zabana de Relizane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Relizane 48000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Algeria  Abstract  We examine non-global and clustering logarithms in the distribution of the azimuthal decorrelation between  two jets in e+e− → dijet events,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' where the jets are defined with E-scheme recombination in the generalized kt  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We calculate at one loop and to all orders the leading global single logarithms in the distribution  of the said observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  We also compute at fixed order up to four loops the non-global and clustering  logarithms, and numerically resum them to all orders in the large-Nc approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  We compare our  results at O(αs) and O(α2  s) with those of the EVENT2 fixed-order Monte Carlo program and find agreement  of the leading singular behavior of the azimuthal decorrelation distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We find that the impact of  non-global logarithms on the resummed distribution in the anti-kt algorithm is substantial, while it is  significantly smaller in the kt algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Furthermore, the combined clustering and non-global logarithms  in the kt algorithm have an even smaller effect on the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Finally, we use the program Gnole  to calculate the resummed distribution at NLL accuracy, thus achieving state-of-the-art accuracy for the  resummation of this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Keywords:  QCD, Jets, Resummation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Introduction  The production of jets in e+e− collisions is a simple and clean environment, yet rich of physics, to test  QCD and the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' It will be used in future colliders such as the ILC and FCC-ee in order to  make precise measurements of QCD-related quantities, which together with detailed theoretical calculations  will pave the way towards potential discovery of new-physics phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  At lowest order two correlated jets are produced back-to-back with a relative azimuthal angle equal to  π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' At higher orders the jets manifest a decorrelation of azimuthal angle δφ which is enhanced near the  back-to-back limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The quantity δφ, being sensitive to soft/collinear QCD effects, is of great interest in  the phenomenology of perturbative and non-perturbative QCD dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' For instance it has been used  to study unintegrated parton distribution functions in deep-inelastic e − p scattering (DIS) [1] and small-x  BFKL effects [2], as well as measurements of the QCD coupling at various scales [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Many studies have  been devoted to the distribution of δφ in various processes, such as dijet production in p − p [4, 5, 6] (and  even p − pb [7]) collisions and DIS [2, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Experimentally, boson-jet (in pp collisions) [9] and lepton-jet or  photon-jet (in DIS) [10, 11] decorrelations have been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Near the back-to-back limit, the distribution of the azimuthal decorrelation is characterized by large  logarithms preventing the convergence of the perturbative series, and thus need to be resummed to all  ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Email addresses: hana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='benslama@univ-batna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='dz (Hana Benslama), yazid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='delenda@univ-batna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='dz (Yazid Delenda,  kamel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='khelifakerfa@univ-relizane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='dz (Kamel Khelifa-Kerfa)  Preprint submitted to Physics Letters B  January 4, 2023  orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Depending on the nature of the algorithm being used to define the jets, the leading logarithms in  this distribution can be double or single logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' For instance, in pt-weighted recombination scheme of  the kt [12, 13, 14], anti-kt [15] and Cambridge/Aachen algorithms [16, 17], the leading logarithms are double,  αn  s L2n, while in E-scheme recombination they are single, αn  s Ln, with L = ln(1/δφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' In the former scheme,  the jets recoil against emissions everywhere in the phase space, and in particular soft and collinear emissions  to these jets, which leads to the double logarithms in δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' On the other hand, in the latter (E-scheme), the  jets recoil only against emissions that do not get clustered to them, and hence only away-from-jets emissions  contribute to δφ, resulting in leading soft wide-angle single logarithmic contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  In addition to this, the classification of the δφ observable, in E-scheme definition, falls in the “non-global”  category, and as a consequence its distribution receives contributions from single non-global (NGLs) [18, 19]  and/or clustering (CLs) [20, 21] logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The resummation of these logarithms is not straightforward,  and is usually performed numerically via Monte Carlo (MC) programs in the planar (large-Nc) limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  In this letter, we are interested in the calculation of NGLs and/or CLs for the δφ distribution both in the  kt and anti-kt algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We compute the coefficients of these logarithms as a function of the jet radius R  up to O(α3  s), and at O(α4  s) in the anti-kt algorithm at small R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We use the fixed-order MC program EVENT2  [22, 23] in order to compare the leading singular behavior of the δφ distribution with our results at O(αs)  and O(α2  s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We also compute the resummed NGLs and CLs at all orders in the large-Nc limit using the MC  code of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [18, 19] as well as the recently-published program Gnole [24, 25] (in the anti-kt algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  The latter program is also used to compute the resummed differential δφ distribution at next-to-leading  logarithmic (NLL) accuracy, in which we additionally control all the sub-leading logarithms αn+1  s  Ln in the  exponent of the resummation, and quantify the corresponding scale uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  This letter is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' In the next section we compute at O(αs) the leading-order distribution  focusing on the logarithmic contribution, and compare with fixed-order MC programs at this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' In  section 3, we present the calculation of NGLs and CLs at O(α2  s) and show plots of the coefficients of these  logarithms as a function of the jet radius and comment on the relative size of these coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We also  compare at this order the calculated δφ distribution with the output of the program EVENT2, thus confirming  our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' In section 4 we extend the calculation to O(α3  s) and (in the anti-kt algorithm and at small R)  O(α4  s), and point out the significantly different color structure of NGLs in kt clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' In section 5 we  present the all-orders resummation of the NGLs and/or CLs in the large-Nc limit up to LL accuracy for the  kt algorithm, and NLL accuracy in the anti-kt clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Finally we draw our conclusions in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' One-loop calculation and the global form factor  In this letter we consider the process of dijet production in e+e− annihilation at centre-of-mass energy  √s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The jets are reconstructed with the kt [14] or anti-kt [15] algorithms, suited for e+e− annihilation, with  merging and stopping distances dij and di defined by  dij = min  �  E2p  i , E2p  j  � 1 − cos θij  1 − cos R ,  di = E2p  i  ,  (1)  where p = +1 for the kt algorithm and p = −1 for the anti-kt algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Here R is the jet radius, Ei  is the energy of the ith parton in the final state, and θij is the opening angle between partons i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  The algorithm sequentially merges objects i and j whenever dij is the smallest of all merging and stopping  distances, and if an object i has its stopping distance as the smallest then it gets admitted to the list of  final inclusive jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The algorithm keeps recursing until all partons are clustered into jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' In this letter,  we assume the jet kinematics to be defined with E-scheme recombination, such that the 4-momentum of a  merged object simply equals the vectorial sum of the momenta of its constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  At the Born level, the two jets are produced back-to-back, and their relative azimuthal angle (with  respect to the beam axis) is exactly π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The observable we are interested in is the deviation from π of this  relative azimuthal angle, δφ, when soft gluons are emitted at higher orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' It is straightforward to obtain  the following expression for δφ in terms of the transverse momenta of the emitted gluons κti and their  2  azimuthal angles ϕi, with respect to the beam axis  δφ =  ������  �  i/∈jets  κti  pt  sin ϕi  ������  ,  (2)  where pt is the jet transverse momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The (algebraic) sum is over all emitted gluons that are not clustered  to any of the two measured (leading) jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' This definition is valid only at single leading logarithmic (LL)  accuracy, and we shall give the proper definition, valid at NLL accuracy, in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Furthermore, the  expression of δφ in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' (2) only applies in E-scheme recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Alternative jet recombination schemes  exist for which the jet kinematics take a different form, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' the pt-weighted scheme, and the resummation  takes an entirely different structure [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  At one loop the cumulative cross-section for events with azimuthal decorrelation δϕ less than some ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  normalized to the Born cross-section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' reads  Σ1(∆) = −2 CF  � αs(kt)  π  dkt  kt  d cos θ  sin2 θ  dφ  2π ωk  q¯q Θout(k) Θ (κt| sin ϕ|/pt − ∆) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (3)  where θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' φ and kt are the polar angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' azimuthal angle and transverse momentum of the emitted soft gluon  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' with respect to the jet (thrust) axis (the back-to-back outgoing jets are aligned along the z axis),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' CF  is the color factor associated with the emission of the gluon off the hard q¯q dipole,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' and αs is the strong  coupling with argument kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The invariant antenna function ωk  q¯q is given by  ωk  q¯q = k2  t  2  pq · p¯q  (pq · k)(p¯q · k) = 1 ,  (4)  with pi denoting the momentum of particle i  pq =  √s  2 (1, 0, 0, 1) ,  (5a)  p¯q =  √s  2 (1, 0, 0, −1) ,  (5b)  k = E (1, cos φ sin θ, sin φ sin θ, cos θ) ,  (5c)  where kt = E sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The constraint Θout(k) restricts the emitted gluon to be outside the jets and forbids it  from being clustered to any of them in order to induce a non-zero azimuthal decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' It depends on  the jet radius R and is given, in the generalized algorithm, by  Θout(k) = Θ (cos R − | cos θ|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (6)  Since the soft emission is restricted to be outside both jets then there are no collinear logarithms as-  sociated with this observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' This means that the leading logarithms are single, which allows us at LL  accuracy to simply change Θ(κt| sin ϕ|/pt − ∆) → Θ(2 kt/√s − ∆), since any factor multiplying kt will only  induce sub-leading logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' 1 We can then perform the integration over kt using the one-loop running  of the coupling (which formally enters the distribution at higher orders), and write the result in terms of  the evolution parameter t defined by  t(∆) ≡  � √s/2 αs(kt)  π  dkt  kt  Θ  �  2 kt/√s − ∆  �  = −  1  2πβ0  ln(1 − 2λ) ,  (7)  where λ = αs(√s/2)β0 ln(1/∆) and β0 is the one-loop coefficient of the QCD beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The angular  integration is straightforward and we obtain  Σ1(∆) = −2 CF t(∆)  � 1  −1  dc  1 − c2 Θ (cos R − |c|) = −2 CF t(∆) ln 1 + cos R  1 − cos R ,  (8)  1Notice that κt and ϕ are different from kt and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  3  with c standing for cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Note that since only emissions in the inter-jet (gap) region are integrated over,  this result may be cast in terms of the rapidity-gap width ∆η  ∆η ≡ ln 1 + cos R  1 − cos R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (9)  The all-orders resummed global form factor is simply the exponential of the one-loop distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' That is  Σglobal(∆) = exp [−2 CF t(∆) ∆η] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (10)  An identical expression was also arrived at in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [26] for jet shapes in e+e− annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  The leading-order result can be verified by comparing it with the output of the MC program EVENT2 at  O(αs) [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Specifically we compare the differential distribution  2π  αs  dΣ1  dL = 4 CF ln 1 + cos R  1 − cos R ,  (11)  where L = ln ∆, with the same MC distribution, for the chosen value of R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We show in figure 1 a  plot of the difference between the MC distribution and the expansion of the resummation at O(αs), where,  as expected, this difference tends to zero in the logarithmically-enhanced region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Figure 1: The difference between the leading-order EVENT2 differential distribution 2π/αs dΣ1/dL and the resummed distribu-  tion expanded at O(αs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The singular behavior of the MC distribution is exactly cancelled by the expanded result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Two-loops calculation: NGLs and CLs  When employing the kt or anti-kt clustering algorithms with E-scheme recombination, the resummation  of the azimuthal decorrelation distribution requires the treatment of NGLs and/or CLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The corresponding  cumulative distribution at O(α2  s) can then be split into three contributions  Σ2(∆) = 1  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [Σ1(∆)]2 + ΣNG  2  (∆) + ΣCL  2 (∆) ,  (12)  with ΣCL  2 (∆) = 0 for anti-kt clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Let us first discuss the NGLs contribution ΣNG  2  (∆) in both algo-  rithms, and then compute the CLs contribution ΣCL  2 (∆) for kt clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Calculation of NGLs  The origin of NGLs at two loops is the emission of a soft gluon k1 inside any of the two outgoing jets  which itself emits a softer gluon k2 outside the jets without being clustered back to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  While this  configuration results in a non-zero δφ, its virtual correction (specifically when k2 is virtual) gives δφ = 0,  4  and thus we have a real-virtual mis-cancellation of the soft singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We express the contribution of the  uncancelled virtual correction to the integrated azimuthal decorrelation distribution as follows  ΣNG  2  (∆) = S2(R) t2  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' ,  (13a)  S2(R) = −2 CF CA  �  dc1  1 − c2  1  dφ1  2π  dc2  1 − c2  2  dφ2  2π A12  q¯q ΞNG  2  (R) ,  (13b)  where CA is the color factor associated with the non-Abelian emission of gluon k2 off k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The irreducible  two-loops antenna function A12  q¯q is given by [27]  A12  q¯q = ω1  q¯q  �  ω2  q1 + ω2  1¯q − ω2  q¯q  �  =  1 − c1c2  1 − c1c2 − s1s2 cos(φ1 − φ2) − 1 ,  (14)  with si ≡ sin θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The function ΞNG  2  restricts the angular phase-space of integration and is given, in the  anti-kt and kt algorithms respectively, by  ΞNG, akt  2  = Θin(k1)Θout(k2) ,  (15a)  ΞNG, kt  2  = Θin(k1)Θout(k2)Θ(d12 − d2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (15b)  The step function Θ(d12 − d2) forbids gluon k2 from being clustered back to the jet in the kt algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' It  is given by Θ(cos R − cos θ12), with cos θ12 = c1c2 + s1s2 cos(φ1 − φ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  In the anti-kt algorithm the integration is simple and its result can be expressed in the same form as  that of the rapidity-gap NGLs coefficient found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [19]  Sakt  2  (R) = −CF CA  �π2  6 + 2∆η2 − 2∆η ln  �  e2∆η − 1  �  − Li2  �  e−2∆η�  − Li2  �  1 − e2∆η��  = −CF CA  �π2  3 − R4  8 − R6  24 − 29 R8  2560 + O  �  R10��  ,  (16)  where ∆η(R) is defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The above formula shows that in the limit R → 0 the two-loops NGLs  coefficient does not vanish, but rather reaches its maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' This feature was observed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [19]  and was ascribed to the fact that NGLs originate from the edge of jets, since this is the phase-space region  where gluons k1 and k2 are collinear and thus the amplitude squared (14) is most singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  In the kt algorithm, and at small values of R (using small angles), we can write the NGLs coefficient as  Skt  2 (R ∼ 0) = −8 CF CA  � 1  0  dθ1  � ∞  1  dθ2  � 2π  0  dφ  2π  1  (θ2  1 + θ2  2) sec φ − 2 θ1 θ2  Θ  �  θ2  1 + θ2  2 − 2 θ1 θ2 cos φ − 1  �  = −2π2  27 CF CA ,  (17)  where we made the following changes of variables: φ = φ1 − φ2 and θi → R θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Away from the small-R  limit one may perform the integration numerically to obtain the full-R result for the two-loops coefficient of  NGLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We present the results in the following subsection together with the CLs coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' CLs with kt clustering  To compute the CLs we consider the Abelian primary emission of two strongly-ordered gluons directly  off the hard q¯q dipole, whereby the harder gluon k1 is inside one of the two jets and the softer k2 is outside  both of them, with the constraint d12 < d2, such that gluon k2 gets clustered into the jet by gluon k1, which  leads to δφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' However, when k1 is virtual then gluon k2 remains in the gap causing the hard jets to  decorrelate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' In this case we obtain a large single logarithmic contribution to the δφ distribution given by  ΣCL  2 (∆) = Ckt  2 (R) t2  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' ,  (18a)  Ckt  2 (R) = 4 C2  F  �  dc1  1 − c2  1  dφ1  2π  dc2  1 − c2  2  dφ2  2π w1  q¯q w2  q¯q Θ(|c1| − cos R) Θ(cos R − |c2|) Θ(cos θ12 − cos R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (18b)  5  Note again that Cakt  2  (R) = 0, as there are no CLs for anti-kt clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  First, let us consider the small-R limit of this integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' In this case we may write  Ckt  2 (R ∼ 0) = 8 C2  F  � 1  0  dθ1  θ1  � ∞  1  dθ2  θ2  � 2π  0  dφ  2π Θ  �  −θ2  1 − θ2  2 + 2 θ1 θ2 cos φ + 1  �  = 5π2  27 C2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (19)  Away from this small-R limit we perform the integration numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We show in figure 2 a plot of the  coefficients of NGLs and CLs at O(α2  s) as a function of the jet radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Also shown is the combined  coefficient of CLs and NGLs in the kt algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Fkt  2 = Skt  2 + Ckt  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Figure 2: CLs and NGLs coefficients at two loops with kt and anti-kt clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Notice that NGLs coefficient in the kt clustering algorithm is significantly smaller than that in the anti-  kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The CLs coefficient is positive and quite small, with the advantage that it cancels the NGLs coefficient,  particularly at small values of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Note that the overall coefficient F2 is less than 1 for values of R smaller  than about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' For small R values, the anti-kt NGLs coefficient computed here is identical to that found  for the single hemisphere mass [28] and jet mass with a jet veto [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Comparison to EVENT2  We compare our two-loops results with the exact MC distribution at O(α2  s) obtained with the EVENT2  program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The latter splits the distribution at NLO into three color contributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' O(C2  F), O(CF CA) and  O(CF Tf nf), with Tf = 1/2 being the normalization constant of the generators of the SU(3) group in the  fundamental representation and nf = 5 is the number of active quark flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The expansion of the resummed  distribution at this order, including running coupling effects, is written, after differentiating with respect to  L, as  �2π  αs  �2 dΣ2  dL =  �  C2  F 4  �  4 ∆η2 + C2  �  − CF CA  4  3 (3 S2 + 11∆η) + CF Tf nf  16  3 ∆η  �  L + O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (20)  At this order this differential distribution is linear in L, but does not capture the O(1) constant, which, in  the cumulative integrated distribution, is an NLL O(α2  s L) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We plot the difference between the MC  distribution and the expansion (20) for each color contribution separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' All curves should tend towards a  constant when L grows large and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The results are shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Here the O(C2  F) part contains  the CLs coefficient in kt clustering, while the O(CF CA) term contains the NGLs coefficient both in kt and  anti-kt algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The O(CF Tf nf) contribution is algorithm independent at LL accuracy, but the NLL  O(1) constant does depend on the jet algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  6  Figure 3: The difference between the NLO EVENT2 differential distribution (2π/αs)2 dΣ2/dL and the resummed distribution  expanded at O(α2  s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The leading logarithmic behavior of the MC distribution O(α2  sL) is cancelled, leaving a constant behavior  at large values of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' NGLs and CLs at three loops  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' NGLs at three loops  At O(α3  s), the cumulative distribution can be written as follows  Σ3(∆) = 1  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [Σ1(∆)]3 + Σ1(∆) × ΣNG  2  (∆) + Σ1(∆) × ΣCL  2 (∆) + ΣNG  3  (∆) + ΣCL  3 (∆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (21)  Focusing first on the pure NGLs contribution ΣNG  3  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=', excluding the cross-talk between the one-loop global  and two-loops non-global logarithms) in the anti-kt algorithm, we may write it in the form  ΣNG,akt  3  (∆) = Sakt  3  (R) t3  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' ,  (22a)  Sakt  3  (R) = 2 CF C2  A  � � 3  �  i=1  dci  1 − c2  i  dφi  2π  �  �  A12  q¯q ¯  A13  q¯q Θin(k1)Θout(k2)Θout(k3) − B123  q¯q Θin(k1)Θin(k2)Θout(k3)  �  ,  (22b)  where we define ¯  A13  q¯q = A13  q¯q/ω1  q¯q, and the 3-loops irreducible cascade antenna function is given by  B123  q¯q = ω1  q¯q  �  A23  q1 + A23  1¯q − A23  q¯q  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (23)  It is worth mentioning that, for the cascade term, there is a non-negligible contribution from the configuration  in which gluon k1 is in one jet emitting gluon k2 in the other jet which itself emits the softest gluon k3 in  the gap between the two jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' At small values of R the integration is quite simple and yields the result  Sakt  3  (R ∼ 0) = CF C2  A 2 ζ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (24)  Notice that this result is twice that found for the hemisphere mass observable [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Away from the small-R  limit the integration can be performed numerically and we shall present the results in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  The pure NGLs contribution in the kt algorithm is given by an identical form to that of the anti-kt (22a)  ΣNG,kt  3  (∆) = Skt  3 (R) t3  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (25)  We perform for the first time in the literature a calculation of NGLs in the kt algorithm beyond two loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Due to non-linearity of kt clustering we shall find a class of NGLs that have a non-standard color factor,  namely C2  F CA at O(α3  s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Such terms usually (in anti-kt clustering) only arise in the cross-talk of the  expansion of the primary-emission global form factor (10) at O(αs) together with NGLs at O(α2  s), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=', the  term Σ1(∆) × ΣNG  2  (∆) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' However, in the kt algorithm, we find them as “pure” irreducible NGLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  That is, they are part of the term ΣNG  3  (∆) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  7  To proceed, we consider three types of emissions at O(α3  s), as shown in figure 4: (a) one primary + two  correlated emissions, (b) ladder emissions, and (c) cascade emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' In each case we consider all possible  virtual-correction Feynman diagrams as well as angular configurations of the emitted gluons that affect the  clustering procedure, and look for a mis-match between the soft divergences of these emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Figure 4: The three types of emissions to consider for NGLs calculation at O(α3  s): (a) one primary + two correlated emissions,  (b) ladder emissions, and (c) cascade emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Starting first with type (a) contributions, there are three possible permutations of the gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  2  For the permutation in which k3 is emitted in correlation with k2, and which has a squared amplitude  4 C2  F CA ω1  q¯q A23  q¯q, we find that the angular phase space of integration that yields a logarithmic contribution  is given by  ΞNG,kt  31  (R) = Θin(k1)Θin(k2)Θout(k3)Θ(d3 − d13)Θ(d23 − d3)+  + Θout(k1)Θin(k2)Θout(k3)Θ(d23 − d3)+  − Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)Θ(d2 − d12) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (26)  To see how one obtains this result let us give one example of angular configurations that result in a logarithmic  contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' There are four Feynman diagrams in this case, shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Consider the situation when  particles k3 and k2 are outside the jet regions, d3j > d3 and d2j > d2, while particle k1 is inside, d1j < d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' In  this scenario, both diagrams in which k1 is virtual (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=', diagrams (3) and (4) in figure 5) yield δφ ̸= 0, since  in both diagrams particle k2 is real and remains in the gap region after applying the clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' However,  these two diagrams contribute equally and with opposite signs, so they cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' For the remaining  two diagrams, (1) and (2), we have a mismatch when particle k2 gets pulled inside the jet by the real particle  k1 while k3 remains in the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' This happens when d12 is smaller than d2, and both d13 and d23 are greater  than d3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' While in diagram (1) we have a real unclustered gluon k3 in the gap region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=', it forms a jet  on its own) giving δφ ̸= 0, in diagram (2) the gap is empty and the hard jets are exactly back-to-back  with δφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The virtual-correction diagram (2) contributes fully to the cumulative distribution while the  real-emission diagram (1) cancels this contribution only up to δφ = ∆, leaving uncancelled virtual-correction  contributions with a negative sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' This is the last term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Figure 5: Feynman diagrams corresponding to the squared amplitude 4 C2  F CA ω1  q¯q A23  q¯q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  2Note that for all permutations the transverse momenta of the three gluons are strongly ordered as follows: kt1 ≫ kt2 ≫ kt3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  8  Similarly, we obtain for the second and third gluon permutations of type (a) diagrams, with squared  amplitudes 4 C2  F CA ω2  q¯q A13  q¯q and 4 C2  F CA ω3  q¯q A12  q¯q, respectively, as well as type (b) (ladder-emission) contri-  butions, with squared amplitude 2 CF C2  A ¯  A12  q¯q A13  q¯q, the same phase space function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' It is given by  ΞNG,kt  32  (R) = Θin(k1)Θin(k2)Θout(k3)Θ(d13 − d3)Θ(d3 − d23)+  + Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d3 − d23)+  + Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)Θ(d12 − d2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (27)  Finally, for type (c) (cascade emission) contributions, corresponding to the squared amplitude 2 CF C2  A B123  q¯q ,  the phase space function reads  ΞNG,kt  33  (R) = − Θin(k1)Θin(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)−  − Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)Θ(d2 − d12) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (28)  Before performing the integration, we subtract off the part of the phase space that produces the interfer-  ence term between the one-loop global primary logarithm and the two-loops NGLs, which only comes from  type (a) emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' For the first permutation of gluons, this part of phase space is identified by the second  term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' (26), Θout(k1)Θin(k2)Θout(k3)Θ(d23 − d3), where gluon k1 reproduces the one-loop global term  (8) and the other correlated gluons, k2 and k3, give the two-loops NGLs (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' (13b) with phase space (15b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  However, for the other two gluon permutations of type (a) emissions, the phase space (27) does not simply  contain such interference terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Strictly speaking, this means that NGLs do not cleanly factorize from the  global form factor in the kt algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Nevertheless, we can manually add and subtract the interference  terms and write the total distribution in the factorizable form (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  We can then write the “pure” NGLs coefficient Skt  3 , given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' (25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' in the following form  Skt  3 (R) = S(a)  3 (R) + S(b)+(c)  3  (R) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (29)  where we split the result according to the color factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' such that for type (a) emissions we have  S(a)  3 (R) = 4 C2  F CA  � � 3  �  i=1  dci  1 − c2  i  dφi  2π  � �  ω1  q¯q A23  q¯q �Ξ31(R) + ω2  q¯q A13  q¯q �Ξ32(R) + ω3  q¯q A12  q¯q �Ξ33(R)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (30)  with modified phase space that subtracts away the interference terms  �Ξ31(R) = ΞNG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='kt  31  (R) − Θout(k1)Θin(k2)Θout(k3)Θ(d23 − d3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (31a)  �Ξ32(R) = ΞNG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='kt  32  (R) − Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (31b)  �Ξ33(R) = ΞNG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='kt  32  (R) − Θin(k1)Θout(k2)Θout(k3)Θ(d12 − d2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (31c)  and for type (b) and (c) emissions  S(b)+(c)  3  (R) = 2 CF C2  A  � � 3  �  i=1  dci  1 − c2  i  dφi  2π  � �  A12  q¯q ¯  A13  q¯q ΞNG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='kt  32  (R) + B123  q¯q ΞNG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='kt  33  (R)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (32)  We are now in a position to perform the integrations numerically as a function of the jet radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We  show the results in the next subsection, in which we also compute the clustering logarithmic contribution  at O(α3  s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' CLs with kt clustering at three loops  Following the same steps as for NGLs calculation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' the phase space clustering function at O(α3  s) for the  CLs contribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' which results from the mismatch of soft singularities between real and virtual emissions  9  of three primary soft gluons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' with a squared amplitude 8 C3  F w1  q¯q w2  q¯q w3  q¯q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' is given by  ΞCL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='kt  3  (R) = − Θin(k1)Θin(k2)Θout(k3)Θ(d3 − d13)Θ(d3 − d23)−  − Θout(k1)Θin(k2)Θout(k3)Θ(d3 − d23)−  − Θin(k1)Θout(k2)Θout(k3)Θ(d3 − d13)−  − Θin(k1)Θout(k2)Θout(k3)Θ(d13 − d3)Θ(d23 − d3)Θ(d2 − d12) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (33)  Note that the above phase-space clustering function is similar (but not exactly identical) to that found for  the jet mass observable [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Extracting the interference terms between the CLs at two loops and the global  logarithm at one loop, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=', the term Σ1(∆) × ΣCL  2 (∆) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' (21), we reduce the above phase space function  to that of the “pure” CLs contribution as  �ΞCL,kt  3  (R) = − Θin(k1)Θin(k2)Θout(k3)Θ(d3 − d13)Θ(d3 − d23)+  + Θin(k1)Θout(k2)Θout(k3)Θ(d2 − d12)[1 − Θ(d13 − d3)Θ(d23 − d3)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (34)  Then, the clustering logarithmic contribution to the cumulative cross-section is given by  ΣCL  3 (∆) = Ckt  3 (R) t3  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' ,  (35a)  Ckt  3 (R) = 8 C3  F  � � 3  �  i=1  dci  1 − c2  i  dφi  2π  �  w1  q¯q w2  q¯q w3  q¯q �ΞCL,kt  3  (R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (35b)  We show in figure 6 a plot of the coefficients of NGLs and CLs in the kt and anti-kt algorithms as a  function of the jet radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Shown also is the combined coefficient Fkt  3 = Skt  3 + Ckt  3 for the kt algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Figure 6: CLs and NGLs coefficients at three loops with kt and anti-kt clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  As in the previous section, we notice that the NGLs coefficient in the anti-kt algorithm at this order is  also quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Clearly, the application of the kt clustering has reduced the significance of NGLs by almost  a factor of 30 for values less than R ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The CLs coefficient is also small such that the overall coefficient  F3 is smaller in magnitude than the Sakt  3  by about a factor of 3 for most values of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Four loops and beyond  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Four-loops NGLs with anti-kt at small R  The calculation of NGLs with anti-kt clustering proceeds in a similar manner at fourth order, and can  easily be deduced from previous calculations of NGLs in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' In fact, the phase space of integration  10  is similar to that of the hemisphere mass distribution reported in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [29], and thus the cumulative cross-  section, at this order, may be cast in the following way  Σakt  4  (∆) = 1  4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [Σ1(∆)]4+ 1  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [Σ1(∆)]2 ΣNG,akt  2  (∆)+Σ1(∆) ΣNG,akt  3  (∆)+ 1  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  �  ΣNG,akt  2  (∆)  �2  +ΣNG,akt  4  (∆) , (36)  with pure NGLs contribution given by  ΣNG,akt  4  (∆) = Sakt  4  (R) t4  4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (37a)  Sakt  4  (R) = 2 CF C3  A  � � 4  �  i=1  dci  1 − c2  i  dφi  2π  � �  − A12  q¯q ¯  A13  q¯q ¯  A14  q¯q Θin(k1)Θout(k2)Θout(k3)Θout(k4)+  + 3 ¯  A12  q¯q B134  q¯q Θin(k1)Θout(k2)Θin(k3)Θout(k4) + U1234  q¯q  Θin(k1)Θin(k2)Θout(k3)Θout(k4)−  − C1234  q¯q  Θin(k1)Θin(k2)Θin(k3)Θout(k4)  �  −  − 2 CF C3  A  �  1 − 2 CF  CA  � � � 4  �  i=1  dci  1 − c2  i  dφi  2π  �  A1234  q¯q  Θin(k1)Θin(k2)Θout(k3)Θout(k4) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (37b)  where the irreducible antenna functions read [27]  C1234  q¯q  = w1  q¯q  �  B234  q1 + B234  1¯q − B234  q¯q  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (38a)  U1234  q¯q  = w1  q¯q  �  A23  q1 ¯  A24  q1 + A23  1¯q ¯  A24  1¯q − A23  q¯q ¯  A24  q¯q  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (38b)  A1234  q¯q  = ω1  q¯q ω2  q1  � ¯  A23  q¯q − ¯  A23  q1  � � ¯  A24  q1 − ¯  A24  1¯q  �  + ω1  q¯q ω2  1¯q  � ¯  A23  q¯q − ¯  A23  1¯q  � � ¯  A24  1¯q − ¯  A24  q1  �  −  − ω1  q¯q ω2  q¯q  � ¯  A23  q1 − ¯  A23  q¯q  � � ¯  A24  1¯q − ¯  A24  q¯q  �  + k3 ↔ k4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (38c)  At small values of R the integration has been performed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [29], and the corresponding result is  given by  Sakt  4  (R ∼ 0) = −CF C3  A ζ4  �29  4 −  �  1 − 2 CF  CA  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (39)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' LL resummation  In this section we present numerical results for the resummation of NGLs and CLs in the large-Nc  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' For this we use the MC code first developed in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [18, 19] with modification of kt clustering  in terms of distances (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The results are shown in figure 7 for the resummed cumulative distribution Σ as  a function of the evolution parameter t (7), for the particular value of the jet radius R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Shown in the  figure are: results for the (primary) global distribution (black curve), obtained by running the MC program  using anti-kt clustering and allowing only for primary emissions, and the full anti-kt distribution (solid pink  curve) which additionally includes NGLs at large Nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We observe the very large impact of NGLs on the  distribution, reducing the global form factor by a factor of 10 for t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  We also show in the same figure the primary-emission distribution obtained by running the above-  mentioned program with kt clustering (solid green curve), which includes the global form factor together  with the resummed CLs, as well as the overall distribution in kt clustering which includes in addition the  resummed NGLs (solid purple curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We observe that the distribution in kt clustering is affected by both  CLs and NGLs, but the combined impact of the two is noticeably small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' This means that CLs tend to cancel  NGLs in kt clustering, as noted with the fixed-order calculations performed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Moreover, we show in figure 7 the analytical results for the resummed distribution in each case (dashed  lines), which are estimated from the observed pattern of exponentiation  Σ(∆) = exp [Σ1(∆)] exp  � ∞  �  i=2  ΣNG  i  (∆)  �  exp  � ∞  �  i=2  ΣCL  i  (∆)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  (40)  11  Figure 7: Numerically resummed NGLs and CLs at large Nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  It is clear that the truncation of the series at i = 3 in the exponent, though quite close to the numerical  result, does not give an accurate fit of the MC distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' This means that higher-order contributions  cannot be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' NLL resummation with anti-kt  We present here a resummed result for the azimuthal decorrelation distribution with anti-kt clustering  at NLL accuracy in the large-Nc limit, obtained using the recently-published program Gnole [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The  distribution can be obtained from this program by defining our observable within the code, using the full  definition rather than the LL approximation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Explicitly written the former reads  δφ =  ������  sin−1 �  i/∈jets  k⊥i  ptr  ������  ,  (41)  where k⊥i is the component of the transverse momentum of the emission i perpendicular to the thrust (or  leading jet) axis, and ptr is the recoiling jet’s transverse momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' For simplicity we take the jets to be  at threshold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=', transverse to the beam direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Figure 8: NLL numerical resummation of the δφ distribution at large Nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  In figure 8 we present the results for the NLL resummed distribution together with the LL one, both  obtained with Gnole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' We note here that the LL distribution is obtained using the definition of the observable  (41), while that obtained with the MC code of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [18, 19] (figure 7) is essentially equivalent to the transverse  12  energy distribution (in other words, definition 2 without the sin φ part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' This does in fact numerically affect  the distribution even at LL accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' 3  We observe that the NLL corrections to the distribution are quite important, just like the transverse  energy distribution shown in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Furthermore, the scale-uncertainty band, obtained by varying the  renormalization and resummation scales by factors of 2 and 1/2 around their central values (µR = √s  and Q0 = √s/2, respectively, with √s = MZ), gets significantly reduced in the NLL curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' It is worth  mentioning that one still needs to account for the matching in order to fully control all sub-leading NLL  logarithms at the tail of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  We note that the actual distribution does not possess a Sudakov peak at low values of δφ, instead it tends  towards a constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' This is explained by the fact that the very low values of δφ are not suppressed  by soft emissions, but are rather enhanced by vectorial cancellation of semi-hard emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Conclusions  In this letter we have presented both fixed-order and all-orders results for the distribution of the azimuthal  decorrelation observable for the specific QCD process e+e− annihilation into two jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The said observable  is of non-global nature and hence its distribution contains, in addition to the usual global logarithms, non-  global and/or clustering logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' These logarithms are jet-algorithm dependent, and start to appear at  two loops and are quite delicate to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  For NGLs, we have calculated the full-R expression analytically at two loops and numerically at three  loops for the anti-kt jet algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' At four loops, they have been determined only for small-R values for  the same said jet algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' For the kt clustering algorithm, calculations have been performed up to three  loops and only numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Moreover, CLs, which are absent in the anti-kt algorithm, have been computed numerically up to three  loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The usual reduction in the significance of NGLs due to kt clustering has been observed confirming  previous findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Furthermore, the combined impact of NGLs and CLs on the distribution is observed to  be very small which has important phenomenological implications in terms of accuracy of the resummed  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Our results at two loops have been checked against the output of the MC program EVENT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Numerical estimates of the all-orders distribution has been presented both at LL and NLL accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  The achievement of the latter accuracy has been made possible by the recently-published Gnole code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' NLL  resummation exhibits a better distribution both in terms of accuracy and scale uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' It is thus worth  investigating the impact of NLL effects with kt clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Of similar worthiness is computing NGLs and  CLs at four loops with full-R dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  Acknowledgements  This work is supported by PRFU research project B00L02UN050120230003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' The authors wish to thank  the Algerian Ministry of Higher Education and Scientific Research and DGRSDT for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  The numerical calculations presented here have been performed in the HPC cluster at the University of  Batna 2 (UB2-HPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content='  We thank Andrea Banfi for clarifications about using Gnole program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
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+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' Aktas, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
+page_content=' (H1), Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAyT4oBgHgl3EQf8voh/content/2301.00860v1.pdf'}
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+arXiv:2301.02947v1  [cond-mat.mes-hall]  7 Jan 2023
+Chiral organic molecular structures supported by multilayer surfaces
+Alexander V. Savin1, 2, ∗ and Yuri S. Kivshar3, †
+1Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow 119991, Russia
+2 Plekhanov Russian University of Economics, Moscow 117997, Russia
+3Nonlinear Physics Center, Department of Fundamental and Theoretical Physics,
+Research School of Physics, Australian National University, Canberra ACT 2601, Australia
+We study numerically the dynamics of acetanilide (ACN) molecules placed on a flat surface of
+a multilayer hexagonal boron nitride structure. We demonstrate that the ACN molecules, being
+achiral in three dimensions, become chiral after being placed on the substrate. Homochirality of the
+ACN molecules leads to stable secondary structures stabilized by hydrogen bonds between peptide
+groups of the molecules. Numerical simulations of systems of such molecules reveal that the structure
+of the resulting hydrogen-bond chains depends on the isomeric composition of the molecules. If all
+molecules are homochiral (i.e. only one isomer is present), they form secondary structures (chains
+of hydrogen bonds in the shapes of arcs, circles, and spirals). If the molecules at the substrate
+form a racemic mixture, then no regular secondary structures appear, and only curvilinear chains of
+hydrogen bonds of random shapes can emerge. A hydrogen-bond chain can form a straight zigzag
+only if it has an alternation of isomers. Such chains can create two-dimensional (2D) regular lattices,
+or 2D crystals. The melting scenarios of such 2D crystals depend on density of its coverage of the
+substrate. At 25% coverage, melting occurs continuously in a certain temperature interval. For a
+complete coverage, melting occurs at 415 ÷ 470 K due to a shift of 11% of all molecules into the
+second layer of the substrate.
+I.
+INTRODUCTION
+Two-dimensional (2D) materials such as graphene (G)
+and hexagonal boron nitride (h-BN) have attracted a lot
+of attention due to their unique electronic [1–3] and me-
+chanical [4–7] properties. Currently, heterogeneous lay-
+ered materials of such 2D materials, which can exhibit
+various novel physical properties compared to their ho-
+mogeneous counterparts, became a special focus of such
+studies [8–10]. For example, the use of hybrid G/h-BN
+structures allows to achieve some desired electronic prop-
+erties [11, 12] and also reduce significantly friction be-
+tween the layers [13].
+In general, such multilayer het-
+erostructures are stabilized by van der Waals (vdW) in-
+teractions between atoms of the neighboring layers.
+The concept of vdW heterostructures can be extended
+to the integration of 2D materials with molecular struc-
+tures of different dimensions, such as nD/2D heterostruc-
+tures, where n stands for the dimension (n = 0, 1, or
+3) [14], describing flat molecules (n = 0), polymer chains
+(n = 1), or three-dimensional molecular objects (n = 3).
+For molecules and molecular chains with benzol rings,
+flat layers of G and h-BN are strong adsorbents [15–19].
+Theoretical studies reveal that molecules adsorbed on G
+and h-BN surfaces through non-covalent interactions can
+modify the properties of the surface as solid-liquid, solid-
+air, or solid-vacuum interfaces [20–22]. A strong stacking
+interaction with a flat substrate allows such molecules
+residing on a surface creating stable 2D supra-molecular
+systems, as shown in the characteristic example of Fig. 1
+∗asavin@chph.ras.ru
+†yuri.kivshar@anu.edu.au
+FIG. 1:
+Example of the molecular structures studied in
+this paper. A spiral structure of 52 R-isomers of the ACN
+molecules C6H5–NHCO–CH3 is stabilized on a h-BN multi-
+layer surface.
+By now,
+the behavior of multifunctional organic
+molecules placed on ideal metal surfaces has been stud-
+ied in detail [23]. Such organometallic systems may ex-
+hibit a variety of different structures induced by the sub-
+strate. In many cases, complex organic molecules (such
+as carboxylic acids, amino acids, anhydrides, and ring
+systems) become self-organized on metal surfaces creat-
+ing ordered super-structures stabilized by inter-molecular
+interactions. Chirality is of a particular interest that can
+appear for initially achiral metal surfaces by adsorbing
+organic molecules [24]. Similar behavior is expected for
+organic molecules adsorbed on flat surfaces of G and h-
+
+2
+BN molecular structures.
+In this paper, we study nu-
+merically the formation of supra-molecular complexes by
+acetanilide molecules placed on the surface of a multilayer
+h-BN sheet, see Fig. 1, serving as an introductory figure
+explaining our problem and results discussed below.
+For poly-cyclic aromatic hydrocarbons (for molecules
+of benzol C6H6, naphthalene C10H8, pyrene C16H10,...),
+graphene is a strong adsorbent [15, 17, 19, 25]. The in-
+teraction of graphene with such molecules often causes
+specific reactions that can be used in new types of sensors
+[26, 27]. Non-covalent functionalization of the graphene
+surface can significantly expand its potential range of
+applications [20, 21].
+It has been shown experimen-
+tally [25, 28] that benzol and pyrene molecules adsorbed
+on graphene form densely packed monolayers.
+For acetanilide (ACN, C6H5NHCOCH3) and parac-
+etamol (PCM, C6H4OHNHCOCH3) molecules, graphene
+and hexagonal boron nitride are also strong adsorbents.
+Due to possible medical applications, much attention has
+been paid to modeling the adsorption of PCM molecules,
+which is a strong analgesic, on h-BN sheets and nan-
+otubes [18, 29]. It has been shown in [30] that function-
+alized graphene can be used as a highly sensitive parac-
+etamol detection sensor.
+An example of a 1D/2D heterostructure is a graphene
+sheet with adsorbed Kevlar chains, kevlar-functionalized
+graphene [31]. The presence of planar C6H4 benzol rings
+and NHCO peptide groups in the polymer chain [–C6H4–
+NHCO–]∞ provides a strong non-covalent (vdW) inter-
+action of the chain with G and h-BN sheets. Such chains
+on the surface of sheets G and h-BN will lie parallel to
+the surface and form chains of hydrogen bonds between
+each other · · ·HNCO· · ·HNCO· · ·.
+Such 3D/2D heterostructures can form thin metal lay-
+ers on the G and h-BN surfaces. In particular, numerical
+modeling suggests that aluminum can form stable two-
+layer structures on the G surface [32].
+It has been shown in Refs. [33–42] that n-alkanes (lin-
+ear polymer chains CH3(CH2)lCH3 with internal units
+2 ≤ l ≤ 388) form a dense ordered monolayer of parallel
+linear chains on the graphite (graphene) surface. Inter-
+est in alkanes is due to the fact that they belong to the
+simplest families of polymer molecules, which members
+of which differ only in their length. Placing linear poly-
+mer chains on a flat graphite surface causes them to self-
+assemble into 2D crystals. The self-assembly mechanism
+depends on the chain length, temperature, and the level
+of coverage of the substrate with chains [36, 40].
+Adsorption by the surface of a long single-chain
+polyethylene molecule leads to its two-dimensional crys-
+tallization – it passes from the form of a three-
+dimensional globule into the form of a parallel folded
+linear chain lying in the plane parallel to the substrate
+surface [43–45].
+Thus, the flat surfaces of the G and h-BN substrates
+create a 2D platform for flat molecules adsorbed on them
+(for poly-cyclic aromatic hydrocarbons, for ACN and
+PCM molecules, Kevlar chains, ...)
+and linear poly-
+mer molecules. At low temperatures, the molecules move
+along the sheet, remaining parallel to its surface. They
+interact with each other and form two-dimensional supra-
+molecular structures. Such molecular adsorbents are con-
+venient systems for studying phase transitions caused by
+freedom restrictions.
+To date, only phase transitions in monolayers of n-
+alkanes have been well studied [33, 42, 46–48]. Modeling
+and experimental studies show that a monolayer always
+undergoes a transition from a solid-crystalline 2D phase
+to a liquid phase (the transition occurs at a temperature
+significantly lower than the desorption temperature of
+molecules). The melting scenario depends on the poly-
+mer chain length.
+The melting temperature increases
+monotonically with chain length, so for pentane, hep-
+tane and nonane (l = 3, 5, 7) the melting temperature
+is Tm = 92, 178 and 255K [46].
+A characteristic fea-
+ture associated with the adsorption of molecules is the
+continuity of melting of a 2D crystal – melting occurs in
+the temperature interval. Thus, for the longest synthe-
+sized monodisperse alkane C390H782 (l = 388), continu-
+ous melting occurs at 393 < T < 484 K [42].
+Despite the large number of theoretical and experimen-
+tal works on phase transitions in adsorbed monolayers of
+alkanes and their derivatives, as far as we know, there
+are no works on modeling phase transitions in adsorbed
+monolayers of ACN, PCM, and Kevlar (para-aramid)
+molecules.
+A detailed description of adsorption simu-
+lation methods is given in [49]. Unlike alkanes, the 2D
+structures of these molecules adsorbed on a flat surface
+are associated with the presence of chains of hydrogen
+bonds. Molecules including amide and hydroxyl groups
+can create 2D lattices and extended hydrogen chains.
+We notice that the ACN molecules are often consid-
+ered as a model system with chains of hydrogen bonds
+between HNCO peptide groups. Acetanilide crystallizes
+into an orthorhombic structure with ribbons of molecules
+linked by hydrogen bonds [50]. The chains of hydrogen
+bonds that stabilize the crystal structure are very similar
+to the chains that stabilize the alpha-helices and beta-
+sheets of proteins. Therefore, ACN was used as a model
+for modeling the energy transfer of vibrations of peptide
+groups along hydrogen bond chains in proteins [51–53].
+Living matter, unlike non-living matter, has chiral pu-
+rity: all proteins consist of left-handed amino acids, while
+DNA and RNA are built on right-handed ribose. In ex-
+periments on abiogenic synthesis, left and right isomers of
+sugars and proteins are formed in equal proportions. It is
+believed that if you try to build proteins from such a mix-
+ture, they will not be able to fold into a stable form and
+therefore will not work as enzymes. In three dimensions,
+the need for chiral purity to form stable protein struc-
+tures requires complex analysis.
+The situation is dra-
+matically simplified if we move from three-dimensional
+space to two-dimensional. Such a transition can be made
+if flat molecules are placed on a flat molecular sheet of
+graphene or hexagonal boron nitride (h-BN). Such a non-
+valent modification of the sheet surface actually creates
+
+3
+FIG. 2: When an ACN molecule is placed on a flat surface
+of a multilayer h-BN structure, it may create two isomers
+with the mirror symmetry (shown by a straight line). Vectors
+connecting the oxygen atom with the hydrogen atom of each
+peptide group show the dipole moments. For the L-isomer,
+the benzol ring is to the left of this vector, for the R-isomer
+it is to the right. Gray balls stand for carbon atoms, white
+balls – hydrogen, blue – nitrogen, red – oxygen, and green –
+bromine atoms.
+a 2D world for the flat molecules placed on it. At low
+temperatures, the molecules move along the sheet all the
+time remaining parallel to its surface. On the surface,
+they can form complex two-dimensional structures.
+An ACN molecule that is achiral in 3D becomes chiral
+after being placed on a flat substrate (the chirality de-
+pends on which side it lays on the surface of the sheet)
+– see Fig. 2. It will be shown that the homochirality of
+ACN molecules leads to the appearance on the surface
+of the sheet of stable secondary structures stabilized by
+hydrogen bonds: cyclic and spiral chains and complexes
+of them. Modeling the formation of such structures will
+make it possible to demonstrate the necessity of homochi-
+rality (chiral purity) of biomolecules for the formation of
+stable secondary molecular structures from them.
+As a flat substrate, we consider a surface of a multilayer
+h-BN structure, and for molecules we consider acetanilide
+(ACN) C6H5NHCOCH3, as shown in Figs. 1 and 2. The
+presence of a planar benzol ring C6H5 and a planar pep-
+tide group (PG) HNCO leads to large interaction energy
+of the molecule with the substrate, Esub = 0.762 eV.
+Molecules can create chains of hydrogen bonds between
+their peptide groups OCNH· · ·OCNH· · ·OCNH· · ·. Such
+chains of hydrogen bonds stabilize the secondary struc-
+tures of the protein molecules.
+The paper is organized as follows. In the next section,
+we describe our model. Section III is devoted to the study
+of secondary structures of the ACN molecules placed on
+a flat substrate. Self-assembly of such structures is simu-
+lated numerically in Sec. IV. Then, in Sec. V we analyze
+melting of 2D crystals. Section VI concludes the paper.
+FIG. 3: Construction of a coarse-grained model of the ACN
+molecule: (a) full-atomic view of the molecule, (b) coarse-
+grained model (the used numbering of the united atoms is
+shown).
+II.
+MODEL
+For modeling of the dynamics of a system of the ACN
+molecules, we will use the united-atoms approximation.
+Let us consider the molecular groups CH and CH3 as
+united atoms whose centers coincide with the centers of
+carbon atoms. In this approximation, the ACN molecule
+is described as a system of 11 united atoms – see Fig. 3.
+The values of the masses of the united atoms are shown
+in the table I.
+To model a ACN molecule, we use the force field in
+which distinct potentials describe the deformation of va-
+lence bonds and valence, torsion and dihedral angles, and
+non-valence atomic interacts [55]. In this model, the de-
+formation energy of the valence bonds C–CH, CH–CH,
+C–N, N–H, C=O and C–CH3 is described by the har-
+monic potential:
+V (ρ) = 1
+2K(ρ − ρ0)2,
+(1)
+where ρ and ρ0 are current and equilibrium bond lengths,
+K is the bond stiffness. The values of potential parame-
+ters for various valence bonds are presented in Table II.
+TABLE I:
+Masses and parameters of interaction potentials
+for united atoms X of the ACN molecule: i – atom number,
+Mi – atom mass (mp = 1.6603 × 10−27 kg – proton mass), εi
+and ri are the energy and radius of the LJ interaction, qi is
+the electric charge of the atom, ǫi and hi are the energy and
+equilibrium distance for the interaction of an atom with a flat
+substrate (with a crystal surface h- BN).
+X
+C
+CH
+N
+H
+C
+O
+CH3
+i
+1
+2,3,4,5,6
+7
+8
+9
+10
+11
+Mi (mp)
+12
+13
+14
+1
+12
+16
+15
+εi (meV)
+4.284
+4.284
+4.080 0.434 4.284 6.344 4.284
+ri (˚A)
+1.861
+1.861
+1.899 0.621 1.861 1.711 1.861
+qi (e)
+0.066
+0
+-0.463 0.286 0.580 -0.504 0.035
+ǫi (meV)
+61.5
+87.3
+47.7
+31.3
+61.5
+42.8
+87.3
+hi (˚A)
+3.52
+3.44
+3.43
+3.08
+3.52
+3.36
+3.44
+
+(a)
+H
+(b)
+CH
+6
+8
+5
+CH3
+4
+C
+N
+1
+9
+11
+3
+2
+10R
+T4
+FIG. 4: Dimer of (a) RR and (b)LR isomers of the ACN
+molecule on a flat substrate (shown in green). The vectors
+show the dipole moments of the peptide groups. Hydrogen
+bond energy of identical isomers Ehb = 0.330 eV, angle be-
+tween dipole moments φhb = 17.05◦, for different isomers
+Ehb = 0.322 eV, φhb = 33.98◦.
+Energies of the deformation of the valence angles X–
+Y–Z are described by the potential
+U(u1, u2, u3) = U(φ) = ǫa(cos φ − cos φ0)2,
+(2)
+where the cosine of the valence angle φ is defined as
+cos φ = −(v1, v2)/ρ1ρ2, with the vectors v1 = u2 − u1,
+v2 = u3 − u2 and bond lengths ρ1 = |v1|, ρ2 = |v2|, the
+vectors u1, u2, u3 specify the coordinates of the atoms
+forming the valence angle φ, φ0 is the value of equilib-
+rium valence angle. The values of potential parameters
+used for various valence angles are presented in Table III.
+Deformation of dihedral angles are described by the
+potential
+W(u1, u2, u3, u4) = ǫd(1 + zd cos kϕ),
+(3)
+where cos ϕ = (w1w2)/|w1||w2|, with the vectors w1 =
+(u2 − u1) × (u3 − u2), w2 = (u3 − u2) × (u4 − u3). The
+values of potential parameters used for various dihedral
+angles are presented in Table IV.
+For pairs of atoms Xi,Xj (i, j are the numbers of atoms
+in the molecule) participating in the formation of the
+dihedral angle Xi–Y–Z–Xj, their non-valence interaction
+is also taken into account described by the Lennard-Jones
+(LJ) potential
+W0(r) = ε0[(r0/r)12 − 2(r0/r)6],
+(4)
+with halved interaction energy ε0 = √εiεj/2, where r is
+current distance between interacting atoms, equilibrium
+distance r0 = ri + rj. The LJ interaction of an oxygen
+atom (i = 10) with two combined atoms CH (i = 2, 6)
+was also taken into account with interaction energy ǫ0 =
+TABLE II: Values of the harmonic potential parameters (1)
+for different valence bonds X—Y.
+X—Y
+C–CH, CH–CH
+C–N
+N–H
+C=O C–CH3
+K (N/m)
+469
+427
+434
+570
+317
+ρ0 (˚A)
+1.39
+1.405
+1.007
+1.222
+1.505
+√ε2ε10 and equilibrium distance r0 = r2+r10. Parameter
+values εi and ri are shown in Table I.
+The interaction of two ACN molecules is described by
+the potential
+U(X1, X2) =
+11
+�
+i=1
+11
+�
+j=1
+{εij[(¯rij/rij)12 − 2(¯rij/rij)6]
++κqiqj/rij},
+(5)
+where the 33-dimensional vector Xk = {uk,i}11
+i=1 (k =
+1, 2) defines the coordinates of atoms of the k-th ACN
+(vector uk,i specifies the coordinates of the i-th atom of
+the k-th molecule), distance between atoms rij = |u1,i −
+u2,j|.
+Here energy εij = √εiεj, equilibrium distances
+¯rij = ri + rj, qi is the electric charge of i-th atom (i, j =
+1, ..., 11), coefficient κ = 14.400611 eV˚A/e2. The values
+of the parameters εi, ri and qi are shown in Table I. All
+values of parameters of interaction potentials (1), (2), (3)
+and (5) are obtained from force field AMBER [55].
+The van der Waals interactions of the atoms of the
+ACN molecule with flat substrate are described by the
+LJ potential (m, l)
+W(X) =
+11
+�
+i=1
+Wi(zi) =
+11
+�
+i=1
+ǫi
+l − m[m(hi/zi)l − n(hi/zi)m],
+(6)
+where zi is the distance from i-th atom to the outer sur-
+face of the substrate, which is plane z = 0. Potential
+Wi(zi) in Eq. (6) is the interaction energy of i-th atom as
+a function of the distance to the substrate. This energy
+was found numerically for different substrates [56, 57].
+The calculations showed that interaction energy with
+substrate Wi(z) can be described with a high accuracy
+by LJ potential (6) with power l > k. Potential Wi(z)
+has the minimum Wi(hi) = −ǫi (ǫi is the binding en-
+ergy of the i-th atom with substrate). For the surface of
+the h-BN crystal l = 10, m = 4.25. The values of the
+parameters ǫi, hi, i = 1, ..., 11, are given in the table I.
+The 10-layer fragment of h-BN crystal was used to find
+values of this parameters. The interaction energy of an
+atom with a substrate was found as the sum of all LJ
+potentials (4) with parameters from the force field UFF
+[58].
+Thus,
+the Hamiltonian of a system of N
+ACN
+molecules located on the flat surface of h-BN crystal has
+the form
+H =
+N
+�
+n=1
+1
+2(M ˙Xn, ˙Xn) + P,
+(7)
+where the first term specifies the kinetic and the second
+– potential energy of the system
+P =
+N
+�
+n=1
+[V (Xn) + W(Xn)] +
+N−1
+�
+n=1
+N
+�
+k=n+1
+U(Xn, Xk). (8)
+Here the vector Xn = {un,i}11
+i=1 specifies the coordinates
+of the atoms of n-th ACN molecule, M is the diago-
+nal matrix of atom masses of the molecule, V (Xn) and
+
+a
+b5
+TABLE III: Values of the parameters of the potential of the valence angle X–Y–Z (2) for different atoms.
+X–Y–Z
+C–C–C
+C–C–N
+C–N–H
+C–N–C
+N–C–O
+N–C–C
+O–C–C
+ǫa (eV)
+3.643
+3.823
+2.781
+4.888
+4.932
+3.758
+4.625
+ϕ0 (◦)
+120
+117
+118
+128
+123
+116
+120
+TABLE IV: Values of the parameters of the potential of the dihedral angle X–Y–Z–W (3) for different atoms.
+X–Y–Z–W
+C–C–C–C
+C–C–C–N
+C–C–N–H
+C–C–N–C
+C–N–C–O
+C–N–C–CH3
+ǫd (eV)
+0.63
+0.63
+0.21
+0.21
+0.42
+0.42
+zd
+-1
+1
+-1
+-1
+-1
+1
+k
+1
+1
+2
+2
+1
+1
+W(Xn) are deformation energy and energy of interac-
+tion with the substrate of n-th molecule, U(Xn, Xk) is
+the interaction energy of n and k molecules.
+III.
+SECONDARY STRUCTURES OF ACN
+MOLECULES ON A FLAT SUBSTRATE
+The ACN molecule is achiral, but it becomes chiral
+when placed on a flat substrate.
+Depending on which
+side it lies on the substrate, it can be either right (when
+the benzol ring is located to the right of the dipole mo-
+ment vector of the peptide group
+⃗
+OH) or left (the ben-
+zol ring is located to the left). Two mirror-symmetrical
+isomers of the molecule are shown in Fig. 2. To trans-
+fer a molecule from one isomer to another, it must be
+partially torn off the substrate and be placed on the sub-
+strate with its other side. All this requires overcoming
+the energy barrier ∆E = 0.466 eV. The total energy of
+interaction with the h-BN substrate (desorption energy)
+Esub = 0.762 eV. Therefore, the spontaneous transition
+of the ACN molecule lying on the substrate from the L
+to the R form and vice versa is possible only at temper-
+atures T > 300 K. At lower temperatures, the molecule
+will always stay on the substrate, adjoining it with the
+same side, i.e. without changing the isomer type.
+To find the stationary state of the system of N ACN
+molecules lying on a flat h-BN substrate, it is necessary
+to find the state of the system with a minimum potential
+energy
+P → min : {Xn}N
+n=1.
+(9)
+The minimization problem (9) is solved numerically by
+the conjugate gradient method. Choosing the starting
+point of the minimization procedure, one can obtain all
+the main stationary states of the molecular system.
+Peptide groups of neighboring molecules can form hy-
+drogen bonds, creating dimers – see Fig. 4. The numeri-
+cal solution of the problem (9) shows that when molecules
+are located on a flat substrate, two types of dimers are
+FIG. 5:
+Typical secondary structures of homochiral ACN
+molecules on a flat substrate:
+(a) arc (number of atoms
+N = 15), (b) circle (N = 21), (c) single-beam spiral (N = 34),
+(d) two-beam spiral (N = 23 + 23), (e) three-beam spiral
+(N = 15 + 15 + 15), (f) nested circles (N = 12 + 24).
+possible: dimers of molecules of the same and different
+chirality. If a dimer is formed by identical isomers, then
+its binding energy is slightly higher: the hydrogen bond
+energy for RR and LL isomers is Ehb = 0.330 eV, and
+for RL and LR isomers Ehb = 0.322 eV. This is due to
+the fact that in this case the benzol rings C6H5 of the
+molecules are on the same side and they make a larger
+contribution to the interaction energy.
+The hydrogen
+bond angle also depends on the chirality of the dimer
+molecules. For dimer molecules of the same chirality, the
+angle between the dipole moments of the peptide groups
+forming a hydrogen bond is φhb = 17◦, and for molecules
+of different chirality φhb = 34◦.
+Chains of hydrogen bonds of molecules of the same chi-
+rality will always have benzol rings on one (outer) side,
+so they will twist in the opposite (inner) direction and
+have approximately the same curvature. On a plane, cir-
+cular arcs, circles, and spirals have such properties. The
+solution of the problem (9) has shown that on a flat sub-
+
+b
+a
+C6
+10
+20
+30
+40
+50
+−0.3
+−0.2
+−0.1
+1
+2
+3
+4
+5
+6
+Es (eV)
+N
+FIG. 6: Dependence of the specific energy of the secondary
+structure of the homochiral ACN molecules Es on the num-
+ber of molecules N for an arc, a circle, one-beam, two-beam,
+three-beam spiral and nested two circles (curves 1, 2, 3, 4,
+5 and 6). For the structure of two nested circles (curve 6),
+the dependence on the number of atoms of the outer circle is
+shown.
+strate molecules of the same chirality form stable shape
+structures with little changing curvature: arcs, spirals,
+circles – see Fig. 1 and 5. Left isomers form structures
+with a twist to the right, right – to the left.
+Hydrogen bond chains of N ≤ 16 ACN molecules of
+the same chirality form circular arcs of the same radius.
+The step of such a chain (the distance between the oxy-
+gen atoms of neighboring peptide groups) is a = 4.76 ˚A,
+the angle between neighboring links is ϕ = 162◦, the ra-
+dius of curvature by oxygen atoms is R = 15.2 ˚A – see
+Fig. 5 (a). The specific energy of the chain Es = E/N
+decreases monotonically with the growth of the number
+of molecules N – see Fig. 6. When the number of links
+is N > 16, the arcs cease to be stable; they either close
+and form circular chains, or touch their ends and form
+flat spirals – see Fig. 5 (b), (c).
+Stable cyclic chains can be formed from N
+≥ 5
+molecules of the same chirality. The dependence of the
+specific energy of the cyclic chain Es on the number of
+its links N is shown in Fig. 6. As can be seen from the
+figure, the most energetically favorable are cyclic chains
+of N = 20, 21, 22 links. Such chains form ring struc-
+tures with inner R1 = 27.3, 28.8, 30.5 and outer radii
+R2 = 40.4, 41.9, 43.6 ˚A.
+The dependence Es(N) for a one-beam spiral actually
+continues the dependence for an arc (see Fig. 6, curves
+1 and 3). Two-beam and three-beam spirals are bound
+states of arc structures.
+The specific energy of helical
+structures decreases monotonically with an increase in
+the number of molecules. For N > 27, helical structures
+are more energy efficient than ring structures (this is due
+to their denser structure).
+The most energy-efficient are nested structures of two
+circles with the number of atoms N = 14 + 26, 15+27,
+17+29 (the first number in the sum corresponds the num-
+ber of atoms in the inner circle, the second – in the outer
+FIG. 7: 2D crystals of ACN molecules on a flat substrate:
+(a) with parallel packing of linear chains of hydrogen bonds
+(periods ax = 9.81, ay = 10.24), (b) with antiparallel packing
+(ax = 9.79, ay = 20.43 ˚A). On the surface of a flat substrate,
+the chain of hydrogen bonds can be linear only if the L and
+R isomers alternate. A crystal is formed by 6 linear chains of
+24 molecules (the total number of molecules is N = 6 × 24).
+The flat substrate is shown in green.
+circle) – see Fig. 6, curve 6.
+If the chain of hydrogen bonds consists of a random
+sequence of isomers, then it will look like an irregular
+broken line. The chain has the shape of a straight zigzag
+only if there is a strict alternation of L and R isomers.
+In this case, the zigzag step (the distance between the
+oxygen atoms of neighboring molecules) is a = 4.85 ˚A,
+the zigzag angle is ϕ = 170◦. Such chains on a flat sub-
+strate surface can form two-dimensional regular lattices
+(2D crystals) with parallel and antiparallel packing of
+neighboring chains – see Fig. 7. With parallel packing,
+the crystal periods are ax = 9.81˚A, ay = 10.24˚A, the
+specific energy is Es = −0.325 eV. With antiparallel
+packing, periods are ax = 9.79˚A, ay = 20.43˚A, energy
+is Es = −0.322 eV.
+IV.
+SELF-ASSEMBLY OF MOLECULAR
+STRUCTURES
+Let us simulate the self-organization of the molecular
+structures of ACN molecules on the flat surface of the h-
+
+a
+b7
+100
+200
+300
+400
+500
+0.4
+0.6
+0.8
+1
+nhb
+(a)
+100
+200
+300
+400
+500
+1
+1.2
+1.4
+T (K)
+c
+(b)
+FIG. 8: Dependence of (a) the normalized number of hydro-
+gen bonds nhb and (b) the dimensionless heat capacity c on
+temperature T for a system of N = 1024 ACN molecules
+located on a flat substrate with the periodic square compu-
+tational cell of size 33 × 33 nm2. Solid (blue) lines show de-
+pendencies for a system of homochiral molecules, dotted (red)
+lines – dependencies for a racemic mixture of molecules. Verti-
+cal dotted straight lines correspond to temperatures T = 250,
+330, and 480 K.
+BN crystal. To do this, we take a square periodic cell of
+size 33 × 33 nm2 on the surface of the substrate and ran-
+domly place N = 1024 ACN molecules into it. Then we
+immerse this molecular system in a Langevin thermostat
+of temperature T and numerically simulate the dynamics
+of the system during the time t = 10 ns. To do this, we
+numerically integrate the system of Langevin equations
+M ¨Xn = −
+∂
+∂Xn
+H − ΓM ˙Xn − Ξn,
+n = 1, ..., N,
+(10)
+where Γ
+=
+1/tr is the friction coefficient,
+Ξn
+=
+{ξn,i,k}11, 3
+i=1,k=1 is 33-dimensional vector of normally dis-
+tributed random Langevin forces with the following cor-
+relations:
+⟨ξn1,i,k(t1)ξn2,j,l(t2)⟩ = 2MikBT Γδn1n2δijδklδ(t1 − t2).
+Here Mi is mass of i-th atom of ACN molecule, kB is
+Boltzmann constant, T is temperature of the Langevin
+thermostat (temperature of the substrate), numbers
+n1, n2 = 1, ..., N, i, j = 1, ..., 11, k, l = 1, 2, 3.
+The parameter tr characterizes the intensity of energy
+exchange between the molecular system and the ther-
+mostat. Simulation of the dynamics of ACN molecules
+on an h-BN sheet, taking into account the mobility of
+the sheet atoms, makes it possible to estimate the relax-
+ation time tr ∼ 100 ps. For the convenience of numer-
+ical integration, we will use a smaller value tr = 10 ps.
+FIG. 9: Structure of N = 1024 homochiral ACN molecules
+appearing on the flat surface of the substrate at T = 240K.
+The straight lines show the boundaries of the periodic calcu-
+lation cell of size 33 × 33 nm2. The substrate surface is not
+shown.
+This makes it possible to significantly reduce the time of
+numerical integration, which is sufficient to obtain reli-
+able average values. After the dynamics of the molec-
+ular system reaches the steady state, we will find the
+time averages of the system energy ¯E(T ) and the num-
+ber of hydrogen bonds ¯Nhb(T ). We assume that two ACN
+molecules form a hydrogen bond if their interaction en-
+ergy is greater than half of the hydrogen bond energy:
+U(X1, X2) < −Ehb/2 = −0.16 eV.
+The state of the system can be conveniently character-
+ized by its dimensionless heat capacity
+c =
+1
+33NkB
+d ¯E(T )
+dT
+,
+(11)
+and the normalized number of hydrogen bonds nhb =
+¯Nhb(T )/N. The dependence of these quantities on tem-
+perature is shown in Fig. 8.
+Numerical simulation shows the existence of three
+characteristic temperature values T1 < T2 < T3.
+At
+T < T1 = 250K the molecules on a flat substrate form a
+stable system of chains of hydrogen bonds. Here, almost
+every molecule participates in the formation of one hy-
+drogen bond (number nhb ≈ 1). The dimensionless heat
+capacity of the system is c = 1. At T1 < T < T2 = 330K,
+a slight decrease in the number of nhb bonds and a
+monotonous increase in heat capacity begin to occur –
+the process of melting of hydrogen bond chains begins.
+At a temperature T > T2, the number of hydrogen bonds
+
+8
+FIG. 10: Structure of N = 1024 homochiral ACN molecules
+appearing on the flat surface of the substrate at T = 300K.
+The straight lines show the boundaries of the periodic calcu-
+lation cell of size 33 × 33 nm2.
+decreases in proportion to the increase in temperature,
+and the heat capacity reaches a constant value c ≈ 1.23.
+Here we have a melt of short chains of hydrogen bonds
+(the average length of the chains decreases proportion-
+ally to the increase in temperature). At T > T3 = 480K,
+individual molecules can already be detached from the
+substrate – desorption of molecules begins.
+The structure of the resulting system of hydrogen
+bond chains depends on the isomeric composition of the
+molecules. If all molecules are homochiral (only one iso-
+mer is present), then secondary structures form on the
+surface of the substrate.
+These structures are circular
+and spiral hydrogen bond chains – see Figs. 9 and 10. The
+substrate surface become optically active, the left isomers
+form chains with a right twist, and the right ones with a
+left twist. As can be seen from Fig. 9 for T = 240K on
+a flat substrate all possible circular secondary structures
+(arcs, circles, spirals) are formed (Fig. 5). An increase in
+temperature leads, first of all, to the destruction of spiral
+structures. As a result, the number of cyclic chains of
+average radius increases since they are the most stable –
+see Fig. 10.
+If the isomers of molecules are taken randomly, we get
+their racemic mixture. In this case, no secondary struc-
+tures are formed. There are only curvilinear chains of
+random shape – see Fig. 11.
+FIG. 11: Structure formed on the flat surface of the sub-
+strate at T = 240K from a racemic mixture of N = 1024
+ACN molecules. The straight lines show the boundaries of
+the periodic calculation cell of size 33 × 33 nm2.
+V.
+MELTING OF 2D CRYSTALS
+To simulate the dynamics of 2D crystals of ACN
+molecules on a flat h-BN substrate, consider a crystal
+formed by 22 linear chains of hydrogen bonds of 48
+molecules (total number of molecules N = 22 × 48 =
+1056). A crystal with parallel chain packing has dimen-
+sions of 23.3 × 22.5 nm2.
+We place the crystal in the
+center of the calculated periodic cell size 23.5×22.6 nm2.
+In this case, the chains of hydrogen bonds located par-
+allel to the x axis are closed, and the first chain begins
+to come into contact with the last one – the 2D crystal
+form a dense packing on the substrate that has no edges
+and has normalized number of hydrogen bonds nhb = 1
+(the number of hydrogen bonds is equal to the number of
+molecules). To simulate a rectangular crystal with edges
+(square crystallite), we take a periodic computational cell
+of size 47 × 45.2 nm2. In this case, the 2D crystal (crys-
+tallite) covers only 25% substrate surface, the chains of
+hydrogen bonds are not closed, the normalized number
+of hydrogen bonds is nhb = (N − 22)/N = 0.979.
+Then we numerically integrate the system of equations
+of motion (10) with the initial condition corresponding to
+the stationary state of the crystal. At different thermo-
+stat temperatures, we find the average values of the sys-
+tem energy ¯E(T ), the normalized number of hydrogen
+bonds nhb, and the fraction of molecules that left the
+substrate from the first layer p (the fraction of molecules
+located on the substrate at distance z > 5 ˚A). Next, using
+
+9
+100
+200
+300
+400
+500
+0.4
+0.6
+0.8
+1
+1
+2
+(a)
+nhb
+100
+200
+300
+400
+500
+1
+1.5
+2
+2.5
+3
+4
+(b)
+c
+100
+200
+300
+400
+500
+0
+5
+10
+6
+5
+T (K)
+p (%)
+(c)
+FIG. 12: Dependence of (a) the normalized number of hy-
+drogen bonds nhb, (b) the dimensionless heat capacity c and
+(c) the fraction of molecules that left the first layer p on the
+temperature T for a 2D crystal of N = 1056 ACN molecules
+located on a flat substrate with periodic computational cell
+of size 46.6 × 45 nm2 (curves 1, 3, 5; 25% substrate coverage)
+and 23.3 × 22.5 nm2 (curves 2, 4 , 6; 100% substrate cover-
+age). Vertical dotted straight lines show temperature values
+T = 295, 335, 365, 415, 445 and 470 K.
+the formula (11), we find the temperature dependence of
+the heat capacity of the molecular system c.
+The dependence of nhb, c, and p on the thermostat
+temperature (on the substrate temperature) T is shown
+in Fig. 12. As can be seen from the figure, at 25% cover-
+age of the substrate, the square crystallite begins to melt
+at a temperature of T1 = 295K. At T < T1, the crystal
+structure is preserved, the number of bonds, heat capac-
+ity, and density of the first layer of the substrate coating
+practically do not change with increasing temperature:
+nhb(T ) ≡ nhb(0), c(T ) ≡ 1, p(T ) ≡ 0. Crystallite melt-
+ing occurs in the temperature range T1 < T < T2 where
+the upper temperature is T2 = 365 K. Here, a slight de-
+crease in the number of bonds and an increase in heat
+capacity begin to occur. The heat capacity reaches its
+maximum value at the temperature Tm = 335 K. In the
+temperature interval [T1, T2], an increasing destruction
+of the edges of the initial crystallite occurs. The ends of
+the chains peel off from the central part of the crystallite,
+100
+200
+300
+400
+500
+1
+1.5
+2
+2.5
+c
+1
+2
+3
+4
+T (K)
+FIG. 13: Dependence of the dimensionless heat capacity c on
+temperature T for a 2D square crystallite of N = 480, 1056,
+2376 of the ACN molecules located on a flat substrate with
+periodic computational cell of the size 31.2× 30.68, 46.6× 45,
+70.2 × 67.5 nm2 (curves 1, 3, 5; 25% substrate coverage) and
+for infinite 2D crystal (curve 4). Vertical dotted straight lines
+mark the temperatures T = 315, 335, 365, and 445 K.
+then break off and go to the free part of the substrate.
+As a result of this ”continuous” melting, the crystallite
+transforms into a melt consisting of short chains of hy-
+drogen bonds, uniformly covering the entire substrate.
+For T > T2, the number of hydrogen bonds decreases
+in proportion to the increase in temperature, while the
+heat capacity remains almost constant: c ≈ 1.4. It can
+be concluded that a complete transition of the molecu-
+lar system from a low-energy and low-entropy crystalline
+state to a high-energy and high-entropy liquid state has
+taken place. As a result of this transition, the substrate
+is uniformly covered with a ”solution” of short chains of
+hydrogen bonds. The chain lengths decreases monotoni-
+cally with increasing temperature (on the Fig. 12 (a) this
+manifests itself in a monotonous decrease in the number
+of bonds). All molecules remain directly adjacent to the
+substrate (p = 0), insignificant desorption is observed
+only at T > 480 K – see Fig. 12 (c).
+With 100% coverage of the substrate, due to the ab-
+sence of edges in the crystal, its melting occurs at higher
+temperatures and happens according to a different sce-
+nario.
+Here melting also occurs ”continuously” in the
+temperature interval [T1, T2], where T1 = 415, T2 =
+470 K. The peak of the heat capacity at Tm = 445 K
+becomes more pronounced. Melting occurs due to the
+expulsion of some molecules from the first layer to the
+second, which manifests itself in a monotonous increase
+in the fraction of displaced molecules p at T1 < T < T2
+– see Fig. 12 (c).
+As a result, the density of the first
+layer during melting decreases by 11%, and after melting
+a dense melt of molecules is formed on the substrate, in
+which 11% of the molecules are located on the second
+layer from the substrate.
+Conventionally, the temperature value Tm, at which
+
+10
+the heat capacity has reached its maximum value, can
+be considered as the melting temperature of a 2D crystal.
+However melting occurs not discretely, but continuously
+in the temperature interval [T1, T2]. The value Tm is in
+the center of this interval.
+To study the dependence of the melting temperature
+on the crystallite size, we analyze numerically the melt-
+ing of 2D square crystallite of N = 480 and 2376 ACN
+molecules placed on a flat substrate with periodic com-
+putational cell of size 31.2 × 30.68 and 70.2 × 67.5 nm2.
+Our results show that melting occurs continuously for all
+crystallite sizes. When the crystallite size is increased,
+the melting interval [T1, T2] shifts to the right and, in
+the limit, coincides with the melting temperature inter-
+val of a 2D crystal with 100% coverage of the substrate,
+as shown in Fig. 13.
+Let us note that the continuum melting scenario also
+takes place for 2D n-alkane crystals lying on a flat sur-
+face of graphite [42].
+This allows us to conclude that
+the quasi-continuous melting scenario is a characteristic
+feature of 2D systems of molecules adsorbed by a flat
+surface.
+VI.
+CONCLUSION
+Our numerical simulations of the dynamics of the sys-
+tem of acetanilide molecules have revealed that the struc-
+tures achiral in three-dimensional space become chiral
+when being placed on a flat substrate: Depending on the
+side it touches the substrate, the molecule has two iso-
+mers L and R. The homochirality of the molecules leads
+to the appearance of stable secondary structures stabi-
+lized by hydrogen bonds on a flat substrate in the form
+of arc, cyclic, and helical chains of hydrogen bonds and
+their complexes.
+Hydrogen-bond chains of N ≤ 16 molecules of the same
+chirality form circular arcs of the same radius.
+When
+the number of molecules in the chain is N > 16, the
+arcs becomes unstable, they either collapse into circles,
+or touch their ends and form flat spirals.
+In addition
+to single-beam spirals, two- and three-beam spirals may
+exist being bound states of arc chains.
+Stable cyclic chains can be formed from N
+≥ 5
+molecules of the same chirality.
+The most energeti-
+cally favorable are cyclic chains of 20, 21 and 22 links.
+The structures of two circles with the number of atoms
+N = 14 + 26, 15 + 27, 17 + 29 (where the first number of
+the sum is the number of atoms in the inner circle, and
+the second number is the number of atoms in the outer
+circle) are more energy efficient.
+If the chain of hydrogen bonds consists of a random
+sequence of isomers, it look like an irregular broken line.
+A chain can take the form of a rectilinear zigzag only if
+there is a strict alternation of the L and R isomers. Such
+chains can form regular two-dimensional crystals with
+parallel and antiparallel packing of the adjacent chains.
+Simulation of the dynamics of a system of molecules
+shows that the homochirality of molecules(the presence
+of only one isomer) leads to the appearance of stable sec-
+ondary structures on the surface of the substrate, i.e. to
+the appearance of chains of hydrogen bonds in the form
+of arcs, circles and spirals.
+As a result, the substrate
+surface becomes optically active, the left isomers form
+chains with a right twist, and the right ones form chain
+with a left twist. At temperature T ≤ 240 K, all possible
+secondary structures are formed on the substrate. An
+increase in temperature leads, first of all, to the disinte-
+gration of spiral structures. As a result, the number of
+more stable circular chains increases.
+If the molecules on the substrate form a racemic mix-
+ture, no regular secondary structures are formed, so only
+curvilinear chains of hydrogen bonds of random shape
+can appear. Thus, our results demonstrate the impor-
+tance of homochirality (chiral purity) of biomolecules for
+the formation of stable secondary molecular structures.
+Numerical simulations of the dynamics of a 2D crystal
+of the ACN molecules shows that the scenario of crystal
+melting depends on the density of its coverage of the
+substrate.
+At 25% coverage of the substrate the melting occurs
+”continuously” in the temperature interval [T1, T2] from
+the edges of the initial crystallite (for crystallite of 1056
+ACN molecules temperatures T1 = 295, T2 = 365 K).
+The ends of the chains peel off from the central part of
+the crystallite, then break off and go to the free part
+of the substrate. As a result, the crystallite transforms
+into a melt consisting of short chains of hydrogen bonds,
+uniformly covering the entire substrate. When the size of
+the crystallite is increased, the melting interval [T1, T2]
+shifts to the right and, in the limit, will coincides with
+the melting temperature interval of infinite crystal with
+100% coverage of the substrate.
+For 100% coverage of the substrate with no crystal
+edges, the melting occurs at higher temperatures 415 ÷
+470 K by a shift of some molecules from the first to the
+second layer in the substrate. As a result, the density of
+the first layer during melting decreases by 11%, and after
+melting, 11% of the molecules move to the second layer
+formed on the substrate.
+A continual melting scenario (the presence of a melting
+temperature interval)has also been found in 2D n-alkane
+crystals lying on a flat surface of graphite [42]. All this
+leads us to conclusion that the quasi-continuous melt-
+ing scenario is a characteristic feature of 2D systems of
+molecules adsorbed by a flat surface.
+ACKNOWLEDGMENTS
+AVS acknowledges the use of the computational facilities
+provided by the Interdepartmental Supercomputer Cen-
+ter of the Russian Academy of Science. YSK acknowl-
+edges a support from the Australian Research Council
+(grant DP210101292) and the Strategic Fund of the Aus-
+tralian National University.
+
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+
diff --git a/WdE1T4oBgHgl3EQfJAOq/content/tmp_files/load_file.txt b/WdE1T4oBgHgl3EQfJAOq/content/tmp_files/load_file.txt
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@@ -0,0 +1,1171 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf,len=1170
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='02947v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='mes-hall]  7 Jan 2023  Chiral organic molecular structures supported by multilayer surfaces  Alexander V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Savin1, 2, ∗ and Yuri S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Kivshar3, †  1Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow 119991, Russia  2 Plekhanov Russian University of Economics, Moscow 117997, Russia  3Nonlinear Physics Center, Department of Fundamental and Theoretical Physics,  Research School of Physics, Australian National University, Canberra ACT 2601, Australia  We study numerically the dynamics of acetanilide (ACN) molecules placed on a flat surface of  a multilayer hexagonal boron nitride structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' We demonstrate that the ACN molecules, being  achiral in three dimensions, become chiral after being placed on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Homochirality of the  ACN molecules leads to stable secondary structures stabilized by hydrogen bonds between peptide  groups of the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Numerical simulations of systems of such molecules reveal that the structure  of the resulting hydrogen-bond chains depends on the isomeric composition of the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' If all  molecules are homochiral (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' only one isomer is present), they form secondary structures (chains  of hydrogen bonds in the shapes of arcs, circles, and spirals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' If the molecules at the substrate  form a racemic mixture, then no regular secondary structures appear, and only curvilinear chains of  hydrogen bonds of random shapes can emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' A hydrogen-bond chain can form a straight zigzag  only if it has an alternation of isomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Such chains can create two-dimensional (2D) regular lattices,  or 2D crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The melting scenarios of such 2D crystals depend on density of its coverage of the  substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' At 25% coverage, melting occurs continuously in a certain temperature interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' For a  complete coverage, melting occurs at 415 ÷ 470 K due to a shift of 11% of all molecules into the  second layer of the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  INTRODUCTION  Two-dimensional (2D) materials such as graphene (G)  and hexagonal boron nitride (h-BN) have attracted a lot  of attention due to their unique electronic [1–3] and me-  chanical [4–7] properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Currently, heterogeneous lay-  ered materials of such 2D materials, which can exhibit  various novel physical properties compared to their ho-  mogeneous counterparts, became a special focus of such  studies [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' For example, the use of hybrid G/h-BN  structures allows to achieve some desired electronic prop-  erties [11, 12] and also reduce significantly friction be-  tween the layers [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  In general, such multilayer het-  erostructures are stabilized by van der Waals (vdW) in-  teractions between atoms of the neighboring layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The concept of vdW heterostructures can be extended  to the integration of 2D materials with molecular struc-  tures of different dimensions, such as nD/2D heterostruc-  tures, where n stands for the dimension (n = 0, 1, or  3) [14], describing flat molecules (n = 0), polymer chains  (n = 1), or three-dimensional molecular objects (n = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  For molecules and molecular chains with benzol rings,  flat layers of G and h-BN are strong adsorbents [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Theoretical studies reveal that molecules adsorbed on G  and h-BN surfaces through non-covalent interactions can  modify the properties of the surface as solid-liquid, solid-  air, or solid-vacuum interfaces [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' A strong stacking  interaction with a flat substrate allows such molecules  residing on a surface creating stable 2D supra-molecular  systems, as shown in the characteristic example of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 1  ∗asavin@chph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='ras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='ru  †yuri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='kivshar@anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='au  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 1:  Example of the molecular structures studied in  this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' A spiral structure of 52 R-isomers of the ACN  molecules C6H5–NHCO–CH3 is stabilized on a h-BN multi-  layer surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  By now,  the behavior of multifunctional organic  molecules placed on ideal metal surfaces has been stud-  ied in detail [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Such organometallic systems may ex-  hibit a variety of different structures induced by the sub-  strate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' In many cases, complex organic molecules (such  as carboxylic acids, amino acids, anhydrides, and ring  systems) become self-organized on metal surfaces creat-  ing ordered super-structures stabilized by inter-molecular  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Chirality is of a particular interest that can  appear for initially achiral metal surfaces by adsorbing  organic molecules [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Similar behavior is expected for  organic molecules adsorbed on flat surfaces of G and h-  2  BN molecular structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  In this paper, we study nu-  merically the formation of supra-molecular complexes by  acetanilide molecules placed on the surface of a multilayer  h-BN sheet, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 1, serving as an introductory figure  explaining our problem and results discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  For poly-cyclic aromatic hydrocarbons (for molecules  of benzol C6H6, naphthalene C10H8, pyrene C16H10,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='),  graphene is a strong adsorbent [15, 17, 19, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The in-  teraction of graphene with such molecules often causes  specific reactions that can be used in new types of sensors  [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Non-covalent functionalization of the graphene  surface can significantly expand its potential range of  applications [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  It has been shown experimen-  tally [25, 28] that benzol and pyrene molecules adsorbed  on graphene form densely packed monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  For acetanilide (ACN, C6H5NHCOCH3) and parac-  etamol (PCM, C6H4OHNHCOCH3) molecules, graphene  and hexagonal boron nitride are also strong adsorbents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Due to possible medical applications, much attention has  been paid to modeling the adsorption of PCM molecules,  which is a strong analgesic, on h-BN sheets and nan-  otubes [18, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' It has been shown in [30] that function-  alized graphene can be used as a highly sensitive parac-  etamol detection sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  An example of a 1D/2D heterostructure is a graphene  sheet with adsorbed Kevlar chains, kevlar-functionalized  graphene [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The presence of planar C6H4 benzol rings  and NHCO peptide groups in the polymer chain [–C6H4–  NHCO–]∞ provides a strong non-covalent (vdW) inter-  action of the chain with G and h-BN sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Such chains  on the surface of sheets G and h-BN will lie parallel to  the surface and form chains of hydrogen bonds between  each other · · ·HNCO· · ·HNCO· · ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Such 3D/2D heterostructures can form thin metal lay-  ers on the G and h-BN surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' In particular, numerical  modeling suggests that aluminum can form stable two-  layer structures on the G surface [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  It has been shown in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' [33–42] that n-alkanes (lin-  ear polymer chains CH3(CH2)lCH3 with internal units  2 ≤ l ≤ 388) form a dense ordered monolayer of parallel  linear chains on the graphite (graphene) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Inter-  est in alkanes is due to the fact that they belong to the  simplest families of polymer molecules, which members  of which differ only in their length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Placing linear poly-  mer chains on a flat graphite surface causes them to self-  assemble into 2D crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The self-assembly mechanism  depends on the chain length, temperature, and the level  of coverage of the substrate with chains [36, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Adsorption by the surface of a long single-chain  polyethylene molecule leads to its two-dimensional crys-  tallization – it passes from the form of a three-  dimensional globule into the form of a parallel folded  linear chain lying in the plane parallel to the substrate  surface [43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Thus, the flat surfaces of the G and h-BN substrates  create a 2D platform for flat molecules adsorbed on them  (for poly-cyclic aromatic hydrocarbons, for ACN and  PCM molecules, Kevlar chains, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=')  and linear poly-  mer molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' At low temperatures, the molecules move  along the sheet, remaining parallel to its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' They  interact with each other and form two-dimensional supra-  molecular structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Such molecular adsorbents are con-  venient systems for studying phase transitions caused by  freedom restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  To date, only phase transitions in monolayers of n-  alkanes have been well studied [33, 42, 46–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Modeling  and experimental studies show that a monolayer always  undergoes a transition from a solid-crystalline 2D phase  to a liquid phase (the transition occurs at a temperature  significantly lower than the desorption temperature of  molecules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The melting scenario depends on the poly-  mer chain length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The melting temperature increases  monotonically with chain length, so for pentane, hep-  tane and nonane (l = 3, 5, 7) the melting temperature  is Tm = 92, 178 and 255K [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  A characteristic fea-  ture associated with the adsorption of molecules is the  continuity of melting of a 2D crystal – melting occurs in  the temperature interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Thus, for the longest synthe-  sized monodisperse alkane C390H782 (l = 388), continu-  ous melting occurs at 393 < T < 484 K [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Despite the large number of theoretical and experimen-  tal works on phase transitions in adsorbed monolayers of  alkanes and their derivatives, as far as we know, there  are no works on modeling phase transitions in adsorbed  monolayers of ACN, PCM, and Kevlar (para-aramid)  molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  A detailed description of adsorption simu-  lation methods is given in [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Unlike alkanes, the 2D  structures of these molecules adsorbed on a flat surface  are associated with the presence of chains of hydrogen  bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Molecules including amide and hydroxyl groups  can create 2D lattices and extended hydrogen chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  We notice that the ACN molecules are often consid-  ered as a model system with chains of hydrogen bonds  between HNCO peptide groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Acetanilide crystallizes  into an orthorhombic structure with ribbons of molecules  linked by hydrogen bonds [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The chains of hydrogen  bonds that stabilize the crystal structure are very similar  to the chains that stabilize the alpha-helices and beta-  sheets of proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Therefore, ACN was used as a model  for modeling the energy transfer of vibrations of peptide  groups along hydrogen bond chains in proteins [51–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Living matter, unlike non-living matter, has chiral pu-  rity: all proteins consist of left-handed amino acids, while  DNA and RNA are built on right-handed ribose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' In ex-  periments on abiogenic synthesis, left and right isomers of  sugars and proteins are formed in equal proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' It is  believed that if you try to build proteins from such a mix-  ture, they will not be able to fold into a stable form and  therefore will not work as enzymes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' In three dimensions,  the need for chiral purity to form stable protein struc-  tures requires complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The situation is dra-  matically simplified if we move from three-dimensional  space to two-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Such a transition can be made  if flat molecules are placed on a flat molecular sheet of  graphene or hexagonal boron nitride (h-BN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Such a non-  valent modification of the sheet surface actually creates  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 2: When an ACN molecule is placed on a flat surface  of a multilayer h-BN structure, it may create two isomers  with the mirror symmetry (shown by a straight line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Vectors  connecting the oxygen atom with the hydrogen atom of each  peptide group show the dipole moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' For the L-isomer,  the benzol ring is to the left of this vector, for the R-isomer  it is to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Gray balls stand for carbon atoms, white  balls – hydrogen, blue – nitrogen, red – oxygen, and green –  bromine atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  a 2D world for the flat molecules placed on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' At low  temperatures, the molecules move along the sheet all the  time remaining parallel to its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' On the surface,  they can form complex two-dimensional structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  An ACN molecule that is achiral in 3D becomes chiral  after being placed on a flat substrate (the chirality de-  pends on which side it lays on the surface of the sheet)  – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' It will be shown that the homochirality of  ACN molecules leads to the appearance on the surface  of the sheet of stable secondary structures stabilized by  hydrogen bonds: cyclic and spiral chains and complexes  of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Modeling the formation of such structures will  make it possible to demonstrate the necessity of homochi-  rality (chiral purity) of biomolecules for the formation of  stable secondary molecular structures from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  As a flat substrate, we consider a surface of a multilayer  h-BN structure, and for molecules we consider acetanilide  (ACN) C6H5NHCOCH3, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The  presence of a planar benzol ring C6H5 and a planar pep-  tide group (PG) HNCO leads to large interaction energy  of the molecule with the substrate, Esub = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='762 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Molecules can create chains of hydrogen bonds between  their peptide groups OCNH· · ·OCNH· · ·OCNH· · ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Such  chains of hydrogen bonds stabilize the secondary struc-  tures of the protein molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' In the next section,  we describe our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Section III is devoted to the study  of secondary structures of the ACN molecules placed on  a flat substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Self-assembly of such structures is simu-  lated numerically in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' V we analyze  melting of 2D crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Section VI concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 3: Construction of a coarse-grained model of the ACN  molecule: (a) full-atomic view of the molecule, (b) coarse-  grained model (the used numbering of the united atoms is  shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  MODEL  For modeling of the dynamics of a system of the ACN  molecules, we will use the united-atoms approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Let us consider the molecular groups CH and CH3 as  united atoms whose centers coincide with the centers of  carbon atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' In this approximation, the ACN molecule  is described as a system of 11 united atoms – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The values of the masses of the united atoms are shown  in the table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  To model a ACN molecule, we use the force field in  which distinct potentials describe the deformation of va-  lence bonds and valence, torsion and dihedral angles, and  non-valence atomic interacts [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' In this model, the de-  formation energy of the valence bonds C–CH, CH–CH,  C–N, N–H, C=O and C–CH3 is described by the har-  monic potential:  V (ρ) = 1  2K(ρ − ρ0)2,  (1)  where ρ and ρ0 are current and equilibrium bond lengths,  K is the bond stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The values of potential parame-  ters for various valence bonds are presented in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  TABLE I:  Masses and parameters of interaction potentials  for united atoms X of the ACN molecule: i – atom number,  Mi – atom mass (mp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='6603 × 10−27 kg – proton mass), εi  and ri are the energy and radius of the LJ interaction, qi is  the electric charge of the atom, ǫi and hi are the energy and  equilibrium distance for the interaction of an atom with a flat  substrate (with a crystal surface h- BN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  X  C  CH  N  H  C  O  CH3  i  1  2,3,4,5,6  7  8  9  10  11  Mi (mp)  12  13  14  1  12  16  15  εi (meV)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='284  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='284  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='434 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='284 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='344 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='284  ri (˚A)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='861  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='861  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='621 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='861 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='711 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='861  qi (e)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='066  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='580 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='035  ǫi (meV)  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='5  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='3  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='7  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='3  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='5  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='8  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='3  hi (˚A)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='52  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='44  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='43  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='08  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='52  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='36  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='44  (a)  H  (b)  CH  6  8  5  CH3  4  C  N  1  9  11  3  2  10R  T4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 4: Dimer of (a) RR and (b)LR isomers of the ACN  molecule on a flat substrate (shown in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The vectors  show the dipole moments of the peptide groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Hydrogen  bond energy of identical isomers Ehb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='330 eV, angle be-  tween dipole moments φhb = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='05◦, for different isomers  Ehb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='322 eV, φhb = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='98◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Energies of the deformation of the valence angles X–  Y–Z are described by the potential  U(u1, u2, u3) = U(φ) = ǫa(cos φ − cos φ0)2,  (2)  where the cosine of the valence angle φ is defined as  cos φ = −(v1, v2)/ρ1ρ2, with the vectors v1 = u2 − u1,  v2 = u3 − u2 and bond lengths ρ1 = |v1|, ρ2 = |v2|, the  vectors u1, u2, u3 specify the coordinates of the atoms  forming the valence angle φ, φ0 is the value of equilib-  rium valence angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The values of potential parameters  used for various valence angles are presented in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Deformation of dihedral angles are described by the  potential  W(u1, u2, u3, u4) = ǫd(1 + zd cos kϕ),  (3)  where cos ϕ = (w1w2)/|w1||w2|, with the vectors w1 =  (u2 − u1) × (u3 − u2), w2 = (u3 − u2) × (u4 − u3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The  values of potential parameters used for various dihedral  angles are presented in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  For pairs of atoms Xi,Xj (i, j are the numbers of atoms  in the molecule) participating in the formation of the  dihedral angle Xi–Y–Z–Xj, their non-valence interaction  is also taken into account described by the Lennard-Jones  (LJ) potential  W0(r) = ε0[(r0/r)12 − 2(r0/r)6],  (4)  with halved interaction energy ε0 = √εiεj/2, where r is  current distance between interacting atoms, equilibrium  distance r0 = ri + rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The LJ interaction of an oxygen  atom (i = 10) with two combined atoms CH (i = 2, 6)  was also taken into account with interaction energy ǫ0 =  TABLE II: Values of the harmonic potential parameters (1)  for different valence bonds X—Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  X—Y  C–CH, CH–CH  C–N  N–H  C=O C–CH3  K (N/m)  469  427  434  570  317  ρ0 (˚A)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='39  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='405  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='007  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='222  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='505  √ε2ε10 and equilibrium distance r0 = r2+r10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Parameter  values εi and ri are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The interaction of two ACN molecules is described by  the potential  U(X1, X2) =  11  �  i=1  11  �  j=1  {εij[(¯rij/rij)12 − 2(¯rij/rij)6]  +κqiqj/rij},  (5)  where the 33-dimensional vector Xk = {uk,i}11  i=1 (k =  1, 2) defines the coordinates of atoms of the k-th ACN  (vector uk,i specifies the coordinates of the i-th atom of  the k-th molecule), distance between atoms rij = |u1,i −  u2,j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Here energy εij = √εiεj, equilibrium distances  ¯rij = ri + rj, qi is the electric charge of i-th atom (i, j =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=', 11), coefficient κ = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='400611 eV˚A/e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The values  of the parameters εi, ri and qi are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' All  values of parameters of interaction potentials (1), (2), (3)  and (5) are obtained from force field AMBER [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The van der Waals interactions of the atoms of the  ACN molecule with flat substrate are described by the  LJ potential (m, l)  W(X) =  11  �  i=1  Wi(zi) =  11  �  i=1  ǫi  l − m[m(hi/zi)l − n(hi/zi)m],  (6)  where zi is the distance from i-th atom to the outer sur-  face of the substrate, which is plane z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Potential  Wi(zi) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' (6) is the interaction energy of i-th atom as  a function of the distance to the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' This energy  was found numerically for different substrates [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The calculations showed that interaction energy with  substrate Wi(z) can be described with a high accuracy  by LJ potential (6) with power l > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Potential Wi(z)  has the minimum Wi(hi) = −ǫi (ǫi is the binding en-  ergy of the i-th atom with substrate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' For the surface of  the h-BN crystal l = 10, m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The values of the  parameters ǫi, hi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=', 11, are given in the table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The 10-layer fragment of h-BN crystal was used to find  values of this parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The interaction energy of an  atom with a substrate was found as the sum of all LJ  potentials (4) with parameters from the force field UFF  [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Thus,  the Hamiltonian of a system of N  ACN  molecules located on the flat surface of h-BN crystal has  the form  H =  N  �  n=1  1  2(M ˙Xn, ˙Xn) + P,  (7)  where the first term specifies the kinetic and the second  – potential energy of the system  P =  N  �  n=1  [V (Xn) + W(Xn)] +  N−1  �  n=1  N  �  k=n+1  U(Xn, Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' (8)  Here the vector Xn = {un,i}11  i=1 specifies the coordinates  of the atoms of n-th ACN molecule, M is the diago-  nal matrix of atom masses of the molecule, V (Xn) and  a  b5  TABLE III: Values of the parameters of the potential of the valence angle X–Y–Z (2) for different atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  X–Y–Z  C–C–C  C–C–N  C–N–H  C–N–C  N–C–O  N–C–C  O–C–C  ǫa (eV)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='643  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='823  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='781  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='888  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='932  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='758  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='625  ϕ0 (◦)  120  117  118  128  123  116  120  TABLE IV: Values of the parameters of the potential of the dihedral angle X–Y–Z–W (3) for different atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  X–Y–Z–W  C–C–C–C  C–C–C–N  C–C–N–H  C–C–N–C  C–N–C–O  C–N–C–CH3  ǫd (eV)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='42  zd  1  1  1  1  1  1  k  1  1  2  2  1  1  W(Xn) are deformation energy and energy of interac-  tion with the substrate of n-th molecule, U(Xn, Xk) is  the interaction energy of n and k molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  SECONDARY STRUCTURES OF ACN  MOLECULES ON A FLAT SUBSTRATE  The ACN molecule is achiral, but it becomes chiral  when placed on a flat substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Depending on which  side it lies on the substrate, it can be either right (when  the benzol ring is located to the right of the dipole mo-  ment vector of the peptide group  ⃗  OH) or left (the ben-  zol ring is located to the left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Two mirror-symmetrical  isomers of the molecule are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' To trans-  fer a molecule from one isomer to another, it must be  partially torn off the substrate and be placed on the sub-  strate with its other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' All this requires overcoming  the energy barrier ∆E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='466 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The total energy of  interaction with the h-BN substrate (desorption energy)  Esub = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='762 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Therefore, the spontaneous transition  of the ACN molecule lying on the substrate from the L  to the R form and vice versa is possible only at temper-  atures T > 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' At lower temperatures, the molecule  will always stay on the substrate, adjoining it with the  same side, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' without changing the isomer type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  To find the stationary state of the system of N ACN  molecules lying on a flat h-BN substrate, it is necessary  to find the state of the system with a minimum potential  energy  P → min : {Xn}N  n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  (9)  The minimization problem (9) is solved numerically by  the conjugate gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Choosing the starting  point of the minimization procedure, one can obtain all  the main stationary states of the molecular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Peptide groups of neighboring molecules can form hy-  drogen bonds, creating dimers – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The numeri-  cal solution of the problem (9) shows that when molecules  are located on a flat substrate, two types of dimers are  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 5:  Typical secondary structures of homochiral ACN  molecules on a flat substrate:  (a) arc (number of atoms  N = 15), (b) circle (N = 21), (c) single-beam spiral (N = 34),  (d) two-beam spiral (N = 23 + 23), (e) three-beam spiral  (N = 15 + 15 + 15), (f) nested circles (N = 12 + 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  possible: dimers of molecules of the same and different  chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' If a dimer is formed by identical isomers, then  its binding energy is slightly higher: the hydrogen bond  energy for RR and LL isomers is Ehb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='330 eV, and  for RL and LR isomers Ehb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='322 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' This is due to  the fact that in this case the benzol rings C6H5 of the  molecules are on the same side and they make a larger  contribution to the interaction energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The hydrogen  bond angle also depends on the chirality of the dimer  molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' For dimer molecules of the same chirality, the  angle between the dipole moments of the peptide groups  forming a hydrogen bond is φhb = 17◦, and for molecules  of different chirality φhb = 34◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Chains of hydrogen bonds of molecules of the same chi-  rality will always have benzol rings on one (outer) side,  so they will twist in the opposite (inner) direction and  have approximately the same curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' On a plane, cir-  cular arcs, circles, and spirals have such properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The  solution of the problem (9) has shown that on a flat sub-  b  a  C6  10  20  30  40  50  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='3  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='1  1  2  3  4  5  6  Es (eV)  N  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 6: Dependence of the specific energy of the secondary  structure of the homochiral ACN molecules Es on the num-  ber of molecules N for an arc, a circle, one-beam, two-beam,  three-beam spiral and nested two circles (curves 1, 2, 3, 4,  5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' For the structure of two nested circles (curve 6),  the dependence on the number of atoms of the outer circle is  shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  strate molecules of the same chirality form stable shape  structures with little changing curvature: arcs, spirals,  circles – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Left isomers form structures  with a twist to the right, right – to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Hydrogen bond chains of N ≤ 16 ACN molecules of  the same chirality form circular arcs of the same radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The step of such a chain (the distance between the oxy-  gen atoms of neighboring peptide groups) is a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='76 ˚A,  the angle between neighboring links is ϕ = 162◦, the ra-  dius of curvature by oxygen atoms is R = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='2 ˚A – see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 5 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The specific energy of the chain Es = E/N  decreases monotonically with the growth of the number  of molecules N – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' When the number of links  is N > 16, the arcs cease to be stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' they either close  and form circular chains, or touch their ends and form  flat spirals – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 5 (b), (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Stable cyclic chains can be formed from N  ≥ 5  molecules of the same chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The dependence of the  specific energy of the cyclic chain Es on the number of  its links N is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' As can be seen from the  figure, the most energetically favorable are cyclic chains  of N = 20, 21, 22 links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Such chains form ring struc-  tures with inner R1 = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='3, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='8, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='5 and outer radii  R2 = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='4, 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='9, 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='6 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The dependence Es(N) for a one-beam spiral actually  continues the dependence for an arc (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 6, curves  1 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Two-beam and three-beam spirals are bound  states of arc structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The specific energy of helical  structures decreases monotonically with an increase in  the number of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' For N > 27, helical structures  are more energy efficient than ring structures (this is due  to their denser structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The most energy-efficient are nested structures of two  circles with the number of atoms N = 14 + 26, 15+27,  17+29 (the first number in the sum corresponds the num-  ber of atoms in the inner circle, the second – in the outer  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 7: 2D crystals of ACN molecules on a flat substrate:  (a) with parallel packing of linear chains of hydrogen bonds  (periods ax = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='81, ay = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='24), (b) with antiparallel packing  (ax = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='79, ay = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='43 ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' On the surface of a flat substrate,  the chain of hydrogen bonds can be linear only if the L and  R isomers alternate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' A crystal is formed by 6 linear chains of  24 molecules (the total number of molecules is N = 6 × 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The flat substrate is shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  circle) – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 6, curve 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  If the chain of hydrogen bonds consists of a random  sequence of isomers, then it will look like an irregular  broken line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The chain has the shape of a straight zigzag  only if there is a strict alternation of L and R isomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  In this case, the zigzag step (the distance between the  oxygen atoms of neighboring molecules) is a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='85 ˚A,  the zigzag angle is ϕ = 170◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Such chains on a flat sub-  strate surface can form two-dimensional regular lattices  (2D crystals) with parallel and antiparallel packing of  neighboring chains – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' With parallel packing,  the crystal periods are ax = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='81˚A, ay = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='24˚A, the  specific energy is Es = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='325 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' With antiparallel  packing, periods are ax = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='79˚A, ay = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='43˚A, energy  is Es = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='322 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  SELF-ASSEMBLY OF MOLECULAR  STRUCTURES  Let us simulate the self-organization of the molecular  structures of ACN molecules on the flat surface of the h-  a  b7  100  200  300  400  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='8  1  nhb  (a)  100  200  300  400  500  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='4  T (K)  c  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 8: Dependence of (a) the normalized number of hydro-  gen bonds nhb and (b) the dimensionless heat capacity c on  temperature T for a system of N = 1024 ACN molecules  located on a flat substrate with the periodic square compu-  tational cell of size 33 × 33 nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Solid (blue) lines show de-  pendencies for a system of homochiral molecules, dotted (red)  lines – dependencies for a racemic mixture of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Verti-  cal dotted straight lines correspond to temperatures T = 250,  330, and 480 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  BN crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' To do this, we take a square periodic cell of  size 33 × 33 nm2 on the surface of the substrate and ran-  domly place N = 1024 ACN molecules into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Then we  immerse this molecular system in a Langevin thermostat  of temperature T and numerically simulate the dynamics  of the system during the time t = 10 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' To do this, we  numerically integrate the system of Langevin equations  M ¨Xn = −  ∂  ∂Xn  H − ΓM ˙Xn − Ξn,  n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=', N,  (10)  where Γ  =  1/tr is the friction coefficient,  Ξn  =  {ξn,i,k}11, 3  i=1,k=1 is 33-dimensional vector of normally dis-  tributed random Langevin forces with the following cor-  relations:  ⟨ξn1,i,k(t1)ξn2,j,l(t2)⟩ = 2MikBT Γδn1n2δijδklδ(t1 − t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Here Mi is mass of i-th atom of ACN molecule, kB is  Boltzmann constant, T is temperature of the Langevin  thermostat (temperature of the substrate), numbers  n1, n2 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=', N, i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=', 11, k, l = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The parameter tr characterizes the intensity of energy  exchange between the molecular system and the ther-  mostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Simulation of the dynamics of ACN molecules  on an h-BN sheet, taking into account the mobility of  the sheet atoms, makes it possible to estimate the relax-  ation time tr ∼ 100 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' For the convenience of numer-  ical integration, we will use a smaller value tr = 10 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 9: Structure of N = 1024 homochiral ACN molecules  appearing on the flat surface of the substrate at T = 240K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The straight lines show the boundaries of the periodic calcu-  lation cell of size 33 × 33 nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The substrate surface is not  shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  This makes it possible to significantly reduce the time of  numerical integration, which is sufficient to obtain reli-  able average values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' After the dynamics of the molec-  ular system reaches the steady state, we will find the  time averages of the system energy ¯E(T ) and the num-  ber of hydrogen bonds ¯Nhb(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' We assume that two ACN  molecules form a hydrogen bond if their interaction en-  ergy is greater than half of the hydrogen bond energy:  U(X1, X2) < −Ehb/2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='16 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The state of the system can be conveniently character-  ized by its dimensionless heat capacity  c =  1  33NkB  d ¯E(T )  dT  ,  (11)  and the normalized number of hydrogen bonds nhb =  ¯Nhb(T )/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The dependence of these quantities on tem-  perature is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Numerical simulation shows the existence of three  characteristic temperature values T1 < T2 < T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  At  T < T1 = 250K the molecules on a flat substrate form a  stable system of chains of hydrogen bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Here, almost  every molecule participates in the formation of one hy-  drogen bond (number nhb ≈ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The dimensionless heat  capacity of the system is c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' At T1 < T < T2 = 330K,  a slight decrease in the number of nhb bonds and a  monotonous increase in heat capacity begin to occur –  the process of melting of hydrogen bond chains begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  At a temperature T > T2, the number of hydrogen bonds  8  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 10: Structure of N = 1024 homochiral ACN molecules  appearing on the flat surface of the substrate at T = 300K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The straight lines show the boundaries of the periodic calcu-  lation cell of size 33 × 33 nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  decreases in proportion to the increase in temperature,  and the heat capacity reaches a constant value c ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Here we have a melt of short chains of hydrogen bonds  (the average length of the chains decreases proportion-  ally to the increase in temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' At T > T3 = 480K,  individual molecules can already be detached from the  substrate – desorption of molecules begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The structure of the resulting system of hydrogen  bond chains depends on the isomeric composition of the  molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' If all molecules are homochiral (only one iso-  mer is present), then secondary structures form on the  surface of the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  These structures are circular  and spiral hydrogen bond chains – see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The  substrate surface become optically active, the left isomers  form chains with a right twist, and the right ones with a  left twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 9 for T = 240K on  a flat substrate all possible circular secondary structures  (arcs, circles, spirals) are formed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' An increase in  temperature leads, first of all, to the destruction of spiral  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' As a result, the number of cyclic chains of  average radius increases since they are the most stable –  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  If the isomers of molecules are taken randomly, we get  their racemic mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' In this case, no secondary struc-  tures are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' There are only curvilinear chains of  random shape – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 11: Structure formed on the flat surface of the sub-  strate at T = 240K from a racemic mixture of N = 1024  ACN molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The straight lines show the boundaries of  the periodic calculation cell of size 33 × 33 nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  MELTING OF 2D CRYSTALS  To simulate the dynamics of 2D crystals of ACN  molecules on a flat h-BN substrate, consider a crystal  formed by 22 linear chains of hydrogen bonds of 48  molecules (total number of molecules N = 22 × 48 =  1056).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' A crystal with parallel chain packing has dimen-  sions of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='3 × 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='5 nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  We place the crystal in the  center of the calculated periodic cell size 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='5×22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='6 nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  In this case, the chains of hydrogen bonds located par-  allel to the x axis are closed, and the first chain begins  to come into contact with the last one – the 2D crystal  form a dense packing on the substrate that has no edges  and has normalized number of hydrogen bonds nhb = 1  (the number of hydrogen bonds is equal to the number of  molecules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' To simulate a rectangular crystal with edges  (square crystallite), we take a periodic computational cell  of size 47 × 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='2 nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' In this case, the 2D crystal (crys-  tallite) covers only 25% substrate surface, the chains of  hydrogen bonds are not closed, the normalized number  of hydrogen bonds is nhb = (N − 22)/N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Then we numerically integrate the system of equations  of motion (10) with the initial condition corresponding to  the stationary state of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' At different thermo-  stat temperatures, we find the average values of the sys-  tem energy ¯E(T ), the normalized number of hydrogen  bonds nhb, and the fraction of molecules that left the  substrate from the first layer p (the fraction of molecules  located on the substrate at distance z > 5 ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Next, using  9  100  200  300  400  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='8  1  1  2  (a)  nhb  100  200  300  400  500  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='5  3  4  (b)  c  100  200  300  400  500  0  5  10  6  5  T (K)  p (%)  (c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 12: Dependence of (a) the normalized number of hy-  drogen bonds nhb, (b) the dimensionless heat capacity c and  (c) the fraction of molecules that left the first layer p on the  temperature T for a 2D crystal of N = 1056 ACN molecules  located on a flat substrate with periodic computational cell  of size 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='6 × 45 nm2 (curves 1, 3, 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 25% substrate coverage)  and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='3 × 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='5 nm2 (curves 2, 4 , 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 100% substrate cover-  age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Vertical dotted straight lines show temperature values  T = 295, 335, 365, 415, 445 and 470 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  the formula (11), we find the temperature dependence of  the heat capacity of the molecular system c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The dependence of nhb, c, and p on the thermostat  temperature (on the substrate temperature) T is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' As can be seen from the figure, at 25% cover-  age of the substrate, the square crystallite begins to melt  at a temperature of T1 = 295K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' At T < T1, the crystal  structure is preserved, the number of bonds, heat capac-  ity, and density of the first layer of the substrate coating  practically do not change with increasing temperature:  nhb(T ) ≡ nhb(0), c(T ) ≡ 1, p(T ) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Crystallite melt-  ing occurs in the temperature range T1 < T < T2 where  the upper temperature is T2 = 365 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Here, a slight de-  crease in the number of bonds and an increase in heat  capacity begin to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The heat capacity reaches its  maximum value at the temperature Tm = 335 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' In the  temperature interval [T1, T2], an increasing destruction  of the edges of the initial crystallite occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The ends of  the chains peel off from the central part of the crystallite,  100  200  300  400  500  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='5  c  1  2  3  4  T (K)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 13: Dependence of the dimensionless heat capacity c on  temperature T for a 2D square crystallite of N = 480, 1056,  2376 of the ACN molecules located on a flat substrate with  periodic computational cell of the size 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='2× 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='68, 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='6× 45,  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='2 × 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='5 nm2 (curves 1, 3, 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 25% substrate coverage) and  for infinite 2D crystal (curve 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Vertical dotted straight lines  mark the temperatures T = 315, 335, 365, and 445 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  then break off and go to the free part of the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  As a result of this ”continuous” melting, the crystallite  transforms into a melt consisting of short chains of hy-  drogen bonds, uniformly covering the entire substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  For T > T2, the number of hydrogen bonds decreases  in proportion to the increase in temperature, while the  heat capacity remains almost constant: c ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' It can  be concluded that a complete transition of the molecu-  lar system from a low-energy and low-entropy crystalline  state to a high-energy and high-entropy liquid state has  taken place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' As a result of this transition, the substrate  is uniformly covered with a ”solution” of short chains of  hydrogen bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The chain lengths decreases monotoni-  cally with increasing temperature (on the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 12 (a) this  manifests itself in a monotonous decrease in the number  of bonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' All molecules remain directly adjacent to the  substrate (p = 0), insignificant desorption is observed  only at T > 480 K – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 12 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  With 100% coverage of the substrate, due to the ab-  sence of edges in the crystal, its melting occurs at higher  temperatures and happens according to a different sce-  nario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Here melting also occurs ”continuously” in the  temperature interval [T1, T2], where T1 = 415, T2 =  470 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The peak of the heat capacity at Tm = 445 K  becomes more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Melting occurs due to the  expulsion of some molecules from the first layer to the  second, which manifests itself in a monotonous increase  in the fraction of displaced molecules p at T1 < T < T2  – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 12 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  As a result, the density of the first  layer during melting decreases by 11%, and after melting  a dense melt of molecules is formed on the substrate, in  which 11% of the molecules are located on the second  layer from the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Conventionally, the temperature value Tm, at which  10  the heat capacity has reached its maximum value, can  be considered as the melting temperature of a 2D crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  However melting occurs not discretely, but continuously  in the temperature interval [T1, T2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The value Tm is in  the center of this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  To study the dependence of the melting temperature  on the crystallite size, we analyze numerically the melt-  ing of 2D square crystallite of N = 480 and 2376 ACN  molecules placed on a flat substrate with periodic com-  putational cell of size 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='2 × 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='68 and 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='2 × 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='5 nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Our results show that melting occurs continuously for all  crystallite sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' When the crystallite size is increased,  the melting interval [T1, T2] shifts to the right and, in  the limit, coincides with the melting temperature inter-  val of a 2D crystal with 100% coverage of the substrate,  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Let us note that the continuum melting scenario also  takes place for 2D n-alkane crystals lying on a flat sur-  face of graphite [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  This allows us to conclude that  the quasi-continuous melting scenario is a characteristic  feature of 2D systems of molecules adsorbed by a flat  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  CONCLUSION  Our numerical simulations of the dynamics of the sys-  tem of acetanilide molecules have revealed that the struc-  tures achiral in three-dimensional space become chiral  when being placed on a flat substrate: Depending on the  side it touches the substrate, the molecule has two iso-  mers L and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' The homochirality of the molecules leads  to the appearance of stable secondary structures stabi-  lized by hydrogen bonds on a flat substrate in the form  of arc, cyclic, and helical chains of hydrogen bonds and  their complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Hydrogen-bond chains of N ≤ 16 molecules of the same  chirality form circular arcs of the same radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  When  the number of molecules in the chain is N > 16, the  arcs becomes unstable, they either collapse into circles,  or touch their ends and form flat spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  In addition  to single-beam spirals, two- and three-beam spirals may  exist being bound states of arc chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Stable cyclic chains can be formed from N  ≥ 5  molecules of the same chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The most energeti-  cally favorable are cyclic chains of 20, 21 and 22 links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The structures of two circles with the number of atoms  N = 14 + 26, 15 + 27, 17 + 29 (where the first number of  the sum is the number of atoms in the inner circle, and  the second number is the number of atoms in the outer  circle) are more energy efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  If the chain of hydrogen bonds consists of a random  sequence of isomers, it look like an irregular broken line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  A chain can take the form of a rectilinear zigzag only if  there is a strict alternation of the L and R isomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Such  chains can form regular two-dimensional crystals with  parallel and antiparallel packing of the adjacent chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Simulation of the dynamics of a system of molecules  shows that the homochirality of molecules(the presence  of only one isomer) leads to the appearance of stable sec-  ondary structures on the surface of the substrate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' to  the appearance of chains of hydrogen bonds in the form  of arcs, circles and spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  As a result, the substrate  surface becomes optically active, the left isomers form  chains with a right twist, and the right ones form chain  with a left twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' At temperature T ≤ 240 K, all possible  secondary structures are formed on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' An  increase in temperature leads, first of all, to the disinte-  gration of spiral structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' As a result, the number of  more stable circular chains increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  If the molecules on the substrate form a racemic mix-  ture, no regular secondary structures are formed, so only  curvilinear chains of hydrogen bonds of random shape  can appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Thus, our results demonstrate the impor-  tance of homochirality (chiral purity) of biomolecules for  the formation of stable secondary molecular structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  Numerical simulations of the dynamics of a 2D crystal  of the ACN molecules shows that the scenario of crystal  melting depends on the density of its coverage of the  substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  At 25% coverage of the substrate the melting occurs  ”continuously” in the temperature interval [T1, T2] from  the edges of the initial crystallite (for crystallite of 1056  ACN molecules temperatures T1 = 295, T2 = 365 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  The ends of the chains peel off from the central part of  the crystallite, then break off and go to the free part  of the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' As a result, the crystallite transforms  into a melt consisting of short chains of hydrogen bonds,  uniformly covering the entire substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' When the size of  the crystallite is increased, the melting interval [T1, T2]  shifts to the right and, in the limit, will coincides with  the melting temperature interval of infinite crystal with  100% coverage of the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  For 100% coverage of the substrate with no crystal  edges, the melting occurs at higher temperatures 415 ÷  470 K by a shift of some molecules from the first to the  second layer in the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' As a result, the density of  the first layer during melting decreases by 11%, and after  melting, 11% of the molecules move to the second layer  formed on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  A continual melting scenario (the presence of a melting  temperature interval)has also been found in 2D n-alkane  crystals lying on a flat surface of graphite [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' All this  leads us to conclusion that the quasi-continuous melt-  ing scenario is a characteristic feature of 2D systems of  molecules adsorbed by a flat surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  AVS acknowledges the use of the computational facilities  provided by the Interdepartmental Supercomputer Cen-  ter of the Russian Academy of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' YSK acknowl-  edges a support from the Australian Research Council  (grant DP210101292) and the Strategic Fund of the Aus-  tralian National University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
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+page_content=' Adsorption of polycyclic  aromatic hydrocarbons by graphene and graphene oxide  nanosheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
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+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfJAOq/content/2301.02947v1.pdf'}
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+Active RISs: Signal Modeling, Asymptotic
+Analysis, and Beamforming Design
+Zijian Zhang∗, Linglong Dai∗, Fellow, IEEE, Xibi Chen∗, Changhao Liu∗, Fan Yang∗, Fellow, IEEE,
+Robert Schober†, Fellow, IEEE, and H. Vincent Poor§, Life Fellow, IEEE
+∗Beijing National Research Center for Information Science and Technology (BNRist)
+Department of Electronic Engineering, Tsinghua University, China
+†Institute for Digital Communications, Friedrich-Alexander University Erlangen-N¨urnberg, Germany
+§Department of Electrical and Computer Engineering, Princeton University, USA
+E-mails: zhangzj20@mails.tsinghua.edu.cn; daill@tsinghua.edu.cn; cxb17@tsinghua.org.cn; liuch17@tsinghua.org.cn;
+fan yang@tsinghua.edu.cn; robert.schober@fau.de; poor@princeton.edu
+Abstract—Reconfigurable
+intelligent
+surfaces
+(RISs)
+have
+emerged as a candidate technology for future 6G networks.
+However, due to the “multiplicative fading” effect, the existing
+passive RISs only achieve a negligible capacity gain in environ-
+ments with strong direct links. In this paper, the concept of
+active RISs is studied to overcome this fundamental limitation.
+Unlike the existing passive RISs that reflect signals without
+amplification, active RISs can amplify the reflected signals via
+amplifiers integrated into their elements. To characterize the
+signal amplification and incorporate the noise introduced by the
+active components, we verify the signal model of active RISs
+through the experimental measurements on a fabricated active
+RIS element. Based on the verified signal model, we formulate
+the sum-rate maximization problem for an active RIS aided
+multi-user multiple-input single-output (MU-MISO) system and
+a joint transmit precoding and reflect beamforming algorithm is
+proposed to solve this problem. Simulation results show that, in
+a typical wireless system, the existing passive RISs can realize
+only a negligible sum-rate gain of 3%, while the active RISs can
+achieve a significant sum-rate gain of 62%, thus overcoming the
+“multiplicative fading” effect. Finally, we develop a 64-element
+active RIS aided wireless communication prototype, and the
+significant gain of active RISs is validated by field test.
+I. INTRODUCTION
+From the first generation (1G) to 5G wireless communica-
+tions, the wireless channel has been considered to be uncon-
+trollable. Recently, due to the advances in meta-materials, re-
+configurable intelligent surfaces (RISs) have been proposed [1]
+for the purpose of intelligently controlling wireless channels
+to achieve improved communication performance. Specifically,
+an RIS is an array composed of a very large number of passive
+elements that reflects electromagnetic signals in a desired
+manner so as to reconfigure the propagation properties of
+wireless environment. As an important advantage of RIS, the
+negligible noise introduced by passive RISs enables a high
+array gain. Benefiting from this advantage, RISs are expected
+to introduce significant capacity gains in wireless systems [2].
+However, in practice, the expected capacity gains are typ-
+ically only observed in communication environments where
+the direct link between transmitter and receiver is completely
+blocked or very weak. By contrast, in many scenarios where
+the direct link is not weak, conventional RISs can only
+achieve negligible capacity gains [3]. The reason behind this
+phenomenon is the “multiplicative fading” effect introduced
+by RISs, i.e., the equivalent path loss of the transmitter-RIS-
+receiver link is the product (instead of the sum) of the path
+losses of the transmitter-RIS and RIS-receiver links, which is
+usually thousands of times larger than that of the direct link
+[1]–[3]. As a result, the “multiplicative fading” effect makes
+it almost impossible for passive RISs to achieve noticeable
+capacity gains in many wireless environments. Therefore,
+to advance the practicability of RISs in future 6G wireless
+networks, a critical issue for RISs to be addressed is: How to
+break the fundamental performance bottleneck caused by the
+“multiplicative fading” effect?
+To overcome the fundamental physical limitation of con-
+ventional passive RISs imposed by the “multiplicative fading”
+effect, in this paper, we investigate the concept of active RISs
+to overcome the “multiplicative fading” effect. Different from
+the existing passive RISs that passively reflect signals without
+amplification, the key feature of active RISs is their ability
+to actively reflect signals with amplification at the expense
+of additional power consumption. Firstly, through the experi-
+mental measurements on a fabricated active RIS element, we
+verify the signal model of active RISs, which characterizes the
+amplification of the incident signal and accounts for the non-
+negligible thermal noise introduced by the active elements.
+Based on the verified signal model, we further analyze the
+asymptotic performance of active RISs and formulate a sum-
+rate maximization problem for an active RIS aided multi-
+user multiple-input single-output (MU-MISO) system. Then,
+a joint transmit precoding and reflect beamforming algorithm
+is proposed to solve this problem. Simulation results show
+that, in a typical wireless system, the existing passive RISs
+achieve only a negligible sum-rate gain of 3%, while the active
+RISs are able to achieve a substantial sum-rate gain of 62%.
+Finally, we develop a 64-element active RIS aided wireless
+communication prototype, and field tests are conducted to
+validate the significant gain of active RISs.
+The rest of this paper is organized as follows. In Section II,
+the concept of RISs and their signal models are introduced.
+In Section III, the asymptotic performance of active RISs is
+analyzed. In Section IV, a sum-rate maximization problem is
+978-1-6654-3540-6/22 © 2022 IEEE
+arXiv:2301.00161v1  [cs.IT]  31 Dec 2022
+
+formulated, and a joint precoding and beamforming design is
+proposed to solve the problem. In Section V, simulation results
+and experimental measurements are presented to validate the
+signal model and evaluate the performance of active RISs.
+Finally, conclusions are drawn in Section VI.
+II. PASSIVE RISS AND ACTIVE RISS
+A. Conventional Passive RISs
+The RISs widely studied in most existing works are passive
+[1]–[3]. In general, each passive RIS element consists of a re-
+flective patch terminated with an impedance-adjustable circuit
+for phase shifting. Thanks to the passive mode of operation,
+the thermal noise at passive RISs is usually negligible [2].
+Thereby, the signal model of an N-element passive RIS widely
+used in the literature is given as follows:
+y = Θx,
+(1)
+where x
+∈
+CN denotes the incident signal, y
+∈
+CN
+denotes
+the
+signal
+reflected
+by
+the
+RIS,
+and
+Θ
+:=
+diag
+�
+ejθ1, · · · , ejθN �
+∈ CN×N denotes the phase shift matrix
+of the RIS with diag(·) being the diagonalization operation.
+By properly adjusting Θ to manipulate the N signals reflected
+by the N RIS elements to coherently add with the same phase
+at the receiver, a high array gain proportional to N 2 can be
+achieved, which is expected to significantly increase the signal-
+to-noise ratio (SNR) [1] at the receiver.
+Unfortunately, this expected high capacity gain often cannot
+be realized in practice, especially in communication scenarios
+where the direct link between the transmitter and the receiver is
+strong. The reason for this negative result is the “multiplicative
+fading” effect introduced by passive RISs. Specifically, the
+equivalent path loss of the transmitter-RIS-receiver reflected
+link is the product (instead of the sum) of the path losses of
+the transmitter-RIS and RIS-receiver links, and therefore, it is
+thousands of times larger than that of the unobstructed direct
+link. Thereby, for an RIS to realize a noticeable capacity gain,
+thousands (or even millions) of RIS elements are required to
+compensate for this extremely large path loss [3]. The resulting
+high signaling overhead for channel estimation and the high
+complexity of real-time beamforming make the application
+of such a large number of passive RIS elements in practical
+wireless networks very challenging.
+B. Concept of Active RISs
+To overcome the fundamental performance bottleneck
+caused by the “multiplicative fading” effect of RISs, we study
+the concept of active RISs as a promising solution1. As shown
+in Fig. 1, similar to the existing passive RISs, active RISs
+can also reflect the incident signals with reconfigurable phase
+shifts. Different from passive RISs that just reflect the incident
+signals without amplification, active RISs can further amplify
+the reflected signals. To achieve this goal, the key component
+1Note that active RISs are fundamentally different from relay-type RISs
+equipped with RF components and relays. Due to space constraints, we refer
+to the journal version of this paper for a more detailed discussion [4, Remark
+1].
+input
+BS
+RIS
+Active RIS
+user 1
+user k
+incident 
+signal
+reflected signal
+with amplification 
+phase-
+shift
+circuit
+patch
+reflection-type 
+amplifier
+output
+active element
+power
+supply
+Fig. 1. The downlink transmission in an active RIS aided MU-MISO system.
+of an active RIS element is the additionally integrated active
+reflection-type amplifier, which can be realized by different
+existing active components, such current-inverting converters
+or some integrated circuits [5].
+With reflection-type amplifiers supported by a power supply,
+the reflected and amplified signal of an N-element active RIS
+can be modeled as follows:
+y =
+PΘx
+� �� �
+Desired signal
++
+PΘv
+� �� �
+Dynamic noise
++
+ns
+����
+Static noise
+,
+(2)
+where P := diag (p1, · · · , pN) ∈ RN×N denotes the amplifi-
+cation factor matrix of the active RIS, wherein each element pn
+can be larger than one thanks to the integrated reflection-type
+amplifier. Due to the use of active components, active RISs
+consume additional power for amplifying the reflected signals,
+and the thermal noise introduced by active RIS elements
+cannot be neglected as is done for passive RISs. Particularly,
+as shown in (2), the introduced noise can be classified into
+dynamic noise and static noise [5]. Specifically, v is related to
+the input noise and the inherent device noise of the active RIS
+elements [5], while the static noise ns is unrelated to P and
+is usually negligible compared to the dynamic noise PΘv, as
+will be verified by experimental results in Section V-A. Thus,
+here we neglect ns and model v as v ∼ CN
+�
+0N, σ2
+vIN
+�
+,
+where CN(µ, Σ) denotes the complex multivariate Gaussian
+distribution with mean µ and variance Σ, IL is an L × L
+identity matrix, and 0L is an L × 1 zero vector.
+C. Active RIS Aided MU-MISO System
+Consider an active RIS aided downlink MU-MISO system
+as shown in Fig. 1, where an M-antenna base station (BS)
+serves K single-antenna users simultaneously with the aid
+of an N-element active RIS. Let s := [s1, · · · , sK]T ∈ CK
+denote the transmitted symbol vector for the K users and let
+wk ∈ CM×1 denote the BS precoding vector for symbol sk.
+According to (2), signal rk ∈ C received at user k can be
+modeled as follows:
+rk = (
+hH
+k
+����
+Direct link
++ f H
+k PΘG
+�
+��
+�
+Reflected link
+)
+�K
+j=1wjsj +
+f H
+k PΘv
+�
+��
+�
+Noise introduced by active RIS
+
++
+zk
+����
+Noise introduced at user k
+,
+(3)
+where [·]H denotes the conjugate-transpose operation; G ∈
+CN×M, hH
+k
+∈ C1×M, and f H
+k
+∈ C1×N characterize the
+channels between the BS and the RIS, between the BS and
+user k, and between the RIS and user k, respectively; and zk
+denotes the additive white Gaussian noise (AWGN) at user k
+with zero mean and variance σ2.
+III. PERFORMANCE ANALYSIS
+In this section, we analyze the performance of active RISs to
+reveal their notable capacity gains compared to passive RISs.
+To this end, in order to make the problem analytically tractable
+and get insightful results, in this section, we consider a single-
+user single-input single-output (SU-SISO) system with M = 1
+BS antenna and K = 1 user, while the general MU-MISO case
+is studied in Section IV.
+A. Asymptotic SNR for Passive RISs and Active RISs
+To illustrate the capacity gain provided by passive/active
+RIS aided reflected links, for the moment, we ignore the
+direct link by setting hk to zero, as was done in, e.g., [6].
+Furthermore, for simplicity, we assume that each active RIS
+element has the same amplification factor (i.e., pn := p). For
+a fair comparison with the asymptotic performance of passive
+RISs, similar to [6], we assume Rayleigh-fading channels.
+We first redefine the BS-RIS channel matrix and the RIS-
+user channel vector as G := g ∈ CN×1 and fk := f ∈ CN×1,
+respectively. Then, we recall the following lemma from [6] for
+the asymptotic SNR achieved by passive RISs.
+Lemma 1 (Asymptotic SNR for passive RISs): Assuming
+f ∼ CN
+�
+0N, ϱ2
+fIN
+�
+, g ∼ CN
+�
+0N, ϱ2
+gIN
+�
+and letting N →
+∞, the asymptotic SNR γpassive of a passive RIS aided SU-
+SISO system is given by
+γpassive → N 2 P max
+BS
+π2ϱ2
+fϱ2
+g
+16σ2
+,
+(4)
+where P max
+BS
+denotes the maximum transmit power at the BS.
+Proof: The proof can be found in [6, Proposition 2].
+For comparison, under the same transmission conditions, we
+provide the asymptotic SNR of an active RIS aided SU-SISO
+system in the following lemma.
+Lemma 2 (Asymptotic SNR for active RISs): Assuming
+f ∼ CN
+�
+0N, ϱ2
+fIN
+�
+, g ∼ CN
+�
+0N, ϱ2
+gIN
+�
+and letting N →
+∞, the asymptotic SNR γactive of an active RIS aided SU-SISO
+system is given by
+γactive → N
+P max
+BS
+P max
+A
+π2ϱ2
+fϱ2
+g
+16
+�
+P max
+A
+σ2vϱ2
+f + P max
+BS
+σ2ϱ2g + σ2σ2v
+�,
+(5)
+where P max
+A
+denotes the maximum reflect power of the active
+RIS.
+Proof: Please see the journal version [4, Appendix A].
+Remark 1: From (5) we observe that, the asymptotic SNR
+of an active RIS aided SU-SISO system depends on both the
+BS transmit power P max
+BS
+and the reflect power of the active
+RIS P max
+A
+. When P max
+BS
+→ ∞, the asymptotic SNR will be
+upper-bounded by γactive → NP max
+A
+π2ϱ2
+f/
+�
+16σ2�
+, which is
+independent of the BS-RIS channel g and the noise power at
+the active RIS σ2
+v. Similarly, if P max
+A
+→ ∞, the asymptotic
+SNR will be upper-bounded by γactive → NP max
+BS π2ϱ2
+g/16σ2
+v,
+which is independent of the RIS-user channel f and the noise
+power at the user σ2. These results reveal that, to increase the
+sum-rate of active RIS aided systems, the negative impact of
+small g and large σ2
+v on system performance can be alleviated
+by increasing the BS transmit power P max
+BS
+, and the negative
+impact of small f and large σ2 can be reduced by increasing
+the reflect power of the active RIS P max
+A
+.
+B. Comparisons between Passive RISs and Active RISs
+We can observe from Lemma 1 and Lemma 2 that, compared
+to the asymptotic SNR for passive RISs γpassive in (4) which is
+proportional to N 2, the asymptotic SNR for active RISs γactive
+in (5) is proportional to N due to the noises introduced by
+the use of active components. At first glance, it seems that the
+SNR achieved by passive RISs γpassive always exceeds the SNR
+achieved by active RISs γactive. However, this is actually not the
+case. The reason behind this counterintuitive behavior is that,
+due to the large path loss caused by the “multiplicative fading”
+effect and thanks to the use of the reflection-type amplifiers
+in active RISs, only when N is unaffordably large can passive
+RISs outperform active RISs.
+To illustrate this claim, let us consider two different SU-
+SISO systems, which are aided by an active RIS and a passive
+RIS, respectively. Then, the following lemma specifies the
+condition that has to be met for passive RISs to outperform
+active RISs.
+Lemma 3 (Case when passive RISs outperform active
+RISs): Assuming the number of RIS elements N is large,
+the required number of elements N for a passive RIS to
+outperform an active RIS has to satisfy
+N ≥ P max
+BS-A
+P max
+BS-P
+P max
+A
+σ2
+�
+P max
+A
+σ2vϱ2
+f + P max
+BS-Aσ2ϱ2g + σ2σ2v
+�,
+(6)
+where P max
+BS-A denotes the maximum BS transmit power for the
+active RIS aided system and P max
+BS-P denotes that for the passive
+RIS aided system.
+Proof: Please see the journal version [4, Appendix B].
+Next, we consider a specific setup to compare the user’s
+achievable SNRs in the above two systems. For a fair com-
+parison, we constrain the total power consumption P max of
+the two systems to 2 W by setting P max
+BS-P = 2 W for the
+passive RIS aided system and P max
+BS-A = P max
+A
+= 1 W for
+the active RIS aided system, respectively. Therefore, when
+σ2 = σ2
+v = −70 dBm and ϱ2
+f = ϱ2
+g = −70 dB, the required
+number of elements N for the passive RIS to outperform the
+active RIS is 2.5 × 106 according to (6), which is impractical
+to realize with current technology. Conversely, for a more
+practical number of elements of N = 256, according to (5)
+and (4), the SNR achieved by the passive RIS is γpassive ≈ 9.0
+dB, while the SNR achieved by the active RIS is γactive ≈ 49.0
+dB, which is about 10, 000 times higher than γpassive.
+
+IV. JOINT TRANSMIT PRECODING AND REFLECT
+BEAMFORMING DESIGN
+To investigate the capacity gain enabled by the use of
+active RISs in typical wireless communication scenarios, in
+this section, we consider more general MU-MISO systems.
+According to the model in (3), the signal-to-interference-plus-
+noise ratio (SINR) at user k can be obtained as
+γk =
+��¯hH
+k wk
+��2
+�K
+j=1,j̸=k
+��¯hH
+k wj
+��2 +
+��f H
+k PΘ
+��2σ2v + σ2 ,
+(7)
+wherein ¯hH
+k = hH
+k + fk
+HPΘG ∈ C1×M is the equivalent
+channel from the BS to user k, which includes both the direct
+link and the reflected link. Therefore, the original problem of
+sum-rate maximization, subject to the power constraints at the
+BS and the active RIS, can be formulated as follows:
+Po : max
+w,P,Θ Rsum(w, P, Θ) =
+�K
+k=1 log2 (1 + γk),
+(8a)
+s.t. C1 :
+�K
+k=1 ∥wk∥2 ≤ P max
+BS
+,
+(8b)
+C2 :
+�K
+k=1∥PΘGwk∥2+∥PΘ∥2 σ2
+v ≤P max
+A
+, (8c)
+where w :=
+�
+wT
+1 , · · · , wT
+K
+�T is the overall transmit precoding
+vector for the K users; C1 and C2 are the power constraints at
+the BS and active RIS, respectively. Due to the non-convexity
+and the highly coupled variables in problem Po in (8), the
+joint design of w, P, and Θ is challenging.
+To efficiently solve the above problem, we develop a joint
+precoding and beamforming algorithm based on alternating
+optimization and fractional programming (FP). Note that P
+and Θ always appear in product form in problem Po in
+(8). Therefore, P and Θ can be merged as Ψ = PΘ =
+diag
+�
+p1ejθ1, · · · , pNejθN �
+∈ CN×N. We refer to Ψ as the
+RIS beamforming matrix. Next, to deal with the non-convex
+sum-of-logarithms and fractions in (8), we exploit the FP
+methods proposed in [7] to decouple the variables in problem
+Po in (8). This leads to the following lemma.
+Lemma 4 (Equivalent problem for sum-rate maximiza-
+tion): By introducing auxiliary variables ρ := [ρ1, · · · , ρK] ∈
+RK and ϖ := [ϖ1, · · · , ϖK] ∈ CK, the original problem Po
+in (8) can be equivalently reformulated as follows
+P1 :
+max
+w,Ψ,ρ,ϖ R′
+sum(w, Ψ, ρ, ϖ) =
+�K
+k=1 ln (1 + ρk)−
+�K
+k=1 ρk +
+�K
+k=1 g(w, Ψ, ρk, ϖk),
+s.t. C1, C2,
+(9)
+where function g(w, Ψ, ρk, ϖk) is defined as
+g(w, Ψ, ρk, ϖk) = 2
+�
+(1 + ρk)R
+�
+ϖ∗
+k¯hH
+k wk
+�
+−
+|ϖk|2
+��K
+j=1
+��¯hH
+k wj
+��2 +
+��f H
+k Ψ
+��2σ2
+v + σ2
+�
+.
+(10)
+Proof: Constructive proof can be found in [7, Subsection
+III-C].
+Strong convergence of the FP methods was proved in [7].
+Thus, a locally optimal solution to (9) can be obtained by
+alternately optimizing the variables. For clarity, we summarize
+the proposed joint precoding and beamforming algorithm in
+Algorithm 1, and the specific solutions for variables w, Ψ,
+ρ, and ϖ are given in the following four steps, respectively.
+Algorithm 1 Proposed joint transmit precoding and reflect
+beamforming algorithm
+Input:
+Channels G, hk, and fk, ∀k ∈ {1, · · · , K}.
+Output:
+Optimized BS precoding vector w, amplification
+factor matrix of active RIS P, phase shift matrix of active
+RIS Θ, and sum-rate Rsum.
+1: Randomly initialize w, P and Θ;
+2: while no convergence of Rsum do
+3:
+Update ρ by (11);
+4:
+Update ϖ by (12);
+5:
+Update w by solving problem P2 in (14);
+6:
+Update Ψ by solving problem P3 in (15);
+7: end while
+8: Obtain P and Θ from Ψ;
+9: return Optimized w, P, Θ, and Rsum.
+1) Fix (w, Ψ, ϖ) and optimize ρ: After fixing precoding
+vector w, beamforming matrix Ψ, and auxiliary variable ϖ,
+the optimal ρ can be obtained by solving ∂R′
+sum
+∂ρk
+= 0 as
+ρopt
+k
+= ξ2
+k + ξk
+�
+ξ2
+k + 4
+2
+,
+∀k ∈ {1, · · · , K},
+(11)
+where ξk = ℜ
+�
+ϖ∗
+k¯hH
+k wk
+�
+.
+2) Fix (w, Ψ, ρ) and optimize ϖ: After fixing the precod-
+ing vector w, beamforming matrix Ψ, and auxiliary variable
+ρ, the optimal ϖ can be derived by solving ∂R′
+sum
+∂ϖk
+= 0 as
+ϖopt
+k
+=
+�
+(1 + ρk)¯hH
+k wk
+�K
+j=1
+��¯hH
+k wj
+��2+
+��f H
+k Ψ
+��2σ2v + σ2 .
+(12)
+3) Fix (Ψ, ρ, ϖ) and optimize w: To simplify the nota-
+tions, we first introduce the following definitions:
+bH
+k =
+�
+(1 + ρk)ϖ∗
+k¯hH
+k , b =
+�
+bT
+1 , bT
+2 , · · · , bT
+N
+�T ,
+(13a)
+A=IK ⊗
+�K
+k=1|ϖk|2¯hk¯hH
+k , Ξ=IK ⊗
+�
+GHΨHΨG
+�
+, (13b)
+P max
+m
+= P max
+A
+− ∥Ψ∥2σ2
+v,
+(13c)
+where ⊗ denotes the Kronecker product. Then, problem P1 in
+(9) can be reformulated as follows:
+P2 :
+max
+w
+R
+�
+2bHw
+�
+− wHAw,
+s.t.
+C1 : ∥w∥2 ≤ P max
+BS
+,
+C2 : wHΞw ≤ P max
+m
+.
+(14)
+Note that P2 in (14) is a standard quadratic constraint
+quadratic programming (QCQP) problem, which can be solved
+by alternating direction method of multipliers (ADMM).
+4) Fix
+(w, ρ, ϖ)
+and
+optimize
+Ψ:
+Define
+ψ
+=
+�
+p1ejθ1, · · · , pNejθN �H as the vectorized RIS beamforming
+matrix Ψ, i.e., diag
+�
+ψH�
+:= Ψ. While fixing w and ρ and
+ϖ, problem P1 in (9) can be reformulated as follows:
+P3 :
+max
+ψ
+R
+�
+2ψHυ
+�
+− ψHΩψ,
+s.t. C2 : ψHΠψ ≤ P max
+A
+,
+(15)
+
+spectrum 
+analyzer
+LNA
+vector network 
+analyzer
+DC source
+active RIS
+element
+circulator
+pump
+source
+noise source
+Fig. 2. The experimental devices and environment used for validating the signal model (2) of active RISs.
+2.359
+2.3595
+2.36
+2.3605
+2.361
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+30
+Fig. 3. Experimental measurement result for reflection gain G versus signal
+frequency f.
+wherein
+υ =
+�K
+k=1
+�
+(1 + ρk)diag
+�
+ϖ∗
+kf H
+k
+�
+Gwk−
+�K
+k=1 |ϖk|2diag
+�
+f H
+k
+�
+G
+�K
+j=1 wjwH
+j hk,
+(16a)
+Ω =
+�K
+k=1 |ϖk|2diag
+�
+f H
+k
+�
+diag (fk) σ2
+v+
+�K
+k=1 |ϖk|2 �K
+j=1diag
+�
+f H
+k
+�
+GwjwH
+j GHdiag (fk), (16b)
+Π =
+�K
+k=1 diag (Gwk) (diag (Gwk))H + σ2
+vIN.
+(16c)
+Note that problem P3 in (15) is also a standard QCQP
+problem. Thus, the optimal solution ψopt can be obtained by
+adopting ADMM.
+V. VALIDATION RESULTS
+A. Validation Results for Signal Model
+To validate the signal model (2), we designed and fabri-
+cated an active RIS element with an integrated reflection-
+type amplifier for experimental measurements in [8]. Note
+that this design can be directly extended to the large-array
+case. Particularly, since the phase-shifting ability of RISs has
+been widely verified, we focus on studying the reflection gain
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+-170
+-165
+-160
+-155
+-150
+-145
+-140
+-135
+-130
+Fig. 4.
+Experimental measurement result for the density of noise power
+Gσ2
+v + σ2
+s versus reflection gain G.
+and the noise introduced by an active RIS element. Thus, the
+validation of signal model (2) is equivalent to validating
+Py =
+GPx
+����
+Desired-signal power
++ Gσ2
+v + σ2
+s
+�
+��
+�
+noise power
+,
+(17)
+where Py is the power of the signals reflected by the active
+RIS element; Px is the power of the incident signal; G := p2 is
+the reflection gain of the active RIS element; Gσ2
+v and σ2
+s are
+the powers of the dynamic noise and static noise introduced
+by the active RIS element, respectively.
+1) Hardware platform: To validate the model in (17), we
+first establish the hardware platform used for our experimental
+measurements in Fig. 2. Due to space constraints, we refer
+the reader to the journal version of this paper [4, Fig. 4] for
+detailed information about the hardware platform.
+2) Reflection gain measurement: Using the measurement
+system for the reflection gain depicted in [4, Fig. 4 (b)], we
+first investigate the reflection gain G of the active RIS element.
+Note that the reflection gain G can be reconfigured by the
+input power of the pump source Pp. By setting the input
+power of the vector network analyzer as Px = −50 dBm, the
+reflection gain G as a function of the signal frequency can be
+directly measured via a vector network analyzer. Then, in Fig.
+
+-10
+-5
+0
+5
+10
+15
+20
+25
+30
+0
+10
+20
+30
+40
+50
+60
+507%
+Fig. 5. Simulation results for the sum-rate versus total power consumption
+P max in scenario 1 with a weak direct link..
+3, we show the measurement results for reflection gain G as
+a function of signal frequency f for different input powers of
+the pump source Pp. We observe that the active RIS element
+can achieve a reflection gain G of more than 25 dB, when
+Pp = 18.24 dBm, which confirms the significant reflection
+gains enabled by active RISs.
+3) Noise power measurement: We further study the noise
+power introduced by the active RIS element, i.e., Gσ2
+v + σ2
+s
+in (17), where Gσ2
+v and σ2
+s are the powers of the dynamic
+noise and the static noise introduced at the active RIS element,
+respectively. Using the noise measurement system in [4, Fig. 4
+(c)], we show the measurement results for the spectral density
+of noise power Gσ2
+v + σ2
+s as a function of G for different
+operating frequencies in Fig. 4. We can observe that the noise
+power increases nearly linearly with G, which verifies the
+noise model Gσ2
+v + σ2
+s in (17). Particularly, for f = 2.3601
+GHz, the spectral density of σ2
+s is about −174 dBm/Hz, while
+that of σ2
+v is about −160 dBm/Hz, which is about 15 dB
+higher. The reason for this is that the input noise is amplified
+by the noise factor, and additional noises are also introduced
+by the other active components such as the DC source used
+to power the active RIS.
+B. Simulation Results for Sum-Rate
+1) Simulation setup: We consider an active RIS aided MU-
+MISO system. Particularly, we consider two scenarios with dif-
+ferent channel conditions. In scenario 1, the direct link is weak
+due to severe obstruction, while the direct link is strong in
+scenario 2. To be specific, two different path loss models from
+the 3GPP TS 36.814 standard are utilized to characterize the
+large-scale fading of the channels: i) PLs = 37.3+22.0 log d;
+ii) PLw = 41.2 + 28.7 log d, where d is the distance between
+two devices. Path loss model PLw is used to generate the weak
+BS-user link in scenario 1, while PLs is used to generate the
+strong BS-user link in scenario 2. For both scenarios, PLs is
+used to generate the BS-RIS and the RIS-user channels. To
+account for small-scale fading, we adopt the Ricean fading
+channel model for all channels involved and we assume the
+Ricean factor as κ = 1.
+-10
+-5
+0
+5
+10
+15
+20
+25
+30
+0
+10
+20
+30
+40
+50
+60
+62%
+Fig. 6. Simulation results for the sum-rate versus total power consumption
+P max in scenario 2 with a strong direct link.
+The BS and the active/passive RIS are located at (0, -60
+m) and (200 m, 30 m), respectively. Four users are randomly
+located in a circle with a radius of 5 m from the center (200
+m, 0). The numbers of BS antennas and RIS elements are set
+as M = 4 and N = 256, respectively. The noise power is set
+as σ2 = σ2
+v = −70 dBm. For fair comparison, we constrain
+the total power consumption P max := P max
+BS
++ P max
+A
+. For the
+active RIS, Algorithm 1 is employed for joint precoding and
+beamforming design, while for the passive RIS, the algorithm
+from [2] is adopted.
+2) Simulation results: In Fig. 5 and Fig. 6, we plot the
+sum-rate versus the total consumed power P max for the two
+considered scenarios, where the direct link is weak and strong,
+respectively. Firstly, in scenario 1 with a weak direct link, the
+passive RIS can indeed achieve a performance improvement,
+while the active RIS achieves a much higher sum-rate gain.
+Secondly, in scenario 2 with a strong direct link, the passive
+RIS achieves only a negligible sum-rate gain, while the active
+RIS still realizes a noticeable sum-rate gain. For example,
+when P max = 10 dBW, the capacities without RIS, with
+passive RIS, and with active RIS in scenario 1 are 5.34 bps/Hz,
+7.00 bps/Hz, and 32.41 bps/Hz respectively, while in scenario
+2, these values are 19.87 bps/Hz, 20.51 bps/Hz, and 32.18
+bps/Hz, respectively. In this case, the passive RIS provides a
+31% gain in scenario 1 and a negligible 3% gain in scenario 2.
+By contrast, the active RIS achieves noticeable sum-rate gains
+of 507% in scenario 1 and 62% in scenario 2, which are much
+higher than those achieved by the passive RIS.
+C. Field Test for a 64-Element Active RIS Aided Wireless
+Communication Prototype
+1) 64-element active RIS aided communication prototype:
+To validate the significant gain of active RISs, we develop a
+64-element active RIS aided wireless communication proto-
+type, as shown in Fig. 7. Specifically, the hardware structure
+of this prototype consists of three parts including a BS, a 64-
+element active RIS, and a user. For the BS and the user, two
+horn antennas with 13 dBi antenna gain are used to transmit
+
+Fig. 7. A photograph of the developed 64-element active RIS aided wireless
+communication system.
+and receive the signals, and the universal software radio
+peripherals (USRPs) are deployed to generate and process the
+baseband and RF signals (hardware version: USRP-2953R).
+By periodically expanding the active RIS elements designed
+in [8], the 64-element active RIS is an 8×8 plane array, of
+which each element has a reflection gain of G = 10 dB.
+2) Experimental environment: Based on the developed pro-
+totype, we establish the experimental environment for further
+validation. To match the transceivers, we configure the oper-
+ating frequency of the active RIS to f = 3.5 GHz and the
+bandwidth to 40 MHz by adjusting the circuit impedance of
+active elements. The polarization of the antenna at the BS
+and that at the user are selected as vertical and horizontal,
+respectively. The transmit power is set to −10 mW. We fix the
+heights of the BS, the RIS, and the user as 1 m. The horizontal
+distance of the BS-RIS link and that of the RIS-user link are
+set to 2 m and 3.5 m, respectively. The angle of arrival (AoA)
+at the active RIS is fixed as 0◦, and the angle of departure
+(AoD) will be specified to evaluate the performance gain of
+active RISs at different orientations. To observe the reflection
+gain of the active RIS, we use a metal plate with the same
+aperture size as the active RIS for performance comparison.
+3) Experimental results: By moving the user at different
+AoDs and configuring the phase shift of the active RIS with
+discrete Fourier transform (DFT) codebook, we obtain the
+experimental results shown in Table I. One can observe that,
+compared with the received power for the metal plate, the
+active RIS can always achieve a gain of about 10 dB. The data
+rate for the active RIS can hold at about 30 Mbps, while that
+for the metal plate only ranges from 1 Mbps to 2Mbps. The
+reason is that, the beamforming at the active RIS can make the
+reflected beam with high array gain and reflection gain, while
+the metal plate can only reflect the incident signals randomly
+without in-phase combination or amplification, which validates
+the significant gain of active RISs.
+VI. CONCLUSIONS
+In this paper, we have studied the concept of active RISs
+to overcome the fundamental limitation of the “multiplicative
+fading” effect. Firstly, we have verified the signal model of
+TABLE I
+EXPERIMENTAL RESULTS FOR THE DEVELOPED PROTOTYPE
+AoD
+Device
+Received Power
+Data Rate
+15◦
+Metal plate
+-110 dBm
+1.2 Mbps
+Active RIS
+-100 dBm
+28.5 Mbps
+30◦
+Metal plate
+-105 dBm
+1.5 Mbps
+Active RIS
+-98 dBm
+30.5 Mbps
+45◦
+Metal plate
+-105 dBm
+1.5 Mbps
+Active RIS
+-95 dBm
+30 Mbps
+60◦
+Metal plate
+-108 dBm
+2 Mbps
+Active RIS
+-90 dBm
+32 Mbps
+active RISs through the experimental measurements on a fab-
+ricated active RIS element. Based on the verified signal model,
+we have formulated the sum-rate maximization problem for
+an active RIS aided MU-MISO system and a joint precoding
+and beamforming algorithm has been proposed to solve this
+problem. Simulation results have shown that, in a typical
+application scenario, the existing passive RIS can realize only
+a negligible sum-rate gain of about 3%, while the active RIS
+can achieve a substantial sum-rate gain of about 62%, thus
+indeed overcoming the “multiplicative fading” effect. Finally,
+we have developed a communication wireless communication
+prototype aided by a 64-element active RIS, and the significant
+gain of active RISs is validated by field test. In the future,
+many research directions for active RISs are worth pursuing,
+such as hardware design, prototype development, channel
+estimation, and energy efficiency analysis.
+ACKNOWLEDGMENT
+This work was supported in part by the National Key
+Research and Development Program of China (Grant No.
+2020YFB1805005), in part by the National Natural Science
+Foundation of China (Grant No. 62031019), and in part by the
+European Commission through the H2020-MSCA-ITN META
+WIRELESS Research Project under Grant 956256.
+REFERENCES
+[1] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and C. Yuen,
+“Reconfigurable intelligent surfaces for energy efficiency in wireless
+communication,” IEEE Trans. Wireless Commun., vol. 18, no. 8, pp.
+4157–4170, Aug. 2019.
+[2] C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan,
+and L. Hanzo, “Multicell MIMO communications relying on intelligent
+reflecting surfaces,” IEEE Trans. Wireless Commun., vol. 19, no. 8, pp.
+5218–5233, Aug. 2020.
+[3] M. Najafi, V. Jamali, R. Schober, and H. V. Poor, “Physics-based mod-
+eling and scalable optimization of large intelligent reflecting surfaces,”
+IEEE Trans. Commun., vol. 69, no. 4, pp. 2673–2691, Apr. 2021.
+[4] Z. Zhang, L. Dai, X. Chen, C. Liu, F. Yang, R. Schober, and H. V. Poor,
+“Active RIS vs. passive RIS: Which will prevail in 6G?” IEEE Trans.
+Commun., Dec. 2022.
+[5] J. Bousquet, S. Magierowski, and G. G. Messier, “A 4-GHz active
+scatterer in 130-nm CMOS for phase sweep amplify-and-forward,” IEEE
+Trans. Circuits Syst. I, vol. 59, no. 3, pp. 529–540, Mar. 2012.
+[6] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless
+network via joint active and passive beamforming,” IEEE Trans. Wireless
+Commun., vol. 18, no. 11, pp. 5394–5409, Nov. 2019.
+[7] K. Shen and W. Yu, “Fractional programming for communication sys-
+tems—part I: Power control and beamforming,” IEEE Trans. Signal
+Process., vol. 66, no. 10, pp. 2616–2630, May 2018.
+[8] X. Chen and F. Yang, “Nonlinear electromagnetic surfaces: Theory,
+design and application,” Master Thesis in Tsinghua University, May 2020,
+[Online] Available: http://etds.lib.tsinghua.edu.cn/Thesis.
+
+3.5GHz
+有源RIS
+8 X 8 Active RIS
+BS
+User
\ No newline at end of file
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf,len=388
+page_content='Active RISs: Signal Modeling, Asymptotic  Analysis, and Beamforming Design  Zijian Zhang∗, Linglong Dai∗, Fellow, IEEE, Xibi Chen∗, Changhao Liu∗, Fan Yang∗, Fellow, IEEE,  Robert Schober†, Fellow, IEEE, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Vincent Poor§, Life Fellow, IEEE  ∗Beijing National Research Center for Information Science and Technology (BNRist)  Department of Electronic Engineering, Tsinghua University, China  †Institute for Digital Communications, Friedrich-Alexander University Erlangen-N¨urnberg, Germany  §Department of Electrical and Computer Engineering, Princeton University, USA  E-mails: zhangzj20@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' daill@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' cxb17@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' liuch17@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  fan yang@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' robert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='schober@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' poor@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='edu  Abstract—Reconfigurable  intelligent  surfaces  (RISs)  have  emerged as a candidate technology for future 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  However, due to the “multiplicative fading” effect, the existing  passive RISs only achieve a negligible capacity gain in environ-  ments with strong direct links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In this paper, the concept of  active RISs is studied to overcome this fundamental limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Unlike the existing passive RISs that reflect signals without  amplification, active RISs can amplify the reflected signals via  amplifiers integrated into their elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' To characterize the  signal amplification and incorporate the noise introduced by the  active components, we verify the signal model of active RISs  through the experimental measurements on a fabricated active  RIS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Based on the verified signal model, we formulate  the sum-rate maximization problem for an active RIS aided  multi-user multiple-input single-output (MU-MISO) system and  a joint transmit precoding and reflect beamforming algorithm is  proposed to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Simulation results show that, in  a typical wireless system, the existing passive RISs can realize  only a negligible sum-rate gain of 3%, while the active RISs can  achieve a significant sum-rate gain of 62%, thus overcoming the  “multiplicative fading” effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Finally, we develop a 64-element  active RIS aided wireless communication prototype, and the  significant gain of active RISs is validated by field test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' INTRODUCTION  From the first generation (1G) to 5G wireless communica-  tions, the wireless channel has been considered to be uncon-  trollable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Recently, due to the advances in meta-materials, re-  configurable intelligent surfaces (RISs) have been proposed [1]  for the purpose of intelligently controlling wireless channels  to achieve improved communication performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Specifically,  an RIS is an array composed of a very large number of passive  elements that reflects electromagnetic signals in a desired  manner so as to reconfigure the propagation properties of  wireless environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' As an important advantage of RIS, the  negligible noise introduced by passive RISs enables a high  array gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Benefiting from this advantage, RISs are expected  to introduce significant capacity gains in wireless systems [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  However, in practice, the expected capacity gains are typ-  ically only observed in communication environments where  the direct link between transmitter and receiver is completely  blocked or very weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' By contrast, in many scenarios where  the direct link is not weak, conventional RISs can only  achieve negligible capacity gains [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The reason behind this  phenomenon is the “multiplicative fading” effect introduced  by RISs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', the equivalent path loss of the transmitter-RIS-  receiver link is the product (instead of the sum) of the path  losses of the transmitter-RIS and RIS-receiver links, which is  usually thousands of times larger than that of the direct link  [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' As a result, the “multiplicative fading” effect makes  it almost impossible for passive RISs to achieve noticeable  capacity gains in many wireless environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Therefore,  to advance the practicability of RISs in future 6G wireless  networks, a critical issue for RISs to be addressed is: How to  break the fundamental performance bottleneck caused by the  “multiplicative fading” effect?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  To overcome the fundamental physical limitation of con-  ventional passive RISs imposed by the “multiplicative fading”  effect, in this paper, we investigate the concept of active RISs  to overcome the “multiplicative fading” effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Different from  the existing passive RISs that passively reflect signals without  amplification, the key feature of active RISs is their ability  to actively reflect signals with amplification at the expense  of additional power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Firstly, through the experi-  mental measurements on a fabricated active RIS element, we  verify the signal model of active RISs, which characterizes the  amplification of the incident signal and accounts for the non-  negligible thermal noise introduced by the active elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Based on the verified signal model, we further analyze the  asymptotic performance of active RISs and formulate a sum-  rate maximization problem for an active RIS aided multi-  user multiple-input single-output (MU-MISO) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Then,  a joint transmit precoding and reflect beamforming algorithm  is proposed to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Simulation results show  that, in a typical wireless system, the existing passive RISs  achieve only a negligible sum-rate gain of 3%, while the active  RISs are able to achieve a substantial sum-rate gain of 62%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Finally, we develop a 64-element active RIS aided wireless  communication prototype, and field tests are conducted to  validate the significant gain of active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In Section II,  the concept of RISs and their signal models are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  In Section III, the asymptotic performance of active RISs is  analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In Section IV, a sum-rate maximization problem is  978-1-6654-3540-6/22 © 2022 IEEE  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='00161v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='IT]  31 Dec 2022  formulated, and a joint precoding and beamforming design is  proposed to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In Section V, simulation results  and experimental measurements are presented to validate the  signal model and evaluate the performance of active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Finally, conclusions are drawn in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' PASSIVE RISS AND ACTIVE RISS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Conventional Passive RISs  The RISs widely studied in most existing works are passive  [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In general, each passive RIS element consists of a re-  flective patch terminated with an impedance-adjustable circuit  for phase shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Thanks to the passive mode of operation,  the thermal noise at passive RISs is usually negligible [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Thereby, the signal model of an N-element passive RIS widely  used in the literature is given as follows:  y = Θx,  (1)  where x  ∈  CN denotes the incident signal, y  ∈  CN  denotes  the  signal  reflected  by  the  RIS,  and  Θ  :=  diag  �  ejθ1, · · · , ejθN �  ∈ CN×N denotes the phase shift matrix  of the RIS with diag(·) being the diagonalization operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  By properly adjusting Θ to manipulate the N signals reflected  by the N RIS elements to coherently add with the same phase  at the receiver, a high array gain proportional to N 2 can be  achieved, which is expected to significantly increase the signal-  to-noise ratio (SNR) [1] at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Unfortunately, this expected high capacity gain often cannot  be realized in practice, especially in communication scenarios  where the direct link between the transmitter and the receiver is  strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The reason for this negative result is the “multiplicative  fading” effect introduced by passive RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Specifically, the  equivalent path loss of the transmitter-RIS-receiver reflected  link is the product (instead of the sum) of the path losses of  the transmitter-RIS and RIS-receiver links, and therefore, it is  thousands of times larger than that of the unobstructed direct  link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Thereby, for an RIS to realize a noticeable capacity gain,  thousands (or even millions) of RIS elements are required to  compensate for this extremely large path loss [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The resulting  high signaling overhead for channel estimation and the high  complexity of real-time beamforming make the application  of such a large number of passive RIS elements in practical  wireless networks very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Concept of Active RISs  To overcome the fundamental performance bottleneck  caused by the “multiplicative fading” effect of RISs, we study  the concept of active RISs as a promising solution1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' As shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 1, similar to the existing passive RISs, active RISs  can also reflect the incident signals with reconfigurable phase  shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Different from passive RISs that just reflect the incident  signals without amplification, active RISs can further amplify  the reflected signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' To achieve this goal, the key component  1Note that active RISs are fundamentally different from relay-type RISs  equipped with RF components and relays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Due to space constraints, we refer  to the journal version of this paper for a more detailed discussion [4, Remark  1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  input  BS  RIS  Active RIS  user 1  user k  incident  signal  reflected signal  with amplification  phase-  shift  circuit  patch  reflection-type  amplifier  output  active element  power  supply  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The downlink transmission in an active RIS aided MU-MISO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  of an active RIS element is the additionally integrated active  reflection-type amplifier, which can be realized by different  existing active components, such current-inverting converters  or some integrated circuits [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  With reflection-type amplifiers supported by a power supply,  the reflected and amplified signal of an N-element active RIS  can be modeled as follows:  y =  PΘx  � �� �  Desired signal  +  PΘv  � �� �  Dynamic noise  +  ns  ����  Static noise  ,  (2)  where P := diag (p1, · · · , pN) ∈ RN×N denotes the amplifi-  cation factor matrix of the active RIS, wherein each element pn  can be larger than one thanks to the integrated reflection-type  amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Due to the use of active components, active RISs  consume additional power for amplifying the reflected signals,  and the thermal noise introduced by active RIS elements  cannot be neglected as is done for passive RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Particularly,  as shown in (2), the introduced noise can be classified into  dynamic noise and static noise [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Specifically, v is related to  the input noise and the inherent device noise of the active RIS  elements [5], while the static noise ns is unrelated to P and  is usually negligible compared to the dynamic noise PΘv, as  will be verified by experimental results in Section V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Thus,  here we neglect ns and model v as v ∼ CN  �  0N, σ2  vIN  �  ,  where CN(µ, Σ) denotes the complex multivariate Gaussian  distribution with mean µ and variance Σ, IL is an L × L  identity matrix, and 0L is an L × 1 zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Active RIS Aided MU-MISO System  Consider an active RIS aided downlink MU-MISO system  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 1, where an M-antenna base station (BS)  serves K single-antenna users simultaneously with the aid  of an N-element active RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Let s := [s1, · · · , sK]T ∈ CK  denote the transmitted symbol vector for the K users and let  wk ∈ CM×1 denote the BS precoding vector for symbol sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  According to (2), signal rk ∈ C received at user k can be  modeled as follows:  rk = (  hH  k  ����  Direct link  + f H  k PΘG  �  ��  �  Reflected link  )  �K  j=1wjsj +  f H  k PΘv  �  ��  �  Noise introduced by active RIS  +  zk  ����  Noise introduced at user k  ,  (3)  where [·]H denotes the conjugate-transpose operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' G ∈  CN×M, hH  k  ∈ C1×M, and f H  k  ∈ C1×N characterize the  channels between the BS and the RIS, between the BS and  user k, and between the RIS and user k, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' and zk  denotes the additive white Gaussian noise (AWGN) at user k  with zero mean and variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' PERFORMANCE ANALYSIS  In this section, we analyze the performance of active RISs to  reveal their notable capacity gains compared to passive RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  To this end, in order to make the problem analytically tractable  and get insightful results, in this section, we consider a single-  user single-input single-output (SU-SISO) system with M = 1  BS antenna and K = 1 user, while the general MU-MISO case  is studied in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Asymptotic SNR for Passive RISs and Active RISs  To illustrate the capacity gain provided by passive/active  RIS aided reflected links, for the moment, we ignore the  direct link by setting hk to zero, as was done in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Furthermore, for simplicity, we assume that each active RIS  element has the same amplification factor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', pn := p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For  a fair comparison with the asymptotic performance of passive  RISs, similar to [6], we assume Rayleigh-fading channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  We first redefine the BS-RIS channel matrix and the RIS-  user channel vector as G := g ∈ CN×1 and fk := f ∈ CN×1,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Then, we recall the following lemma from [6] for  the asymptotic SNR achieved by passive RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Lemma 1 (Asymptotic SNR for passive RISs): Assuming  f ∼ CN  �  0N, ϱ2  fIN  �  , g ∼ CN  �  0N, ϱ2  gIN  �  and letting N →  ∞, the asymptotic SNR γpassive of a passive RIS aided SU-  SISO system is given by  γpassive → N 2 P max  BS  π2ϱ2  fϱ2  g  16σ2  ,  (4)  where P max  BS  denotes the maximum transmit power at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Proof: The proof can be found in [6, Proposition 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  For comparison, under the same transmission conditions, we  provide the asymptotic SNR of an active RIS aided SU-SISO  system in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Lemma 2 (Asymptotic SNR for active RISs): Assuming  f ∼ CN  �  0N, ϱ2  fIN  �  , g ∼ CN  �  0N, ϱ2  gIN  �  and letting N →  ∞, the asymptotic SNR γactive of an active RIS aided SU-SISO  system is given by  γactive → N  P max  BS  P max  A  π2ϱ2  fϱ2  g  16  �  P max  A  σ2vϱ2  f + P max  BS  σ2ϱ2g + σ2σ2v  �,  (5)  where P max  A  denotes the maximum reflect power of the active  RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Proof: Please see the journal version [4, Appendix A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Remark 1: From (5) we observe that, the asymptotic SNR  of an active RIS aided SU-SISO system depends on both the  BS transmit power P max  BS  and the reflect power of the active  RIS P max  A  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' When P max  BS  → ∞, the asymptotic SNR will be  upper-bounded by γactive → NP max  A  π2ϱ2  f/  �  16σ2�  , which is  independent of the BS-RIS channel g and the noise power at  the active RIS σ2  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Similarly, if P max  A  → ∞, the asymptotic  SNR will be upper-bounded by γactive → NP max  BS π2ϱ2  g/16σ2  v,  which is independent of the RIS-user channel f and the noise  power at the user σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' These results reveal that, to increase the  sum-rate of active RIS aided systems, the negative impact of  small g and large σ2  v on system performance can be alleviated  by increasing the BS transmit power P max  BS  , and the negative  impact of small f and large σ2 can be reduced by increasing  the reflect power of the active RIS P max  A  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Comparisons between Passive RISs and Active RISs  We can observe from Lemma 1 and Lemma 2 that, compared  to the asymptotic SNR for passive RISs γpassive in (4) which is  proportional to N 2, the asymptotic SNR for active RISs γactive  in (5) is proportional to N due to the noises introduced by  the use of active components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' At first glance, it seems that the  SNR achieved by passive RISs γpassive always exceeds the SNR  achieved by active RISs γactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' However, this is actually not the  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The reason behind this counterintuitive behavior is that,  due to the large path loss caused by the “multiplicative fading”  effect and thanks to the use of the reflection-type amplifiers  in active RISs, only when N is unaffordably large can passive  RISs outperform active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  To illustrate this claim, let us consider two different SU-  SISO systems, which are aided by an active RIS and a passive  RIS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Then, the following lemma specifies the  condition that has to be met for passive RISs to outperform  active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Lemma 3 (Case when passive RISs outperform active  RISs): Assuming the number of RIS elements N is large,  the required number of elements N for a passive RIS to  outperform an active RIS has to satisfy  N ≥ P max  BS-A  P max  BS-P  P max  A  σ2  �  P max  A  σ2vϱ2  f + P max  BS-Aσ2ϱ2g + σ2σ2v  �,  (6)  where P max  BS-A denotes the maximum BS transmit power for the  active RIS aided system and P max  BS-P denotes that for the passive  RIS aided system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Proof: Please see the journal version [4, Appendix B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Next, we consider a specific setup to compare the user’s  achievable SNRs in the above two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For a fair com-  parison, we constrain the total power consumption P max of  the two systems to 2 W by setting P max  BS-P = 2 W for the  passive RIS aided system and P max  BS-A = P max  A  = 1 W for  the active RIS aided system, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Therefore, when  σ2 = σ2  v = −70 dBm and ϱ2  f = ϱ2  g = −70 dB, the required  number of elements N for the passive RIS to outperform the  active RIS is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 × 106 according to (6), which is impractical  to realize with current technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Conversely, for a more  practical number of elements of N = 256, according to (5)  and (4), the SNR achieved by the passive RIS is γpassive ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='0  dB, while the SNR achieved by the active RIS is γactive ≈ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='0  dB, which is about 10, 000 times higher than γpassive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' JOINT TRANSMIT PRECODING AND REFLECT  BEAMFORMING DESIGN  To investigate the capacity gain enabled by the use of  active RISs in typical wireless communication scenarios, in  this section, we consider more general MU-MISO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  According to the model in (3), the signal-to-interference-plus-  noise ratio (SINR) at user k can be obtained as  γk =  ��¯hH  k wk  ��2  �K  j=1,j̸=k  ��¯hH  k wj  ��2 +  ��f H  k PΘ  ��2σ2v + σ2 ,  (7)  wherein ¯hH  k = hH  k + fk  HPΘG ∈ C1×M is the equivalent  channel from the BS to user k, which includes both the direct  link and the reflected link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Therefore, the original problem of  sum-rate maximization, subject to the power constraints at the  BS and the active RIS, can be formulated as follows:  Po : max  w,P,Θ Rsum(w, P, Θ) =  �K  k=1 log2 (1 + γk),  (8a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' C1 :  �K  k=1 ∥wk∥2 ≤ P max  BS  ,  (8b)  C2 :  �K  k=1∥PΘGwk∥2+∥PΘ∥2 σ2  v ≤P max  A  , (8c)  where w :=  �  wT  1 , · · · , wT  K  �T is the overall transmit precoding  vector for the K users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' C1 and C2 are the power constraints at  the BS and active RIS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Due to the non-convexity  and the highly coupled variables in problem Po in (8), the  joint design of w, P, and Θ is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  To efficiently solve the above problem, we develop a joint  precoding and beamforming algorithm based on alternating  optimization and fractional programming (FP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Note that P  and Θ always appear in product form in problem Po in  (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Therefore, P and Θ can be merged as Ψ = PΘ =  diag  �  p1ejθ1, · · · , pNejθN �  ∈ CN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' We refer to Ψ as the  RIS beamforming matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Next, to deal with the non-convex  sum-of-logarithms and fractions in (8), we exploit the FP  methods proposed in [7] to decouple the variables in problem  Po in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' This leads to the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Lemma 4 (Equivalent problem for sum-rate maximiza-  tion): By introducing auxiliary variables ρ := [ρ1, · · · , ρK] ∈  RK and ϖ := [ϖ1, · · · , ϖK] ∈ CK, the original problem Po  in (8) can be equivalently reformulated as follows  P1 :  max  w,Ψ,ρ,ϖ R′  sum(w, Ψ, ρ, ϖ) =  �K  k=1 ln (1 + ρk)−  �K  k=1 ρk +  �K  k=1 g(w, Ψ, ρk, ϖk),  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' C1, C2,  (9)  where function g(w, Ψ, ρk, ϖk) is defined as  g(w, Ψ, ρk, ϖk) = 2  �  (1 + ρk)R  �  ϖ∗  k¯hH  k wk  �  −  |ϖk|2  ��K  j=1  ��¯hH  k wj  ��2 +  ��f H  k Ψ  ��2σ2  v + σ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  (10)  Proof: Constructive proof can be found in [7, Subsection  III-C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Strong convergence of the FP methods was proved in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Thus, a locally optimal solution to (9) can be obtained by  alternately optimizing the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For clarity, we summarize  the proposed joint precoding and beamforming algorithm in  Algorithm 1, and the specific solutions for variables w, Ψ,  ρ, and ϖ are given in the following four steps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Algorithm 1 Proposed joint transmit precoding and reflect  beamforming algorithm  Input:  Channels G, hk, and fk, ∀k ∈ {1, · · · , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Output:  Optimized BS precoding vector w, amplification  factor matrix of active RIS P, phase shift matrix of active  RIS Θ, and sum-rate Rsum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  1: Randomly initialize w, P and Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2: while no convergence of Rsum do  3:  Update ρ by (11);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  4:  Update ϖ by (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  5:  Update w by solving problem P2 in (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  6:  Update Ψ by solving problem P3 in (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  7: end while  8: Obtain P and Θ from Ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  9: return Optimized w, P, Θ, and Rsum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  1) Fix (w, Ψ, ϖ) and optimize ρ: After fixing precoding  vector w, beamforming matrix Ψ, and auxiliary variable ϖ,  the optimal ρ can be obtained by solving ∂R′  sum  ∂ρk  = 0 as  ρopt  k  = ξ2  k + ξk  �  ξ2  k + 4  2  ,  ∀k ∈ {1, · · · , K},  (11)  where ξk = ℜ  �  ϖ∗  k¯hH  k wk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2) Fix (w, Ψ, ρ) and optimize ϖ: After fixing the precod-  ing vector w, beamforming matrix Ψ, and auxiliary variable  ρ, the optimal ϖ can be derived by solving ∂R′  sum  ∂ϖk  = 0 as  ϖopt  k  =  �  (1 + ρk)¯hH  k wk  �K  j=1  ��¯hH  k wj  ��2+  ��f H  k Ψ  ��2σ2v + σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  (12)  3) Fix (Ψ, ρ, ϖ) and optimize w: To simplify the nota-  tions, we first introduce the following definitions:  bH  k =  �  (1 + ρk)ϖ∗  k¯hH  k , b =  �  bT  1 , bT  2 , · · · , bT  N  �T ,  (13a)  A=IK ⊗  �K  k=1|ϖk|2¯hk¯hH  k , Ξ=IK ⊗  �  GHΨHΨG  �  , (13b)  P max  m  = P max  A  − ∥Ψ∥2σ2  v,  (13c)  where ⊗ denotes the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Then, problem P1 in  (9) can be reformulated as follows:  P2 :  max  w  R  �  2bHw  �  − wHAw,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  C1 : ∥w∥2 ≤ P max  BS  ,  C2 : wHΞw ≤ P max  m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  (14)  Note that P2 in (14) is a standard quadratic constraint  quadratic programming (QCQP) problem, which can be solved  by alternating direction method of multipliers (ADMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  4) Fix  (w, ρ, ϖ)  and  optimize  Ψ:  Define  ψ  =  �  p1ejθ1, · · · , pNejθN �H as the vectorized RIS beamforming  matrix Ψ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', diag  �  ψH�  := Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' While fixing w and ρ and  ϖ, problem P1 in (9) can be reformulated as follows:  P3 :  max  ψ  R  �  2ψHυ  �  − ψHΩψ,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' C2 : ψHΠψ ≤ P max  A  ,  (15)  spectrum  analyzer  LNA  vector network  analyzer  DC source  active RIS  element  circulator  pump  source  noise source  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The experimental devices and environment used for validating the signal model (2) of active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='359  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='3595  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='36  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='3605  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='361  15  10  5  0  5  10  15  20  25  30  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Experimental measurement result for reflection gain G versus signal  frequency f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  wherein  υ =  �K  k=1  �  (1 + ρk)diag  �  ϖ∗  kf H  k  �  Gwk−  �K  k=1 |ϖk|2diag  �  f H  k  �  G  �K  j=1 wjwH  j hk,  (16a)  Ω =  �K  k=1 |ϖk|2diag  �  f H  k  �  diag (fk) σ2  v+  �K  k=1 |ϖk|2 �K  j=1diag  �  f H  k  �  GwjwH  j GHdiag (fk), (16b)  Π =  �K  k=1 diag (Gwk) (diag (Gwk))H + σ2  vIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  (16c)  Note that problem P3 in (15) is also a standard QCQP  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Thus, the optimal solution ψopt can be obtained by  adopting ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' VALIDATION RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Validation Results for Signal Model  To validate the signal model (2), we designed and fabri-  cated an active RIS element with an integrated reflection-  type amplifier for experimental measurements in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Note  that this design can be directly extended to the large-array  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Particularly, since the phase-shifting ability of RISs has  been widely verified, we focus on studying the reflection gain  15  10  5  0  5  10  15  20  25  170  165  160  155  150  145  140  135  130  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Experimental measurement result for the density of noise power  Gσ2  v + σ2  s versus reflection gain G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  and the noise introduced by an active RIS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Thus, the  validation of signal model (2) is equivalent to validating  Py =  GPx  ����  Desired-signal power  + Gσ2  v + σ2  s  �  ��  �  noise power  ,  (17)  where Py is the power of the signals reflected by the active  RIS element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Px is the power of the incident signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' G := p2 is  the reflection gain of the active RIS element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Gσ2  v and σ2  s are  the powers of the dynamic noise and static noise introduced  by the active RIS element, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  1) Hardware platform: To validate the model in (17), we  first establish the hardware platform used for our experimental  measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Due to space constraints, we refer  the reader to the journal version of this paper [4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 4] for  detailed information about the hardware platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2) Reflection gain measurement: Using the measurement  system for the reflection gain depicted in [4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 4 (b)], we  first investigate the reflection gain G of the active RIS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Note that the reflection gain G can be reconfigured by the  input power of the pump source Pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' By setting the input  power of the vector network analyzer as Px = −50 dBm, the  reflection gain G as a function of the signal frequency can be  directly measured via a vector network analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Then, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  10  5  0  5  10  15  20  25  30  0  10  20  30  40  50  60  507%  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Simulation results for the sum-rate versus total power consumption  P max in scenario 1 with a weak direct link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='.  3, we show the measurement results for reflection gain G as  a function of signal frequency f for different input powers of  the pump source Pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' We observe that the active RIS element  can achieve a reflection gain G of more than 25 dB, when  Pp = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='24 dBm, which confirms the significant reflection  gains enabled by active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  3) Noise power measurement: We further study the noise  power introduced by the active RIS element, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=', Gσ2  v + σ2  s  in (17), where Gσ2  v and σ2  s are the powers of the dynamic  noise and the static noise introduced at the active RIS element,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Using the noise measurement system in [4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 4  (c)], we show the measurement results for the spectral density  of noise power Gσ2  v + σ2  s as a function of G for different  operating frequencies in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' We can observe that the noise  power increases nearly linearly with G, which verifies the  noise model Gσ2  v + σ2  s in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Particularly, for f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='3601  GHz, the spectral density of σ2  s is about −174 dBm/Hz, while  that of σ2  v is about −160 dBm/Hz, which is about 15 dB  higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The reason for this is that the input noise is amplified  by the noise factor, and additional noises are also introduced  by the other active components such as the DC source used  to power the active RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Simulation Results for Sum-Rate  1) Simulation setup: We consider an active RIS aided MU-  MISO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Particularly, we consider two scenarios with dif-  ferent channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In scenario 1, the direct link is weak  due to severe obstruction, while the direct link is strong in  scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' To be specific, two different path loss models from  the 3GPP TS 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='814 standard are utilized to characterize the  large-scale fading of the channels: i) PLs = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='3+22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='0 log d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  ii) PLw = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='2 + 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='7 log d, where d is the distance between  two devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Path loss model PLw is used to generate the weak  BS-user link in scenario 1, while PLs is used to generate the  strong BS-user link in scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For both scenarios, PLs is  used to generate the BS-RIS and the RIS-user channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' To  account for small-scale fading, we adopt the Ricean fading  channel model for all channels involved and we assume the  Ricean factor as κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  10  5  0  5  10  15  20  25  30  0  10  20  30  40  50  60  62%  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Simulation results for the sum-rate versus total power consumption  P max in scenario 2 with a strong direct link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  The BS and the active/passive RIS are located at (0, -60  m) and (200 m, 30 m), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Four users are randomly  located in a circle with a radius of 5 m from the center (200  m, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The numbers of BS antennas and RIS elements are set  as M = 4 and N = 256, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The noise power is set  as σ2 = σ2  v = −70 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For fair comparison, we constrain  the total power consumption P max := P max  BS  + P max  A  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For the  active RIS, Algorithm 1 is employed for joint precoding and  beamforming design, while for the passive RIS, the algorithm  from [2] is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2) Simulation results: In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 6, we plot the  sum-rate versus the total consumed power P max for the two  considered scenarios, where the direct link is weak and strong,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Firstly, in scenario 1 with a weak direct link, the  passive RIS can indeed achieve a performance improvement,  while the active RIS achieves a much higher sum-rate gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  Secondly, in scenario 2 with a strong direct link, the passive  RIS achieves only a negligible sum-rate gain, while the active  RIS still realizes a noticeable sum-rate gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For example,  when P max = 10 dBW, the capacities without RIS, with  passive RIS, and with active RIS in scenario 1 are 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='34 bps/Hz,  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='00 bps/Hz, and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='41 bps/Hz respectively, while in scenario  2, these values are 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='87 bps/Hz, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='51 bps/Hz, and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='18  bps/Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In this case, the passive RIS provides a  31% gain in scenario 1 and a negligible 3% gain in scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  By contrast, the active RIS achieves noticeable sum-rate gains  of 507% in scenario 1 and 62% in scenario 2, which are much  higher than those achieved by the passive RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Field Test for a 64-Element Active RIS Aided Wireless  Communication Prototype  1) 64-element active RIS aided communication prototype:  To validate the significant gain of active RISs, we develop a  64-element active RIS aided wireless communication proto-  type, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Specifically, the hardware structure  of this prototype consists of three parts including a BS, a 64-  element active RIS, and a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' For the BS and the user, two  horn antennas with 13 dBi antenna gain are used to transmit  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' A photograph of the developed 64-element active RIS aided wireless  communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  and receive the signals, and the universal software radio  peripherals (USRPs) are deployed to generate and process the  baseband and RF signals (hardware version: USRP-2953R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  By periodically expanding the active RIS elements designed  in [8], the 64-element active RIS is an 8×8 plane array, of  which each element has a reflection gain of G = 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2) Experimental environment: Based on the developed pro-  totype, we establish the experimental environment for further  validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' To match the transceivers, we configure the oper-  ating frequency of the active RIS to f = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 GHz and the  bandwidth to 40 MHz by adjusting the circuit impedance of  active elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The polarization of the antenna at the BS  and that at the user are selected as vertical and horizontal,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The transmit power is set to −10 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' We fix the  heights of the BS, the RIS, and the user as 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The horizontal  distance of the BS-RIS link and that of the RIS-user link are  set to 2 m and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The angle of arrival (AoA)  at the active RIS is fixed as 0◦, and the angle of departure  (AoD) will be specified to evaluate the performance gain of  active RISs at different orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' To observe the reflection  gain of the active RIS, we use a metal plate with the same  aperture size as the active RIS for performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  3) Experimental results: By moving the user at different  AoDs and configuring the phase shift of the active RIS with  discrete Fourier transform (DFT) codebook, we obtain the  experimental results shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' One can observe that,  compared with the received power for the metal plate, the  active RIS can always achieve a gain of about 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The data  rate for the active RIS can hold at about 30 Mbps, while that  for the metal plate only ranges from 1 Mbps to 2Mbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' The  reason is that, the beamforming at the active RIS can make the  reflected beam with high array gain and reflection gain, while  the metal plate can only reflect the incident signals randomly  without in-phase combination or amplification, which validates  the significant gain of active RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' CONCLUSIONS  In this paper, we have studied the concept of active RISs  to overcome the fundamental limitation of the “multiplicative  fading” effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Firstly, we have verified the signal model of  TABLE I  EXPERIMENTAL RESULTS FOR THE DEVELOPED PROTOTYPE  AoD  Device  Received Power  Data Rate  15◦  Metal plate  110 dBm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='2 Mbps  Active RIS  100 dBm  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 Mbps  30◦  Metal plate  105 dBm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 Mbps  Active RIS  98 dBm  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 Mbps  45◦  Metal plate  105 dBm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='5 Mbps  Active RIS  95 dBm  30 Mbps  60◦  Metal plate  108 dBm  2 Mbps  Active RIS  90 dBm  32 Mbps  active RISs through the experimental measurements on a fab-  ricated active RIS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Based on the verified signal model,  we have formulated the sum-rate maximization problem for  an active RIS aided MU-MISO system and a joint precoding  and beamforming algorithm has been proposed to solve this  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Simulation results have shown that, in a typical  application scenario, the existing passive RIS can realize only  a negligible sum-rate gain of about 3%, while the active RIS  can achieve a substantial sum-rate gain of about 62%, thus  indeed overcoming the “multiplicative fading” effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' Finally,  we have developed a communication wireless communication  prototype aided by a 64-element active RIS, and the significant  gain of active RISs is validated by field test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' In the future,  many research directions for active RISs are worth pursuing,  such as hardware design, prototype development, channel  estimation, and energy efficiency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  ACKNOWLEDGMENT  This work was supported in part by the National Key  Research and Development Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content='  2020YFB1805005), in part by the National Natural Science  Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
+page_content=' 62031019), and in part by the  European Commission through the H2020-MSCA-ITN META  WIRELESS Research Project under Grant 956256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfWPfd/content/2301.00161v1.pdf'}
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diff --git a/ZdE5T4oBgHgl3EQfeA8r/content/tmp_files/2301.05615v1.pdf.txt b/ZdE5T4oBgHgl3EQfeA8r/content/tmp_files/2301.05615v1.pdf.txt
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@@ -0,0 +1,475 @@
+Linear Computation Coding:
+Exponential Search and Reduced-State Algorithms
+Hans Rosenberger, Johanna S. Fr¨ohlich, Ali Bereyhi and Ralf R. M¨uller
+Institute for Digital Communications (IDC)
+Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg
+Erlangen, Germany
+{hans.rosenberger, johanna.froehlich, ali.bereyhi, ralf.r.mueller}@fau.de
+Abstract
+Linear computation coding is concerned with the compression of multidimensional linear
+functions, i.e. with reducing the computational effort of multiplying an arbitrary vector to
+an arbitrary, but known, constant matrix. This paper advances over the state-of-the art,
+that is based on a discrete matching pursuit (DMP) algorithm, by a step-wise optimal search.
+Offering significant performance gains over DMP, it is however computationally infeasible
+for large matrices and high accuracy. Therefore, a reduced-state algorithm is introduced
+that offers performance superior to DMP, while still being computationally feasible even for
+large matrices. Depending on the matrix size, the performance gain over DMP is on the
+order of at least 10 %.
+Introduction
+Multiplying a vector by a constant matrix is an ubiquitous task performed in various
+technical and scientific applications. The main body of earlier work is focused on
+speeding up the calculation of matrix-vector multiplications in a structure-oriented
+fashion. A well-known example is the fast implementation of the discrete fourier
+transform (DFT). Here, the structure of the DFT matrix is exploited to eliminate
+redundant computations and reduce the number of required operations as compared
+to a naive implementation. For arbitrary constant matrices, redundancies within the
+finite-precision representation of the matrix entries can be exploited as well, a method
+that is typically known as common subexpression sharing/elimination. Earlier work
+in this respect has either targeted special cases of constant multiplication [1, 2] or has
+proposed schemes with high computational complexity, such that their implementation
+in practice is difficult for medium to large size matrices [3, 4].
+Recently, linear computation coding (LCC) has been proposed in [5, 6, 7]. This
+framework develops an information-theoretic scheme for the efficient calculation
+of matrix-vector products that is especially well-suited for the implementation on
+reconfigurable hardware, such as field programmable gate arrays (FPGAs) [8]. Similar
+to rate-distortion theory, LCC is concerned with the tradeoff between distortion
+and compression. However, instead of compressing data, LCC deals with the lossy
+compression of multidimensional linear functions under a given fidelity constraint. An
+This work was supported by Deutsche Forschungsgemeinschaft (DFG) under the project Compu-
+tation Coding (MU-3735/8-1).
+arXiv:2301.05615v1  [cs.IT]  13 Jan 2023
+
+instance can be found in [7], where an optimal decomposition scheme is first defined
+in terms of classical metrics for computation and distortion. A greedy approach is
+then developed to approximate the proposed scheme sub-optimally with tractable
+complexity.
+Contributions
+In this paper, we develop a new LCC scheme. Similar to earlier approaches discussed
+in [7], the optimal decomposition deals with an exponentially complex problem. We first
+address this problem via an exhaustive search procedure with a careful optimization.
+This enables us to evaluate the performance of the optimal scheme for reasonable
+matrix sizes. We then present a computationally tractable scheme by proposing a
+reduced-state algorithm for the underlying search problem. Our investigations show
+that the proposed algorithm can achieve a computation-distortion tradeoff close to the
+exponentially-complex optimal scheme while drastically reducing the decomposition
+complexity.
+Notation
+Vectors are denoted as lower-case boldface letters x and matrices as upper-case boldface
+letters X. The Euclidean and the Frobenius norm are denoted by ∥ · ∥2 and ∥ · ∥F,
+respectively. The symbol 0N×K denotes an N × K matrix with all zero elements,
+IN×K denotes the augmented identity matrix of dimension N × K and 1j,K denotes
+the j-th row unit vector in K dimensions.
+Problem Formulation
+We consider the problem of matrix-vector multiplication, i.e the calculation
+y = Ax
+(1)
+for an arbitrary input vector x ∈ RK×1 and a constant matrix A ∈ RN×K. Commonly,
+matrices are approximated by quantizing their entries independently. By using the
+canonically signed digit (CSD) binary representation the quantization error can be
+decreased on average by a factor of
+√
+28 per CSD [9]. This still leaves room for
+improvement. LCC instead suggests to approximate A by a product of matrices, i.e.
+finding W and C such that
+A ≈ W C.
+(2)
+The matrix C ∈ AN×K is termed the codebook matrix and W ∈ AN×N is termed the
+wiring matrix in the sequel1. The entries of the wiring matrix are restricted to the set
+of zero and signed powers of two (A ⊆ {0, ±2Z}).
+Obtaining the wiring and codebook matrix jointly is typically NP-hard and infea-
+sible. To overcome this computational intractability, [7] proposes a scheme where the
+1In [6] the multiplication order of the decomposed matrices is reversed. Please note that this
+change makes no difference to the general idea of the decomposition and to the following algorithms.
+It is equal to the transposed version of the algorithm presented in [7].
+
+n-th row of the wiring matrix is determined by solving the following sparse recovery
+problem for some design parameter S < N controlling the cost between distortion and
+computation effort [10]
+wn =
+argmin
+ω∈{ω=�S
+s=1 is1js,N: is∈A, js∈{1,...,N} ∀s}
+∥an − ωC∥2
+∀n.
+(3)
+The new scheme is still NP-hard, but not in N, anymore, but in S. Thus, small values
+of S are required, in practice.
+In order to have a high accuracy despite small values of S, the factorization
+procedure can be applied multiple times. Then the product Ci = W iCi−1 of the
+previous wiring step acts as the new codebook for obtaining the following matrix
+factor W i of the current wiring step. Hence, by setting2 C0 = IN×K, we obtain the
+approximated matrix P after I wiring steps:
+A ≈ P =
+� I�
+i=1
+W i
+�
+C0.
+(4)
+To quantify the accuracy of a given approximation P we use the signal-to-
+quantization-noise-ratio (SQNR)
+SQNR(A, P ) =
+∥A∥2
+F
+∥A − P ∥2
+F
+.
+(5)
+Computational Cost
+In a binary number representation the multiplication by a signed power of two
+corresponds only to a bitshift. On reconfigurable hardware, this shift can be realized
+simply by appropriate wiring without the need for dedicated processing elements
+such as adders [8]. The parameter S in (3) determines the number of vectors from
+the codebook to be used in forming the linear combination to approximate a row
+an of A. It therefore directly controls the computational cost, as in computing the
+linear combination, exactly S − 1 additions are required. No multiplications, except
+by signed powers of two, are necessary due to the specific structure of the wiring
+matrix. Therefore, the separate product of the decomposed matrices with the input
+vector y ≈ W (Cx) is much simpler to compute than calculating the product in (1)
+straightforwardly.
+The total computational cost Cadd of a decomposition in (4) is given by the number
+of additions (or subtractions) required to form the linear combinations
+Cadd = IN(S − 1).
+(6)
+2In [6] this choice is termed the self-designing codebook. It was found to work very well for a
+wide range of matrices.
+
+Algorithms
+In this section we will briefly look at the state of the art for solving the optimization
+problem in (3) to obtain the wiring matrices and then introduce two improved novel
+algorithms.
+State-of-the-Art: Discrete Matching Pursuit
+The discrete matching pursuit (DMP) follows the matching pursuit approach to
+successively determine the wiring coefficients. The algorithm can be summarized in
+the following key steps; for details see [6].
+1. Start with iteration s ← 0. Initialize ω ← 01×N
+2. Update ω in at most a single component, such that ∥an − ωC∥2 is minimized.
+3. Increment s.
+4. If s ≤ S, go to step 2, otherwise the procedure terminates.
+It is straightforward to show that the time complexity of the DMP algorithm for
+computing a single matrix factor scales with O(N 3S).
+Exponential Search Algorithm
+The row-wise optimization problem in (3) is NP-hard. However, for small S, reasonable
+matrix sizes and some careful optimization it can be solved in a tractable timeframe. We
+limit the set of scaling factors to a finite set of signed powers of two (Aexp ⊂ {0, ±2Z}),
+as an exhaustive search over the whole set is infeasible. As the search procedure has
+to be performed for each row of the target matrix individually, the time complexity
+for the computation of each wiring step is given by O(N S|Aexp|S).
+Generally, which and how many coefficients are included in the subset Aexp is a
+design parameter and needs to be adapted to each specific decomposition. It depends
+primarily on two factors. First, the current wiring step plays a crucial role. For
+each additional wiring layer, the error between each row of the target matrix and
+the approximated matrix decreases. Hence, for any subsequent wiring step, smaller
+coefficients are needed to scale the rows of the newly found codebook matrix to
+appropriately approximate the residual error. This also means that for high desired
+accuracy the coefficient set needs to be chosen large, i.e. to include also many small
+coefficients, to accurately approximate the error. Furthermore, relative variations
+in the length of the row vectors of the target matrix require a larger coefficient
+set to compensate for differences. Still, to keep the decomposition computationally
+feasible, the number of elements in Aexp needs to be chosen as small as possible, as
+the computational complexity scales exponentially in S with the product of the size
+of the coefficient set |Aexp| and N as base.
+A promising approach for further research is to adapt the coefficient set for each
+wiring step dynamically based on the current fidelity of the approximation. The
+coefficient set may then be determined from the probability distribution of the likely
+entries of the wiring matrix.
+
+Reduced-State Algorithm
+The runtime of the exponential algorithm for a matrix with N = 64 rows is on the
+order of an hour3, for larger matrices the runtime scales up accordingly and can
+become infeasible. Hence, for some applications the computational burden of the
+exponential algorithm might consequently not be feasible. A reasonable compromise
+between complexity and performance is desirable.
+Unlike in DMP, we do not immediately update the components of the wiring
+vector for the reduced-state algorithm. Instead, we keep updating in each iteration a
+list of the M best vectors, which minimize (3) and select at the termination of the
+algorithm the vector with minimum error from the list. Specifically, we apply the
+following successive, greedy procedure for the optimization problem in (3):
+1. Start with s ← 0. Initialize the set Ω with M all-zero vectors.
+2. For each ω ∈ Ω find the set Ωm of M mutually distinct vectors ω ˜m with
+˜m ∈ {1, ..., M} that minimize ∥an − ω ˜mC∥2 and differ from ω in at most a
+single component.
+3. Update the set Ω by selecting from �M
+m=1 Ωm the M distinct vectors ω with
+minimum ∥an − ωC∥2
+4. Increment s.
+5. If s ≤ S, go to step 2, else, continue
+6. Return argmin
+ω∈Ω
+∥an − ωC∥2
+By tuning M, we are able to adjust the space of possible combinations that the
+algorithm explores. For each wiring step, we have to evaluate O(SN 3M 2) combinations.
+Compared to DMP the complexity is increased by a factor of M 2. Note that for
+M = 1 the algorithm reduces to the DMP.
+Numerical Evaluation
+In this section, we compare the performance of the proposed algorithms to the baseline
+DMP [7]. We decompose matrices whose entries are drawn i.i.d. from a Gaussian
+distribution with zero mean and unit variance. Similar to DMP, the performance of the
+improved versions of the algorithm hardly depends on the distribution of the matrix
+elements. The algorithms are invariant to scaling of the variance, however it is crucial
+that for the exponential search algorithm the coefficient set Aexp is scaled appropriately
+as well, as to not compromise performance. Throughout the simulations, we select the
+coefficient set for the exponential search algorithm to Aexp = {±2−40, .., ±23}.
+As a first experiment, we compare all three algorithms for fixed matrix sizes in
+Figure 1. We choose matrices with N = 64 rows and vary the number of columns K
+3For a multithreaded implementation in Python with Numba acceleration executed on an Intel
+i9-12900@2.40 GHz and parameters S = 3, Aexp =
+�
+±2−40, . . . , ±23�
+.
+
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Cumulative number of additions
+0
+20
+40
+60
+80
+100
+120
+SQNR [dB]
+64x4
+64x6
+64x8
+Figure 1: Performance comparison for matrices with 64 rows and different aspect ratios.
+Matrices of dimension 64×4 are indicated by crosses, 64×6 is indicated by squares and 64×8
+is indicated by triangles. The solid lines refer to the exponential search algorithm (S = 3),
+the dashed lines to the DMP [7] (S = 2) and the dashed-dotted lines to the reduced state
+algorithm with memory size M = 10 and S = 3. Results are averaged over 104 matrix entries
+for the exponential search algorithm and over 105 matrix entries for the other algorithms.
+from four to eight. Hence, we can compare the performance for different aspect ratios
+of the matrices4. To quantify performance, we plot the tradeoff between distortion
+and computational cost. The latter being measured by the number of cumulative
+additions Cadd required for a given wiring step.
+As Figure 1 shows, for all three matrix sizes there is a performance gain by both
+proposed algorithms against DMP. Further, for memory size M = 10, the reduced
+state algorithm performs only slightly worse than exhaustive search.
+For the first two wiring steps, the performance of all three algorithms is equal
+for a given matrix size. This is due to the fact that for the first two steps we use
+DMP with S = 2 for an initial refinement of the codebook5. Applying any of the
+4LCC works best for matrices with an exponential aspect ratio, i.e. for K ≈ log2 N. For square
+matrices it is beneficial to cut these into multiple tall matrices and decompose each slice independently,
+see [8] for details.
+5Using any of the novel algorithms with S = 2 is possible as well with very similar performance, i.e.
+a slight increase in SQNR by 0.2 dB to 0.5 dB for the novel algorithms and matrix sizes considered.
+
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Cumulative number of additions
+0
+10
+20
+30
+40
+50
+60
+70
+80
+SQNR [dB]
+Increasing M
+(2, 5, 10, 50)
+DMP
+Exponential Search
+Reduced State (Variable M)
+Figure 2: Performance comparison for different memory sizes M of the reduced state
+algorithm of a 64 × 6 matrix. The solid line refers to the exponential search algorithm
+(S = 3), the dashed line to the DMP [7] (S = 2) and the dashed-dotted lines to the reduced
+state algorithm with 4 different memory sizes and S = 3. Results are averaged over 104
+matrix entries for the exponential search algorithm and over 105 matrix entries for the other
+algorithms.
+algorithms with S ≥ 3 directly to the initial codebook C0 = IN×K would lead to
+degraded performance for the first few wiring steps. Instead it is beneficial to set S = 2
+to allow for more frequent updates of the codebook in the beginning. From empirical
+investigations it seems that two wiring iterations with S = 2 are most beneficial for
+the overall performance.
+As the next experiment, we compare the performance of the reduced state algorithm
+for different choices of the memory parameter M for given matrix size of 64 × 6 in
+Figure 2.
+From the figure we observe that even for small M the reduced state
+algorithm offers a noticeable performance gain over DMP. As M grows, the reduced
+state algorithm approaches the performance of the exponential search. The large
+advantage of the reduced state algorithm is that the computation time is reduced
+drastically6. For M = 1 the reduced state algorithm reduces to DMP and both lines
+6For the considered matrix size the execution time differs from an hour for the exponential
+algorithm to a few seconds for the reduced state algorithm in practice.
+
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Cumulative number of additions
+0
+20
+40
+60
+80
+100
+120
+SQNR [dB]
+Increasing S
+(3, 4, 8)
+DMP
+Exponential search (S = 3)
+Reduced State (S = 3, 4, 8)
+Figure 3: Performance comparison for different choices of the parameter S of a matrix
+with dimension 64 × 4. The solid line refers to the exponential search algorithm (S = 3),
+the dashed line to the DMP [7] (S = 2) and the dashed-dotted lines to the reduced state
+algorithm with different choices of the parameter S and M = 10. Results are averaged over
+104 matrix entries for the exponential search algorithm and over 105 matrix entries for the
+other algorithms.
+coincide for any given matrix size.
+Table 1 lists the relative performance gains over DMP for various matrix sizes and
+configurations of the algorithms.
+Practical Considerations for the Choice of S
+Figure 3 shows the performance of the reduced state algorithm for varying S. Due to
+the exponentially growing complexity in S, the exponential algorithm is not feasible
+for S > 3, except for very small matrices. We can observe, that by choosing S = 4
+for the reduced state algorithm, we approximately achieve the same distortion-cost
+tradeoff as for the exponential algorithm with S = 3. For choosing S even larger the
+gains increase likewise.
+However for a practicable implementation S should not be chosen arbitrarily. In [8],
+the performance of DMP is validated in an implementation on reconfigurable hardware
+with S = 2. This means that on an FPGA exactly N adders are required per wiring
+matrix. With the inputs depending only on the outputs of the previous wiring matrix,
+
+Table 1: Relative average gain in terms of SQNR of the novel algorithms over the baseline
+DMP algorithm with S = 2 (For at least 8 bit signed integer accuracy (10 log(SQNR) ≥
+47 dB)). Results are averaged over 104 matrix entries for the exponential search algorithm
+and over 105 matrix entries for the other algorithms.
+Exponential
+Reduced State
+search
+S = 3
+S = 4
+S = 8
+Matrix size
+S = 3
+M = 5
+M = 10
+M = 5
+M = 10
+M = 5
+M = 10
+16 × 2
+17.8 %
+10.4 %
+13.4 %
+14.1 %
+17.8 %
+16.7 %
+21.9 %
+16 × 4
+34.5 %
+16.0 %
+24.5 %
+25.8 %
+32.7 %
+25.4 %
+34.4 %
+32 × 4
+15.7 %
+10.5 %
+12.9 %
+14.0 %
+17.2 %
+18.4 %
+22.3 %
+32 × 6
+25.3 %
+15.5 %
+19.0 %
+19.7 %
+24.5 %
+19.4 %
+26.0 %
+64 × 4
+12.9 %
+7.5 %
+9.8 %
+11.0 %
+13.8 %
+13.5 %
+16.8 %
+64 × 6
+14.9 %
+9.6 %
+11.4 %
+13.6 %
+16.5 %
+15.6 %
+19.0 %
+the decomposition is well suited for parallel execution and pipelining [8]. Due to the
+greedy, step-wise nature of DMP S > 2 does not offer significant performance gains
+over S = 2. Both novel algorithms behave differently.
+From an implementation point of view, S = 3 is even more suitable than S = 2
+for an effective implementation in hardware due to the availability of efficient adders
+with three inputs [11]. Interestingly, on modern FPGAs, these adders do not require
+more hardware resources, in terms of Lookup-Tables (LUTs), than an adder with two
+inputs. Any powers of two and three (S = 4, 8, 9, . . . ) can be realized efficiently as well
+by the use of adder trees. However, it is questionable if choosing S > 4 is beneficial,
+as performance gains over S = 3 or S = 4 are small and the desired fidelity of the
+approximation cannot be chosen in a fine granularity anymore7.
+Conclusion
+In this paper, we have proposed two new algorithms for LCC, a framework for
+the lossy compression of multidimensional linear functions. While the exponential
+search algorithm shows the best performance, it is generally infeasible especially for
+large matrices. The proposed reduced-state algorithm, performs close to exponential
+search at a fraction of the computational complexity. The time complexity of the
+decomposition compared to the baseline algorithm from earlier works is only mildly
+increased, while the performance gains over the baseline DMP algorithm are on the
+order of at least 10 %.
+References
+[1] Jason Thong and Nicola Nicolici, “An optimal and practical approach to single constant
+multiplication,” IEEE Transactions on Computer-Aided Design of Integrated Circuits
+and Systems, vol. 30, no. 9, pp. 1373–1386, Sep 2011.
+[2] Yevgen Voronenko and Markus P¨uschel, “Multiplierless multiple constant multiplication,”
+ACM Transactions on Algorithms, vol. 3, no. 2, May 2007.
+7For a matrix of dimension 64 × 4 the reduced state algorithm with S = 8 improves the SQNR
+approximately by 70 dB per matrix factor.
+
+[3] N. Boullis and A. Tisserand, “Some optimizations of hardware multiplication by constant
+matrices,” in Proceedings 2003 16th IEEE Symposium on Computer Arithmetic, 2003,
+pp. 20–27.
+[4] Levent Aksoy, Paulo Flores, and Jose Monteiro, “A novel method for the approximation
+of multiplierless constant matrix vector multiplication,” EURASIP Journal on Embedded
+Systems, , no. 12, May 2016.
+[5] Ralf R. M¨uller, Bernhard G¨ade, and Ali Bereyhi, “Efficient matrix multiplication:
+The sparse power-of-2 factorization,” in 2020 Information Theory and Applications
+Workshop (ITA), 2020, pp. 1–6.
+[6] Ralf R. M¨uller, Bernhard G¨ade, and Ali Bereyhi, “Linear computation coding,” 2021,
+arXiv:2102.00398.
+[7] Ralf R. M¨uller, Bernhard M. W. G¨ade, and Ali Bereyhi, “Linear computation coding:
+A framework for joint quantization and computing,” Algorithms, vol. 15, no. 7, 2022.
+[8] Alexander Lehnert, Philipp Holzinger, Simon Pfenning, Ralf R. M¨uller, and Marc
+Reichenbach, “Most ressource efficient matrix vector multiplication on FPGA,” IEEE
+Access, 2022, Early Access.
+[9] A. D. Booth, “A signed binary mutliplication technique,” The Quarterly Journal of
+Mechanics and Applied Mathematics, vol. 4, no. 2, pp. 236 – 240, Jan 1951.
+[10] Simon Foucart and Holger Rauhut,
+A Mathematical Introduction to Compressive
+Sensing, Springer New York, 2013.
+[11] James M. Simkins and Brian D. Philofsky, “Structures and methods for implementing
+ternary adders/subtractors in programmable logic devices,” Sep 2007,
+US Patent
+7,274,211.
+
diff --git a/ZdE5T4oBgHgl3EQfeA8r/content/tmp_files/load_file.txt b/ZdE5T4oBgHgl3EQfeA8r/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f3019e4a3e86d39d4534ca31e1f75f4e9060a798
--- /dev/null
+++ b/ZdE5T4oBgHgl3EQfeA8r/content/tmp_files/load_file.txt
@@ -0,0 +1,277 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf,len=276
+page_content='Linear Computation Coding:  Exponential Search and Reduced-State Algorithms  Hans Rosenberger, Johanna S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Fr¨ohlich, Ali Bereyhi and Ralf R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' M¨uller  Institute for Digital Communications (IDC)  Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg  Erlangen, Germany  {hans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='rosenberger, johanna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='froehlich, ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='bereyhi, ralf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='mueller}@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='de  Abstract  Linear computation coding is concerned with the compression of multidimensional linear  functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' with reducing the computational effort of multiplying an arbitrary vector to  an arbitrary, but known, constant matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' This paper advances over the state-of-the art,  that is based on a discrete matching pursuit (DMP) algorithm, by a step-wise optimal search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Offering significant performance gains over DMP, it is however computationally infeasible  for large matrices and high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Therefore, a reduced-state algorithm is introduced  that offers performance superior to DMP, while still being computationally feasible even for  large matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Depending on the matrix size, the performance gain over DMP is on the  order of at least 10 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Introduction  Multiplying a vector by a constant matrix is an ubiquitous task performed in various  technical and scientific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The main body of earlier work is focused on  speeding up the calculation of matrix-vector multiplications in a structure-oriented  fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' A well-known example is the fast implementation of the discrete fourier  transform (DFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Here, the structure of the DFT matrix is exploited to eliminate  redundant computations and reduce the number of required operations as compared  to a naive implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For arbitrary constant matrices, redundancies within the  finite-precision representation of the matrix entries can be exploited as well, a method  that is typically known as common subexpression sharing/elimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Earlier work  in this respect has either targeted special cases of constant multiplication [1, 2] or has  proposed schemes with high computational complexity, such that their implementation  in practice is difficult for medium to large size matrices [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Recently, linear computation coding (LCC) has been proposed in [5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' This  framework develops an information-theoretic scheme for the efficient calculation  of matrix-vector products that is especially well-suited for the implementation on  reconfigurable hardware, such as field programmable gate arrays (FPGAs) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Similar  to rate-distortion theory, LCC is concerned with the tradeoff between distortion  and compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' However, instead of compressing data, LCC deals with the lossy  compression of multidimensional linear functions under a given fidelity constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' An  This work was supported by Deutsche Forschungsgemeinschaft (DFG) under the project Compu-  tation Coding (MU-3735/8-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='05615v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='IT]  13 Jan 2023  instance can be found in [7], where an optimal decomposition scheme is first defined  in terms of classical metrics for computation and distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' A greedy approach is  then developed to approximate the proposed scheme sub-optimally with tractable  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Contributions  In this paper, we develop a new LCC scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Similar to earlier approaches discussed  in [7], the optimal decomposition deals with an exponentially complex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' We first  address this problem via an exhaustive search procedure with a careful optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  This enables us to evaluate the performance of the optimal scheme for reasonable  matrix sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' We then present a computationally tractable scheme by proposing a  reduced-state algorithm for the underlying search problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Our investigations show  that the proposed algorithm can achieve a computation-distortion tradeoff close to the  exponentially-complex optimal scheme while drastically reducing the decomposition  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Notation  Vectors are denoted as lower-case boldface letters x and matrices as upper-case boldface  letters X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The Euclidean and the Frobenius norm are denoted by ∥ · ∥2 and ∥ · ∥F,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The symbol 0N×K denotes an N × K matrix with all zero elements,  IN×K denotes the augmented identity matrix of dimension N × K and 1j,K denotes  the j-th row unit vector in K dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Problem Formulation  We consider the problem of matrix-vector multiplication, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='e the calculation  y = Ax  (1)  for an arbitrary input vector x ∈ RK×1 and a constant matrix A ∈ RN×K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Commonly,  matrices are approximated by quantizing their entries independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' By using the  canonically signed digit (CSD) binary representation the quantization error can be  decreased on average by a factor of  √  28 per CSD [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' This still leaves room for  improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' LCC instead suggests to approximate A by a product of matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  finding W and C such that  A ≈ W C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  (2)  The matrix C ∈ AN×K is termed the codebook matrix and W ∈ AN×N is termed the  wiring matrix in the sequel1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The entries of the wiring matrix are restricted to the set  of zero and signed powers of two (A ⊆ {0, ±2Z}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Obtaining the wiring and codebook matrix jointly is typically NP-hard and infea-  sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' To overcome this computational intractability, [7] proposes a scheme where the  1In [6] the multiplication order of the decomposed matrices is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Please note that this  change makes no difference to the general idea of the decomposition and to the following algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  It is equal to the transposed version of the algorithm presented in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  n-th row of the wiring matrix is determined by solving the following sparse recovery  problem for some design parameter S < N controlling the cost between distortion and  computation effort [10]  wn =  argmin  ω∈{ω=�S  s=1 is1js,N: is∈A, js∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=',N} ∀s}  ∥an − ωC∥2  ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  (3)  The new scheme is still NP-hard, but not in N, anymore, but in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Thus, small values  of S are required, in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  In order to have a high accuracy despite small values of S, the factorization  procedure can be applied multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Then the product Ci = W iCi−1 of the  previous wiring step acts as the new codebook for obtaining the following matrix  factor W i of the current wiring step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Hence, by setting2 C0 = IN×K, we obtain the  approximated matrix P after I wiring steps:  A ≈ P =  � I�  i=1  W i  �  C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  (4)  To quantify the accuracy of a given approximation P we use the signal-to-  quantization-noise-ratio (SQNR)  SQNR(A, P ) =  ∥A∥2  F  ∥A − P ∥2  F  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  (5)  Computational Cost  In a binary number representation the multiplication by a signed power of two  corresponds only to a bitshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' On reconfigurable hardware, this shift can be realized  simply by appropriate wiring without the need for dedicated processing elements  such as adders [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The parameter S in (3) determines the number of vectors from  the codebook to be used in forming the linear combination to approximate a row  an of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' It therefore directly controls the computational cost, as in computing the  linear combination, exactly S − 1 additions are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' No multiplications, except  by signed powers of two, are necessary due to the specific structure of the wiring  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Therefore, the separate product of the decomposed matrices with the input  vector y ≈ W (Cx) is much simpler to compute than calculating the product in (1)  straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  The total computational cost Cadd of a decomposition in (4) is given by the number  of additions (or subtractions) required to form the linear combinations  Cadd = IN(S − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  (6)  2In [6] this choice is termed the self-designing codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' It was found to work very well for a  wide range of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Algorithms  In this section we will briefly look at the state of the art for solving the optimization  problem in (3) to obtain the wiring matrices and then introduce two improved novel  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  State-of-the-Art: Discrete Matching Pursuit  The discrete matching pursuit (DMP) follows the matching pursuit approach to  successively determine the wiring coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The algorithm can be summarized in  the following key steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' for details see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Start with iteration s ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Initialize ω ← 01×N  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Update ω in at most a single component, such that ∥an − ωC∥2 is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Increment s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' If s ≤ S, go to step 2, otherwise the procedure terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  It is straightforward to show that the time complexity of the DMP algorithm for  computing a single matrix factor scales with O(N 3S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Exponential Search Algorithm  The row-wise optimization problem in (3) is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' However, for small S, reasonable  matrix sizes and some careful optimization it can be solved in a tractable timeframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' We  limit the set of scaling factors to a finite set of signed powers of two (Aexp ⊂ {0, ±2Z}),  as an exhaustive search over the whole set is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' As the search procedure has  to be performed for each row of the target matrix individually, the time complexity  for the computation of each wiring step is given by O(N S|Aexp|S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Generally, which and how many coefficients are included in the subset Aexp is a  design parameter and needs to be adapted to each specific decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' It depends  primarily on two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' First, the current wiring step plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For  each additional wiring layer, the error between each row of the target matrix and  the approximated matrix decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Hence, for any subsequent wiring step, smaller  coefficients are needed to scale the rows of the newly found codebook matrix to  appropriately approximate the residual error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' This also means that for high desired  accuracy the coefficient set needs to be chosen large, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' to include also many small  coefficients, to accurately approximate the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Furthermore, relative variations  in the length of the row vectors of the target matrix require a larger coefficient  set to compensate for differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Still, to keep the decomposition computationally  feasible, the number of elements in Aexp needs to be chosen as small as possible, as  the computational complexity scales exponentially in S with the product of the size  of the coefficient set |Aexp| and N as base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  A promising approach for further research is to adapt the coefficient set for each  wiring step dynamically based on the current fidelity of the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The  coefficient set may then be determined from the probability distribution of the likely  entries of the wiring matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Reduced-State Algorithm  The runtime of the exponential algorithm for a matrix with N = 64 rows is on the  order of an hour3, for larger matrices the runtime scales up accordingly and can  become infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Hence, for some applications the computational burden of the  exponential algorithm might consequently not be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' A reasonable compromise  between complexity and performance is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Unlike in DMP, we do not immediately update the components of the wiring  vector for the reduced-state algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Instead, we keep updating in each iteration a  list of the M best vectors, which minimize (3) and select at the termination of the  algorithm the vector with minimum error from the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Specifically, we apply the  following successive, greedy procedure for the optimization problem in (3):  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Start with s ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Initialize the set Ω with M all-zero vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For each ω ∈ Ω find the set Ωm of M mutually distinct vectors ω ˜m with  ˜m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=', M} that minimize ∥an − ω ˜mC∥2 and differ from ω in at most a  single component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Update the set Ω by selecting from �M  m=1 Ωm the M distinct vectors ω with  minimum ∥an − ωC∥2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Increment s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' If s ≤ S, go to step 2, else, continue  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Return argmin  ω∈Ω  ∥an − ωC∥2  By tuning M, we are able to adjust the space of possible combinations that the  algorithm explores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For each wiring step, we have to evaluate O(SN 3M 2) combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Compared to DMP the complexity is increased by a factor of M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Note that for  M = 1 the algorithm reduces to the DMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Numerical Evaluation  In this section, we compare the performance of the proposed algorithms to the baseline  DMP [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' We decompose matrices whose entries are drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' from a Gaussian  distribution with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Similar to DMP, the performance of the  improved versions of the algorithm hardly depends on the distribution of the matrix  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The algorithms are invariant to scaling of the variance, however it is crucial  that for the exponential search algorithm the coefficient set Aexp is scaled appropriately  as well, as to not compromise performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Throughout the simulations, we select the  coefficient set for the exponential search algorithm to Aexp = {±2−40, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='., ±23}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  As a first experiment, we compare all three algorithms for fixed matrix sizes in  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' We choose matrices with N = 64 rows and vary the number of columns K  3For a multithreaded implementation in Python with Numba acceleration executed on an Intel  i9-12900@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='40 GHz and parameters S = 3, Aexp =  �  ±2−40, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' , ±23�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  0  100  200  300  400  500  600  700  800  Cumulative number of additions  0  20  40  60  80  100  120  SQNR [dB]  64x4  64x6  64x8  Figure 1: Performance comparison for matrices with 64 rows and different aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Matrices of dimension 64×4 are indicated by crosses, 64×6 is indicated by squares and 64×8  is indicated by triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The solid lines refer to the exponential search algorithm (S = 3),  the dashed lines to the DMP [7] (S = 2) and the dashed-dotted lines to the reduced state  algorithm with memory size M = 10 and S = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Results are averaged over 104 matrix entries  for the exponential search algorithm and over 105 matrix entries for the other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  from four to eight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Hence, we can compare the performance for different aspect ratios  of the matrices4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' To quantify performance, we plot the tradeoff between distortion  and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The latter being measured by the number of cumulative  additions Cadd required for a given wiring step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  As Figure 1 shows, for all three matrix sizes there is a performance gain by both  proposed algorithms against DMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Further, for memory size M = 10, the reduced  state algorithm performs only slightly worse than exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  For the first two wiring steps, the performance of all three algorithms is equal  for a given matrix size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' This is due to the fact that for the first two steps we use  DMP with S = 2 for an initial refinement of the codebook5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Applying any of the  4LCC works best for matrices with an exponential aspect ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' for K ≈ log2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For square  matrices it is beneficial to cut these into multiple tall matrices and decompose each slice independently,  see [8] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  5Using any of the novel algorithms with S = 2 is possible as well with very similar performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  a slight increase in SQNR by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='2 dB to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 dB for the novel algorithms and matrix sizes considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  0  100  200  300  400  500  600  700  800  Cumulative number of additions  0  10  20  30  40  50  60  70  80  SQNR [dB]  Increasing M  (2, 5, 10, 50)  DMP  Exponential Search  Reduced State (Variable M)  Figure 2: Performance comparison for different memory sizes M of the reduced state  algorithm of a 64 × 6 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The solid line refers to the exponential search algorithm  (S = 3), the dashed line to the DMP [7] (S = 2) and the dashed-dotted lines to the reduced  state algorithm with 4 different memory sizes and S = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Results are averaged over 104  matrix entries for the exponential search algorithm and over 105 matrix entries for the other  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  algorithms with S ≥ 3 directly to the initial codebook C0 = IN×K would lead to  degraded performance for the first few wiring steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Instead it is beneficial to set S = 2  to allow for more frequent updates of the codebook in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' From empirical  investigations it seems that two wiring iterations with S = 2 are most beneficial for  the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  As the next experiment, we compare the performance of the reduced state algorithm  for different choices of the memory parameter M for given matrix size of 64 × 6 in  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  From the figure we observe that even for small M the reduced state  algorithm offers a noticeable performance gain over DMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' As M grows, the reduced  state algorithm approaches the performance of the exponential search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The large  advantage of the reduced state algorithm is that the computation time is reduced  drastically6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For M = 1 the reduced state algorithm reduces to DMP and both lines  6For the considered matrix size the execution time differs from an hour for the exponential  algorithm to a few seconds for the reduced state algorithm in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  0  100  200  300  400  500  600  700  800  Cumulative number of additions  0  20  40  60  80  100  120  SQNR [dB]  Increasing S  (3, 4, 8)  DMP  Exponential search (S = 3)  Reduced State (S = 3, 4, 8)  Figure 3: Performance comparison for different choices of the parameter S of a matrix  with dimension 64 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The solid line refers to the exponential search algorithm (S = 3),  the dashed line to the DMP [7] (S = 2) and the dashed-dotted lines to the reduced state  algorithm with different choices of the parameter S and M = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Results are averaged over  104 matrix entries for the exponential search algorithm and over 105 matrix entries for the  other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  coincide for any given matrix size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Table 1 lists the relative performance gains over DMP for various matrix sizes and  configurations of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Practical Considerations for the Choice of S  Figure 3 shows the performance of the reduced state algorithm for varying S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Due to  the exponentially growing complexity in S, the exponential algorithm is not feasible  for S > 3, except for very small matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' We can observe, that by choosing S = 4  for the reduced state algorithm, we approximately achieve the same distortion-cost  tradeoff as for the exponential algorithm with S = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' For choosing S even larger the  gains increase likewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  However for a practicable implementation S should not be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' In [8],  the performance of DMP is validated in an implementation on reconfigurable hardware  with S = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' This means that on an FPGA exactly N adders are required per wiring  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' With the inputs depending only on the outputs of the previous wiring matrix,  Table 1: Relative average gain in terms of SQNR of the novel algorithms over the baseline  DMP algorithm with S = 2 (For at least 8 bit signed integer accuracy (10 log(SQNR) ≥  47 dB)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Results are averaged over 104 matrix entries for the exponential search algorithm  and over 105 matrix entries for the other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Exponential  Reduced State  search  S = 3  S = 4  S = 8  Matrix size  S = 3  M = 5  M = 10  M = 5  M = 10  M = 5  M = 10  16 × 2  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='8 %  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='1 %  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='8 %  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='7 %  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='9 %  16 × 4  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='0 %  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='8 %  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='7 %  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  32 × 4  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='7 %  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='9 %  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='0 %  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='2 %  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='3 %  32 × 6  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='3 %  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='0 %  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='7 %  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='0 %  64 × 4  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='9 %  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='8 %  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='0 %  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='8 %  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='8 %  64 × 6  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='9 %  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='6 %  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='4 %  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='6 %  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='5 %  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='6 %  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='0 %  the decomposition is well suited for parallel execution and pipelining [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Due to the  greedy, step-wise nature of DMP S > 2 does not offer significant performance gains  over S = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Both novel algorithms behave differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  From an implementation point of view, S = 3 is even more suitable than S = 2  for an effective implementation in hardware due to the availability of efficient adders  with three inputs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Interestingly, on modern FPGAs, these adders do not require  more hardware resources, in terms of Lookup-Tables (LUTs), than an adder with two  inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' Any powers of two and three (S = 4, 8, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' ) can be realized efficiently as well  by the use of adder trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' However, it is questionable if choosing S > 4 is beneficial,  as performance gains over S = 3 or S = 4 are small and the desired fidelity of the  approximation cannot be chosen in a fine granularity anymore7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content='  Conclusion  In this paper, we have proposed two new algorithms for LCC, a framework for  the lossy compression of multidimensional linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' While the exponential  search algorithm shows the best performance, it is generally infeasible especially for  large matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The proposed reduced-state algorithm, performs close to exponential  search at a fraction of the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
+page_content=' The time complexity of the  decomposition compared to the baseline algorithm from earlier works is only mildly  increased, while the performance gains over the baseline DMP algorithm are on the  order of at least 10 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
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+page_content='  7For a matrix of dimension 64 × 4 the reduced state algorithm with S = 8 improves the SQNR  approximately by 70 dB per matrix factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
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+page_content='  [6] Ralf R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
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+page_content='  [8] Alexander Lehnert, Philipp Holzinger, Simon Pfenning, Ralf R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE5T4oBgHgl3EQfeA8r/content/2301.05615v1.pdf'}
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+Harold Costales  et al.,  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 
+7076 
+ 
+ 
+ 
+ABSTRACT 
+ 
+Rice is the number one staple food in the country, as this 
+serves as the primary livelihood for thousands of Filipino 
+households. However, as the tradition continues, farmers are 
+not familiar with the different types of rice leaf diseases that 
+might compromise the entire rice crop. The need to address 
+the common bacterial leaf blight in rice is a serious disease 
+that can lead to reduced yields and even crop loss of up to 
+75%. This paper is a design and development of a rice leaf 
+disease detection mobile application prototype using an 
+algorithm used for image analysis. The researchers also used 
+the Rice Disease Image Dataset by Huy Minh Do available at 
+https://www.kaggle.com/ 
+to 
+train 
+state-of-the-art 
+convolutional neural networks using transfer learning. 
+Moreover, we used image augmentation to increase the 
+number of image samples and the accuracy of the neural 
+networks as well. 
+ 
+Key words : deep neural networks, convolutional neural 
+networks, agriculture, transfer learning, rice diseases 
+ 
+1 INTRODUCTION 
+ 
+Across the 20th century, our prospects of engineering have 
+rapidly changed. It was once thought of as impossible for 
+everyone, as all we had seen on earth showed us a manual 
+process that is uneasy for us; we would recognize as 
+technology. However, in the 21st century, our machines 
+continue to improve in what we conceive "Artificial 
+Intelligence," which makes it more even feasible for devices 
+to learn from experience, adjust to a high new level input and 
+perform human-like tasks. Methods such as the use of 
+convolutional neural networks may hold the key to 
+developing software applications, particularly in the field of 
+agriculture. 
+ 
+Portability, efficiency, and affordability of agricultural 
+technology and information continue to be a major 
+interference for improving agricultural productivity among 
+small enterprises in the country. The Department of 
+Agriculture, in partnership with the Department of 
+Information and Communications Technology (DICT), has 
+furnished a possible solution to enhance this kind of situation. 
+ 
+ 
+Recently, they have launched the HACKATON to address 
+further innovations that integrate software applications and 
+systems into the development of agricultural technology, 
+especially to farmers, which have been conducted nationwide. 
+They have currently established outstanding software 
+applications and systems to aid the farmers for better 
+understanding when it comes to e-agriculture. 
+ 
+This paper emphasizes the role of ICT and the functional 
+benefaction of software development to agriculture in the 
+Philippines. Data from Rice Disease Image Dataset by Huy 
+Minh Do available at https://www.kaggle.com are used to 
+train data using a convolutional neural network algorithm 
+using transfer learning. Moreover, we used image 
+augmentation to increase the number of image samples and 
+the neural network accuracy as well. It turns out that the 
+modified neural networks achieved a state-of-the-art result 
+with an accuracy closed to human-level performance. The 
+researchers have developed a prototype application for Rice 
+Leaf Disease Detection using Convolutional Neural Networks 
+(CNNs) for farmers in the local community. This application 
+will primarily install and use the said application on their 
+smartphone, simply took a picture of the infected area of the 
+rice leaf. Then the app gives a percentage of accuracy of rice 
+disease infected in rice leaf. 
+ 
+Farmers were also able to grasp the knowledge of the different 
+rice leaf diseases because of the information that the 
+application is providing. This paper recommends the adoption 
+of such software applications by institutions such as the 
+Department of Agriculture to improve provision for 
+appropriate decision making by agricultural farmers in the 
+country.  
+ 
+Related studies on rice leaf disease detection using neural 
+networks have been on-trend. According to [1] , "the use of 
+computational intelligence-based techniques has proven 
+successful in recent times for automated rice-disease detection 
+(p.21)". Another related study stated that [3] "an automated 
+system could have a feature on detection of diseases present in 
+a rice leaf using color image analysis". Furthermore, [4] cited 
+that "the management of perennial fruit crops requires close 
+monitoring especially for the management of diseases that can 
+affect production significantly and subsequently the 
+post-harvest life (p. 1)". Corollary to the contexts presented, 
+the researchers came up with a rice disease detection that 
+ 
+Development of a Prototype Application for Rice Disease 
+Detection Using Convolutional Neural Networks 
+Harold Costales1, Arpee Callejo-Arruejo2, Noel Rafanan3 
+1 University of Northern Philippines, Philippines, hlcostales@unp.edu.ph 
+2 University of Northern Philippines, Philippines, arpee.callejo@unp.edu.ph 
+3 University of Northern Philippines, Philippines, noel.rafanan@unp.edu.ph 
+        ISSN  2347 - 3983 
+Volume 8. No. 10, October 2020 
+International Journal of Emerging Trends in Engineering Research 
+Available Online at http://www.warse.org/IJETER/static/pdf/file/ijeter708102020.pdf 
+https://doi.org/10.30534/ijeter/2020/708102020 
+ 
+ 
+ 
+
+WARSEHarold Costales  et al.,  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 
+7077 
+ 
+ 
+integrates the use of an algorithm, specifically CNNs and an 
+image analysis, for immediate monitoring of the rice crops. 
+ 
+Thus, this mobile application is needed to bring virtual IT 
+experts into the field to determine, examine, and to give 
+accurate results (on rice leaf diseases) that would inform the 
+farmers what to do next without any further expenses. 
+1.1 Objectives of the Study 
+This study aimed to design and develop a mobile application 
+for rice disease detection. Specifically, it sought to answer the 
+following objectives: 
+1. Propose an application to address the problem, issues 
+and challenges encountered; 
+2. Identify an appropriate algorithm for the application; and 
+3. Features for the Mobile application. 
+ 
+1.2 Conceptual Framework 
+ 
+ 
+ 
+Figure 1: Paradigm of the Study 
+ 
+Figure 1 shows researcher used the input-process-output (IPO) 
+model as a guide in conducting the study. The Application is 
+proposed for the problems and issues encountered, 
+appropriate algorithm to be incorporated, and features for the 
+mobile application are the input of the study (input), which 
+served as the guide in designing the application. The software 
+development methodology, which is the Feature-Driven 
+Development Methodology, is the process for the 
+"Development of a Prototype Application for Rice Disease 
+Detection Using Convolutional Neural Networks"(output). 
+ 
+2 METHODS 
+ 
+This part presents the algorithm and the software development 
+methodology used in the development of the mobile 
+application.  
+ 
+For the algorithm, the researchers chose Convolutional Neural 
+Networks (CNNs) for the mobile application. As stated by 
+[12], technological advancements in Computer Vision and 
+Deep Learning a subset of the Artificial Intelligence gains 
+more importance in the last decade especially in the field of 
+object detection using Convolutional Neural Networks (CNN) 
+became popular in many fields to address the current societal 
+issues. 
+2.1 Convolutional Neural Networks Deep Learning 
+Algorithm for the Prototype Mobile App 
+ 
+A Convolutional Neural Network (CNN) is a deep learning 
+algorithm in which images serve as input to a learnable 
+process to analyze various aspects of the image and be able to 
+differentiate one from the other. A rice leaf is an input, and 
+images of diseases of rice leaves are stored in the database to 
+match the input. As cited by [11], many researchers use deep 
+learning method to classify an image automatically. The 
+purpose of classification is to arrange objects that will be 
+observed into categories that have been defined. Furthermore, 
+[14] cited that the detection technique assisted by simple 
+image processing in the evidence is very interesting to be 
+further researched. The researcher also utilized the method of 
+[16] for data processing for the realization of the study.  
+ 
+Convolutional Neural Networks (CNNs) is the most 
+appropriate model of Rice Disease Image Dataset since it uses 
+image analysis. It was proven by [6] that convolutional neural 
+networks (CNNs) have used in the field of computer vision for 
+decades. Moreover, [5] proposes a convolutional autoencoder 
+deep learning framework to support unsupervised image 
+features learning for lung nodule through unlabeled data, 
+which only needs a small amount of labeled data for efficient 
+feature learning. Below are the steps on how the researchers 
+came up with the integration of the CNNs for the developed 
+application following the process of the CNNs in the study of 
+[6]. 
+ 
+ 
+ 
+Figure 2. A Typical Architecture of CNNs by Tajbakhsh 
+(2016) 
+ 
+ 
+Figure 2 presents a typical architecture of a CNN. It consists 
+of an input layer (images), convolutional layer, pooling layer, 
+a fully connected layer, a classifier and an output layer.  
+ 
+The convolutional layer is the most important layer in the 
+CNN hence the name Convolutional Neural Networks. It acts 
+as an automatic feature that extracts meaningful features of 
+the object in the image. In mathematics, convolution is the 
+operation of two functions to produce a third modified 
+function. 
+ 
+               
+                      (1) 
+
+WNPOI
+PROCESS
+OUIPUTAplicationcanbeproposedtoaddress
+Development
+icationttheproblem,issues
+Methodology
+eafDiseasepuoue
+·DevelopanOverallencounterea
+Model
+CIgorithmforthe
+·BuildaFeatureListdevelopedapplication;
+PlanbyFeatureand
+·DesignbyFeatureeFeaturesofthe
+·BuildbyFeatureproposedsystembird
+Prird
+sunset
+Psunset
+dog
+cat
+Pol
+o
+convolution+
+maxpooling
+vec
+nonlinearity
+convolution+pooling layers
+fully connected layers
+Nxbinaryclassification(f*g)(t) ≤
+f(T)g(t - T) dT
+00Harold Costales  et al.,  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 
+7078 
+ 
+ 
+ 
+In the context of CNNs, convolutional operation is simply a 
+matrix operation, specifically matrix multiplication and 
+addition. The byproduct of this operation called the feature 
+map. 
+ 
+The pooling layer is used to down sample the feature map to 
+reduce computational complexity. It is common in the 
+literature to add the pooling layer after the convolution layer. 
+There are two main types of pooling the max pooling and the 
+average pooling. In the literature max pooling is preferred. It 
+calculates the maximum value for each patch of the feature 
+map. 
+ 
+ 
+                  (2) 
+ 
+The fully connected layer, abbreviated FC is responsible for 
+vectorizing the features from the convolution and pooling 
+layers. This transforms the multi-dimensional feature map 
+into a row vector to be fed into a classifier.  
+ 
+The classifier is responsible in outputing the probibilities of 
+the output layer. There are two commonly used activation 
+functions used in classifiers, the sigmoid activation function 
+and the softmax activation function. Since the problem of 
+detecting disease on rice leaf images is a multi-class 
+classification problem, the appropriate function to be used is 
+the softmax activation function.  
+ 
+ 
+                             
+                                     (3) 
+ 
+ 
+Another is the use of a software development methodology 
+which helped the researchers identify the problems, created an 
+overview of the concept for the mobile application and built 
+the application. 
+ 
+2.2 Software Development Methodology 
+ 
+The researcher used the Feature-driven Development (FDD) 
+under the family of Agile methodology for the software 
+development because it is the most suitable and it has a 
+customer-centric process. Its iterative feature allowed the 
+researchers to develop the application while it is being tested. 
+In the study of [7] stated that, “the iterative feature of the 
+methodology allowed the proponents to capitalize on the 
+learning that was accumulated during the development of 
+earlier parts or versions of the solution”. The figure below 
+presents the phases of FDD methodology. 
+ 
+Figure 3: Feature-Driven Development Methodology 
+ 
+Figure 
+3 
+shows 
+the 
+Feature-Driven 
+Development 
+Methodology which has five phases, which include the (a) 
+Development of an Overall Model, (b) Build a Feature List, 
+(c) Plan by Feature, (d) Design by Feature, and (e) Build by 
+Feature. The phases of development are presented as follows. 
+ 
+2.2.1 Develop Overall Model 
+ 
+A stage where the researchers created a fundamental 
+foundation of the application and the variables it required. 
+This stage also identified the initial development of the 
+application. The researchers designed the application's overall 
+model and served it as a guideline in developing the 
+application. As the primary objective, the researchers develop 
+the mobile app for the farmers for the early detection of rice 
+diseases of rice-crops. 
+ 
+2.2.2 Build Feature List 
+ 
+The researchers created concrete plans for each feature of the 
+mobile application. The researchers created a prototype of the 
+user interfaces for the conceptualization of the mobile 
+application. 
+ 
+2.2.3 Plan by Feature 
+ 
+The researchers created concrete plans for each feature of the 
+mobile application. The researchers created a prototype of the 
+user interfaces for the conceptualization of the mobile 
+application. 
+ 
+2.2.4 Design by Feature 
+ 
+The researchers designed the features individually using the 
+needed preferences. To fulfill this stage, the researchers 
+designed the identified features with the help of the needed 
+tools and processes. 
+ 
+2.2.5 Build by Feature 
+The last stage of the methodology wherein the researchers 
+created the planned and designed features for the application. 
+With the previous step as a guide, the researchers turned the 
+details of the features into a working application, with the help 
+of the tools and processes. After an inspection via a test run by 
+the researchers, the researchers created the planned and 
+designed features for the application. 
+ 
+ 
+
+Mf(p) =max(f(q)-dmax(p-q))
+52berj
+1xa7Modeling
+Initial
+Model
+Storming
+DevelopanOverall
+Bulld a Features
+Planby
+Designby
+Buildby
+Model
+List
+Feature
+Feature
+Feature
+(moreshape
+Alistoffeatures
+A developmentplan
+Adesign package
+Completed
+thancontant)
+grouped into sets
+Class owners
+client-valued
+and subjectareas
+Features setowners
+function
+Anobjectmodel
+(addmorecontent
++notes
+to the object model)Harold Costales  et al.,  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 
+7079 
+ 
+ 
+3 RESULTS 
+3.1 Prototype Mobile Application for Rice Disease 
+Detection 
+ 
+ 
+Figure 4: The Prototype of the Mobile Application 
+ 
+Figure 4 shows the rice disease detection of the mobile 
+application. The researchers designed and develop a mobile 
+application to assist the farmers in disease detection. It aims to 
+help the farmers who are the backbone of the country to 
+increase rice crop productivity. The lack of knowledge of 
+farmers in the detection of rice diseases made the researchers 
+conceptualize the integration of the field of agriculture and 
+medicine in detecting diseases of rice crops. Also, Duterte's 
+Administration supports Research and Development along 
+with agriculture. It is stated in the Philippine Development 
+Plan (PDP) 2017-2022 that there is a need to conduct 
+researches in the field because of rice-crop yield losses. Thus, 
+the PDP 2017-2022 established a plan to support the 
+initiatives of the agriculture sector along with R&D. 
+ 
+Moreover, the application is different from other similar apps 
+available in the market because it was developed specifically 
+for the Philippine agriculture setting. The application will be 
+stored with rice disease data exclusively for agriculture in the 
+northern Philippines and further validated by pathologists as 
+the researchers continue to gather data that will be for the 
+database of the system. 
+3.2 Convolutional Neural Networks Deep Learning 
+Algorithm for the Prototype Application 
+ 
+The researchers utilized an Image Processing typical machine 
+learning workflow in the development of the model which 
+consists of the following Data Acquisition, Data Preparation, 
+and Training and Validation.  As cited by [10] Image 
+processing is the method of using different manipulation 
+techniques and algorithms so that a desired features can be 
+extracted such as morphology, color and texture from an 
+image. Another study stated that [13] CNN proved to be best 
+among the results in terms of accuracy when compared to 
+other classifiers 
+ 
+3.2.1. Image Acquisition 
+ 
+The 
+dataset 
+by 
+Huy 
+Minh 
+Do 
+available 
+at 
+https://www.kaggle.com/ containing 1,260 labeled rice leaf 
+images was used for creating the model. The images were 
+labeled as Leaf Blast, Brown Spot, and Hispa, which are the 
+(3) three rice leaf diseases that the model will try to classify. 
+ 
+3.2.2. Image Preprocessing 
+ 
+The images were downscaled to 500 x 500 pixels to reduce 
+computational complexity. Image dataset were split into train 
+dataset (80% of the entire samples) and validation dataset 
+(20% of the entire samples). Image augmentation such as 
+flipping, shearing, and rescaling in order to increase dataset 
+samples as neural networks are data crunchers. 
+ 
+3.2.3. Training and Validation 
+ 
+To fast track the training process the researchers employed 
+transfer learning. Transfer learning, in general, is the process 
+of taking a previously trained model used in a problem and 
+apply in to another related problem. It is referring to the 
+knowledge transfer from pretrained network in one domain to 
+your own problem in a different domain. .[8] 
+ 
+The researcher used MobileNet 2 [9] as the base model. It is a 
+state-of-the-art CNN model trained from ImageNet and one of 
+the winners of the ImageNet Large Scale Visual Recognition 
+Challenge (ILSVRC), a prestigious computer vision 
+competition. Since the target dataset is small and somewhat 
+different from the dataset where MobileNet 2 was trained, the 
+top layer of the network needs to be frozen. 
+ 
+First Iteration 
+Base model mobile net Version 2.0 
+Hyper-parameters: 
+Top layer = false 
+Initialweights 
+ 
+ 
+
+.a58%12:57.PM
+agDetect
+blast
+50.13%
+brownspol
+bacterial_leaf_blight
+0.14%Harold Costales  et al.,  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 
+7080 
+ 
+ 
+Second Iteration 
+Base model mobile net Version 2.0 
+Hyper-parameters: 
+Top layer = false 
+Initialweights = imagenet 
+Loss = categorical cross entropy 
+Metrics = accuracy 
+Optimization = ADAM  
+EPOCHS= 10 
+TRAINABLE = FALSE 
+Result= 97% trained data set, 94% validation 
+Suffered from overfitting 
+ 
+Third Iteration 
+Base model mobile net Version 2.0,  
+image augmentation 
+Hyper-parameters: 
+Top layer = false 
+Initialweights = imagenet 
+Loss = categorical cross entropy 
+Metrics = accuracy 
+Optimization = ADAM  
+EPOCHS= 20 
+TRAINABLE = true 
+Result= 98.9% trained data set, 98% validation 
+Overfitting was solved 
+ 
+The third iteration proved that the process for the image 
+analysis on the training and testing for the data sets was 
+validated with an accuracy rate of 98% from 94% from the 
+second iteration. This proved that the CNNs algorithm of the 
+mobile application was proven for use. 
+3.3 Features of the Prototype Mobile Application 
+ 
+The features of the application include (1) real-time detection, 
+(2) artificial intelligence, and (3) image analysis. 
+ 
+3.3.1 Real-time Detection 
+ 
+One of the features of the mobile application is a real-time 
+detection. The application helps in the identification of the 
+rice leaf diseases using convolutional neural networks that 
+match the input data and the data stored in the database of the 
+application.  
+ 
+The embed camera of the mobile application can detect the 
+diseases in real-time. This feature helps farmers to detect the 
+diseases even without the instruction of plant pathologists. 
+ 
+3.3.2 Artificial Intelligence 
+ 
+Experts in the field of Agriculture are marginal, especially in 
+rice leaf disease detection. This feature of the mobile 
+application replicates the knowledge about rice leaf disease 
+detection in the field, which is one of the features of the 
+mobile application. As data grows on the database of the 
+mobile application, the more accurate results it can give.   
+ 
+3.3.3 Image Analysis 
+ 
+Another feature is Image Analysis. This feature uses the 
+CNNs algorithm aspects and features of images were stored, 
+enabling the mobile application to analyze the rice leaf for 
+disease detection through the following steps of the algorithm. 
+4 CONCLUSION 
+In summary, the researchers have developed a mobile 
+application, which could help the agricultural sector. 
+Specifically, the researchers conclude that the mobile 
+application will help in the realization of the high-yield of rice 
+crops against rice diseases for preventive measures. Thus, this 
+study is helpful to the country, particularly the agricultural 
+sector. 
+5 RECOMMENDATION 
+ 
+The 
+researchers 
+came 
+up 
+with 
+the 
+following 
+recommendations: 
+(1) The output of the Mobile Application will have its 
+validation of experts in rice disease for accuracy of the 
+information from the mobile application, (2) the mobile 
+application will have a series of tests for the implementation 
+for future use. Lastly, (3) the researchers highly recommend 
+that the application shall be introduced to the Department of 
+Agriculture for dissemination. 
+ 
+ACKNOWLEDGEMENT 
+ 
+The researchers would like to thank the following people: the 
+active UNP Research Office- Science and Technology 
+Coordinator, Prof. Redentor S. Rojas, the ever-supportive 
+UNP Research Director, Dr. Edelyn Cadorna and the dynamic 
+UNP President, Dr. Erwin F. Cadorna for pushing us to 
+enhance our research knowledge and capabilities. Also, the 
+researchers would like to thank the IMPACT HACKATON 
+2050 for organizing the IMPACT HACKATON 2019, where 
+they conceptualized this research study. Above all, thanks to 
+Almighty God for all the blessings for our families and the 
+society. 
+REFERENCES 
+1. E. G. Emberda, D. L. Dumas, and T. M. Rentillo, 
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+5. Talingdan, J. A., & Llanda, C. R., Assessment of the 
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+12. Devi Devani, Alethea Suryadibrata, Julio Christian, 
+Young, Implementation of Siamese Convolutional 
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+Liutfani, Ninda, Real Time Multi-Scale Facial Mask 
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+10.22266/ijies2018.0831.02 
+
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+page_content='Harold Costales  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=',  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 7076  \uf020  ABSTRACT  Rice is the number one staple food in the country, as this serves as the primary livelihood for thousands of Filipino households.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' However, as the tradition continues, farmers are not familiar with the different types of rice leaf diseases that might compromise the entire rice crop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The need to address the common bacterial leaf blight in rice is a serious disease that can lead to reduced yields and even crop loss of up to 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This paper is a design and development of a rice leaf disease detection mobile application prototype using an algorithm used for image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The researchers also used the Rice Disease Image Dataset by Huy Minh Do available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='com/ to train state-of-the-art convolutional neural networks using transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Moreover, we used image augmentation to increase the number of image samples and the accuracy of the neural networks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Key words : deep neural networks, convolutional neural networks, agriculture, transfer learning, rice diseases  1 INTRODUCTION  Across the 20th century, our prospects of engineering have rapidly changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It was once thought of as impossible for everyone, as all we had seen on earth showed us a manual process that is uneasy for us;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' we would recognize as technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' However, in the 21st century, our machines continue to improve in what we conceive "Artificial Intelligence," which makes it more even feasible for devices to learn from experience, adjust to a high new level input and perform human-like tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Methods such as the use of convolutional neural networks may hold the key to developing software applications, particularly in the field of agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Portability, efficiency, and affordability of agricultural technology and information continue to be a major interference for improving agricultural productivity among small enterprises in the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The Department of Agriculture, in partnership with the Department of Information and Communications Technology (DICT), has furnished a possible solution to enhance this kind of situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Recently, they have launched the HACKATON to address further innovations that integrate software applications and systems into the development of agricultural technology, especially to farmers, which have been conducted nationwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' They have currently established outstanding software applications and systems to aid the farmers for better understanding when it comes to e-agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  This paper emphasizes the role of ICT and the functional benefaction of software development to agriculture in the Philippines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Data from Rice Disease Image Dataset by Huy Minh Do available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='com are used to train data using a convolutional neural network algorithm using transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Moreover, we used image augmentation to increase the number of image samples and the neural network accuracy as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It turns out that the modified neural networks achieved a state-of-the-art result with an accuracy closed to human-level performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The researchers have developed a prototype application for Rice Leaf Disease Detection using Convolutional Neural Networks (CNNs) for farmers in the local community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This application will primarily install and use the said application on their smartphone, simply took a picture of the infected area of the rice leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Then the app gives a percentage of accuracy of rice disease infected in rice leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Farmers were also able to grasp the knowledge of the different rice leaf diseases because of the information that the application is providing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This paper recommends the adoption of such software applications by institutions such as the Department of Agriculture to improve provision for appropriate decision making by agricultural farmers in the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Related studies on rice leaf disease detection using neural networks have been on-trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' According to [1] , "the use of computational intelligence-based techniques has proven successful in recent times for automated rice-disease detection (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='21)".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Another related study stated that [3] "an automated system could have a feature on detection of diseases present in a rice leaf using color image analysis".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Furthermore, [4] cited that "the management of perennial fruit crops requires close monitoring especially for the management of diseases that can affect production significantly and subsequently the post-harvest life (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 1)".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Corollary to the contexts presented, the researchers came up with a rice disease detection that  Development of a Prototype Application for Rice Disease Detection Using Convolutional Neural Networks Harold Costales1, Arpee Callejo-Arruejo2, Noel Rafanan3 1 University of Northern Philippines, Philippines, hlcostales@unp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
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+page_content='ph 2 University of Northern Philippines, Philippines, arpee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='30534/ijeter/2020/708102020  WARSEHarold Costales  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=',  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 7077  integrates the use of an algorithm, specifically CNNs and an image analysis, for immediate monitoring of the rice crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Thus, this mobile application is needed to bring virtual IT experts into the field to determine, examine, and to give accurate results (on rice leaf diseases) that would inform the farmers what to do next without any further expenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='1 Objectives of the Study This study aimed to design and develop a mobile application for rice disease detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Specifically, it sought to answer the following objectives: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Propose an application to address the problem, issues and challenges encountered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Identify an appropriate algorithm for the application;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Features for the Mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2 Conceptual Framework  Figure 1: Paradigm of the Study  Figure 1 shows researcher used the input-process-output (IPO) model as a guide in conducting the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The Application is proposed for the problems and issues encountered, appropriate algorithm to be incorporated, and features for the mobile application are the input of the study (input), which served as the guide in designing the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The software development methodology, which is the Feature-Driven Development Methodology, is the process for the "Development of a Prototype Application for Rice Disease Detection Using Convolutional Neural Networks"(output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2 METHODS  This part presents the algorithm and the software development methodology used in the development of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  For the algorithm, the researchers chose Convolutional Neural Networks (CNNs) for the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' As stated by [12], technological advancements in Computer Vision and Deep Learning a subset of the Artificial Intelligence gains more importance in the last decade especially in the field of object detection using Convolutional Neural Networks (CNN) became popular in many fields to address the current societal issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='1 Convolutional Neural Networks Deep Learning Algorithm for the Prototype Mobile App  A Convolutional Neural Network (CNN) is a deep learning algorithm in which images serve as input to a learnable process to analyze various aspects of the image and be able to differentiate one from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' A rice leaf is an input, and images of diseases of rice leaves are stored in the database to match the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' As cited by [11], many researchers use deep learning method to classify an image automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The purpose of classification is to arrange objects that will be observed into categories that have been defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Furthermore, [14] cited that the detection technique assisted by simple image processing in the evidence is very interesting to be further researched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The researcher also utilized the method of [16] for data processing for the realization of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Convolutional Neural Networks (CNNs) is the most appropriate model of Rice Disease Image Dataset since it uses image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It was proven by [6] that convolutional neural networks (CNNs) have used in the field of computer vision for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Moreover, [5] proposes a convolutional autoencoder deep learning framework to support unsupervised image features learning for lung nodule through unlabeled data, which only needs a small amount of labeled data for efficient feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Below are the steps on how the researchers came up with the integration of the CNNs for the developed application following the process of the CNNs in the study of [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' A Typical Architecture of CNNs by Tajbakhsh (2016)  Figure 2 presents a typical architecture of a CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It consists of an input layer (images), convolutional layer, pooling layer, a fully connected layer, a classifier and an output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  The convolutional layer is the most important layer in the CNN hence the name Convolutional Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It acts as an automatic feature that extracts meaningful features of the object in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' In mathematics, convolution is the operation of two functions to produce a third modified function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  (1)  WNPOI PROCESS OUIPUTAplicationcanbeproposedtoaddress Development icationttheproblem,issues Methodology eafDiseasepuoue ·DevelopanOverallencounterea Model CIgorithmforthe ·BuildaFeatureListdevelopedapplication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' PlanbyFeatureand ·DesignbyFeatureeFeaturesofthe ·BuildbyFeatureproposedsystembird Prird sunset Psunset dog cat Pol o convolution+ maxpooling vec nonlinearity convolution+pooling layers fully connected layers Nxbinaryclassification(f*g)(t) ≤ f(T)g(t - T) dT 00Harold Costales  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=',  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 7078  In the context of CNNs, convolutional operation is simply a matrix operation, specifically matrix multiplication and addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The byproduct of this operation called the feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  The pooling layer is used to down sample the feature map to reduce computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It is common in the literature to add the pooling layer after the convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' There are two main types of pooling the max pooling and the average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' In the literature max pooling is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It calculates the maximum value for each patch of the feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  (2)  The fully connected layer, abbreviated FC is responsible for vectorizing the features from the convolution and pooling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This transforms the multi-dimensional feature map into a row vector to be fed into a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  The classifier is responsible in outputing the probibilities of the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' There are two commonly used activation functions used in classifiers, the sigmoid activation function and the softmax activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Since the problem of detecting disease on rice leaf images is a multi-class classification problem, the appropriate function to be used is the softmax activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  (3)  Another is the use of a software development methodology which helped the researchers identify the problems, created an overview of the concept for the mobile application and built the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2 Software Development Methodology  The researcher used the Feature-driven Development (FDD) under the family of Agile methodology for the software development because it is the most suitable and it has a customer-centric process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Its iterative feature allowed the researchers to develop the application while it is being tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' In the study of [7] stated that, “the iterative feature of the methodology allowed the proponents to capitalize on the learning that was accumulated during the development of earlier parts or versions of the solution”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The figure below presents the phases of FDD methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Figure 3: Feature Driven Development Methodology  Figure 3 shows the Feature-Driven Development Methodology which has five phases, which include the (a) Development of an Overall Model, (b) Build a Feature List, (c) Plan by Feature, (d) Design by Feature, and (e) Build by Feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The phases of development are presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='1 Develop Overall Model  A stage where the researchers created a fundamental foundation of the application and the variables it required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This stage also identified the initial development of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=" The researchers designed the application's overall model and served it as a guideline in developing the application." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' As the primary objective, the researchers develop the mobile app for the farmers for the early detection of rice diseases of rice-crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2 Build Feature List  The researchers created concrete plans for each feature of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The researchers created a prototype of the user interfaces for the conceptualization of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3 Plan by Feature  The researchers created concrete plans for each feature of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The researchers created a prototype of the user interfaces for the conceptualization of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='4 Design by Feature  The researchers designed the features individually using the needed preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' To fulfill this stage, the researchers designed the identified features with the help of the needed tools and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='5 Build by Feature The last stage of the methodology wherein the researchers created the planned and designed features for the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' With the previous step as a guide, the researchers turned the details of the features into a working application, with the help of the tools and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' After an inspection via a test run by the researchers, the researchers created the planned and designed features for the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Mf(p) =max(f(q)-dmax(p-q)) 52berj 1xa7Modeling Initial Model Storming DevelopanOverall Bulld a Features Planby Designby Buildby Model List Feature Feature Feature (moreshape Alistoffeatures A developmentplan Adesign package Completed thancontant) grouped into sets Class owners client-valued and subjectareas Features setowners function Anobjectmodel (addmorecontent +notes to the object model)Harold Costales  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=',  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 7079  3 RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='1 Prototype Mobile Application for Rice Disease Detection  Figure 4: The Prototype of the Mobile Application  Figure 4 shows the rice disease detection of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The researchers designed and develop a mobile application to assist the farmers in disease detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It aims to help the farmers who are the backbone of the country to increase rice crop productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The lack of knowledge of farmers in the detection of rice diseases made the researchers conceptualize the integration of the field of agriculture and medicine in detecting diseases of rice crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=" Also, Duterte's Administration supports Research and Development along with agriculture." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It is stated in the Philippine Development Plan (PDP) 2017-2022 that there is a need to conduct researches in the field because of rice-crop yield losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Thus, the PDP 2017-2022 established a plan to support the initiatives of the agriculture sector along with R&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Moreover, the application is different from other similar apps available in the market because it was developed specifically for the Philippine agriculture setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The application will be stored with rice disease data exclusively for agriculture in the northern Philippines and further validated by pathologists as the researchers continue to gather data that will be for the database of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2 Convolutional Neural Networks Deep Learning Algorithm for the Prototype Application  The researchers utilized an Image Processing typical machine learning workflow in the development of the model which consists of the following Data Acquisition, Data Preparation, and Training and Validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' As cited by [10] Image processing is the method of using different manipulation techniques and algorithms so that a desired features can be extracted such as morphology, color and texture from an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Another study stated that [13] CNN proved to be best among the results in terms of accuracy when compared to other classifiers  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Image Acquisition  The dataset by Huy Minh Do available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='com/ containing 1,260 labeled rice leaf images was used for creating the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The images were labeled as Leaf Blast, Brown Spot, and Hispa, which are the (3) three rice leaf diseases that the model will try to classify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Image Preprocessing  The images were downscaled to 500 x 500 pixels to reduce computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Image dataset were split into train dataset (80% of the entire samples) and validation dataset (20% of the entire samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Image augmentation such as flipping, shearing, and rescaling in order to increase dataset samples as neural networks are data crunchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Training and Validation  To fast track the training process the researchers employed transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Transfer learning, in general, is the process of taking a previously trained model used in a problem and apply in to another related problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It is referring to the knowledge transfer from pretrained network in one domain to your own problem in a different domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' [8]  The researcher used MobileNet 2 [9] as the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' It is a state-of-the-art CNN model trained from ImageNet and one of the winners of the ImageNet Large Scale Visual Recognition Challenge (ILSVRC), a prestigious computer vision competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Since the target dataset is small and somewhat different from the dataset where MobileNet 2 was trained, the top layer of the network needs to be frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  First Iteration Base model mobile net Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='0 Hyper-parameters: Top layer = false Initialweights  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='a58%12:57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='PM agDetect blast 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='13% brownspol bacterial_leaf_blight 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='14%Harold Costales  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=',  International Journal of Emerging Trends in Engineering Research, 8(10), October 2020,  7076  -  7081 7080  Second Iteration Base model mobile net Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='0 Hyper-parameters: Top layer = false Initialweights = imagenet Loss = categorical cross entropy Metrics = accuracy Optimization = ADAM EPOCHS= 10 TRAINABLE = FALSE Result= 97% trained data set, 94% validation Suffered from overfitting  Third Iteration Base model mobile net Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='0, image augmentation Hyper-parameters: Top layer = false Initialweights = imagenet Loss = categorical cross entropy Metrics = accuracy Optimization = ADAM EPOCHS= 20 TRAINABLE = true Result= 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='9% trained data set, 98% validation Overfitting was solved  The third iteration proved that the process for the image analysis on the training and testing for the data sets was validated with an accuracy rate of 98% from 94% from the second iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This proved that the CNNs algorithm of the mobile application was proven for use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3 Features of the Prototype Mobile Application  The features of the application include (1) real-time detection, (2) artificial intelligence, and (3) image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='1 Real time Detection  One of the features of the mobile application is a real-time detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' The application helps in the identification of the rice leaf diseases using convolutional neural networks that match the input data and the data stored in the database of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  The embed camera of the mobile application can detect the diseases in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This feature helps farmers to detect the diseases even without the instruction of plant pathologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='2 Artificial Intelligence  Experts in the field of Agriculture are marginal, especially in rice leaf disease detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This feature of the mobile application replicates the knowledge about rice leaf disease detection in the field, which is one of the features of the mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' As data grows on the database of the mobile application, the more accurate results it can give.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='3 Image Analysis  Another feature is Image Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' This feature uses the CNNs algorithm aspects and features of images were stored, enabling the mobile application to analyze the rice leaf for disease detection through the following steps of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 4 CONCLUSION In summary, the researchers have developed a mobile application, which could help the agricultural sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Specifically, the researchers conclude that the mobile application will help in the realization of the high-yield of rice crops against rice diseases for preventive measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Thus, this study is helpful to the country, particularly the agricultural sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' 5 RECOMMENDATION  The researchers came up with the following recommendations: (1) The output of the Mobile Application will have its validation of experts in rice disease for accuracy of the information from the mobile application, (2) the mobile application will have a series of tests for the implementation for future use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Lastly, (3) the researchers highly recommend that the application shall be introduced to the Department of Agriculture for dissemination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  The researchers would like to thank the following people: the active UNP Research Office- Science and Technology Coordinator, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Redentor S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Rojas, the ever-supportive UNP Research Director, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Edelyn Cadorna and the dynamic UNP President, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Erwin F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Cadorna for pushing us to enhance our research knowledge and capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Also, the researchers would like to thank the IMPACT HACKATON 2050 for organizing the IMPACT HACKATON 2019, where they conceptualized this research study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Above all, thanks to Almighty God for all the blessings for our families and the society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
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+page_content='30534/ijeter/2020/63882020 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content=' Fitria Hidayanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
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+page_content='30534/ijeter/2020/62882020 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='  Lamarca, Bryan Irvin J, et al, The Development of a Performance Appraisal System Using Decision Tree Analysis and Fuzzy Logic, International Journal of Intelligent Engineering and Systems, 11-4, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
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+page_content='22266/ijies2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='0831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
+page_content='02' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E5T4oBgHgl3EQfSQ7q/content/2301.05528v1.pdf'}
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+arXiv:2301.04531v1  [physics.optics]  11 Jan 2023
+Unified Model for Nonlinear Pulse Propagation in Composites and Optimization of
+THz Generation
+A. Husakou∗
+Max Born Institute, Max Born Str.
+2a, D-12489 Berlin, Germany
+O. Fedotova, R. Rusetsky, and O. Khasanov
+Scientific-Practical Materials Research Centre of National Academy
+of Sciences of Belarus, P.Brovki str.
+19, 220072 Minsk, Belarus
+T. Smirnova and A. Fedotov
+Belarus State University, Niezalieˇznasci ave 4, 220030 Minsk, Belarus
+T. Apostolova
+Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,
+Tsarigradsko Chausse 72, 1784 Sofia, Bulgaria and
+Institute for Advanced Physical Studies, New Bulgarian University, 1618 Sofia, Bulgaria
+I. Babushkin
+Institute of Quantum Optics, Leibniz University Hannover, Welfengarten 1, 30167, Hannover, Germany
+Cluster of Excellence PhoenixD (Photonics, Optics,
+and Engineering-Innovation Across Disciplines), 30167, Hannover, Germany and
+Max Born Institute, Max-Born-Str. 2a, 12489, Berlin, Germany
+U. Sapaev
+Tashkent State Technical University, University street 2, 100097 Tashkent, Uzbekistan
+We describe a unified numerical model which allows fast and accurate simulation of nonlinear light
+propagation in nanoparticle composites, including various effects such as group velocity dispersion,
+second- and third-order nonlinearity, quasi-free-carrier formation and plasma contribution, exciton
+dynamics, scattering and so on. The developed software package SOLPIC is made available for
+the community. Using this model, we analyze and optimize efficient generation of THz radiation
+by two-color pulses in ZnO/fused silica composite, predicting an efficiency of 3%. We compare the
+role of various nonlinear effects contributing to the frequency conversion, and show that optimum
+conditions of THz generation differ from those expected intuitively.
+I.
+INTRODUCTION
+THz technology has attracted a lot of attention in
+the recent years, since it provides unique experimental
+tools and techniques in nonlinear and time-domain spec-
+troscopy, biology and medicine, remote sensing, security
+screening, as well as information and communication sys-
+tems (see e.g. [1–4]). For generation of THz radiation,
+different techniques were proposed, such as two-color ion-
+izing femtosecond pulses in gases [5–10] and plasmas,
+produced at the surface of solids (e.g. metal foil) with
+high intensities [11] that are free from optical damage
+threshold and phase-matching drawbacks; optical rectifi-
+cation of the transmitted intense ultrashort laser pulses
+in non-linear crystals [12–16], which provide a basis for
+compact low-intensity devices.
+The needs of the THz
+technology require, however, extension of the range of
+the available techniques and materials, in order to pro-
+vide flexible designs required in multifarious applications.
+∗ gusakov@mbi-berlin.de
+Following this line, investigations of THz generation in
+various media such as water [17] and strongly magnetized
+plasmas [18] were performed. Emission of terahertz radi-
+ation with broad bandwidth by femtosecond photoexcita-
+tion of spintronic materials (ferromagnetic and synthetic
+multiferroic heterostructures) was also reported recently
+[19, 20].
+Nanoparticle composites were actively investigated in
+the past as a nonlinear material, e.g. [21–23], and their
+particular strength lies in the flexibility of their design
+leading to unusual properties such as e.g. negative re-
+fractive index [24].
+However, surprisingly, up to our
+knowledge they have not attracted attention as a medium
+for THz generation. In this paper, we close this gap by
+developing a numerical model suitable for simulation of
+THz generation in nanocomposites.
+A range of linear
+and nonlinear effects such as group velocity dispersion,
+second- and third-order nonlinearity, quasi-free-carrier
+formation, exciton dynamics and so on are encompassed
+by the developed model. We use it to explore THz gen-
+eration by two-color pulses in nanoparticle composites,
+to elucidate the contributions of different frequency con-
+version mechanisms, and to predict efficiencies in few-
+
+2
+percent range.
+The applicability of the above model is, in fact, much
+broader than mere simulation of THz generation; a wide
+range of nonlinear effects such as soliton dynamics and
+supercontinuum generation, frequency conversion, multi-
+level dynamics and electromagnetically-induced trans-
+parency and so on can be studied using this unified ap-
+proach. With this is mind, we have created the extensive
+documentation of the code and made the code publicly
+available [25], in a hope that it will be useful to the optical
+community for investigations of the nonlinear processes
+in nanocomposites and other materials.
+The paper is organized as follows. In Section 2, we
+present the numerical model, including detailed formal-
+ism for all the relevant mechanisms.
+In Section 3, we
+optimize the THz generation by two-color pulses, and
+analyze the role of different parameters. A summary of
+the paper is given in the conclusion.
+II.
+THEORETICAL MODEL
+We consider a composite consisting of two components,
+a homogeneous host material and spherical nanoparticles
+(inclusions) randomly distributed in space. We assume
+a sufficiently low (typically few percent or below) filling
+fraction of the inclusions so that neither percolation nor
+interaction between the inclusions play a role. We con-
+sider homogeneous inclusions to be sufficiently small with
+diameter below the light wavelength so that effective-
+medium theory can be applied.
+Note that we do not
+place any limitations on the nature of host and inclu-
+sion materials, i.e., either of them could be a dielectric,
+a metal, or a semiconductor. We do not require point
+symmetry in host material or in the inclusions, so that
+the second-order susceptibility can be non-zero in either
+material.
+The model is designed to simulate propaga-
+tion over relatively short distances of few millimeters, be-
+low the damage threshold, and without back-reflection,
+therefore (1+1)D treatment using unidirectional propa-
+gation equation [26] is the most suitable.
+Under these conditions, the following effects have to
+be taken into account: linear dispersion including intrin-
+sic and scattering losses, second- and third-order optical
+nonlinearities and photoionization accompanied by ion-
+ization losses and plasma dynamics. In addition, transi-
+tions between excitonic states can play a significant role
+in the inclusion response, in particular for the genera-
+tion of new frequencies in the THz range. Among the
+effects which were neglected in this treatment are ther-
+mal effects (due to slow ns-scale response), coupling to
+phonons (because of relatively slow ps-scale response),
+Raman scattering (which is typically weaker than in-
+stantaneous nonlinearities), anisotropy of the host ma-
+terial (due to manufacturing limitations for composites),
+deviations of the inclusion form from a sphere (because
+of typical manufacturing conditions), and generation of
+high-order harmonics (because of the considered inten-
+sity ranges).
+The following unidirectional propagation equation is
+used to model the light propagation in a homogeneous
+medium [26, 27]:
+∂E(z, ω)
+∂z
+= −i
+�
+[
+�
+ǫ(ω) − ng]ω
+c
+− β(ω0)
+�
+E(z, ω)
+−
+iω
+2c
+�
+ǫ(ω)
+PNL(z, ω),
+(1)
+where E(z, ω) = ˆFE(z, t) =
+� ∞
+−∞ E(z, t) exp(−iωt)dt
+is the Fourier transform ˆF of the electric field E(z, t), z
+is the propagation coordinate, ǫ(ω) is the linear dielec-
+tric permittivity (generally speaking, complex-valued to
+include loss mechanisms), ng is the group refractive in-
+dex, ω0 is a characteristic frequency of the pulse spec-
+trum, β(ω) =
+�
+ǫ(ω)ω/c, and PNL(z, ω) is the Fourier
+transform of the nonlinear part of the polarization. We
+would like to empathise that no slowly-varying envelope
+approximation is used, and E(z, t) represents the real-
+valued field including the carrier oscillations. This ap-
+proach yields a unified treatment for a pulse with arbi-
+trary spectral content, which is particularly important
+for extremely broad spectra.
+A.
+Linear dispersion
+The effective-medium theory allows to substitute the
+composite material by a homogenised medium with ap-
+propriately defined effective material parameters.
+The
+effective refractive index of a composite can be expressed
+as [27]
+neff =
+�
+(1 − f)ǫh + fǫi
+3ǫh
+2ǫh + ǫi
++ 2i
+� ǫh − ǫi
+2ǫh + ǫi
+�2 �rNPω√ǫh
+c
+�3�1/2
+,
+(2)
+where f is the volume filling factor of the inclusions,
+rNP is their radius, and ǫh,i are the frequency-dependent
+dielectric functions of the host and of the inclusions, cor-
+respondingly. The last term in the square brackets de-
+scribes scattering losses.
+B.
+Second- and third-order nonlinearities
+The second- and third-order nonlinear processes can
+also be described in the framework of the effective-
+medium theory. The expressions for the effective second-
+order susceptibility looks like [28]
+
+3
+χ(2)
+eff (ω1 = ω2 + ω3; ω2, ω3) = (1 − f)χ(2)
+h
++
++ fx(ω1)x(ω2)x(ω3)χ(2)
+i ,
+(3)
+where χ(2)
+h
+and χ(2)
+i
+are the susceptibilities of host and
+inclusion materials, correspondingly. Note that we ne-
+glected the frequency dependence of the susceptibilities
+of host and inclusions, which is a good assumption far
+from resonances. Quantity x(ω) is the ratio of local field
+inside the inclusion and the incident field:
+x(ω) =
+3ǫh(ω)
+2ǫh(ω) + ǫi(ω).
+(4)
+Here we note that, due to photoionization as described
+below, the ǫi(ω) and therefore x, strictly speaking, de-
+pend on time due to buildup of plasma during the pulse.
+However, in the current simulation we neglect this de-
+pendence, assuming that corresponding change of ǫi(ω)
+is small and that we are far from the plasmonic resonance
+given by 2ǫh(ω) = −ǫi(ω).
+Similarly, for the effective third-order susceptibility we
+write [28]
+χ(3)
+eff (ω1 = ω2 + ω3 + ω4; ω2, ω3, ω4) = (1 − f)χ(3)
+h
++
++ fx(ω1)x(ω2)x(ω3)x(ω4)χ(3)
+i ,
+(5)
+where χ(3)
+h
+and χ(3)
+i
+are the susceptibilities of host and
+inclusion materials, correspondingly. The final expres-
+sions which were used to calculate the corresponding po-
+larizations look like
+Pχ(2)(z, ω) = (1 − f)ǫ0χ(2)
+h
+ˆFE(z, t)2 +
++ fǫ0χ(2)
+i
+ˆF[ ˆF −1{E(z, ω)x(ω)}2],
+(6)
+Pχ(3)(z, ω) = (1 − f)ǫ0χ(3)
+h
+ˆFE(z, t)3 +
++ fǫ0χ(3)
+i
+ˆF[ ˆF −1{E(z, ω)x(ω)}3].
+(7)
+C.
+Plasma dynamics
+Let us turn to the description of plasma formation and
+dynamics. In the framework of SOLPIC, we consider a
+case when the ionization potential Ip of the inclusions
+is lower than that of the host material, so that due to
+the sensitive dependence of the polarization rate on the
+ionization potential we can neglect plasma formation in
+host material.
+The contribution from the plasma is determined by the
+average displacement ⟨d⟩(z, t) of the electron from the
+equilibrium position in the parent ”molecule”, whereby
+by a ”molecule” we denote an atom or a group of atoms of
+the solid-state material which can provide a single ioniza-
+tion event. Furthermore, it is determined by the relative
+ionization of the solid state ρ(z, t), which is the ratio of
+the conduction-band electron density to the density of
+”molecules”:
+Pplasma(z, ω) = −Nmole ˆF[⟨d⟩(z, t)ρ(z, t)]
+(8)
+Here Nmol is the concentration of the molecules and
+e = 1.6×10−19 is the electron charge. The above expres-
+sion would be valid in a homogeneous medium; however,
+as it refers to a polarization which occurs inside of inclu-
+sions, in contrast to averaged macroscopic polarization,
+in the case of effective-medium theory it has to be addi-
+tionally multiplied by x(ω). For the origin of this factor
+and further details see Ref. [28].
+The dynamics of the quantity ⟨d⟩(z, t)ρ(z, t) is given
+by [33]
+∂(⟨d⟩(z, t)ρ(z, t))
+∂t
+= ⟨v⟩(z, t) + x0Γ(t),
+(9)
+where ⟨v⟩ is the average velocity of electrons and
+x0 ≃ −Ip/eE(t) is the initial displacement of the elec-
+tron immediately after the ionization event, Ip being the
+bandgap. It can be shown that the second term describes
+the energy loss of the pulse due to photoionization. The
+dynamics of the ⟨v⟩(z, t)ρ(z, t) is given by second New-
+ton’s law as
+∂(⟨d⟩(z, t)ρ(z, t))
+∂t
+= −eE(z, t)
+me
+ρ,
+(10)
+where me is the effective electron mass near the bot-
+tom of the conduction band. Here we neglect the initial
+displacement and velocity of electron just after the ion-
+ization.
+The dynamics of the relative plasma density ρ is given
+by
+∂ρ
+∂t = Γ( ˆF −1[x(ω)E(z, ω)]),
+(11)
+where x(ω)E(z, ω) is the local field inside of inclusions
+which determines the photoionization rate Γ.
+D.
+Ionization rate
+Depending on the relation between the frequency of
+pump light and the ionization potential of inclusions, we
+consider two models for the ionization rate. For the case
+when the energy of pump photons is much smaller than
+the ionization potential, the photoionization occurs ei-
+ther by multiphoton regime or by tunneling regime, as
+determined by intensity and Keldysh parameter. Here we
+
+4
+10-4
+10-3
+10-2
+0.1
+1
+ 1
+ 2
+ 3
+ 4
+ 5
+(a)
+I(ω) (arb. units)
+ω (fs-1)
+-6
+-4
+-2
+ 0
+ 2
+ 4
+ 6
+-40
+-20
+ 0
+ 20
+ 40
+ 0
+ 0.005
+ 0.01
+(b)
+E (GV/m)
+ρ
+t (fs)
+FIG. 1. Dependence of the spectra (a) and the electric field
+(b) on the propagation length. 15-fs pulses at 2.26 fs−1 and
+4.58 fs−1 are considered, with intensity of 1 TW/cm2 and
+propagation length of 0.75 µm (magenta curves), 2.25 µm
+(green curves), and 6.75 µm (bluecurves). In (b), additionally
+the relative plasma density is shown for propagation length
+of 6.75 µm. A composite of ZnO inclusions with f = 0.03 in
+fused silica is considered.
+utilize so-called Yudin-Ivanov model [29], which provides
+a formalism for both of these regimes in a unified way.
+This model was initially developed for isolated atoms; its
+use for solid state is justified in a case a negligible an-
+harmonicity of the bands in the center of the Brillouin
+zone.
+The cycle-resolved ionization rate Γ is given (in atomic
+units, that is, with frequency ω, time t and field E mea-
+sured in the corresponding Hartree units ωa = 0.26
+rad/as, ta = 24.2 as, xa = 0.0529 nm, and Ea = 514.2
+V/nm) by
+Γ(z, t) = π
+τT
+exp
+�
+−σ0
+⟨2E(z, t)2⟩
+ω3
+� �
+2κ3
+�
+⟨2E(z, t)2⟩
+�2Z/κ
+× exp
+�
+−E(z, t)2
+2ω3
+σ1
+�
+.
+(12)
+Here τT = κ/E(z, t), κ =
+�
+Ip/(ℏωa), σ0 =
+1
+2(γ2 +
+1
+2) ln C − 1
+2γ
+�
+1 + γ2, γ = ωτT , Z is the effective atomic
+charge, C = 1 + 2γ
+�
+1 + γ2 + 2γ2, and σ1 = ln C-
+2γ/
+�
+1 + γ2.
+The quantity ⟨E(z, t)2⟩ is the averaged
+value of the squared electric field over few past periods
+(5 fs is assumed in this work).
+The Yudin-Ivanov model was initially derived for gases;
+its applicability for solid state, while generally justi-
+fied for materials with tight binding, is not strictly es-
+tablished.
+We have benchmarked Yudin-Ivanov model
+by comparing it to the numerical solution of the time-
+dependent Schrodinger equation in single active electron
+approximation [30]. In this approach the empirical pseu-
+dopotential method was used for describing the electron
+band structure of ZnO [31]. We have found that the dif-
+ference of the ionization rate does not typically exceed
+one order of magnitude. This difference is, in fact, not
+very significant: because of the threshold-like behavior of
+the ionization rate, it leads to only a slight shift of the
+intensity at which a strong plasma generation is reached.
+For the specal case when the energy of pump photons
+is around two ionization potentials, it is preferable to
+use the two-photon formalism [32] and write the cycle-
+resolved ionization rate Γ (in SI units) as
+Γ(z, t) =
+2e4x4
+aν
+ℏ4ω2
+0[(2ω0 − Ip/ℏ)2 + ν2 ⟨E(z, t)2⟩E(z, t)2,
+(13)
+where ν is the relaxation constant of the two-photon
+transition.
+E.
+Contribution by excitons
+Finally, we include the nonlinear polarization due to
+excitons into treatment. We consider multiple excitonic
+levels and utilize the standard Bloch equations for the de-
+scription of the ionization. The dynamics of the density
+matrix ρe is given by (see e.g. [32])
+iℏ∂ρe
+∂t = [H, ρe],
+(14)
+where H = H0 + Hint, H0 is the Hamiltonian of the
+system in the absence of excitation, Hint is the interaction
+Hamiltonian, which components Hij are related with the
+corresponding dipole transition moments eMij:
+Hij = eMij ˆF −1[E(z, ω)x(ω)].
+(15)
+In addition, polarization decay (decay of the off-
+diagonal elements of ρe) with decay time T2 and decay
+of the population to the ground state with decay time T1
+are included. In order to avoid numerical instabilities,
+the normalization of the density matrix ρ is performed
+each few steps in time, by a) enforcing 0 ≤ ρe,ii ≤ 1, b)
+enforcing T r(ρe) = 1, and c) adjusting the non-diagonal
+elements which exceed the maximum possible value de-
+termined by the corresponding level populations.
+The excitonic polarization is then defined in a standard
+way as
+Pexc(z, ω) = x(ω) ˆF[fTr(ρeM)].
+(16)
+We solve the propagation equation by an extended
+split-step method, whereby each of the contributions to
+
+5
+the polarization is treated subsequently, which allows to
+reduce the accumulation of numerical error. Nonlinear
+steps are performed using the Runge-Kutta approach,
+the order of which can be selected between 1,2, and 4.
+Fixed step of the grid both in time and in the propa-
+gation coordinate is used. The appearance of numerical
+artifacts during the propagation is monitored by tracing
+the total pulse energy as well as the total energy absorbed
+at the boundaries of the numerical time window.
+III.
+NUMERICAL RESULTS AND DISCUSSION
+In order to exemplify the above model and function-
+ing of SOLPIC, we present in this section a simulation of
+THz generation. We consider a composite of ZnO inclu-
+sions in SiO2 matrix. Phenomenological Sellmeyer-type
+expressions were used to describe dispersion on ZnO [35]
+and SiO2 [36]. Similarly, experimental data on second-
+order [37] and third-order [38] susceptibility of bulk ZnO
+and third-order susceptibility of SiO2 [40] were used. We
+estimated the value of T2 as 50 fs from Ref. [41] and used
+T1 = 2T2. For ZnO, the typical exciton size is larger than
+interatomic distance, meaning that we are dealing with
+Wannier-Mott type of excitons. In a case of sufficiently
+small inclusions, the exciton is bounded by the inclu-
+sion boundaries, therefore its wavefunctions (as well as
+energy levels and dipole momenta) are better described,
+instead of hydrogen-like potential, by a constant poten-
+tial inside a sphere [42] with a step on its boundary. We
+have taken into account 5 lowest excitonic levels, and
+typical values of the off-diagonal dipole momenta, as cal-
+culated by this approach, are around 3×10−28 C·m, for
+same-size nanoparticles with a radius of 2.5 nm which
+are considered here and hereafter.
+For the permanent
+dipole momenta of ZnO, we have adopted a typical value
+of 6.66×10−30 C·m per ZnO molecule, which was used
+to define the on-diagonal elements of the dipole matrix.
+We used the ionization potential of 3.37 eV equal to the
+bandgap of ZnO to characterize the transition from va-
+lence band to conduction band, and all the presented
+numerical results correspond to the conditions below the
+damage threshold of ZnO [43].
+The evolution of the field profile and spectra with prop-
+agation is illustrated in Fig. 1, for two-color pulsed exci-
+tation with pump pulses around 800 nm and 400 nm, for
+conditions given in the caption. In Fig. 1(a) one can see
+that initial stages of the propagation are characterized by
+self-phase modulation with typical spectral side lobes. At
+later stages, spectrum becomes irregular and transform
+into a supercontinuum extending up to the absorption
+edge given by the bandgap. The evolution of the tempo-
+ral profile, shown in Fig. 1(b), shows gradual reduction
+of the enegry of electric field, as well as significantly ir-
+regular envelope for longer propagation. This reduction
+of the maximum field determines the saturation of the
+THz generation efficiency and is caused both by strong
+group-velocity dispersion for broad spectrum and energy
+absorption due to transition to conduction band. One
+can see from the red curve in Fig.
+1(b) that relative
+plasma density reaches values of roughly 0.01 after the
+pulse, which is sufficient to induce significant energy ab-
+sorption.
+10-4
+10-3
+10-2
+0.1
+1
+ 10
+ 20
+ 30
+I(ν) (arb. units)
+ ν (THz)
+FIG. 2. Dependence of the spectra in the THz range on the
+propagation distance. We consider 1-TW/cm2, 15-fs pump
+pulses at 2.26 fs−1 and 4.58 fs−1, in a composite of ZnO
+nanoparticles with filling fraction of f = 0.03 in a fused-silica
+matrix, after propagation length of 5 µm (magenta curve),
+15 µm (green curve), 45 µm (blue curve), and 50 µm (yellow
+curve).
+In Fig. 2 the evolution of the spectrum in the THz
+range is shown. One can see that while the spectrum is
+flat at early stages of propagation, for larger propagation
+lengths the spectrum is localized around 28 THz, proba-
+bly due to phase-matching effects with a phase-mismatch
+length of 10.5 µm for the four-wave mixing between the
+two photons at 2.26 fs−1, one photon at 4.58 fs−1, and a
+THz photon. Losses around 15 THz and below can also
+contribute to saturation of generation. After 45 µm prop-
+agation length, the efficiency of the generation reaches
+3.05%, which is sufficiently high for practical applica-
+tions.
+In order to determine the optimum conditions of the
+THz generation, in Fig. 3 we plot the dependence of the
+generation efficiency on the distribution of energy be-
+tween the 830-nm pulse and 412-nm pulse (a), intensity
+of pulses (b), and wavelength of the short-wavelength
+pulse (c). One can see that the efficiency of THz gen-
+eration is non-zero but very small for the cases when
+only one of the pulses is present (energy fraction of 0
+or 1). This indicates that the optical rectification based
+the second-order susceptibility of ZnO cannot efficiently
+generate THz for the considered conditions, and that
+the dominant contribution comes from the third-order
+susceptibility of ZnO nanoparticles, third-order suscep-
+tibility of SiO2 being comparatively weak. In an ideal
+case without pump pulses modification, the efficiency of
+the THz generation is proportional to E2
+830(Etot − E830),
+where E830 is the energy of the pulse at 830 nm and
+Etot = E830 + E412 is the total energy of the pulses. The
+maximum efficiency is then reached at E830/Etot = 1/3,
+
+6
+0%
+0.5%
+1%
+1.5%
+2%
+2.5%
+3%
+ 0
+ 0.25
+ 0.5
+ 0.75
+ 1
+(a)
+THz efficiency
+Energy fraction of 800-nm pulse
+0%
+0.5%
+1%
+1.5%
+2%
+2.5%
+3%
+ 0
+ 0.25
+ 0.5
+ 0.75
+ 1
+ 1.25
+ 1.5
+(b)
+THz efficiency
+I, TW/cm2
+0%
+0.5%
+1%
+1.5%
+2%
+2.5%
+3%
+ 405
+ 410
+ 415
+ 420
+ 425
+ 430
+(c)
+THz efficiency
+λ (nm)
+FIG. 3. Dependence of the THz generation efficiency on en-
+ergy fraction of the 800-nm pulse (a), intensity of each of the
+pump pulses (b), and the wavelength of the second-harmonic
+pulse (c). A composite of ZnO nanoparticles with f = 0.03 in
+fused silica is considered. In (a), 15-fs pulses at 2.26 fs−1 and
+4.58 fs−1 are considered, with total intensity of 2 TW/cm2
+and propagation length of 50 µm. In (b) we consider 15-fs
+(red curve) and 150-fs (green curve) pulses at 2.26 fs−1 and
+4.58 fs−1. In (c), 1 TW/cm2, 15-fs pulses are considered, with
+IR pulse frequency of 2.26 fs−1 and propagation length of 50
+µm.
+however, as shown in Fig. 3(a), maximum numerical ef-
+ficiency is achieved for E830/Etot = 0.5. This could be
+due to strong SPM-induced spectral spreading of high-
+frequency pulse during the propagation, which needs to
+be compensated by relatively higher value of E412. In
+Fig. 3(b), the dependence of the efficiency on the pulse
+intensity is shown, exhibiting saturation and decrease
+after a certain intensity as well as lower efficiencies for
+longer pulses. We attribute these features to detrimen-
+tal contribution of the accumulated plasma, which grows
+with intensity and pulse duration [cf. Fig. 4(a)]. In Fig.
+3(c), the dependence of the efficiency on the wavelength
+of the short-wavelength pulse exhibits several maxima.
+Note that while one might expect an optimum THz gen-
+eration for 415 nm, which would correspond to generation
+of frequencies near zero, our simulation in fact predict a
+minimum around this value, determined most probably
+by phase mismatch and losses below 15 THz.
+Finally, in order to access the role of plasma and ex-
+citons in the THz generation in composites, in Fig. 4
+we compare the spectra for plasma contribution (a) and
+exciton contribution (b) switched on/off. One can see
+that the plasma contribution is significant, both due to
+contribution to refractive index and due to losses, and
+absence of plasma contribution leads to a notable (more
+than twofold) increase of the efficiency.
+On the other
+hand, from Fig.
+4(b) one can see that exciton polar-
+ization do not provide a strong contribution to the effi-
+ciency for the considered parameters. Also, additionally
+including the permanent dipole momenta, described in
+10-3
+10-2
+0.1
+1
+ 10
+ 20
+ 30
+(a)
+I(ν) (arb. units)
+ ν (THz)
+10-3
+10-2
+0.1
+1
+ 10
+ 20
+ 30
+(b)
+I(ν) (arb. units)
+ ν (THz)
+FIG. 4. Spectra in the THz range with (a) plasma contri-
+bution on (red) and off (green) and (b) exciton contribution
+on (red) and off (green), as well including both exciton con-
+tribution and permanent dipole moment (blue). We consider
+1-TW/cm2, 15-fs pump pulses at 2.26 fs−1 and 4.58 fs−1, in a
+composite of ZnO inclusions with filling fraction of f = 0.03
+in a fused-silica matrix, after propagation length of 10 µm (a)
+and 0.75 µm (b).
+the model above, does not significantly increase the effi-
+ciency of THz generation, as indicated by the blue curve
+in Fig. 4(b) which is close to the red and green curves.
+We note, however, that this conclusion is of limited gen-
+erality; for other parameters of the medium excitons can
+provide the key mechanism of THz generation (see e.g.
+[44, 45] and references therein).
+IV.
+CONCLUSION
+In this paper we have established a comprehensive nu-
+merical model for the simulation of light propagation in
+composites, including all the relevant physical effects for
+a broad range of parameters, such as linear dispersion
+of the composite, second- and third-order nonlinear ef-
+fects, plasma contribution, excitons contribution and so
+on. The model was applied to simulate the generation of
+THz radiation in a ZnO-SiO2 composite. We have per-
+formed optimization of the frequency conversion process,
+predicting an efficiency of 3.05%.
+We show that sim-
+ulations provide insights into the optimization, such as
+the power distribution between the pump pulses, which
+would not be accessible intuitively. We hope that the nu-
+merical model and the corresponding software solution,
+which we make available for the community, will con-
+tribute to the capacity of the simulations in the area of
+nonlinear optics.
+ACKNOWLEDGMENTS
+Authors acknowledge financial support
+from Eu-
+ropean Union project H2020-MSCA-RISE-2018-823897
+”Atlantic”. I.B. thanks Cluster of Excellence PhoenixD
+(EXC 2122, project ID 390833453) for financial support.
+
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+tations”, Phys. Rev. B 77, 075330 (2008).
+[42] I. Babushkin, L. Shi, A. Demircan, U. Morgner, J. Her-
+rmann, and A. Husakou, ”Metallic nanostructures as
+electronic billiards”, https://arxiv.org/abs/2104.14637
+[43] C. Li, D. Feng, T. Jia, H. Sun, X. Li, S. Xu, X. Wang,
+and Z. Xu, ”Ultrafast dynamics in ZnO thin films irradi-
+ated by femtosecond lasers”, Solid State Communications
+136, 389-394 (2005).
+[44] O. Khasanov, O. Fedotova, G. Rusetsky and V. Niki-
+forov, ”Echo-Spectroscopy of Nanocomposites”, Laser
+Physics 29, 124011 (2019).
+[45] O. Fedotova, A. Husakou, G. Rusetsky, O. Khasanov,
+A. Fedotov, T. Smirnova, U. Sapaev, and I. Babushkin,
+”Mechanisms of Terahertz Generation under Femtosec-
+ond Pulses propagation in Nanocomposites”, jsii 1 4,
+CLEO-Europe - European Quantum Electronics Confer-
+ence, Munich, Germany (2021).
+
diff --git a/adE3T4oBgHgl3EQfdAqs/content/tmp_files/load_file.txt b/adE3T4oBgHgl3EQfdAqs/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf,len=645
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='04531v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='optics]  11 Jan 2023  Unified Model for Nonlinear Pulse Propagation in Composites and Optimization of  THz Generation  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Husakou∗  Max Born Institute, Max Born Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  2a, D-12489 Berlin, Germany  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Fedotova, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Rusetsky, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Khasanov  Scientific-Practical Materials Research Centre of National Academy  of Sciences of Belarus, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='Brovki str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  19, 220072 Minsk, Belarus  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Smirnova and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Fedotov  Belarus State University, Niezalieˇznasci ave 4, 220030 Minsk, Belarus  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Apostolova  Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,  Tsarigradsko Chausse 72, 1784 Sofia, Bulgaria and  Institute for Advanced Physical Studies, New Bulgarian University, 1618 Sofia, Bulgaria  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Babushkin  Institute of Quantum Optics, Leibniz University Hannover, Welfengarten 1, 30167, Hannover, Germany  Cluster of Excellence PhoenixD (Photonics, Optics,  and Engineering-Innovation Across Disciplines), 30167, Hannover, Germany and  Max Born Institute, Max-Born-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 2a, 12489, Berlin, Germany  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Sapaev  Tashkent State Technical University, University street 2, 100097 Tashkent, Uzbekistan  We describe a unified numerical model which allows fast and accurate simulation of nonlinear light  propagation in nanoparticle composites, including various effects such as group velocity dispersion,  second- and third-order nonlinearity, quasi-free-carrier formation and plasma contribution, exciton  dynamics, scattering and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' The developed software package SOLPIC is made available for  the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Using this model, we analyze and optimize efficient generation of THz radiation  by two-color pulses in ZnO/fused silica composite, predicting an efficiency of 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We compare the  role of various nonlinear effects contributing to the frequency conversion, and show that optimum  conditions of THz generation differ from those expected intuitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  INTRODUCTION  THz technology has attracted a lot of attention in  the recent years, since it provides unique experimental  tools and techniques in nonlinear and time-domain spec-  troscopy, biology and medicine, remote sensing, security  screening, as well as information and communication sys-  tems (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' [1–4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' For generation of THz radiation,  different techniques were proposed, such as two-color ion-  izing femtosecond pulses in gases [5–10] and plasmas,  produced at the surface of solids (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' metal foil) with  high intensities [11] that are free from optical damage  threshold and phase-matching drawbacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' optical rectifi-  cation of the transmitted intense ultrashort laser pulses  in non-linear crystals [12–16], which provide a basis for  compact low-intensity devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The needs of the THz  technology require, however, extension of the range of  the available techniques and materials, in order to pro-  vide flexible designs required in multifarious applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  ∗ gusakov@mbi-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='de  Following this line, investigations of THz generation in  various media such as water [17] and strongly magnetized  plasmas [18] were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Emission of terahertz radi-  ation with broad bandwidth by femtosecond photoexcita-  tion of spintronic materials (ferromagnetic and synthetic  multiferroic heterostructures) was also reported recently  [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Nanoparticle composites were actively investigated in  the past as a nonlinear material, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' [21–23], and their  particular strength lies in the flexibility of their design  leading to unusual properties such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' negative re-  fractive index [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  However, surprisingly, up to our  knowledge they have not attracted attention as a medium  for THz generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In this paper, we close this gap by  developing a numerical model suitable for simulation of  THz generation in nanocomposites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  A range of linear  and nonlinear effects such as group velocity dispersion,  second- and third-order nonlinearity, quasi-free-carrier  formation, exciton dynamics and so on are encompassed  by the developed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We use it to explore THz gen-  eration by two-color pulses in nanoparticle composites,  to elucidate the contributions of different frequency con-  version mechanisms, and to predict efficiencies in few-  2  percent range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The applicability of the above model is, in fact, much  broader than mere simulation of THz generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' a wide  range of nonlinear effects such as soliton dynamics and  supercontinuum generation, frequency conversion, multi-  level dynamics and electromagnetically-induced trans-  parency and so on can be studied using this unified ap-  proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' With this is mind, we have created the extensive  documentation of the code and made the code publicly  available [25], in a hope that it will be useful to the optical  community for investigations of the nonlinear processes  in nanocomposites and other materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In Section 2, we  present the numerical model, including detailed formal-  ism for all the relevant mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  In Section 3, we  optimize the THz generation by two-color pulses, and  analyze the role of different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' A summary of  the paper is given in the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  THEORETICAL MODEL  We consider a composite consisting of two components,  a homogeneous host material and spherical nanoparticles  (inclusions) randomly distributed in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We assume  a sufficiently low (typically few percent or below) filling  fraction of the inclusions so that neither percolation nor  interaction between the inclusions play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We con-  sider homogeneous inclusions to be sufficiently small with  diameter below the light wavelength so that effective-  medium theory can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Note that we do not  place any limitations on the nature of host and inclu-  sion materials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=', either of them could be a dielectric,  a metal, or a semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We do not require point  symmetry in host material or in the inclusions, so that  the second-order susceptibility can be non-zero in either  material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The model is designed to simulate propaga-  tion over relatively short distances of few millimeters, be-  low the damage threshold, and without back-reflection,  therefore (1+1)D treatment using unidirectional propa-  gation equation [26] is the most suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Under these conditions, the following effects have to  be taken into account: linear dispersion including intrin-  sic and scattering losses, second- and third-order optical  nonlinearities and photoionization accompanied by ion-  ization losses and plasma dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In addition, transi-  tions between excitonic states can play a significant role  in the inclusion response, in particular for the genera-  tion of new frequencies in the THz range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Among the  effects which were neglected in this treatment are ther-  mal effects (due to slow ns-scale response),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' coupling to  phonons (because of relatively slow ps-scale response),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Raman scattering (which is typically weaker than in-  stantaneous nonlinearities),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' anisotropy of the host ma-  terial (due to manufacturing limitations for composites),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  deviations of the inclusion form from a sphere (because  of typical manufacturing conditions),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' and generation of  high-order harmonics (because of the considered inten-  sity ranges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The following unidirectional propagation equation is  used to model the light propagation in a homogeneous  medium [26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 27]:  ∂E(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' ω)  ∂z  = −i  �  [  �  ǫ(ω) − ng]ω  c  − β(ω0)  �  E(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' ω)  −  iω  2c  �  ǫ(ω)  PNL(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  (1)  where E(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' ω) = ˆFE(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' t) =  � ∞  −∞ E(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' t) exp(−iωt)dt  is the Fourier transform ˆF of the electric field E(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' z  is the propagation coordinate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' ǫ(ω) is the linear dielec-  tric permittivity (generally speaking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' complex-valued to  include loss mechanisms),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' ng is the group refractive in-  dex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' ω0 is a characteristic frequency of the pulse spec-  trum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' β(ω) =  �  ǫ(ω)ω/c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' and PNL(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' ω) is the Fourier  transform of the nonlinear part of the polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We  would like to empathise that no slowly-varying envelope  approximation is used, and E(z, t) represents the real-  valued field including the carrier oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' This ap-  proach yields a unified treatment for a pulse with arbi-  trary spectral content, which is particularly important  for extremely broad spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Linear dispersion  The effective-medium theory allows to substitute the  composite material by a homogenised medium with ap-  propriately defined effective material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The  effective refractive index of a composite can be expressed  as [27]  neff =  �  (1 − f)ǫh + fǫi  3ǫh  2ǫh + ǫi  + 2i  � ǫh − ǫi  2ǫh + ǫi  �2 �rNPω√ǫh  c  �3�1/2  ,  (2)  where f is the volume filling factor of the inclusions,  rNP is their radius, and ǫh,i are the frequency-dependent  dielectric functions of the host and of the inclusions, cor-  respondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' The last term in the square brackets de-  scribes scattering losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Second- and third-order nonlinearities  The second- and third-order nonlinear processes can  also be described in the framework of the effective-  medium theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' The expressions for the effective second-  order susceptibility looks like [28]  3  χ(2)  eff (ω1 = ω2 + ω3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' ω2, ω3) = (1 − f)χ(2)  h  +  + fx(ω1)x(ω2)x(ω3)χ(2)  i ,  (3)  where χ(2)  h  and χ(2)  i  are the susceptibilities of host and  inclusion materials, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Note that we ne-  glected the frequency dependence of the susceptibilities  of host and inclusions, which is a good assumption far  from resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Quantity x(ω) is the ratio of local field  inside the inclusion and the incident field:  x(ω) =  3ǫh(ω)  2ǫh(ω) + ǫi(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  (4)  Here we note that, due to photoionization as described  below, the ǫi(ω) and therefore x, strictly speaking, de-  pend on time due to buildup of plasma during the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  However, in the current simulation we neglect this de-  pendence, assuming that corresponding change of ǫi(ω)  is small and that we are far from the plasmonic resonance  given by 2ǫh(ω) = −ǫi(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Similarly, for the effective third-order susceptibility we  write [28]  χ(3)  eff (ω1 = ω2 + ω3 + ω4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' ω2, ω3, ω4) = (1 − f)χ(3)  h  +  + fx(ω1)x(ω2)x(ω3)x(ω4)χ(3)  i ,  (5)  where χ(3)  h  and χ(3)  i  are the susceptibilities of host and  inclusion materials, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' The final expres-  sions which were used to calculate the corresponding po-  larizations look like  Pχ(2)(z, ω) = (1 − f)ǫ0χ(2)  h  ˆFE(z, t)2 +  + fǫ0χ(2)  i  ˆF[ ˆF −1{E(z, ω)x(ω)}2],  (6)  Pχ(3)(z, ω) = (1 − f)ǫ0χ(3)  h  ˆFE(z, t)3 +  + fǫ0χ(3)  i  ˆF[ ˆF −1{E(z, ω)x(ω)}3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  (7)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Plasma dynamics  Let us turn to the description of plasma formation and  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In the framework of SOLPIC, we consider a  case when the ionization potential Ip of the inclusions  is lower than that of the host material, so that due to  the sensitive dependence of the polarization rate on the  ionization potential we can neglect plasma formation in  host material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The contribution from the plasma is determined by the  average displacement ⟨d⟩(z, t) of the electron from the  equilibrium position in the parent ”molecule”, whereby  by a ”molecule” we denote an atom or a group of atoms of  the solid-state material which can provide a single ioniza-  tion event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Furthermore, it is determined by the relative  ionization of the solid state ρ(z, t), which is the ratio of  the conduction-band electron density to the density of  ”molecules”:  Pplasma(z, ω) = −Nmole ˆF[⟨d⟩(z, t)ρ(z, t)]  (8)  Here Nmol is the concentration of the molecules and  e = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='6×10−19 is the electron charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' The above expres-  sion would be valid in a homogeneous medium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' however,  as it refers to a polarization which occurs inside of inclu-  sions, in contrast to averaged macroscopic polarization,  in the case of effective-medium theory it has to be addi-  tionally multiplied by x(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' For the origin of this factor  and further details see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The dynamics of the quantity ⟨d⟩(z, t)ρ(z, t) is given  by [33]  ∂(⟨d⟩(z, t)ρ(z, t))  ∂t  = ⟨v⟩(z, t) + x0Γ(t),  (9)  where ⟨v⟩ is the average velocity of electrons and  x0 ≃ −Ip/eE(t) is the initial displacement of the elec-  tron immediately after the ionization event, Ip being the  bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' It can be shown that the second term describes  the energy loss of the pulse due to photoionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' The  dynamics of the ⟨v⟩(z, t)ρ(z, t) is given by second New-  ton’s law as  ∂(⟨d⟩(z, t)ρ(z, t))  ∂t  = −eE(z, t)  me  ρ,  (10)  where me is the effective electron mass near the bot-  tom of the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Here we neglect the initial  displacement and velocity of electron just after the ion-  ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The dynamics of the relative plasma density ρ is given  by  ∂ρ  ∂t = Γ( ˆF −1[x(ω)E(z, ω)]),  (11)  where x(ω)E(z, ω) is the local field inside of inclusions  which determines the photoionization rate Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Ionization rate  Depending on the relation between the frequency of  pump light and the ionization potential of inclusions, we  consider two models for the ionization rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' For the case  when the energy of pump photons is much smaller than  the ionization potential, the photoionization occurs ei-  ther by multiphoton regime or by tunneling regime, as  determined by intensity and Keldysh parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Here we  4  10-4  10-3  10-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='1  1  1  2  3  4  5  (a)  I(ω) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' units)  ω (fs-1)  6  4  2  0  2  4  6  40  20  0  20  40  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='01  (b)  E (GV/m)  ρ  t (fs)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Dependence of the spectra (a) and the electric field  (b) on the propagation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 15-fs pulses at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='26 fs−1 and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='58 fs−1 are considered, with intensity of 1 TW/cm2 and  propagation length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='75 µm (magenta curves), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='25 µm  (green curves), and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='75 µm (bluecurves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In (b), additionally  the relative plasma density is shown for propagation length  of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='75 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' A composite of ZnO inclusions with f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='03 in  fused silica is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  utilize so-called Yudin-Ivanov model [29], which provides  a formalism for both of these regimes in a unified way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  This model was initially developed for isolated atoms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' its  use for solid state is justified in a case a negligible an-  harmonicity of the bands in the center of the Brillouin  zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The cycle-resolved ionization rate Γ is given (in atomic  units, that is, with frequency ω, time t and field E mea-  sured in the corresponding Hartree units ωa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='26  rad/as, ta = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='2 as, xa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='0529 nm, and Ea = 514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='2  V/nm) by  Γ(z, t) = π  τT  exp  �  −σ0  ⟨2E(z, t)2⟩  ω3  � �  2κ3  �  ⟨2E(z, t)2⟩  �2Z/κ  × exp  �  −E(z, t)2  2ω3  σ1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  (12)  Here τT = κ/E(z, t), κ =  �  Ip/(ℏωa), σ0 =  1  2(γ2 +  1  2) ln C − 1  2γ  �  1 + γ2, γ = ωτT , Z is the effective atomic  charge, C = 1 + 2γ  �  1 + γ2 + 2γ2, and σ1 = ln C-  2γ/  �  1 + γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The quantity ⟨E(z, t)2⟩ is the averaged  value of the squared electric field over few past periods  (5 fs is assumed in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The Yudin-Ivanov model was initially derived for gases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  its applicability for solid state, while generally justi-  fied for materials with tight binding, is not strictly es-  tablished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  We have benchmarked Yudin-Ivanov model  by comparing it to the numerical solution of the time-  dependent Schrodinger equation in single active electron  approximation [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In this approach the empirical pseu-  dopotential method was used for describing the electron  band structure of ZnO [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We have found that the dif-  ference of the ionization rate does not typically exceed  one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' This difference is, in fact, not  very significant: because of the threshold-like behavior of  the ionization rate, it leads to only a slight shift of the  intensity at which a strong plasma generation is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  For the specal case when the energy of pump photons  is around two ionization potentials, it is preferable to  use the two-photon formalism [32] and write the cycle-  resolved ionization rate Γ (in SI units) as  Γ(z, t) =  2e4x4  aν  ℏ4ω2  0[(2ω0 − Ip/ℏ)2 + ν2 ⟨E(z, t)2⟩E(z, t)2,  (13)  where ν is the relaxation constant of the two-photon  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Contribution by excitons  Finally, we include the nonlinear polarization due to  excitons into treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We consider multiple excitonic  levels and utilize the standard Bloch equations for the de-  scription of the ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' The dynamics of the density  matrix ρe is given by (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' [32])  iℏ∂ρe  ∂t = [H, ρe],  (14)  where H = H0 + Hint, H0 is the Hamiltonian of the  system in the absence of excitation, Hint is the interaction  Hamiltonian, which components Hij are related with the  corresponding dipole transition moments eMij:  Hij = eMij ˆF −1[E(z, ω)x(ω)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  (15)  In addition, polarization decay (decay of the off-  diagonal elements of ρe) with decay time T2 and decay  of the population to the ground state with decay time T1  are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In order to avoid numerical instabilities,  the normalization of the density matrix ρ is performed  each few steps in time, by a) enforcing 0 ≤ ρe,ii ≤ 1, b)  enforcing T r(ρe) = 1, and c) adjusting the non-diagonal  elements which exceed the maximum possible value de-  termined by the corresponding level populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The excitonic polarization is then defined in a standard  way as  Pexc(z, ω) = x(ω) ˆF[fTr(ρeM)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  (16)  We solve the propagation equation by an extended  split-step method, whereby each of the contributions to  5  the polarization is treated subsequently, which allows to  reduce the accumulation of numerical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Nonlinear  steps are performed using the Runge-Kutta approach,  the order of which can be selected between 1,2, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Fixed step of the grid both in time and in the propa-  gation coordinate is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' The appearance of numerical  artifacts during the propagation is monitored by tracing  the total pulse energy as well as the total energy absorbed  at the boundaries of the numerical time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  NUMERICAL RESULTS AND DISCUSSION  In order to exemplify the above model and function-  ing of SOLPIC, we present in this section a simulation of  THz generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We consider a composite of ZnO inclu-  sions in SiO2 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Phenomenological Sellmeyer-type  expressions were used to describe dispersion on ZnO [35]  and SiO2 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Similarly, experimental data on second-  order [37] and third-order [38] susceptibility of bulk ZnO  and third-order susceptibility of SiO2 [40] were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We  estimated the value of T2 as 50 fs from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' [41] and used  T1 = 2T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' For ZnO, the typical exciton size is larger than  interatomic distance, meaning that we are dealing with  Wannier-Mott type of excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In a case of sufficiently  small inclusions, the exciton is bounded by the inclu-  sion boundaries, therefore its wavefunctions (as well as  energy levels and dipole momenta) are better described,  instead of hydrogen-like potential, by a constant poten-  tial inside a sphere [42] with a step on its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We  have taken into account 5 lowest excitonic levels, and  typical values of the off-diagonal dipole momenta, as cal-  culated by this approach, are around 3×10−28 C·m, for  same-size nanoparticles with a radius of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5 nm which  are considered here and hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  For the permanent  dipole momenta of ZnO, we have adopted a typical value  of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='66×10−30 C·m per ZnO molecule, which was used  to define the on-diagonal elements of the dipole matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  We used the ionization potential of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='37 eV equal to the  bandgap of ZnO to characterize the transition from va-  lence band to conduction band, and all the presented  numerical results correspond to the conditions below the  damage threshold of ZnO [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  The evolution of the field profile and spectra with prop-  agation is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 1, for two-color pulsed exci-  tation with pump pulses around 800 nm and 400 nm, for  conditions given in the caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 1(a) one can see  that initial stages of the propagation are characterized by  self-phase modulation with typical spectral side lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' At  later stages, spectrum becomes irregular and transform  into a supercontinuum extending up to the absorption  edge given by the bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' The evolution of the tempo-  ral profile, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 1(b), shows gradual reduction  of the enegry of electric field, as well as significantly ir-  regular envelope for longer propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' This reduction  of the maximum field determines the saturation of the  THz generation efficiency and is caused both by strong  group-velocity dispersion for broad spectrum and energy  absorption due to transition to conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' One  can see from the red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  1(b) that relative  plasma density reaches values of roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='01 after the  pulse, which is sufficient to induce significant energy ab-  sorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  10-4  10-3  10-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='1  1  10  20  30  I(ν) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' units)  ν (THz)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Dependence of the spectra in the THz range on the  propagation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We consider 1-TW/cm2, 15-fs pump  pulses at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='26 fs−1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='58 fs−1, in a composite of ZnO  nanoparticles with filling fraction of f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='03 in a fused-silica  matrix, after propagation length of 5 µm (magenta curve),  15 µm (green curve), 45 µm (blue curve), and 50 µm (yellow  curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 2 the evolution of the spectrum in the THz  range is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' One can see that while the spectrum is  flat at early stages of propagation, for larger propagation  lengths the spectrum is localized around 28 THz, proba-  bly due to phase-matching effects with a phase-mismatch  length of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5 µm for the four-wave mixing between the  two photons at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='26 fs−1, one photon at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='58 fs−1, and a  THz photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Losses around 15 THz and below can also  contribute to saturation of generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' After 45 µm prop-  agation length, the efficiency of the generation reaches  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='05%, which is sufficiently high for practical applica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  In order to determine the optimum conditions of the  THz generation, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 3 we plot the dependence of the  generation efficiency on the distribution of energy be-  tween the 830-nm pulse and 412-nm pulse (a), intensity  of pulses (b), and wavelength of the short-wavelength  pulse (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' One can see that the efficiency of THz gen-  eration is non-zero but very small for the cases when  only one of the pulses is present (energy fraction of 0  or 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' This indicates that the optical rectification based  the second-order susceptibility of ZnO cannot efficiently  generate THz for the considered conditions, and that  the dominant contribution comes from the third-order  susceptibility of ZnO nanoparticles, third-order suscep-  tibility of SiO2 being comparatively weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In an ideal  case without pump pulses modification, the efficiency of  the THz generation is proportional to E2  830(Etot − E830),  where E830 is the energy of the pulse at 830 nm and  Etot = E830 + E412 is the total energy of the pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' The  maximum efficiency is then reached at E830/Etot = 1/3,  6  0%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5%  1%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5%  2%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5%  3%  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='75  1  (a)  THz efficiency  Energy fraction of 800-nm pulse  0%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5%  1%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5%  2%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5%  3%  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='75  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5  (b)  THz efficiency  I, TW/cm2  0%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5%  1%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5%  2%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5%  3%  405  410  415  420  425  430  (c)  THz efficiency  λ (nm)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Dependence of the THz generation efficiency on en-  ergy fraction of the 800-nm pulse (a), intensity of each of the  pump pulses (b), and the wavelength of the second-harmonic  pulse (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' A composite of ZnO nanoparticles with f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='03 in  fused silica is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In (a), 15-fs pulses at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='26 fs−1 and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='58 fs−1 are considered, with total intensity of 2 TW/cm2  and propagation length of 50 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In (b) we consider 15-fs  (red curve) and 150-fs (green curve) pulses at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='26 fs−1 and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='58 fs−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In (c), 1 TW/cm2, 15-fs pulses are considered, with  IR pulse frequency of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='26 fs−1 and propagation length of 50  µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  however, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 3(a), maximum numerical ef-  ficiency is achieved for E830/Etot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' This could be  due to strong SPM-induced spectral spreading of high-  frequency pulse during the propagation, which needs to  be compensated by relatively higher value of E412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 3(b), the dependence of the efficiency on the pulse  intensity is shown, exhibiting saturation and decrease  after a certain intensity as well as lower efficiencies for  longer pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We attribute these features to detrimen-  tal contribution of the accumulated plasma, which grows  with intensity and pulse duration [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 4(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  3(c), the dependence of the efficiency on the wavelength  of the short-wavelength pulse exhibits several maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Note that while one might expect an optimum THz gen-  eration for 415 nm, which would correspond to generation  of frequencies near zero, our simulation in fact predict a  minimum around this value, determined most probably  by phase mismatch and losses below 15 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  Finally, in order to access the role of plasma and ex-  citons in the THz generation in composites, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 4  we compare the spectra for plasma contribution (a) and  exciton contribution (b) switched on/off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' One can see  that the plasma contribution is significant, both due to  contribution to refractive index and due to losses, and  absence of plasma contribution leads to a notable (more  than twofold) increase of the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  On the other  hand, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  4(b) one can see that exciton polar-  ization do not provide a strong contribution to the effi-  ciency for the considered parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Also, additionally  including the permanent dipole momenta, described in  10-3  10-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='1  1  10  20  30  (a)  I(ν) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' units)  ν (THz)  10-3  10-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='1  1  10  20  30  (b)  I(ν) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' units)  ν (THz)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' Spectra in the THz range with (a) plasma contri-  bution on (red) and off (green) and (b) exciton contribution  on (red) and off (green), as well including both exciton con-  tribution and permanent dipole moment (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We consider  1-TW/cm2, 15-fs pump pulses at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='26 fs−1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='58 fs−1, in a  composite of ZnO inclusions with filling fraction of f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='03  in a fused-silica matrix, after propagation length of 10 µm (a)  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='75 µm (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  the model above, does not significantly increase the effi-  ciency of THz generation, as indicated by the blue curve  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' 4(b) which is close to the red and green curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  We note, however, that this conclusion is of limited gen-  erality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' for other parameters of the medium excitons can  provide the key mechanism of THz generation (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  [44, 45] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  CONCLUSION  In this paper we have established a comprehensive nu-  merical model for the simulation of light propagation in  composites, including all the relevant physical effects for  a broad range of parameters, such as linear dispersion  of the composite, second- and third-order nonlinear ef-  fects, plasma contribution, excitons contribution and so  on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' The model was applied to simulate the generation of  THz radiation in a ZnO-SiO2 composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We have per-  formed optimization of the frequency conversion process,  predicting an efficiency of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='05%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  We show that sim-  ulations provide insights into the optimization, such as  the power distribution between the pump pulses, which  would not be accessible intuitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' We hope that the nu-  merical model and the corresponding software solution,  which we make available for the community, will con-  tribute to the capacity of the simulations in the area of  nonlinear optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  Authors acknowledge financial support  from Eu-  ropean Union project H2020-MSCA-RISE-2018-823897  ”Atlantic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
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+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content=' thanks Cluster of Excellence PhoenixD  (EXC 2122, project ID 390833453) for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
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+page_content=' Xu, ”Ultrafast dynamics in ZnO thin films irradi-  ated by femtosecond lasers”, Solid State Communications  136, 389-394 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
+page_content='  [44] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
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+page_content=' Rusetsky and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
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+page_content='  [45] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
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+page_content=' Khasanov,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
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+page_content=' Babushkin,  ”Mechanisms of Terahertz Generation under Femtosec-  ond Pulses propagation in Nanocomposites”, jsii 1 4,  CLEO-Europe - European Quantum Electronics Confer-  ence, Munich, Germany (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf'}
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+Advanced Scaling Methods for VNF deployment with
+Reinforcement Learning
+Namjin Seoa, DongNyeong Heoa, Heeyoul Choia
+aHandong Global University, Pohang, 37554, Gyeongbuk, South Korea
+Abstract
+Network function virtualization (NFV) and software-defined network (SDN)
+have become emerging network paradigms, allowing virtualized network func-
+tion (VNF) deployment at a low cost. Even though VNF deployment can be
+flexible, it is still challenging to optimize VNF deployment due to its high
+complexity. Several studies have approached the task as dynamic program-
+ming, e.g., integer linear programming (ILP). However, optimizing VNF de-
+ployment for highly complex networks remains a challenge. Alternatively, re-
+inforcement learning (RL) based approaches have been proposed to optimize
+this task, especially to employ a scaling action-based method which can de-
+ploy VNFs within less computational time. However, the model architecture
+can be improved further to generalize to the different networking settings.
+In this paper, we propose an enhanced model which can be adapted to more
+general network settings. We adopt the improved GNN architecture and a
+few techniques to obtain a better node representation for the VNF deploy-
+ment task. Furthermore, we apply a recently proposed RL method, phasic
+policy gradient (PPG), to leverage the shared representation of the service
+function chain (SFC) generation model from the value function. We evaluate
+the proposed method in various scenarios, achieving a better QoS with min-
+imum resource utilization compared to the previous methods. Finally, as a
+qualitative evaluation, we analyze our proposed encoder’s representation for
+the nodes, which shows a more disentangled representation.
+Keywords:
+Network Function Virtualization, Software Defined Networking,
+Reinforcement Learning, Graph Neural Network
+Preprint submitted to
+January 23, 2023
+arXiv:2301.08325v1  [cs.NI]  19 Jan 2023
+
+1. Introduction
+Softwarization of the Internet network, such as software-defined network-
+ing (SDN) and network function virtualization (NFV), has emerged as a
+new network paradigm. In this paradigm, the network functions provided
+by hardware-based middleboxes (e.g., Network Address Translation (NAT),
+Firewall, Proxy) are replaced with virtualized network functions (VNFs),
+running on virtual machines as VNF instances. By decoupling the network
+functions from the hardware, NFV allows the network service providers to
+deploy VNFs with low capital expenses (CAPEX) and operating expenses
+(OPEX). In addition, the traffics and network devices are managed and
+monitored by NFV orchestration (NFVO) in the centralized NFV frame-
+work. Hence, the VNF instances can be deployed by NFVO adaptable to the
+traffic requirements, which is the VNF deployment task.
+Even though VNF deployment can be accomplished dynamically with
+flexibility, requirements for the VNF deployment task are also getting more
+complex.
+The first requirement is that the service function chain (SFC)
+should be generated efficiently while maintaining an acceptable quality of
+service (QoS). SFC requires the traffics of a request to be routed through
+multiple stages of VNFs in NFV. Fig. 1 shows an example of SFC where the
+SFC request (Req1) demands an SFC: “NAT (‘N’) → Firewall(‘F’)”. This
+SFC request should be sequentially processed at these VNF instances as it
+travels from its ingress to its egress.
+These types of VNF should be de-
+ployed, taking into account the path of the requests, to meet the pre-defined
+service-level agreement (SLA) for QoS. Another requirement is to optimize
+resource utilization while satisfying the QoS. Redundantly deployed VNF in-
+stances could preserve high QoS, but it incurs unnecessary operating costs
+and energy consumption. Therefore, it is essential to improve the QoS while
+maintaining minimized resource consumption for the efficient management
+of VNF deployment.
+To achieve such efficient management of VNF deployment, the existing
+works have exploited dynamic programming algorithms, like integer linear
+programming (ILP) [1, 2, 3]. However, even though the ILP-based approach
+exhibits acceptable performance in networks with a low level of complex-
+ity, its computational cost becomes too expensive as the network scales up.
+Therefore, the ILP-based approach is not suitable for large-scale networks
+and the dynamical adjustment for traffic lifespans.
+As an alternative solution, ML-based approaches have been proposed
+2
+
+Figure 1: Overview of VNF Deployment task: SFC requests are required to pass through
+the VNFs sequentially as well as satisfy the SLA. Optimal deployment (Right) takes
+into account the paths of SFC requests, while sub-optimal deployment (Left) is deployed
+regardless of the paths of SFC request. Red boxes indicate inefficiently deployed VNF
+instances, which are on the wrong path and redundant.
+employing deep learning models, and reinforcement learning (RL) for VNF
+deployment [4, 5, 6, 7]. Especially, an RL-based method for VNF deployment
+was proposed to scale in and out VNFs on the network nodes [8].
+They
+adapted the graph neural network (GNN) model and RL algorithms on top
+of previously deployed VNF instances. Their action controls the number of
+VNFs on each node by scaling in, scaling out, or doing nothing. It showed
+that the agent was able to make appropriate scaling decisions for all the nodes
+and VNF types with a single forwarding process, involving fewer computation
+times for VNF deployment compared to other ML-based VNF deployment
+approaches. However, their model could not obtain the node representations
+in a general deployment setting where the networks include some nodes which
+cannot have VNF instances (e.g., switches) deployed on them due to its model
+setting for the node features.
+In this paper, we propose to enhance the neural network model and adapt
+new learning techniques so that the model can be adapted to the general
+deployment setting.
+First, we redesign the GNN architecture, and adapt
+a few techniques motivated from other domains, such as natural language
+processing and image processing [9, 10, 11, 12, 13], to effectively obtain a
+node representation of the networking information for the VNF deployment
+task. As the new GNN-based architecture, we employ the graph attention
+network (GAT) [14] and separately process different types of nodes: VNF
+deployable or non-deployable nodes.
+This architecture allows an effective
+propagation of node information from the network with different types of
+nodes. In addition, we adapt the positional encoding [11] to preserve the
+3
+
+SFC
+Qos
+SFC
+Qos
+Delay > SLA
+SRC→NF→DST
+Delay < SLA
+F
+N
+F
+Npositional information in the node representations.
+Furthermore, we apply a recently proposed RL algorithm, phasic policy
+gradient (PPG) [15], which is a variation of the policy gradient algorithm to
+optimize the policy network efficiently. By jointly optimizing the objectives
+of the policy network and the value network, the policy network can share
+representation with the value network, which helps the policy network obtain
+a more effective representation for the VNF deployment task.
+In the experiment, our approach optimizes the policy network to obtain a
+higher reward with rapid convergence compared to the previous approaches.
+Also, it proves that our approach can work in various scenarios. In addition,
+we analyze the result of our approach to show the robustness in the various
+topology and compare the processing time with ILP to show the practicality
+for the dynamic adaptation of the traffic lifespans. Finally, as a qualitative
+evaluation, we analyze the representation from the encoder of the policy
+network, and it demonstrates that our method obtains more disentangled
+representations of nodes.
+2. Background
+In this section, we briefly review graph neural networks and reinforcement
+learning for VNF deployment.
+2.1. Graph Neural Network
+Graph neural networks (GNN) [16] were proposed to effectively handle
+the graph-structured data consisting of a set of nodes V = {v1, · · · , v|V|} and
+edges E = {ˆeij|vi, vj ∈ V} [17, 18, 19, 14, 20, 21]. GNNs aim to address
+graph-related tasks (e.g., node/edge/graph classification) in an end-to-end
+manner with neural network model [17]. In the training process, GNNs are
+trained to extract nodes’ representations which reflect the graph information
+for the target tasks.
+In GNNs, a graph is represented as the following two matrix forms: the
+adjacency matrix A and the node feature matrix X. The adjacency matrix
+A ∈ R|V|×|V| represents the connections between pairs of nodes. The node
+feature matrix X contains node features xi ∈ R|V|×d. With A, X is trans-
+formed into node representation H = [h1, ..., h|V|]. Finally, these node repre-
+sentations are adapted to the target task in an end-to-end manner, which is
+implemented with a multi-layer perceptron (MLP) and a softmax layer.
+4
+
+GNN models are designed under the assumption that a node represen-
+tation for the target task is trained by propagating node information to its
+neighbors [17]. There are many different types of GNN models depending on
+the way how the node information propagates, like recurrent connection, con-
+volution, or attention. Among many GNN models, we review and compare
+gated graph neural network (GGNN) and graph attention network (GAT)
+which are used in the baseline and our approach, respectively.
+2.1.1. Gated Graph Neural Network
+In gated graph neural network (GGNN) [19], propagation between neigh-
+bors is implemented as recurrent connection as in recurrent neural networks
+(RNNs), and the node representation is calculated iteratively by RNN-like
+updates. For the recurrent connection, they use gated recurrent unit (GRU)
+[9], and the node representations are updated as follows:
+h′
+i = GRU(hi,
+�
+j∈Ni
+Whj),
+(1)
+where GRU is the GRU-like update function, Ni are the neighbors of node
+vi, W is a weight matrix, and the initial node representation is set to X.
+The recurrent updates are repeated a fixed number of times. Then, node
+representations are fed into the output layers.
+2.1.2. Graph Attention Network
+Graph attention network (GAT) [14] was proposed to update node repre-
+sentations with attention mechanism [10], while node information is propa-
+gated through recurrent steps in GGNN. GAT computes the attention scores
+of its neighbors so that the node representations can be updated according
+to the different importance of its neighbors as follows:
+h′
+i =
+�
+j∈Ni∪{i}
+ˆαijWhi,
+(2)
+ˆαij =
+exp(σ(a⊤ [Whi||Whj||Weˆeij]))
+�
+j′∈Ni∪{i} exp(σ(a⊤ [Whi||Whj′||Weˆeij′])),
+(3)
+where αij is the attention score of node vj for node vi, and ˆeij is edge at-
+tributes between nodes vi and vj. W, We, and a are weight matrices, σ is a
+non-linear function, and [·||·] indicates concatenation.
+5
+
+2.2. Reinforcement Learning
+Reinforcement learning (RL) trains an agent to maximize the expected
+reward by interacting with the environment. It formulates the task as Markov
+decision process (MDP) described as a sequence of state st, action at, and
+reward rt over discrete time steps. At discrete time-step t, the agent observes
+the state st of the environment and takes an action at. Subsequently, the
+current state st transitions into the next state st+1 along with the transition
+probability P(st+1|st, at), and finally, the agent receives a reward rt+1. This
+process is repeated until the agent reaches the terminal state which ends one
+episode. As the agent experiences many episodes, the agent could maximize
+the return, Gt = rt+1 + γrt+2 + γ2rt+3 + · · · , where γ is the discount factor.
+To maximize Gt over an episode, the RL agent needs to update policy
+π(at|st), a probability distribution of possible actions given state st.
+To
+estimate how good the action is at discrete time t, the agent should learn the
+state value V (st) = E[Gt|st] (or action-state value Q(st, at) at action-state
+pair (at, st)). In other words, the goal of an RL agent is to optimize its π(a|s)
+based on the trained state value V (s) (or action value Q(s, a)) to maximize
+the expected return in any episode.
+2.2.1. Policy Gradient
+Policy gradient (PG) is an RL method to model policy π(a|s) as a param-
+eterized function with neural networks called the policy network πθ(a|s). It
+trains the parameter of the policy network optimizing the objective function
+defined as follows:
+LPG(θ) = Et
+�
+log πθ(at|st) ˆAt
+�
+,
+(4)
+where ˆAt is the advantage estimator. For example, the REINFORCE algo-
+rithm [22] trains the parameters of the policy network defining advantage
+estimator ˆAt as return Gt.
+2.2.2. Proximal Policy Optimization
+Proximal policy optimization (PPO) [23] is one of the actor-critic (AC)
+based RL algorithms [24] with the trust-region method [25].
+In addition
+to the policy network, the AC algorithm employs the value network Vθ(s)
+estimating the state value V (st) in order to mitigate the variance in the
+training process. Furthermore, PPO has an additional constraint on the step
+size of the policy update to prevent inappropriate update of parameters,
+6
+
+called the trust-region method [25, 26]. Let ˆrt(θ) be the probability ratio
+ˆrt(θ) =
+πθ(at|st)
+πθold(at|st), then PPO maximizes
+LPPO(θ) = Et
+�
+min(ˆrt(θ) ˆAt, clip(ˆrt(θ), 1 − ϵ, 1 + ϵ) ˆAt)
+�
+,
+(5)
+where clip(·, x, y) is an operation clamping in range over x and y, and ϵ
+is a hyper-parameter of the clipping. The clipping operation constrains the
+probability ratio ˆrt to be close to 1 so that the parameters cannot be updated
+radically.
+2.2.3. Phasic Policy Gradient
+The phasic policy gradient (PPG) [15] algorithm was proposed to share
+parameters between the policy and value networks in the AC algorithms.
+Even though parameter sharing allows both networks to learn features jointly
+with a higher level of sample reuses, it is unclear whether it optimizes effi-
+ciently both networks jointly or not. Thus, they employ two value networks
+to avoid conflicts between competing objectives of the policy and value net-
+works. The one is value network VθV (s) not sharing parameters, and the
+other shares the parameter, called the auxiliary value head Vθπ(s).
+The PPG algorithm divides the policy and value network training phases
+into policy and auxiliary phases. In the policy phase, the policy is optimized
+in the same manner as PPO. Then, in every NPPG
+th iterations, an auxiliary
+loss is minimized in the auxiliary phase, and the auxiliary loss is defined by
+Laux = Et
+�1
+2(Vθπ(st) − VθV (st))2
+�
++ βcloneEt [KL [πθold(·|st), πθ(·|st)]] ,
+(6)
+where βclone is the hyper-parameter controlling how much the original policy
+is preserved.
+2.3. Scaling Agent for VNF Deployment
+Scaling of VNF deployment monitors VNF instances deployed in a net-
+work and adjusts the number of instances to maintain reliable QoS at min-
+imized cost by scaling in (removing instances), keeping, and scaling out
+(adding new instances). An RL-based method was proposed for VNF scaling
+[8] to optimize QoS and resource utilization, simultaneously. They adapted
+the graph neural network (GNN) models and RL algorithms on top of pre-
+viously deployed VNF instances. The GNN-based model inputs the network
+7
+
+information, and then produces the probabilities of scaling actions: scale
+in/out and keep. The parameters of the model were updated by PG-based
+RL algorithms, such as REINFORCE [22] and PPO [23]. In this section, we
+briefly review the model of the policy network and the RL formulation in [8].
+2.3.1. GNN based Policy Network
+The policy network has an encoder-decoder architecture. The encoder re-
+ceives the network information (e.g., network topology, deployed VNFs, and
+SFC requests.) and outputs the node representation reflecting the network
+information.
+The decoder reads the representations and produces action
+probabilities for ‘scaling in/out and keep’ of nodes and VNF types. To com-
+pute node representations that reflect diverse network topologies, the encoder
+is designed based on GNN [27], which is implemented with Gated Graph
+Neural Network (GGNN) [19]. Additionally, the network topology-related
+knowledge is transferred from the trained model for the SFC task, and the
+parameters are frozen so that the model could obtain proper representations
+of the network information.
+2.3.2. RL formulation for VNF scaling
+In the RL formulation, the RL agent observes the VNF deployment and
+a set of SFC requests as the initial state and produces the scaling actions for
+all the nodes and VNF types from the policy network. Then, after scaling
+from the initial VNF deployment, the reward is computed 1, which is defined
+as follows:
+R = − 1
+Nreq
+�
+k
+δk
+δk
+SLA
+− α
+�
+vnf
+Nvnf,
+(7)
+where Nreq and Nvnf are the number of SFC requests and the instances of
+VNF type vnf ∈ {Firewall, NAT, · · · }. δk and δk
+SLA are the traversing delay
+and SLA of request k, and α is the coefficient of penalty for restriction of
+resource utilization. The RL scenario is illustrated in Fig. 2.
+3. Proposed Method
+In this paper, we propose to enhance the neural network model and adapt
+new learning techniques. First, we enhance the previous VNF scaling model
+1The traversing path of the requests is generated by GNN-based SFC model [27].
+8
+
+Figure 2: Overview of the RL pipeline for VNF scaling: 1) The initial deployment is ob-
+served as the state from the environment. 2) Then, the RL agent updates the deployment
+by scaling action. 3) Finally, the reward is given, and one episode ends.
+[8] so that the model can be adaptable to more general deployment settings
+which includes nodes which cannot have any VNF instances deployed on
+them. Then, we apply new RL algorithms like PPG to train the model more
+efficiently.
+3.1. Model Architecture
+As in [8], the encoder-decoder based model reads the network information
+and produces the action probabilities for all the nodes and VNF types. The
+pipeline of the model architecture is presented in Fig. 3.
+The encoder computes the node representations H(k) by forwarding the
+pair of the adjacency matrix A and node feature matrix X(k) for each SFC
+request, where X(k) = [x(k)
+1 , · · · , x(k)
+|V|] is the node feature matrix for SFC
+request k. Specifically, node vi’s feature is defined by
+x(k)
+i
+= [src(k)(i), N (k)
+i,Firewall, N (k)
+i,NAT, · · · , dst(k)(i)],
+(8)
+where src(k)(i) and dst(k)(i) indicate whether node vi is the ingress or egress
+of request k. N (k)
+i,Firewall, N (k)
+i,NAT, · · · are the instance numbers of VNF type
+vnf ∈ {Firewall, NAT, · · · } deployed on node vi if needed for request k,
+otherwise set to 0. After encoding all the SFC requests, the encoder outputs
+the sets of node representations {H(k)}. Then, {H(k)} are averaged into a
+single representation ¯H, which is broadcasted and concatenated with each
+VNF embeddings evnf ∈ {eFirewall, eNAT, · · · } to make target VNF represen-
+tations ¯hi,vnf at the target nodes.
+The RNN-based decoder consisting of GRU and MLP layers reads the
+target representation ¯hi,vnf at each decoding step i, then outputs the action
+probabilities of the target VNF type vnf on target node vi. The actions
+9
+
+GNN-based
+RNN-based
+Encoder
+Decoder
+Initial
+Scaled
+Compute
+Deployment
+Deployment
+QoS&Resource(a) Encoding
+(b) Decoding (Example of Firewall as target VNF type)
+Figure 3: The proposed model architecture and the pipeline. (a) Node representations
+{H(k)} are obtained from all the pairs of an SFC request and the VNF deployment,
+and the target representation ¯hi,vnf for the target VNF type vnf on target node vi is
+obtained by averaging the node representations {H(k)} and concatenating with target
+VNF embeddings evnf as well as a positional encoding li (“P.E.”). (b) The decoder reads
+the target representation ¯hi,vnf to make a decision for scaling action of the target at the
+decoding step i. Over every decoding step, the decoder generates actions for the target
+VNF type, and the iteration stops after the number of target nodes. Scaling for all the
+VNF types can be performed in parallel.
+include three types of scaling actions for the target: ‘scale-in’, ‘scale-keep’,
+and ‘scale-out’.
+Over every decoding step, the decoder generates actions
+for the target VNF type, and the iteration stops after the number of target
+nodes. Scaling for all the VNF types can be performed in parallel, with this
+single process.
+3.1.1. GAT-based Encoder
+We apply the graph attention network (GAT) for the encoder instead of
+GGNN [27]. GAT has several training advantages compared to GGNN for
+VNF deployment. First, GAT could effectively propagate the node informa-
+tion from the previous layers through the attention mechanism, while GGNN
+10
+
+Ex.)
+Reqk5→NEP→1
+X
+N
+X
+GAT-based
+Encoder
+d
+Auxiliary
+ModuleDecoder
+Decoder
+Decoder
+N
+N
+Erelies on RNN-like updates. Next, GAT updates the node representation with
+different importance of the neighbor nodes as the attention score, which is
+helpful to train the model for requests with multiple hops between ingress
+and egress. Due to the fact that the VNF deployment is highly related to
+the paths of the requests, it is essential to train the importance of neighbors
+on the paths.
+The input of the GAT encoder is the adjacency matrix A and the in-
+put node feature matrix X given the input pair of the VNF deployment
+and an SFC request.
+The encoder produces node representation matrix
+H =
+�
+h1, ..., h|V|
+�
+.
+The input node features and edge weights defined as
+link latency are first forwarded through a shared linear transformation, pa-
+rameterized by weight matrices W, We, where W, We are weight matrices
+for node features and edge latency.
+Then, self-attention is performed on
+the nodes, where attention scores αij between node vi and vj are computed
+only for neighbor nodes, which is defined by Eq. 3. In addition, we employ
+multi-head attention to stabilize the learning process [14]. Finally, the node
+representation hi is updated as a weighted sum with the attention scores over
+nodes.
+3.1.2. Auxiliary Encoding Module with Node Embedding
+In the general deployment setting, the nodes in the network can be classi-
+fied into 2 different types: non-resource-allocating nodes (“non-deployable”)
+and resource-allocating nodes (“deployable”). As defined in Eq. 8, node fea-
+ture xi contains the number of instances for each VNF type, which are always
+zero-valued for the non-deployable nodes (N (k)
+i,vnf = 0). Fig. 4a represents an
+example of node features.
+Given the different node types, the model cannot update enough the
+representation of non-deployable nodes since the zero-valued features cannot
+have enough attention, which could cause a bottleneck when propagating
+node information through these nodes. To overcome the issue, we designed
+an auxiliary encoding module that provides additional neighbor information
+to the GNN-based encoder, as illustrated in Fig. 4b. This information is then
+utilized by the next module of the encoder to complement for information
+propagation when updating the representation of deployable nodes.
+The auxiliary encoding module consists of vectorized embeddings and a
+GAT layer. We first initialize vectorized embeddings {n1, n2, · · · } randomly
+and assign them to each non-deployable node, and then update them along
+with other parameters. As these embeddings are optimized during the train-
+11
+
+(a)
+Illustrative
+example
+of
+how
+non-
+deployable nodes could be bottlenecks,
+which hinder the information propagation
+from node#5 to node#1.
+(b) The auxiliary module provides the neighboring infor-
+mation of non-deployable nodes so that the GNN-based
+encoder could update node representation efficiently.
+Figure 4: Example of the general deployment setting and architecture of the encoder.
+ing process, the model could provide the non-deployable nodes’ neighboring
+information to mitigate the bottleneck effect. We refer to these embeddings
+as node embeddings (‘N.E.’). In addition, we define and assign the same node
+embedding n0 to all deployable nodes, allowing the module to focus on non-
+deployable cases. Subsequently, we process these node embeddings through
+a GAT layer to further reflect relations between multi-hops neighbors. We
+implement the GAT layer of this auxiliary module with less parameters than
+the encoder’s GAT layers to avoid over-fitting.
+Finally, we concatenate the output of the auxiliary module and the output
+of the GNN-based encoder’s first layer (refer to Fig. 4b) and forward to the
+second layer of the encoder to get a final nodes representations Hk.
+3.1.3. Positional Encoding
+To pass spatial information of nodes, we employ positional encoding, as in
+the Transformer [11]. Positional encoding could alleviate the decoder’s bur-
+den of transferring knowledge from previous nodes to decode hidden states.
+Before forwarding the aggregated representation ¯H, the location vector li is
+computed with sine and cosine functions of different frequencies as in the
+Transformer, then concatenated with ¯hi.
+Finally, the decoder inputs the
+representation of node vi partitioned into three parts: aggregated node vi’s
+representation ¯hi, target VNF embeddings evnf, and location vector li.
+12
+
+Ex.)
+Reqk
+5 
+N
+1→1
+Bottleneck
+N
+P
+XE.
+#1
+0
+0
+1
+0
+0
+X#2
+0
+0
+0
+0
+0
+0
+GAT
+GAT
+H
+...
+1#5
+0
+0
+0
+0
+0
+Additional Info
+Node Feature Mat. Xk
+forefficientpropagation
+#1
+n 
+X#2 -
+ni
+X#3 >
+n2 
+GAT
+#5一
+no
+NodeEmb3.2. RL for VNF Deployment
+In this section, we discuss how we optimize the policy network with RL
+algorithms. We follow the setting for RL formation and the perturbation
+scenario as in [8]. In the perturbation scenario where the VNF deployment
+is perturbed from the optimal deployment, the RL agent needs to remove
+the redundant VNFs or add necessary VNF instances to reconstruct optimal
+deployment by making scaling decisions for all the nodes and VNF types in
+the current deployment.
+In the perturbation scenario, we optimize the proposed model as the
+policy network. Moreover, we propose an architecture of the value network
+to apply the PPG algorithm, with which the policy network can leverage the
+shared representation of the SFC generation from the value network. The
+new value network architecture is described in the next section.
+3.2.1. Auxiliary Value Head for PPG
+The PPG algorithm jointly optimizes the objectives of the policy network
+and value function, where the policy network and value function share the
+parameters. In the optimization process, the objective of the value function
+is to minimize the error of the estimation for the state value, and the value
+function extracts the useful representation for the estimation of the SFC
+generation. The policy network can incorporate the representation from the
+value function to achieve a better policy.
+However, to avoid a conflict between both objectives of the policy and
+value network, PPG utilizes the auxiliary value head Vaux(s) as well as the
+value network V (s) to estimate the state value. The value head shares the
+parameters with the policy network, while the value network is implemented
+with the separated parameters. Thus, we design these two value functions
+and then apply the PPG to update the policy network.
+Fig. 5 presents the architectures of the value network and the auxiliary
+value head. The value network is implemented as a 2-layer MLP forwarding
+the mean of the aggregated representations ¯H to produce the state value
+V (st). For the auxiliary value head, we design the architecture to reflect the
+action’s state value for the node at each decoding step. The value head is
+implemented as a 2-layer MLP which is connected to the GRU layer of the
+decoder of the policy network. The value head estimates Vi when the policy
+network produces the node vi’s action probabilities. Finally, the state value
+13
+
+Vaux(st) is computed by averaging ¯Vi over nodes. This can be formulated by
+Vaux(s) = 1
+|V|
+�
+vi∈V
+Vi,
+(9)
+where Vi = f aux(�
+vnf zi,vnf), zi,vnf is the output of the decoder’s GRU layer
+from target VNF representation ¯hi,vnf, and f aux is the MLP layer of the
+auxiliary value head.
+Figure 5: Architectures of the value network and the auxiliary value head: PPO optimizes
+the policy and the value networks (orange box). In addition, PPG optimizes the auxiliary
+value head (green box) sharing parameters with the policy network.
+We apply the PPG algorithms with the value network and the auxiliary
+value head. After one episode, the agent gets reward R from the updated
+deployment, and then it stores the tuple (R, st, πθ(at|st)) in the buffer. Then,
+for NPPO episodes, the agent updates the parameter of the policy network
+to optimize the surrogate objective defined in Eq. 5. At the same time,
+the RL agent updates the parameter of the value network to minimize the
+mean squared error (MSE) between the return and the estimated state value.
+Lastly, for NPPG episodes, the parameter of the auxiliary value head is up-
+dated to minimize the auxiliary loss, defined in Eq. 6, which jointly updates
+the parameter of the policy network.
+14
+
+GRU4. Experiments
+4.1. Dataset and Configuration
+In this section, we describe the experiment settings and configurations
+to train the proposed model. The dataset was created from two networks:
+Internet2 [28] and Mobile Edge Computing (MEC), represented in Fig. 6.
+In the Internet2 network, the topology consists of 12 nodes with 15 links.
+SFC requests are created by normalizing Abilene traffic matrices proposed
+by [4]. In the MEC network, the topology consists of 14 nodes with 13 links.
+We followed [4] for other configurations, and specifically for the generation
+of SFC requests, we set the SLA as 95% of the latency computed from the
+SFC path generated by ILP [1]. The SFC requests contain the ingress and
+egress nodes, and the SFC path includes 3 or 4 services out of 5 VNF types.
+(a) Internet2
+(b) MEC
+Figure 6: Internet2 and MEC networks in the experiment settings. The nodes with ‘X’
+are non-deployable nodes like switches.
+First, we generated the ILP-based deployment to determine the optimal
+number and location of VNF instances for the set of active requests at each
+interval. Then, we created a VNF deployment dataset of which each entry
+contains an ILP-based deployment and a set of SFC requests. The dataset
+was then divided into training, validation, and testing sets with the ratio
+of 8:1:1.
+Lastly, we implemented a simulation environment for the same
+topology with each network and calculated the latency of the requests in the
+same manner as [27].
+Then, we trained the models on perturbed deployments by perturbing the
+ILP-based deployment. To perturb the deployment, an integer noise -1, 0 or
++1 is added on nodes and VNF types, and these deployments are set as the
+15
+
+X
+70
+23
+86
+209
+26
+129
+54
+63
+58
+84
+36
+90
+X
+117
+6812000
+1000
+1000
+5000
+5000
+1000
+X
+100
+100
+100
+100
+100
+100
+100initial states. Furthermore, we made random and zero deployments to evalu-
+ate the generalizability of the models. We set the number of VNF instances
+as 0 or 1 for the random deployment and only 0 for the zero deployment.
+For the details of hyper-parameters, we mainly followed the setting of [27]
+to pre-train the encoder and the configurations shown in Table 1 to train the
+decoder. The decoder architecture is implemented with layer normalization,
+1-layer GRU with 64 hidden units.
+We use three linear layers with the
+same number of hidden units in the decoder for the value function. As the
+activation function, ReLU (Rectified Linear Unit) is used. On each epoch,
+we measured the average reward in the validation set, and after the training,
+we selected the final model with the best reward in the validation set.
+Table 1: Hyper-Parameters for Training.
+Parameter
+Value
+Learning rate (LR)2
+3e-4
+Decoder dim. (GRU)
+32
+Decoder dim. (decoder-layer1)
+32
+Decoder dim. (decoder-layer2)
+32
+Decoder dim. (VNF embedding)
+5
+Decoder dim. (Positional Encoding)
+4
+Discount factor (γ)
+0.995
+PPO value network dim. (layer1)
+128
+PPO value network dim. (layer2)
+64
+PPO aux. value head dim. (layer1)
+32
+PPO aux. value head dim. (layer2)
+32
+PPO epsilon (ϵ)
+0.2
+PPO epoch (K)
+4
+PPO minibatch size (M)
+4
+PPO interval & PPG interval (policy phase) (NP P O)
+16
+PPG interval (auxiliary phase) (NP P G)
+64
+PPG hyper-parameter (βclone)
+1
+4.2. Quantitative Results
+We evaluate the performance of the method in both networks: Internet2
+and MEC.
+Internet2 Network:
+Table 2 shows an ablation study on Internet2 where we trained the models
+with PPO and tested them. For the metric, we measured the average number
+16
+
+of VNF instances (‘Avg. #VNF’), the average of delay time, and the average
+SLA violation ratio (‘Avg. SLAV’), as well as a reward (Eq. 7). Furthermore,
+we train the agent with different coefficient value (α) of instance penalty in
+Eq. 7 to show the effect of restriction for the level of the resource utilization,
+which controls the trade-off between the resource utilization and the QoS. We
+trained the agent with different seeds three times and reported the averages.
+Models (SFC)
+α
+Reward
+Avg. #VNF
+Avg. Delay
+Avg. SLAV
+ILP (GGNN)
+-
+-0.922
+12.86
+575.57
+0.197
+ILP (GAT)
+-
+-0.881
+12.86
+545.42
+0.167
+ILP (GAT+N.E.)
+-
+-0.876
+12.86
+542.11
+0.162
+GGNN (GGNN)
+0.15
+-1.796
+19.89
+1158.01
+0.279
+GGNN (GGNN)
+0.20
+-1.877
+18.44
+1226.94
+0.260
+GGNN (GGNN)
+0.25
+-2.345
+16.57
+1580.09
+0.354
+GAT(GAT)
+0.15
+-1.376
+17.81
+868.16
+0.141
+GAT(GAT)
+0.20
+-1.580
+17.58
+1018.22
+0.179
+GAT(GAT)
+0.25
+-2.018
+16.40
+1343.65
+0.255
+GAT+N.E. (GAT+N.E.)
+0.15
+-1.062
+17.71
+641.21
+0.086
+GAT+N.E. (GAT+N.E.)
+0.20
+-1.314
+16.04
+836.31
+0.117
+GAT+N.E. (GAT+N.E.)
+0.25
+-1.345
+15.75
+860.86
+0.122
+GAT+P.E. (GAT)
+0.15
+-1.084
+16.59
+665.61
+0.093
+GAT+P.E. (GAT)
+0.20
+-1.283
+15.89
+815.00
+0.122
+GAT+P.E. (GAT)
+0.25
+-1.409
+15.17
+911.44
+0.135
+GAT+P.E.+N.E. (GAT+N.E.)
+0.15
+-1.029
+17.39
+619.91
+0.081
+GAT+P.E.+N.E. (GAT+N.E.)
+0.20
+-1.240
+15.85
+784.09
+0.108
+GAT+P.E.+N.E. (GAT+N.E.)
+0.25
+-1.499
+14.87
+978.89
+0.139
+Table 2: Comparison of different models in the Internet2 network. The models are used
+to adjust VNF deployment based on scaling in/out/keep, while the methods in the paren-
+theses indicate how the SFC path was created for evaluation.
+Each row indicates the method to make VNF deployments and the method
+to generate SFC paths. For example, the row “ILP (GAT)” shows the re-
+sults when VNFs were deployed by the ILP method and the deployments
+were evaluated on the path generated by GAT for SFC. For the RL models,
+the same encoders were used for the policy networks and for the SFC path
+generation, though positional encoding was not used for SFC.
+From the table, we can see that the GAT-based models (‘GAT’) outper-
+form the baseline models (i.e., GGNN-based models). For example, ‘GAT
+(GAT)’ decreases the SLA violation rates from 0.279 (by ‘GGNN(GGNN)’)
+to 0.141 when the coefficient α for the instance penalty is 0.15. Furthermore,
+it improved the level of QoS and decreased the delay time with less number of
+instances. We believe that GAT models can be trained more efficiently, while
+GGNN-based models are hard to be optimized on our deployment settings,
+including many non-deployable nodes.
+17
+
+Moreover, ‘GAT+N.E.’ models outperform GGNN as well as GAT mod-
+els.
+They decrease the SLA violation rate, the number of instances and
+the delay time. This means that the node embedding-based approach, de-
+scribed in Section 3.1.2 can get better representations separating the VNF
+deployable and non-deployable nodes. In addition, the positional encoding
+(‘GAT+P.E.’) shows an even further improvement on SLAV compared to
+the corresponding methods. It implies that the positional encoding provides
+more distinguishable information for the policy network to find a better pol-
+icy for deployment. Lastly, we trained GAT models with both positional en-
+coding and node embedding (GAT+P.E.+N.E.). This method increases the
+reward and SLAV, while keeping the same level of the other metrics (‘Avg.
+#VNF’ and ‘Avg. Delay’), compared to the GAT-based models using only
+the positional encoding.
+MEC network:
+We experimented in the MEC network, where 5 nodes out of a total of
+12 nodes are non-deployable.
+We train the models with PPG, and com-
+pare GAT-based models to ILP approaches, which is summarized in Table
+3. The reward is computed using the SFC model, which employs the same
+architecture as the policy network. In the table, we can see that Positional
+Encoding (‘GAT+P.E.’) significantly decreases the SLA violation rate, the
+number of instances as well as the delay time. For the node embedding-
+based approach, ‘GAT+P.E.+N.E.’ has similar performance as ‘GAT+P.E.’.
+It means node embedding does not have additional information compared
+to positional embedding. Actually, as shown in Fig. 6, MEC has a regu-
+lar structure separating deployable and non-deployable nodes, which might
+be a possible explanation of why ‘N.E.’ does not additionally increase the
+performance on top of ‘GAT+P.E.’
+For the GGNN-based approach, it does not optimize the reward effec-
+tively, because links of the MEC network contain high latency3 and high
+variance, which hinders updating node representation in the model. Espe-
+cially, GGNN uses a weighted sum of its node representation with the edge
+attribute when updating the node representation, which causes it to under-
+estimate of the information from nodes connected with high latency. On the
+other hand, the GAT-based approach employs the attention mechanism to
+3The edge attribute is set as the reciprocal of latency.
+18
+
+Models (SFC)
+α
+Reward
+Avg. #VNF
+Avg. Delay
+Avg. SLAV
+ILP (GAT)
+0
+-0.702
+13.00
+29097.86
+0.046
+ILP (GAT+N.E.)
+-
+-0.707
+13.00
+29282.68
+0.045
+GAT (GAT)
+0.15
+-1.658
+20.33
+74682.66
+0.121
+GAT (GAT)
+0.20
+-1.867
+18.75
+87157.00
+0.146
+GAT (GAT)
+0.25
+-2.136
+18.75
+97987.34
+0.174
+GAT+P.E. (GAT)
+0.15
+-0.887
+16.84
+37044.12
+0.052
+GAT+P.E. (GAT)
+0.20
+-0.956
+16.25
+41100.82
+0.060
+GAT+P.E. (GAT)
+0.25
+-1.316
+15.41
+58091.53
+0.096
+GAT+P.E.+N.E. (GAT+N.E.)
+0.15
+-0.915
+16.55
+38930.56
+0.055
+GAT+P.E.+N.E. (GAT+N.E.)
+0.20
+-1.119
+16.29
+48965.09
+0.075
+GAT+P.E.+N.E. (GAT+N.E.)
+0.25
+-1.350
+15.95
+60878.52
+0.099
+Table 3: Comparison of different models in the MEC network. The positional encoding
+helps the training process more effective, while the node embedding does not additionally
+increase the performance on top of ‘GAT+P.E.’.
+compute the different importance of its neighbors and the edges so that it
+could mitigate the underestimation problem.
+4.2.1. Comparison of Various RL algorithms
+Since we propose to use the PPG algorithms, we compare them to other
+RL algorithms in the Internet2 network. In the experiment, we trained the
+GAT-based models with positional encoding with the various RL algorithms.
+As a result, the PPG algorithms can obtain better rewards, compared to
+DQN and PPO. PPG decreases the SLA violation rate and the delay time
+with a similar number of VNFs as presented in Table 4.
+Method
+α
+Reward
+Avg. #VNF
+Avg. Delay
+Avg. SLAV
+DQN
+0.15
+-1.317
+32.00
+718.56
+0.112
+DQN
+0.20
+-1.311
+25.00
+765.22
+0.174
+DQN
+0.25
+-2.701
+21.90
+1793.06
+0.285
+PPO
+0.15
+-1.084
+16.59
+665.62
+0.093
+PPO
+0.20
+-1.283
+15.89
+815.00
+0.122
+PPO
+0.25
+-1.409
+15.17
+911.44
+0.135
+PPG
+0.15
+-1.053
+16.30
+644.77
+0.091
+PPG
+0.20
+-1.164
+16.19
+726.24
+0.101
+PPG
+0.25
+-1.353
+15.51
+868.61
+0.130
+Table 4: Comparison of the various RL algorithms. PPG can optimize the reward more
+effectively compared to DQN and PPO.
+Moreover, as shown in Fig.
+7, PPG can optimize the average loss of
+the policy network more effectively than PPO. We believe that the auxiliary
+19
+
+Figure 7: Comparison of policy loss (α = 0.2) in training process.
+The loss of PPG
+decreases faster than PPO after around 40k iterations.
+Initial Deployment
+Reward
+Avg. #VNF
+Avg. Delay
+Avg. SLAV
+ILP-perturbed
+-1.029
+17.03
+622.59
+0.087
+Random
+-1.060
+16.95
+646.12
+0.090
+Zero
+-1.060
+16.94
+646.00
+0.090
+Table 5: Evaluations from different initial deployments in testing for the trained GAT
+model with α = 0.2.
+value head helps the policy network converge more efficiently, even though
+parameter sharing might make the training slow at the beginning of the
+training process.
+4.2.2. Analysis with Random and Zero Deployments
+To analyze further the performance of the trained models, we trained the
+GAT-based model (α = 0.2) optimized with PPG as before, and tested the
+trained model on different settings, where initial deployment is random or no
+VNFs are deployed (random deployment or zero deployment). The experi-
+ment results are presented in Table 5, where “Random” and “Zero” initial
+deployments are compared to the ILP-perturbed initial deployment. As the
+result, the performance on both initial deployments shows the same level of
+QoS and resource utilization as the ILP-perturbed case. It demonstrates that
+our approach can work on any sub-optimal initial deployment. Furthermore,
+our method can deploy VNFs without the ILP-based initial deployment.
+20
+
+Avg. Policy Loss
+PPG
+ agent: PPO
+0
+-0.01
+-0.02
+-0.03
+-0.04
+-0.05
+Valid/iter
+20k
+40k
+60k
+80kMethod
+Avg. Time (sec)
+ILP
+14.06
+GAT
+0.431
+Table 6: Average of execution times of 10 VNF-deployments.
+Figure 8: The VNF deployment generated from the model (Left), target SFC requests
+(Middle), and generated paths (Right). Given the target SFC request, VNF deployment
+and path are generated by the proposed method and the SFC model.
+4.2.3. Execution Time
+To show how fast our method works, we measured the average execution
+times to make decisions for VNF deployments and compared the proposed
+GAT-based model to ILP. The execution time includes the VNF deployment
+actions by the agent as well as SFC path generation by the SFC model [27].
+Each approach makes 10 deployments on a machine with 24 x 12-Core AMD
+Ryzen 9 3900X CPUs. Table 6 presents the average results of each approach.
+Our method can process VNF deployment about 30 times faster than ILP.
+4.3. Qualitative Evaluation
+In this section, we discuss the quality of the results by our approach. We
+analyze the VNF deployment generated from the agent and plot t-SNE [29]
+of the representation computed from the encoder of the policy networks.
+4.3.1. Generated Deployment
+We plot the VNF deployment generated from the GAT-based model
+trained with PPG as presented in Fig. 8, which shows VNF deployment
+generated from the SFC requests, and paths generated for the requests by
+the SFC model [27]. The VNF deployment is generated with more than 20
+requests, so we omitted the other requests in the figure for simplicity.
+21
+
+Req1
+6 →
+个N
+个
+6→1→1→11→8 (275.0 / 735)
+NW
+Req2
+←m
+N
+个
+个山
+7
+3→10→4→4→7 (765.0 / 722)
+23
+86
+209
+26
+N
+Req3
+←
+个N
+H个山
+个
+7
+5→1→4→4→7 (420.0 /723)
+129
+63
+Req4
+2 →
+个N
+8个
+2→ 11→ 11→ 11→ 8 (172.0 / 729)
+主E
+W
+36
+90
+WNPN
+117
+189
+Req23
+S个d个个
+4→1→1→5 (225.0 /700)
+Req24
+个个M个F个N个
+0→1→1→1→4→4(179.0 /705)The most VNF instances are deployed in the middle of the shortest paths
+for the traffics. For example, the first traffic (‘Req1’) has node 6 and node 8
+as the ingress and egress, and is required to pass through NAT (‘N’), Firewall
+(‘F’), and IDS (‘I’). In the network, the traffic goes through NAT at node 1,
+Firewall at node 11, and IDS at node 8. This traffic takes 275 ms while SLA
+for the SFC path is 735 ms. That is, all the VNFs of ‘Req1’ are deployed on
+the shortest path of its ingress and its egress.
+In addition, the generated deployment needs to meet the QoS with the
+optimized amount of resources. As shown in Fig. 8, VNF instances are de-
+ployed on the intersection of the shortest paths for the traffics. For example,
+the VNF instances at node 1 process more than four requests, like NAT for
+‘Req1’, ‘Req2’, ‘Req23’ and ‘Req24’. That is, our approach can deploy the
+instances at the shared nodes of the paths for many requests, which can
+reduce the number of VNF instances.
+4.3.2. Node Representations by t-SNE
+In the section, to analyze how the network information is represented by
+the encoder, we plot the 2D representation of node embeddings by t-SNE [29]
+as in Fig. 9, which shows the representations from the GGNN-based encoder
+and our proposed GAT-based encoder.
+Each dot represents one node of
+network, and the nodes are located in the Internet2 network as shown in Fig.
+6a.
+(a) GGNN
+(b) GAT
+Figure 9: Node representations by t-SNE. The GAT-based model gets more disentangled
+representations, compared to the GGNN-based model.
+Even though the GGNN-based encoder has multiple clusters, the nodes
+within each cluster are not distinguishable. However, the GAT-based en-
+coder’s representations are disentangled so that each cluster contains only
+22
+
+0
+60
+1
+2
+40
+3
+4
+5
+20
+6
+0
+8
+9
+-20
+10
+-40
+-60
+-80
+-60
+-40
+-20
+0
+20
+40
+600
+60
+1
+2
+40
+3
+4
+20
+5
+6
+7
+0
+8
+9
+-20
+10
+11
+-40
+-60
+-80
+-60
+-40
+-20
+0
+20
+40
+60
+80one or two nodes. Furthermore, the representations of neighbors in the net-
+work are close to each other in the plot.
+For example, clusters for node
+8 (‘brown’) and node 2 (‘green’) are close to each other, which reflects its
+neighborhood with the same node type as presented in Fig. 6a. This demon-
+strates the representation by GAT-based model reflects the network topology
+effectively.
+5. Conclusion
+In this paper, we proposed enhanced models which can be adapted to
+more general network settings. We proposed an improved GNN architecture
+and a few techniques to obtain a better node representation for the VNF
+deployment task. Furthermore, we optimized the model with PPG, a variant
+policy gradient-based algorithm. In the experiment, we evaluated the pro-
+posed method in various scenarios, achieving a better QoS with minimum
+resource utilization compared to the previous methods. Finally, we analyzed
+the generated VNF deployment and compare the node representations of our
+model to the baseline.
+Acknowledgement
+This research was supported by Basic Science Research Program through
+the National Research Foundation of Korea funded by the Ministry of Edu-
+cation (NRF-2022R1A2C1012633), and by Institute for Information & com-
+munications Technology Promotion (IITP) grant funded by the Korea gov-
+ernment(MSIT) (No. 2018-0-00749, Development of virtual network man-
+agement technology based on artificial intelligence).
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diff --git a/b9E_T4oBgHgl3EQfzxyC/content/tmp_files/load_file.txt b/b9E_T4oBgHgl3EQfzxyC/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5c4972f96bba24517202fd9cf0e1891e553697e7
--- /dev/null
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@@ -0,0 +1,1058 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf,len=1057
+page_content='Advanced Scaling Methods for VNF deployment with  Reinforcement Learning  Namjin Seoa, DongNyeong Heoa, Heeyoul Choia  aHandong Global University, Pohang, 37554, Gyeongbuk, South Korea  Abstract  Network function virtualization (NFV) and software-defined network (SDN)  have become emerging network paradigms, allowing virtualized network func-  tion (VNF) deployment at a low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Even though VNF deployment can be  flexible, it is still challenging to optimize VNF deployment due to its high  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Several studies have approached the task as dynamic program-  ming, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=', integer linear programming (ILP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' However, optimizing VNF de-  ployment for highly complex networks remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Alternatively, re-  inforcement learning (RL) based approaches have been proposed to optimize  this task, especially to employ a scaling action-based method which can de-  ploy VNFs within less computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' However, the model architecture  can be improved further to generalize to the different networking settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  In this paper, we propose an enhanced model which can be adapted to more  general network settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' We adopt the improved GNN architecture and a  few techniques to obtain a better node representation for the VNF deploy-  ment task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Furthermore, we apply a recently proposed RL method, phasic  policy gradient (PPG), to leverage the shared representation of the service  function chain (SFC) generation model from the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' We evaluate  the proposed method in various scenarios, achieving a better QoS with min-  imum resource utilization compared to the previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Finally, as a  qualitative evaluation, we analyze our proposed encoder’s representation for  the nodes, which shows a more disentangled representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Keywords:  Network Function Virtualization, Software Defined Networking,  Reinforcement Learning, Graph Neural Network  Preprint submitted to  January 23, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='08325v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='NI]  19 Jan 2023  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Introduction  Softwarization of the Internet network, such as software-defined network-  ing (SDN) and network function virtualization (NFV), has emerged as a  new network paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In this paradigm, the network functions provided  by hardware-based middleboxes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=', Network Address Translation (NAT),  Firewall, Proxy) are replaced with virtualized network functions (VNFs),  running on virtual machines as VNF instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' By decoupling the network  functions from the hardware, NFV allows the network service providers to  deploy VNFs with low capital expenses (CAPEX) and operating expenses  (OPEX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In addition, the traffics and network devices are managed and  monitored by NFV orchestration (NFVO) in the centralized NFV frame-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Hence, the VNF instances can be deployed by NFVO adaptable to the  traffic requirements, which is the VNF deployment task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Even though VNF deployment can be accomplished dynamically with  flexibility, requirements for the VNF deployment task are also getting more  complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  The first requirement is that the service function chain (SFC)  should be generated efficiently while maintaining an acceptable quality of  service (QoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' SFC requires the traffics of a request to be routed through  multiple stages of VNFs in NFV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 1 shows an example of SFC where the  SFC request (Req1) demands an SFC: “NAT (‘N’) → Firewall(‘F’)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' This  SFC request should be sequentially processed at these VNF instances as it  travels from its ingress to its egress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  These types of VNF should be de-  ployed, taking into account the path of the requests, to meet the pre-defined  service-level agreement (SLA) for QoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Another requirement is to optimize  resource utilization while satisfying the QoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Redundantly deployed VNF in-  stances could preserve high QoS, but it incurs unnecessary operating costs  and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Therefore, it is essential to improve the QoS while  maintaining minimized resource consumption for the efficient management  of VNF deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  To achieve such efficient management of VNF deployment, the existing  works have exploited dynamic programming algorithms, like integer linear  programming (ILP) [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' However, even though the ILP-based approach  exhibits acceptable performance in networks with a low level of complex-  ity, its computational cost becomes too expensive as the network scales up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Therefore, the ILP-based approach is not suitable for large-scale networks  and the dynamical adjustment for traffic lifespans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  As an alternative solution, ML-based approaches have been proposed  2  Figure 1: Overview of VNF Deployment task: SFC requests are required to pass through  the VNFs sequentially as well as satisfy the SLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Optimal deployment (Right) takes  into account the paths of SFC requests, while sub-optimal deployment (Left) is deployed  regardless of the paths of SFC request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Red boxes indicate inefficiently deployed VNF  instances, which are on the wrong path and redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  employing deep learning models, and reinforcement learning (RL) for VNF  deployment [4, 5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Especially, an RL-based method for VNF deployment  was proposed to scale in and out VNFs on the network nodes [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  They  adapted the graph neural network (GNN) model and RL algorithms on top  of previously deployed VNF instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Their action controls the number of  VNFs on each node by scaling in, scaling out, or doing nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' It showed  that the agent was able to make appropriate scaling decisions for all the nodes  and VNF types with a single forwarding process, involving fewer computation  times for VNF deployment compared to other ML-based VNF deployment  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' However, their model could not obtain the node representations  in a general deployment setting where the networks include some nodes which  cannot have VNF instances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=', switches) deployed on them due to its model  setting for the node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  In this paper, we propose to enhance the neural network model and adapt  new learning techniques so that the model can be adapted to the general  deployment setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  First, we redesign the GNN architecture, and adapt  a few techniques motivated from other domains, such as natural language  processing and image processing [9, 10, 11, 12, 13], to effectively obtain a  node representation of the networking information for the VNF deployment  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' As the new GNN-based architecture, we employ the graph attention  network (GAT) [14] and separately process different types of nodes: VNF  deployable or non-deployable nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  This architecture allows an effective  propagation of node information from the network with different types of  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In addition, we adapt the positional encoding [11] to preserve the  3  SFC  Qos  SFC  Qos  Delay > SLA  SRC→NF→DST  Delay < SLA  F  N  F  Npositional information in the node representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Furthermore, we apply a recently proposed RL algorithm, phasic policy  gradient (PPG) [15], which is a variation of the policy gradient algorithm to  optimize the policy network efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' By jointly optimizing the objectives  of the policy network and the value network, the policy network can share  representation with the value network, which helps the policy network obtain  a more effective representation for the VNF deployment task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  In the experiment, our approach optimizes the policy network to obtain a  higher reward with rapid convergence compared to the previous approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Also, it proves that our approach can work in various scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In addition,  we analyze the result of our approach to show the robustness in the various  topology and compare the processing time with ILP to show the practicality  for the dynamic adaptation of the traffic lifespans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Finally, as a qualitative  evaluation, we analyze the representation from the encoder of the policy  network, and it demonstrates that our method obtains more disentangled  representations of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Background  In this section, we briefly review graph neural networks and reinforcement  learning for VNF deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Graph Neural Network  Graph neural networks (GNN) [16] were proposed to effectively handle  the graph-structured data consisting of a set of nodes V = {v1, · · · , v|V|} and  edges E = {ˆeij|vi, vj ∈ V} [17, 18, 19, 14, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' GNNs aim to address  graph-related tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=', node/edge/graph classification) in an end-to-end  manner with neural network model [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In the training process, GNNs are  trained to extract nodes’ representations which reflect the graph information  for the target tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  In GNNs, a graph is represented as the following two matrix forms: the  adjacency matrix A and the node feature matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The adjacency matrix  A ∈ R|V|×|V| represents the connections between pairs of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The node  feature matrix X contains node features xi ∈ R|V|×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' With A, X is trans-  formed into node representation H = [h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=', h|V|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Finally, these node repre-  sentations are adapted to the target task in an end-to-end manner, which is  implemented with a multi-layer perceptron (MLP) and a softmax layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  4  GNN models are designed under the assumption that a node represen-  tation for the target task is trained by propagating node information to its  neighbors [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' There are many different types of GNN models depending on  the way how the node information propagates, like recurrent connection, con-  volution, or attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Among many GNN models, we review and compare  gated graph neural network (GGNN) and graph attention network (GAT)  which are used in the baseline and our approach, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Gated Graph Neural Network  In gated graph neural network (GGNN) [19], propagation between neigh-  bors is implemented as recurrent connection as in recurrent neural networks  (RNNs), and the node representation is calculated iteratively by RNN-like  updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' For the recurrent connection, they use gated recurrent unit (GRU)  [9], and the node representations are updated as follows:  h′  i = GRU(hi,  �  j∈Ni  Whj),  (1)  where GRU is the GRU-like update function, Ni are the neighbors of node  vi, W is a weight matrix, and the initial node representation is set to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  The recurrent updates are repeated a fixed number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Then, node  representations are fed into the output layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Graph Attention Network  Graph attention network (GAT) [14] was proposed to update node repre-  sentations with attention mechanism [10], while node information is propa-  gated through recurrent steps in GGNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' GAT computes the attention scores  of its neighbors so that the node representations can be updated according  to the different importance of its neighbors as follows:  h′  i =  �  j∈Ni∪{i}  ˆαijWhi,  (2)  ˆαij =  exp(σ(a⊤ [Whi||Whj||Weˆeij]))  �  j′∈Ni∪{i} exp(σ(a⊤ [Whi||Whj′||Weˆeij′])),  (3)  where αij is the attention score of node vj for node vi, and ˆeij is edge at-  tributes between nodes vi and vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' W, We, and a are weight matrices, σ is a  non-linear function, and [·||·] indicates concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Reinforcement Learning  Reinforcement learning (RL) trains an agent to maximize the expected  reward by interacting with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' It formulates the task as Markov  decision process (MDP) described as a sequence of state st, action at, and  reward rt over discrete time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' At discrete time-step t, the agent observes  the state st of the environment and takes an action at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Subsequently, the  current state st transitions into the next state st+1 along with the transition  probability P(st+1|st, at), and finally, the agent receives a reward rt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' This  process is repeated until the agent reaches the terminal state which ends one  episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' As the agent experiences many episodes, the agent could maximize  the return, Gt = rt+1 + γrt+2 + γ2rt+3 + · · · , where γ is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  To maximize Gt over an episode, the RL agent needs to update policy  π(at|st), a probability distribution of possible actions given state st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  To  estimate how good the action is at discrete time t, the agent should learn the  state value V (st) = E[Gt|st] (or action-state value Q(st, at) at action-state  pair (at, st)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In other words, the goal of an RL agent is to optimize its π(a|s)  based on the trained state value V (s) (or action value Q(s, a)) to maximize  the expected return in any episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Policy Gradient  Policy gradient (PG) is an RL method to model policy π(a|s) as a param-  eterized function with neural networks called the policy network πθ(a|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' It  trains the parameter of the policy network optimizing the objective function  defined as follows:  LPG(θ) = Et  �  log πθ(at|st) ˆAt  �  ,  (4)  where ˆAt is the advantage estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' For example, the REINFORCE algo-  rithm [22] trains the parameters of the policy network defining advantage  estimator ˆAt as return Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Proximal Policy Optimization  Proximal policy optimization (PPO) [23] is one of the actor-critic (AC)  based RL algorithms [24] with the trust-region method [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  In addition  to the policy network, the AC algorithm employs the value network Vθ(s)  estimating the state value V (st) in order to mitigate the variance in the  training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Furthermore, PPO has an additional constraint on the step  size of the policy update to prevent inappropriate update of parameters,  6  called the trust-region method [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Let ˆrt(θ) be the probability ratio  ˆrt(θ) =  πθ(at|st)  πθold(at|st), then PPO maximizes  LPPO(θ) = Et  �  min(ˆrt(θ) ˆAt, clip(ˆrt(θ), 1 − ϵ, 1 + ϵ) ˆAt)  �  ,  (5)  where clip(·, x, y) is an operation clamping in range over x and y, and ϵ  is a hyper-parameter of the clipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The clipping operation constrains the  probability ratio ˆrt to be close to 1 so that the parameters cannot be updated  radically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Phasic Policy Gradient  The phasic policy gradient (PPG) [15] algorithm was proposed to share  parameters between the policy and value networks in the AC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Even though parameter sharing allows both networks to learn features jointly  with a higher level of sample reuses, it is unclear whether it optimizes effi-  ciently both networks jointly or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Thus, they employ two value networks  to avoid conflicts between competing objectives of the policy and value net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The one is value network VθV (s) not sharing parameters, and the  other shares the parameter, called the auxiliary value head Vθπ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  The PPG algorithm divides the policy and value network training phases  into policy and auxiliary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In the policy phase, the policy is optimized  in the same manner as PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Then, in every NPPG  th iterations, an auxiliary  loss is minimized in the auxiliary phase, and the auxiliary loss is defined by  Laux = Et  �1  2(Vθπ(st) − VθV (st))2  �  + βcloneEt [KL [πθold(·|st), πθ(·|st)]] ,  (6)  where βclone is the hyper-parameter controlling how much the original policy  is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Scaling Agent for VNF Deployment  Scaling of VNF deployment monitors VNF instances deployed in a net-  work and adjusts the number of instances to maintain reliable QoS at min-  imized cost by scaling in (removing instances), keeping, and scaling out  (adding new instances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' An RL-based method was proposed for VNF scaling  [8] to optimize QoS and resource utilization, simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' They adapted  the graph neural network (GNN) models and RL algorithms on top of pre-  viously deployed VNF instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The GNN-based model inputs the network  7  information, and then produces the probabilities of scaling actions: scale  in/out and keep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The parameters of the model were updated by PG-based  RL algorithms, such as REINFORCE [22] and PPO [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In this section, we  briefly review the model of the policy network and the RL formulation in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' GNN based Policy Network  The policy network has an encoder-decoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The encoder re-  ceives the network information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=', network topology, deployed VNFs, and  SFC requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=') and outputs the node representation reflecting the network  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  The decoder reads the representations and produces action  probabilities for ‘scaling in/out and keep’ of nodes and VNF types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' To com-  pute node representations that reflect diverse network topologies, the encoder  is designed based on GNN [27], which is implemented with Gated Graph  Neural Network (GGNN) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Additionally, the network topology-related  knowledge is transferred from the trained model for the SFC task, and the  parameters are frozen so that the model could obtain proper representations  of the network information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' RL formulation for VNF scaling  In the RL formulation, the RL agent observes the VNF deployment and  a set of SFC requests as the initial state and produces the scaling actions for  all the nodes and VNF types from the policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Then, after scaling  from the initial VNF deployment, the reward is computed 1, which is defined  as follows:  R = − 1  Nreq  �  k  δk  δk  SLA  − α  �  vnf  Nvnf,  (7)  where Nreq and Nvnf are the number of SFC requests and the instances of  VNF type vnf ∈ {Firewall, NAT, · · · }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' δk and δk  SLA are the traversing delay  and SLA of request k, and α is the coefficient of penalty for restriction of  resource utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The RL scenario is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Proposed Method  In this paper, we propose to enhance the neural network model and adapt  new learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' First, we enhance the previous VNF scaling model  1The traversing path of the requests is generated by GNN-based SFC model [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  8  Figure 2: Overview of the RL pipeline for VNF scaling: 1) The initial deployment is ob-  served as the state from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 2) Then, the RL agent updates the deployment  by scaling action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 3) Finally, the reward is given, and one episode ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  [8] so that the model can be adaptable to more general deployment settings  which includes nodes which cannot have any VNF instances deployed on  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Then, we apply new RL algorithms like PPG to train the model more  efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Model Architecture  As in [8], the encoder-decoder based model reads the network information  and produces the action probabilities for all the nodes and VNF types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The  pipeline of the model architecture is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  The encoder computes the node representations H(k) by forwarding the  pair of the adjacency matrix A and node feature matrix X(k) for each SFC  request, where X(k) = [x(k)  1 , · · · , x(k)  |V|] is the node feature matrix for SFC  request k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Specifically, node vi’s feature is defined by  x(k)  i  = [src(k)(i), N (k)  i,Firewall, N (k)  i,NAT, · · · , dst(k)(i)],  (8)  where src(k)(i) and dst(k)(i) indicate whether node vi is the ingress or egress  of request k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' N (k)  i,Firewall, N (k)  i,NAT, · · · are the instance numbers of VNF type  vnf ∈ {Firewall, NAT, · · · } deployed on node vi if needed for request k,  otherwise set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' After encoding all the SFC requests, the encoder outputs  the sets of node representations {H(k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Then, {H(k)} are averaged into a  single representation ¯H, which is broadcasted and concatenated with each  VNF embeddings evnf ∈ {eFirewall, eNAT, · · · } to make target VNF represen-  tations ¯hi,vnf at the target nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  The RNN-based decoder consisting of GRU and MLP layers reads the  target representation ¯hi,vnf at each decoding step i, then outputs the action  probabilities of the target VNF type vnf on target node vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The actions  9  GNN-based  RNN-based  Encoder  Decoder  Initial  Scaled  Compute  Deployment  Deployment  QoS&Resource(a) Encoding  (b) Decoding (Example of Firewall as target VNF type)  Figure 3: The proposed model architecture and the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' (a) Node representations  {H(k)} are obtained from all the pairs of an SFC request and the VNF deployment,  and the target representation ¯hi,vnf for the target VNF type vnf on target node vi is  obtained by averaging the node representations {H(k)} and concatenating with target  VNF embeddings evnf as well as a positional encoding li (“P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' (b) The decoder reads  the target representation ¯hi,vnf to make a decision for scaling action of the target at the  decoding step i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Over every decoding step, the decoder generates actions for the target  VNF type, and the iteration stops after the number of target nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Scaling for all the  VNF types can be performed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  include three types of scaling actions for the target: ‘scale-in’, ‘scale-keep’,  and ‘scale-out’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Over every decoding step, the decoder generates actions  for the target VNF type, and the iteration stops after the number of target  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Scaling for all the VNF types can be performed in parallel, with this  single process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' GAT-based Encoder  We apply the graph attention network (GAT) for the encoder instead of  GGNN [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' GAT has several training advantages compared to GGNN for  VNF deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' First, GAT could effectively propagate the node informa-  tion from the previous layers through the attention mechanism, while GGNN  10  Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=')  Reqk5→NEP→1  X  N  X  GAT-based  Encoder  d  Auxiliary  ModuleDecoder  Decoder  Decoder  N  N  Erelies on RNN-like updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Next, GAT updates the node representation with  different importance of the neighbor nodes as the attention score, which is  helpful to train the model for requests with multiple hops between ingress  and egress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Due to the fact that the VNF deployment is highly related to  the paths of the requests, it is essential to train the importance of neighbors  on the paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  The input of the GAT encoder is the adjacency matrix A and the in-  put node feature matrix X given the input pair of the VNF deployment  and an SFC request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  The encoder produces node representation matrix  H =  �  h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=', h|V|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  The input node features and edge weights defined as  link latency are first forwarded through a shared linear transformation, pa-  rameterized by weight matrices W, We, where W, We are weight matrices  for node features and edge latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Then, self-attention is performed on  the nodes, where attention scores αij between node vi and vj are computed  only for neighbor nodes, which is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In addition, we employ  multi-head attention to stabilize the learning process [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Finally, the node  representation hi is updated as a weighted sum with the attention scores over  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Auxiliary Encoding Module with Node Embedding  In the general deployment setting, the nodes in the network can be classi-  fied into 2 different types: non-resource-allocating nodes (“non-deployable”)  and resource-allocating nodes (“deployable”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' As defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 8, node fea-  ture xi contains the number of instances for each VNF type, which are always  zero-valued for the non-deployable nodes (N (k)  i,vnf = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 4a represents an  example of node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Given the different node types, the model cannot update enough the  representation of non-deployable nodes since the zero-valued features cannot  have enough attention, which could cause a bottleneck when propagating  node information through these nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' To overcome the issue, we designed  an auxiliary encoding module that provides additional neighbor information  to the GNN-based encoder, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' This information is then  utilized by the next module of the encoder to complement for information  propagation when updating the representation of deployable nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  The auxiliary encoding module consists of vectorized embeddings and a  GAT layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' We first initialize vectorized embeddings {n1, n2, · · · } randomly  and assign them to each non-deployable node, and then update them along  with other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' As these embeddings are optimized during the train-  11  (a)  Illustrative  example  of  how  non-  deployable nodes could be bottlenecks,  which hinder the information propagation  from node#5 to node#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  (b) The auxiliary module provides the neighboring infor-  mation of non-deployable nodes so that the GNN-based  encoder could update node representation efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Figure 4: Example of the general deployment setting and architecture of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  ing process, the model could provide the non-deployable nodes’ neighboring  information to mitigate the bottleneck effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' We refer to these embeddings  as node embeddings (‘N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In addition, we define and assign the same node  embedding n0 to all deployable nodes, allowing the module to focus on non-  deployable cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Subsequently, we process these node embeddings through  a GAT layer to further reflect relations between multi-hops neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' We  implement the GAT layer of this auxiliary module with less parameters than  the encoder’s GAT layers to avoid over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Finally, we concatenate the output of the auxiliary module and the output  of the GNN-based encoder’s first layer (refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 4b) and forward to the  second layer of the encoder to get a final nodes representations Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Positional Encoding  To pass spatial information of nodes, we employ positional encoding, as in  the Transformer [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Positional encoding could alleviate the decoder’s bur-  den of transferring knowledge from previous nodes to decode hidden states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Before forwarding the aggregated representation ¯H, the location vector li is  computed with sine and cosine functions of different frequencies as in the  Transformer, then concatenated with ¯hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Finally, the decoder inputs the  representation of node vi partitioned into three parts: aggregated node vi’s  representation ¯hi, target VNF embeddings evnf, and location vector li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  12  Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=')  Reqk  5  N  1→1  Bottleneck  N  P  XE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  #1  0  0  1  0  0  X#2  0  0  0  0  0  0  GAT  GAT  H  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  1#5  0  0  0  0  0  Additional Info  Node Feature Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Xk  forefficientpropagation  #1  n  X#2 -  ni  X#3 >  n2  GAT  #5一  no  NodeEmb3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' RL for VNF Deployment  In this section, we discuss how we optimize the policy network with RL  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' We follow the setting for RL formation and the perturbation  scenario as in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In the perturbation scenario where the VNF deployment  is perturbed from the optimal deployment, the RL agent needs to remove  the redundant VNFs or add necessary VNF instances to reconstruct optimal  deployment by making scaling decisions for all the nodes and VNF types in  the current deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  In the perturbation scenario, we optimize the proposed model as the  policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Moreover, we propose an architecture of the value network  to apply the PPG algorithm, with which the policy network can leverage the  shared representation of the SFC generation from the value network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The  new value network architecture is described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Auxiliary Value Head for PPG  The PPG algorithm jointly optimizes the objectives of the policy network  and value function, where the policy network and value function share the  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In the optimization process, the objective of the value function  is to minimize the error of the estimation for the state value, and the value  function extracts the useful representation for the estimation of the SFC  generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The policy network can incorporate the representation from the  value function to achieve a better policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  However, to avoid a conflict between both objectives of the policy and  value network, PPG utilizes the auxiliary value head Vaux(s) as well as the  value network V (s) to estimate the state value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The value head shares the  parameters with the policy network, while the value network is implemented  with the separated parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Thus, we design these two value functions  and then apply the PPG to update the policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 5 presents the architectures of the value network and the auxiliary  value head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The value network is implemented as a 2-layer MLP forwarding  the mean of the aggregated representations ¯H to produce the state value  V (st).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' For the auxiliary value head, we design the architecture to reflect the  action’s state value for the node at each decoding step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The value head is  implemented as a 2-layer MLP which is connected to the GRU layer of the  decoder of the policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The value head estimates Vi when the policy  network produces the node vi’s action probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Finally, the state value  13  Vaux(st) is computed by averaging ¯Vi over nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' This can be formulated by  Vaux(s) = 1  |V|  �  vi∈V  Vi,  (9)  where Vi = f aux(�  vnf zi,vnf), zi,vnf is the output of the decoder’s GRU layer  from target VNF representation ¯hi,vnf, and f aux is the MLP layer of the  auxiliary value head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Figure 5: Architectures of the value network and the auxiliary value head: PPO optimizes  the policy and the value networks (orange box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In addition, PPG optimizes the auxiliary  value head (green box) sharing parameters with the policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  We apply the PPG algorithms with the value network and the auxiliary  value head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' After one episode, the agent gets reward R from the updated  deployment, and then it stores the tuple (R, st, πθ(at|st)) in the buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Then,  for NPPO episodes, the agent updates the parameter of the policy network  to optimize the surrogate objective defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' At the same time,  the RL agent updates the parameter of the value network to minimize the  mean squared error (MSE) between the return and the estimated state value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Lastly, for NPPG episodes, the parameter of the auxiliary value head is up-  dated to minimize the auxiliary loss, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 6, which jointly updates  the parameter of the policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  14  GRU4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Dataset and Configuration  In this section, we describe the experiment settings and configurations  to train the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The dataset was created from two networks:  Internet2 [28] and Mobile Edge Computing (MEC), represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  In the Internet2 network, the topology consists of 12 nodes with 15 links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  SFC requests are created by normalizing Abilene traffic matrices proposed  by [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In the MEC network, the topology consists of 14 nodes with 13 links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  We followed [4] for other configurations, and specifically for the generation  of SFC requests, we set the SLA as 95% of the latency computed from the  SFC path generated by ILP [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The SFC requests contain the ingress and  egress nodes, and the SFC path includes 3 or 4 services out of 5 VNF types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  (a) Internet2  (b) MEC  Figure 6: Internet2 and MEC networks in the experiment settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The nodes with ‘X’  are non-deployable nodes like switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  First, we generated the ILP-based deployment to determine the optimal  number and location of VNF instances for the set of active requests at each  interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Then, we created a VNF deployment dataset of which each entry  contains an ILP-based deployment and a set of SFC requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The dataset  was then divided into training, validation, and testing sets with the ratio  of 8:1:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Lastly, we implemented a simulation environment for the same  topology with each network and calculated the latency of the requests in the  same manner as [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Then, we trained the models on perturbed deployments by perturbing the  ILP-based deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' To perturb the deployment, an integer noise -1, 0 or  +1 is added on nodes and VNF types, and these deployments are set as the  15  X  70  23  86  209  26  129  54  63  58  84  36  90  X  117  6812000  1000  1000  5000  5000  1000  X  100  100  100  100  100  100  100initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Furthermore, we made random and zero deployments to evalu-  ate the generalizability of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' We set the number of VNF instances  as 0 or 1 for the random deployment and only 0 for the zero deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  For the details of hyper-parameters, we mainly followed the setting of [27]  to pre-train the encoder and the configurations shown in Table 1 to train the  decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The decoder architecture is implemented with layer normalization,  1-layer GRU with 64 hidden units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  We use three linear layers with the  same number of hidden units in the decoder for the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' As the  activation function, ReLU (Rectified Linear Unit) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' On each epoch,  we measured the average reward in the validation set, and after the training,  we selected the final model with the best reward in the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Table 1: Hyper-Parameters for Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Parameter  Value  Learning rate (LR)2  3e-4  Decoder dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' (GRU)  32  Decoder dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' (decoder-layer1)  32  Decoder dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' (decoder-layer2)  32  Decoder dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' (VNF embedding)  5  Decoder dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' (Positional Encoding)  4  Discount factor (γ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='995  PPO value network dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' (layer1)  128  PPO value network dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' (layer2)  64  PPO aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' value head dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' (layer1)  32  PPO aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' value head dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' (layer2)  32  PPO epsilon (ϵ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2  PPO epoch (K)  4  PPO minibatch size (M)  4  PPO interval & PPG interval (policy phase) (NP P O)  16  PPG interval (auxiliary phase) (NP P G)  64  PPG hyper-parameter (βclone)  1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Quantitative Results  We evaluate the performance of the method in both networks: Internet2  and MEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Internet2 Network:  Table 2 shows an ablation study on Internet2 where we trained the models  with PPO and tested them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' For the metric, we measured the average number  16  of VNF instances (‘Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' #VNF’), the average of delay time, and the average  SLA violation ratio (‘Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' SLAV’), as well as a reward (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Furthermore,  we train the agent with different coefficient value (α) of instance penalty in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 7 to show the effect of restriction for the level of the resource utilization,  which controls the trade-off between the resource utilization and the QoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' We  trained the agent with different seeds three times and reported the averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Models (SFC)  α  Reward  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' #VNF  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Delay  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' SLAV  ILP (GGNN) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='922  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='86  575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='197  ILP (GAT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='881  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='86  545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='167  ILP (GAT+N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='876  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='86  542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='162  GGNN (GGNN)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
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+page_content=' The models are used  to adjust VNF deployment based on scaling in/out/keep, while the methods in the paren-  theses indicate how the SFC path was created for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Each row indicates the method to make VNF deployments and the method  to generate SFC paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' For example, the row “ILP (GAT)” shows the re-  sults when VNFs were deployed by the ILP method and the deployments  were evaluated on the path generated by GAT for SFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' For the RL models,  the same encoders were used for the policy networks and for the SFC path  generation, though positional encoding was not used for SFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  From the table, we can see that the GAT-based models (‘GAT’) outper-  form the baseline models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
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+page_content=', GGNN-based models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' For example, ‘GAT  (GAT)’ decreases the SLA violation rates from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='279 (by ‘GGNN(GGNN)’)  to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='141 when the coefficient α for the instance penalty is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Furthermore,  it improved the level of QoS and decreased the delay time with less number of  instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' We believe that GAT models can be trained more efficiently, while  GGNN-based models are hard to be optimized on our deployment settings,  including many non-deployable nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  17  Moreover, ‘GAT+N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.’ models outperform GGNN as well as GAT mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  They decrease the SLA violation rate, the number of instances and  the delay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' This means that the node embedding-based approach, de-  scribed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2 can get better representations separating the VNF  deployable and non-deployable nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In addition, the positional encoding  (‘GAT+P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.’) shows an even further improvement on SLAV compared to  the corresponding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' It implies that the positional encoding provides  more distinguishable information for the policy network to find a better pol-  icy for deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Lastly, we trained GAT models with both positional en-  coding and node embedding (GAT+P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='+N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' This method increases the  reward and SLAV, while keeping the same level of the other metrics (‘Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  #VNF’ and ‘Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Delay’), compared to the GAT-based models using only  the positional encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  MEC network:  We experimented in the MEC network, where 5 nodes out of a total of  12 nodes are non-deployable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  We train the models with PPG, and com-  pare GAT-based models to ILP approaches, which is summarized in Table  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The reward is computed using the SFC model, which employs the same  architecture as the policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In the table, we can see that Positional  Encoding (‘GAT+P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.’) significantly decreases the SLA violation rate, the  number of instances as well as the delay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' For the node embedding-  based approach, ‘GAT+P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='+N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.’ has similar performance as ‘GAT+P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  It means node embedding does not have additional information compared  to positional embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Actually, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 6, MEC has a regu-  lar structure separating deployable and non-deployable nodes, which might  be a possible explanation of why ‘N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.’ does not additionally increase the  performance on top of ‘GAT+P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.’  For the GGNN-based approach, it does not optimize the reward effec-  tively, because links of the MEC network contain high latency3 and high  variance, which hinders updating node representation in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Espe-  cially, GGNN uses a weighted sum of its node representation with the edge  attribute when updating the node representation, which causes it to under-  estimate of the information from nodes connected with high latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' On the  other hand, the GAT-based approach employs the attention mechanism to  3The edge attribute is set as the reciprocal of latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  18  Models (SFC)  α  Reward  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' #VNF  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Delay  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' SLAV  ILP (GAT)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
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+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' (GAT+N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='350  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='95  60878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='099  Table 3: Comparison of different models in the MEC network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The positional encoding  helps the training process more effective, while the node embedding does not additionally  increase the performance on top of ‘GAT+P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='E.’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  compute the different importance of its neighbors and the edges so that it  could mitigate the underestimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Comparison of Various RL algorithms  Since we propose to use the PPG algorithms, we compare them to other  RL algorithms in the Internet2 network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In the experiment, we trained the  GAT-based models with positional encoding with the various RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  As a result, the PPG algorithms can obtain better rewards, compared to  DQN and PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' PPG decreases the SLA violation rate and the delay time  with a similar number of VNFs as presented in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Method  α  Reward  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' #VNF  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Delay  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' SLAV  DQN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='317  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='00  718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='112  DQN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='311  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='00  765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='174  DQN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='701  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='90  1793.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='285  PPO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='084  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='59  665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='093  PPO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='283  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='89  815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='122  PPO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='409  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='17  911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='135  PPG  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='053  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='30  644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='091  PPG  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='164  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='19  726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='101  PPG  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='353  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='51  868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='130  Table 4: Comparison of the various RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' PPG can optimize the reward more  effectively compared to DQN and PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Moreover, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  7, PPG can optimize the average loss of  the policy network more effectively than PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' We believe that the auxiliary  19  Figure 7: Comparison of policy loss (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2) in training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  The loss of PPG  decreases faster than PPO after around 40k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Initial Deployment  Reward  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' #VNF  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Delay  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' SLAV  ILP-perturbed  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='029  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='03  622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='087  Random  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='060  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='95  646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='090  Zero  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='060  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='94  646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='090  Table 5: Evaluations from different initial deployments in testing for the trained GAT  model with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  value head helps the policy network converge more efficiently, even though  parameter sharing might make the training slow at the beginning of the  training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Analysis with Random and Zero Deployments  To analyze further the performance of the trained models, we trained the  GAT-based model (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2) optimized with PPG as before, and tested the  trained model on different settings, where initial deployment is random or no  VNFs are deployed (random deployment or zero deployment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The experi-  ment results are presented in Table 5, where “Random” and “Zero” initial  deployments are compared to the ILP-perturbed initial deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' As the  result, the performance on both initial deployments shows the same level of  QoS and resource utilization as the ILP-perturbed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' It demonstrates that  our approach can work on any sub-optimal initial deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Furthermore,  our method can deploy VNFs without the ILP-based initial deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  20  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Policy Loss  PPG  agent: PPO  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='05  Valid/iter  20k  40k  60k  80kMethod  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Time (sec)  ILP  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='06  GAT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='431  Table 6: Average of execution times of 10 VNF-deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Figure 8: The VNF deployment generated from the model (Left), target SFC requests  (Middle), and generated paths (Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Given the target SFC request, VNF deployment  and path are generated by the proposed method and the SFC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Execution Time  To show how fast our method works, we measured the average execution  times to make decisions for VNF deployments and compared the proposed  GAT-based model to ILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The execution time includes the VNF deployment  actions by the agent as well as SFC path generation by the SFC model [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Each approach makes 10 deployments on a machine with 24 x 12-Core AMD  Ryzen 9 3900X CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Table 6 presents the average results of each approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Our method can process VNF deployment about 30 times faster than ILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Qualitative Evaluation  In this section, we discuss the quality of the results by our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' We  analyze the VNF deployment generated from the agent and plot t-SNE [29]  of the representation computed from the encoder of the policy networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Generated Deployment  We plot the VNF deployment generated from the GAT-based model  trained with PPG as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 8, which shows VNF deployment  generated from the SFC requests, and paths generated for the requests by  the SFC model [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The VNF deployment is generated with more than 20  requests, so we omitted the other requests in the figure for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  21  Req1  6 →  个N  个  6→1→1→11→8 (275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='0 / 735)  NW  Req2  ←m  N  个  个山  7  3→10→4→4→7 (765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='0 / 722)  23  86  209  26  N  Req3  ←  个N  H个山  个  7  5→1→4→4→7 (420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='0 /723)  129  63  Req4  2 →  个N  8个  2→ 11→ 11→ 11→ 8 (172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='0 / 729)  主E  W  36  90  WNPN  117  189  Req23  S个d个个  4→1→1→5 (225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='0 /700)  Req24  个个M个F个N个  0→1→1→1→4→4(179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='0 /705)The most VNF instances are deployed in the middle of the shortest paths  for the traffics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' For example, the first traffic (‘Req1’) has node 6 and node 8  as the ingress and egress, and is required to pass through NAT (‘N’), Firewall  (‘F’), and IDS (‘I’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In the network, the traffic goes through NAT at node 1,  Firewall at node 11, and IDS at node 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' This traffic takes 275 ms while SLA  for the SFC path is 735 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' That is, all the VNFs of ‘Req1’ are deployed on  the shortest path of its ingress and its egress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  In addition, the generated deployment needs to meet the QoS with the  optimized amount of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 8, VNF instances are de-  ployed on the intersection of the shortest paths for the traffics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' For example,  the VNF instances at node 1 process more than four requests, like NAT for  ‘Req1’, ‘Req2’, ‘Req23’ and ‘Req24’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' That is, our approach can deploy the  instances at the shared nodes of the paths for many requests, which can  reduce the number of VNF instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Node Representations by t-SNE  In the section, to analyze how the network information is represented by  the encoder, we plot the 2D representation of node embeddings by t-SNE [29]  as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 9, which shows the representations from the GGNN-based encoder  and our proposed GAT-based encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Each dot represents one node of  network, and the nodes are located in the Internet2 network as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  (a) GGNN  (b) GAT  Figure 9: Node representations by t-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' The GAT-based model gets more disentangled  representations, compared to the GGNN-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Even though the GGNN-based encoder has multiple clusters, the nodes  within each cluster are not distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' However, the GAT-based en-  coder’s representations are disentangled so that each cluster contains only  22  0  60  1  2  40  3  4  5  20  6  0  8  9  20  10  40  60  80  60  40  20  0  20  40  600  60  1  2  40  3  4  20  5  6  7  0  8  9  20  10  11  40  60  80  60  40  20  0  20  40  60  80one or two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Furthermore, the representations of neighbors in the net-  work are close to each other in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  For example, clusters for node  8 (‘brown’) and node 2 (‘green’) are close to each other, which reflects its  neighborhood with the same node type as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' This demon-  strates the representation by GAT-based model reflects the network topology  effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Conclusion  In this paper, we proposed enhanced models which can be adapted to  more general network settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' We proposed an improved GNN architecture  and a few techniques to obtain a better node representation for the VNF  deployment task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Furthermore, we optimized the model with PPG, a variant  policy gradient-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' In the experiment, we evaluated the pro-  posed method in various scenarios, achieving a better QoS with minimum  resource utilization compared to the previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Finally, we analyzed  the generated VNF deployment and compare the node representations of our  model to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content='  Acknowledgement  This research was supported by Basic Science Research Program through  the National Research Foundation of Korea funded by the Ministry of Edu-  cation (NRF-2022R1A2C1012633), and by Institute for Information & com-  munications Technology Promotion (IITP) grant funded by the Korea gov-  ernment(MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' 2018-0-00749, Development of virtual network man-  agement technology based on artificial intelligence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
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+page_content=' Jordan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
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+page_content='05477  [27] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Heo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
+page_content=' Lange, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
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+page_content=' Choi, Graph neural network based  service function chaining for automatic network control (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
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+page_content='05240  26  [28] Internet2 research network topology and traffic matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
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+page_content='edu/$\\sim$yzhang/research/  AbileneTM/  [29] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
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+page_content=' Hinton, Visualizing data using t-SNE, Journal of  Machine Learning Research 9 (86) (2008) 2579–2605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E_T4oBgHgl3EQfzxyC/content/2301.08325v1.pdf'}
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+arXiv:2301.03874v1  [math.OA]  10 Jan 2023
+INNERNESS OF DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC
+SPACES IS DETERMINED COMMUTATIVELY
+JINGHAO HUANG AND FEDOR SUKOCHEV
+Abstract. Let E = E(0, ∞) be a symmetric function space and E(M, τ) be a symmet-
+ric operator space associated with a semifinite von Neumann algebra with a faithful normal
+semifinite trace.
+Our main result identifies the class of spaces E for which every derivation
+δ : A → E(M, τ) is necessarily inner for each C∗-subalgebra A in the class of all semifinite von
+Neumann algebras M as those with the Levi property.
+1. Introduction
+Let A be a C∗-algebra and let J be an A-bimodule [80]. A derivation δ : A → J is a linear
+mapping satisfying δ(xy) = δ(x)y + xδ(y), x, y ∈ A. In particular, if a ∈ J, then δa(x) := xa − ax
+is a derivation. Such derivations implemented by elements in J are said to be inner. One of the
+classical problems in operator algebra theory is the question whether every derivation from A into
+J is automatically inner.
+The celebrated Kadison–Sakai theorem [50,76] states that derivations are always inner when A
+is a von Neumann algebra and the A-bimodule J coincides with the algebra A itself. Further, it was
+proved that every derivation from a von Neumann algebra into any of its ideal s is automatically
+inner [15,16]. However, when one considers more general C∗-algebras A and A-bimodules J, there
+are examples of non-inner derivations for some specific A and J (see e.g. [8, 11, 36–38, 72, 77]).
+We quote the following from the memoir by Johnson [47, Section 10.11]): “It would be desirable
+to identify those spaces X with H1(G, X∗) = 0 for all G ”. A similar statement appeared in [43]
+(see also [80, p.60]): “Here again it can be asked if such derivations are inner; that is, are they
+induced by an element of J as above? In fancier language, the question asks if the cohomology
+group H1(A, J) is trivial.” During the past decades, a number of important special cases have
+been studied (see e.g. [5,13,21,22,26,47,48,53,54,63,69,71,74]).
+Due to the rapid development of noncommutative analysis and motivated by questions due
+to Johnson et al., there are a number of papers concerning various versions of the following
+question [5,7,10,18,83]:
+Question 1. Assume that M is a von Neumann algebra equipped with a faithful normal semifinite
+trace τ. Let E(M, τ) be a symmetric space of τ-compact operators affiliated with M. How can
+one identity those E(M, τ) such that derivations from an arbitrary C∗-subalgebra A of M into
+E(M, τ) are necessarily inner?
+Experts in the operator theory are probably more familiar with symmetrically normed ideals
+in B(H), which are a special case of noncommutative symmetric spaces.
+Various versions of
+Question 1 for derivations with values in ideals of a von Neumann algebra were asked and discussed
+in [11,12,43,44,48,55,71,72].
+It was a long-standing open question whether every derivation a C∗-subalgebra of a semifinite
+von Neumann algebra M into M∗ must be inner (see e.g. [18, p.247]), which in our setting
+is equivalent to the special version of Question 1 with E = L1.
+This question was resolved
+completely by Bader, Gelander and Monod in 2012 [5] (see also [70] for a slightly different proof
+2010 Mathematics Subject Classification. 46L57, 47A56, 46L52, 46L10, 46E30.
+Key words and phrases. derivation; symmetric space; measurable operator.
+J. Huang was supported by the NNSF of China (No.12031004). F. Sukochev’s research was supported by the
+Australian Research Council (FL170100052).
+1
+
+2
+J. HUANG AND F. SUKOCHEV
+due to Pfitzner). The method used in [5] (or [70]) does not have any chance to deliver the full
+answer on Question 1. This fact was emphaiszed in [5, Section 3], where the following points were
+raised:
+a. In marked contrast to the classical fixed point theorems, there is no hope to
+find a fixed point inside a general bounded closed convex subset of L1 · · · the
+weak compactness · · · seems almost unavoidable · · ·
+b. · · · a canonical norm one projection V ∗∗ → V is not enough.
+c. It would be interesting to find a purely geometric version of the proposition
+· · ·
+The fact that the “fixed point” obtained in [5] is not inside a general bounded closed convex subset
+of L1 leads to extra difficulties in the general case.
+In this paper, we completely resolve Question 1 above, see Theorem 4.8 and Corollary 5.4 below.
+We show that the Levi property1 of a symmetric space is a sufficient and necessary condition for
+the Question 1 above having an affirmative answer for every semifinite von Neumann algebra
+M. Thus, the derivation theorem for preduals of von Neumann algebras in [5] (in the semifinite
+setting) becomes a trivial corollary of our Theorem 4.8.
+The new approach devised in this paper answers to points (a), (b) and (c) above raised
+in [5], which provides an alternative proof for the resolution of the question raised by Bunce
+and Paschke [18], without involving weak compactness of a subset in a L-embedded space as [5]
+and [70] did. This enables us to find a “fixed point” (implementing the derivation) from a not
+necessarily weakly compact closed convex subset of a noncommutative symmetric space which is
+1-complemented subspace of its bidual but not necessarily an L-embedded Banach space (i.e., a
+symmetric space having the Levi property [30, 31]2), see Theorem 4.8. On the other hand, the
+Levi property of the space E(M, τ) means that E(M, τ) coincides with its second K¨othe dual
+and this geometrical condition is the only one required in Theorem 4.8 thus delivering (at least
+spiritually) an answer to the question suggested in [5, Comment c] above. We believe that the
+method developed in this work is of interest in its own right.
+It is important to emphasize that the Fatou/Levi property was hatched in the theory of Banach
+lattices [1,17], and was even included into the original definition of Banach function spaces over
+σ-finite measure spaces (see [6, 65]). The property is somewhat analogous to the so-called “dual
+normal” property3. The importance of the Fatou/Levi property in the theory of Banach function
+spaces and symmetric operator spaces is hard to overestimate [29–31]. It seems appropriate to
+recall here that every derivation from a hyperfinite von Neumann algebra A into a dual normal
+A-bimodule is inner (see e.g. [80, Theorem 2.4.3], [23] and [49]). Recall also, that derivations from
+a nuclear C∗-algebra A into a dual Banach A-module are inner [24, 42]. However, Theorem 4.8
+below holds for arbitrary C∗-subalgebras A of M and for symmetric spaces which may not have
+a predual space.
+The reflexive gate type result (see e.g. [68, Corollary 3.2.3]) is relatively unknown but plays
+a significant role in our approach. In Section 3, we establish a noncommutative version of this
+result and lay the groundwork for its usage in the derivation problem. In Section 4, using the
+weak compactness criteria for noncommutative symmetric spaces obtained in [28,35], we show that
+the Ryll-Nardzewski fixed point theorem is applicable to any noncommutative strongly symmetric
+KB-space (or a Kantorovich–Banach space, see Section 2.2) whose bounded part does not coincides
+with C1(M, τ) = L1(M, τ) ∩ M. This is the key technical step in the proof of the main result
+of the present paper, Theorem 4.8. In Section 5, we demonstrate why the Fatou/Levi property of
+1A symmetric function space E(0, ∞) having the Levi property has an equivalent symmetric norm such that
+E(0, ∞) has the Fatou property, see Remark 4.7. The Soviet school on Banach lattices used the term monotone
+complete norm or property (B) [58, Chapter X.4], see also [1, p.89]. In the theory of operator algebras, a similar
+property is called ‘monotone closed’ [81, Chapter III, Definition 3.13].
+2Indeed, Theorem 4.8 holds for the case when the projection constant is not necessarily 1.
+3Let M be a von Neumann algebra. An M-bimodule X is said to be a dual normal X-bimodule if X is a dual
+space and the maps m �→ mx and m �→ xm are both ultraweak-weak∗ continuous from M into X for each fixed
+element x ∈ X [80, p.6].
+
+DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES
+3
+the space E(M, τ) is a necessary condition for an affirmative answer to Question 1 in the class of
+all semifinite von Neumann algebras.
+2. Preliminaries
+In this section, we recall some notions of the theory of noncommutative integration.
+In what follows, H is a Hilbert space and B(H) is the ∗-algebra of all bounded linear operators
+on H equipped with the uniform norm ∥·∥∞, and 1 is the identity operator on H. Let M be a
+von Neumann algebra on H. We denote by P(M) the collection of all projections in M, by M′
+the commutant of M and by Z(M) the center of M. For details on von Neumann algebra theory,
+the reader is referred to e.g. [51,52] or [81]. General facts concerning measurable operators may
+be found in [67], [78] (see also [82, Chapter IX] and the forthcoming book [34]). For convenience
+of the reader, some of the basic definitions are recalled.
+2.1. τ-measurable operators and generalized singular value functions. A closed, densely
+defined operator x : D (x) → H with the domain D (x) is said to be affiliated with M if yx ⊆ xy for
+all y ∈ M′, where M′ is the commutant of M. A closed, densely defined operator x : D (x) → H
+affiliated with M is said to be measurable if there exists a sequence {pn}∞
+n=1 ⊂ P (M), such that
+pn ↑ 1, pn(H) ⊆ D (x) and 1−pn is a finite projection (with respect to M) for all n. The collection
+of all measurable operators with respect to M is denoted by S (M), which is a unital ∗-algebra
+with respect to strong sums and products (denoted simply by x + y and xy for all x, y ∈ S (M)).
+Let x be a self-adjoint operator affiliated with M. We denote its spectral measure by {ex}. It is
+well known that if x is an operator affiliated with M with the polar decomposition x = u|x|, then
+u ∈ M and e ∈ M for all projections e ∈ {e|x|}. Moreover, x ∈ S(M) if and only if e|x|(λ, ∞) is a
+finite projection for some λ > 0. It follows immediately that in the case when M is a von Neumann
+algebra of type III or a type I factor, we have S(M) = M. For type II von Neumann algebras,
+this is no longer true. From now on, let M be a semifinite von Neumann algebra equipped with
+a faithful normal semifinite trace τ.
+An operator x ∈ S (M) is called τ-measurable if there exists a sequence {pn}∞
+n=1 in P (M)
+such that pn ↑ 1, pn(H) ⊆ D (x) and τ(1 − pn) < ∞ for all n. The collection S (M, τ) of all
+τ-measurable operators is a unital ∗-subalgebra of S (M). It is well known that a linear operator
+x belongs to S (M, τ) if and only if x ∈ S(M) and there exists λ > 0 such that τ(e|x|(λ, ∞)) < ∞.
+Alternatively, an unbounded operator x affiliated with M is τ-measurable (see [39]) if and only if
+τ
+�
+e|x|�
+n, ∞
+��
+→ 0,
+n → ∞.
+For convenience of the reader, we also recall the definition of the measure topology tτ on the
+algebra S(M, τ). For every ε, δ > 0, we define the set
+V (ε, δ) = {x ∈ S(M, τ) : ∃p ∈ P (M) such that ∥x(1 − p)∥∞ ≤ ε, τ(p) ≤ δ} .
+The topology generated by the sets V (ε, δ), ε, δ > 0, is called the measure topology tτ on S(M, τ)
+[34, 39, 67]. It is well known that the algebra S(M, τ) equipped with the measure topology is a
+complete metrizable topological algebra. A sequence {xn}∞
+n=1 ⊂ S(M, τ) converges to zero with
+respect to measure topology tτ if and only if τ
+�
+e|xn|(ε, ∞)
+�
+→ 0 as n → ∞ for all ε > 0 [34].
+Another important vector topology on S(M, τ) is the local measure topology. For convenience
+we denote by Pf(M) the collection of all τ-finite projections in M, that is, the set of all e ∈ P(M)
+satisfying τ(e) < ∞. A neighbourhood base for this topology is given by the sets V (ε, δ; e), ε, δ > 0,
+e ∈ Pf(M), where
+V (ε, δ; e) = {x ∈ S(M, τ) : exe ∈ V (ε, δ)} .
+Obviously, local measure topology is weaker than measure topology [32]. We note here, that the
+local measure topology used in the present paper differs from the local measure topology defined
+in e.g. [9,10].
+Definition 2.1. Let M be a von Neumann algebra equipped with a faithful normal semi-finite
+trace τ and let x ∈ S(M, τ). The generalized singular value function µ(x) : t �→ µ(t; x) of the
+
+4
+J. HUANG AND F. SUKOCHEV
+operator x is defined by setting
+µ(s; x) = inf{∥xp∥∞ : p ∈ P(M) with τ(1 − p) ≤ s}, ∀s ∈ (0, ∞).
+An equivalent definition in terms of the distribution function of the operator |x| is the following.
+for every operator x ∈ S(M, τ), setting d|x|(t) = τ(e|x|(t, ∞)),
+t > 0, we have (see e.g. [39])
+µ(t; x) = inf{s ≥ 0 : d|x|(s) ≤ t}.
+(1)
+Note that µ(x) is a function defined on (0, ∞) even if the trace τ is finite.
+In particular,
+µ(s; x) = 0 when s ≥ τ(1).
+Consider the algebra M = L∞(0, ∞) of all Lebesgue measurable essentially bounded functions
+on (0, ∞). The algebra M can be seen as an abelian von Neumann algebra acting via multiplication
+on the Hilbert space H = L2(0, ∞), with the trace given by integration with respect to Lebesgue
+measure m. It is easy to see that the algebra of all τ-measurable operators affiliated with M can be
+identified with the subalgebra S(0, ∞) of the algebra of Lebesgue measurable functions L0(0, ∞)
+which consists of all functions x such that m({|x| > s}) is finite for some s > 0.
+It should
+also be pointed out that the generalized singular value function µ(x) is precisely the decreasing
+rearrangement µ(x) of the function |x| (see e.g. [60]) defined by
+µ(t; x) = inf{s ≥ 0 : m({|x| ≥ s}) ≤ t}.
+If M = B(H) (respectively, ℓ∞) and τ is the standard trace Tr (respectively, the counting
+measure on N), then it is not difficult to see that S(M) = S(M, τ) = M. In this case, for
+x ∈ S(M, τ) we have
+µ(n; x) = µ(t; x),
+t ∈ [n, n + 1),
+n ≥ 0.
+The sequence {µ(n; x)}n≥0 is just the sequence of singular values of the operator x.
+If x, y ∈ S(M, τ), then x is said to be submajorized by y, denoted by x ≺≺ y, if
+� t
+0
+µ(s; x)ds ≤
+� t
+0
+µ(s; y)ds for all t ≥ 0.
+In particular, for x, y ∈ S(0, ∞), x ≺≺ y if and only if
+� t
+0 µ(s; x)ds ≤
+� t
+0 µ(s; y)ds, t ≥ 0.
+2.2. Symmetric spaces.
+Definition 2.2. A linear subspace E of S(M, τ) equipped with a complete norm ∥·∥E, is called a
+symmetric space (of τ-measurable operators) if x ∈ S(M, τ), y ∈ E and µ(x) ≤ µ(y) imply that
+x ∈ E and ∥x∥E ≤ ∥y∥E.
+It is well-known that any symmetric space E is a normed M-bimodule, that is, axb ∈ E for any
+x ∈ E, a, b ∈ M and ∥axb∥E ≤ ∥a∥∞ ∥b∥∞ ∥x∥E [32,34]. A symmetric space E(M, τ) ⊂ S(M, τ)
+is called strongly symmetric if its norm ∥·∥E has the additional property that ∥x∥E ≤ ∥y∥E
+whenever x, y ∈ E(M, τ) satisfy x ≺≺ y. In addition, if x ∈ S(M, τ), y ∈ E(M, τ) and x ≺≺ y
+imply that x ∈ E(M, τ) and ∥x∥E ≤ ∥y∥E, then E(M, τ) is called fully symmetric space (of
+τ-measurable operators).
+If E ⊂ S(M, τ) is a symmetric space, then the norm ∥·∥E is called order continuous if ∥xα∥E →
+0 whenever {xα} is a downwards directed net in E+ satisfying xα ↓ 0. A symmetric space E(M, τ)
+is said to have the Fatou property if for every upwards directed net {xβ} in E(M, τ)+, satisfying
+supβ ∥xβ∥E < ∞, there exists an element x ∈ E(M, τ)+ such that xβ ↑ x in E(M, τ) and
+∥x∥E = supβ ∥xβ∥E. Examples such as Schatten–von Neumann operator ideals, Lorentz operator
+ideals, etc. all have symmetric norms which have the Fatou property [34,60,61]. If E has the Fatou
+property and order continuous norm, then it is said to be a KB-space (or Kantorovich–Banach
+space) [32].
+If E(M, τ) is a symmetric space, then the carrier projection cE ∈ P(M) is defined by setting
+cE =
+�
+{p : p ∈ P(M), p ∈ E(M, τ)}.
+We remark that, replacing the von Neumann algebra M by the reduced von Neumann algebra
+McE (note that E(M, τ) = cEE(M, τ)cE, see e.g. [32, Corollary 6]), it is often assumed that the
+carrier projection of E(M, τ) is equal to 1.
+
+DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES
+5
+If E(M, τ) is a symmetric space, then the K¨othe dual E(M, τ)× of E(M, τ) is defined by
+E(M, τ)× = {x ∈ S(M, τ) :
+sup
+∥y∥E≤1,y∈E
+τ(|xy|) < ∞},
+and for every x ∈ E(M, τ)×, we set ∥x∥E× = sup {τ(|yx|) : y ∈ E(M, τ), ∥y∥E ≤ 1} (see e.g. [32,
+Section 5.2], see also [28,62]). It is well-known that ∥·∥E× is a norm on E(M, τ)× if and only if
+the carrier projection cE of E(M, τ) is equal to 1. In this case, for a strongly symmetric space
+E(M, τ), the following statements are equivalent [31,32].
+• E(M, τ) has the Fatou property.
+• E(M, τ)×× = E(M, τ) and ∥x∥E = ∥x∥E×× for all X ∈ E(M, τ).
+• The norm closed unit ball BE of E(M, τ) is closed in S(M, τ) with respect to the local
+measure topology .
+If E(M, τ) is a strongly symmetric space with cE = 1 (or a symmetric space affiliated with a
+semifinite von Neumann algebra which is either atomless or atomic with all minimal projections
+having equal trace), then [32, Lemma 30] (see also [34, Theorem IV.5.7] and [60])
+∥x∥E = ∥x∥E×× , x ∈ (L1 ∩ L∞)(M, τ),
+(2)
+and
+L1 ∩ L∞(M, τ) ⊂ E(M, τ) ⊂ (L1 + L∞)(M, τ).
+(3)
+A wide class of symmetric operator spaces associated with the von Neumann algebra M can
+be constructed from concrete symmetric function spaces studied extensively in e.g. [60].
+Let
+(E(0, ∞), ∥·∥E(0,∞)) be a symmetric function space on the semi-axis (0, ∞). The pair
+E(M, τ) = {x ∈ S(M, τ) : µ(x) ∈ E(0, ∞)},
+∥x∥E(M,τ) := ∥µ(x)∥E(0,∞)
+is a symmetric operator space affiliated with M with cE = 1 [56] (see also [62]). For convenience,
+we denote ∥·∥E(M,τ) by ∥·∥E.
+Let E(0, ∞) be a symmetric function space. We define a symmetric space [60, Chapter I.3]
+(L1 + E)(0, ∞) := {f ∈ S(0, ∞) : ∥f∥L1+E :=
+inf
+x=u+v,u∈L1(0,∞),v∈E(0,∞) {∥u∥1 + ∥v∥E} < ∞}.
+2.3. The ideal of τ-compact operators. For a self-adjoint operator x ∈ S(M, τ), we denote
+by s(x) its support. The two-sided ideal F(M, τ) in M consisting of all elements of τ-finite range
+is defined by setting
+F(M, τ) = {x ∈ M : τ(s(|x|)) < ∞}.
+The C∗-algebra C0(M, τ) of all τ-compact bounded operators can be described as the closure
+in the norm ∥ · ∥∞ of the linear span of all τ-finite projections [62, Definition 2.6.8]. Equivalently,
+C0(M, τ) the set of all elements x ∈ M such that τ(E|x|(λ, ∞)) < ∞ for every λ > 0 (see
+e.g. [34, Chapter II, Section 4]).
+The space C0(M, τ) is associated to the ideal of essentially
+bounded functions vanishing at infinity (see [62, Lemma 2.6.9]), that is,
+C0(M, τ) =
+�
+a ∈ S(M, τ) : µ(a) ∈ L∞(0, ∞), µ(∞; a) := lim
+t→∞ µ(t; a) = 0
+�
+.
+In particular, if τ is finite, then M = C0(M, τ) (see e.g. [62, Page 64]). The space S0(M, τ) of
+τ-compact operators is the space associated to the algebra of functions from S(0, ∞) vanishing at
+infinity, that is,
+S0(M, τ) = {x ∈ S(M, τ) : µ(∞; x) = 0}.
+This is a two-sided ideal in S(M, τ) and, clearly, C0(M, τ) = S0(M, τ) ∩ M.
+We denote
+C0(0, ∞) := {f ∈ L∞(0, ∞) : µ(∞; f) = 0},
+and
+(L1 + C0)(0, ∞) := {f ∈ L1 + L∞(0, ∞) : µ(∞; f) = 0}.
+
+6
+J. HUANG AND F. SUKOCHEV
+3. A noncommutative reflexive gate type theorem
+Recall the reflexive gate type result for symmetric function spaces E(0, 1) [68, Corollary 3.2.3],
+i.e., if E(0, 1) ̸= L1(0, 1) and E(0, 1) ̸= L∞(0, 1), then there exist two reflexive symmetric space
+F1(0, 1) and F2(0, 1) such that F2(0, 1) ⊂ E(0, 1) ⊂ F1(0, 1). For a symmetric sequence space
+ℓE such that ℓE ⫋ c0, by the well-known construction of Davis–Figiel–Johnson–Pe�lczy´nski [27]
+(see [4, Theorem 5.37], [25, Chapter VI, Lemma 5.3] or [57, p. 255] for detailed expositions),
+there exists a reflexive sequence space ℓF ⫋ ℓE, see e.g. the proof of [57, Proposition 3.4]. In the
+following lemma, we show that this reflexive sequence is actually symmetric, which is a discrete
+analogue of the reflexive gate result [68, Corollary 3.2.3] for symmetric function spaces on the unit
+interval.
+Lemma 3.1. Let ℓE be a symmetric sequence space which does not coincide with ℓ1 (up to equiv-
+alent norms). Then, there exists a reflexive symmetric sequence space ℓF such that ℓF ⊂ ℓE.
+Proof. The proof relies on the construction of Davis–Figiel–Johnson–Pe�lczy´nski [27, p.313].
+Let {en}n≥1 be the standard symmetric basis of ℓE. Since ℓE ̸= ℓ1, it follows that {en}n≥1
+is weakly null. By the Krein–Smulian theorem [4, Theorem 3.42], the convex circled4 hull W of
+{en}n≥1 is weakly compact.
+Let Un = 2nW +2−nU, where U stands for the unit ball of ℓE, and ∥·∥n denotes the Minkowski
+functional of Un, i.e.,
+∥x∥n := inf {λ > 0 : x ∈ λUn} .
+Put ℓF :=
+�
+x ∈ ℓE : ∥x∥F :=
+��∞
+n=1 ∥x∥2
+n
+�1/2
+< ∞
+�
+. By [27, Lemma 1] (see also [4, Theorem
+5.37]), ℓF is reflexive.
+The ideal property of ℓF was asserted in [57, Proposition 3.4]. For the sake of completeness, we
+include a short proof below. Let y = (y(k))k≥1 ∈ ℓF and x = (x(k))k≥1 ∈ ℓ∞ such that |x| ≤ |y|.
+We have y ∈ (∥y∥n + ε)Un for any ε > 0, i.e.,
+y =
+N
+�
+k=1
+skek + yn,ε,
+where �N
+k=1 |sk| ≤ 2n (∥y∥n + ε) , 1 ≤ N < ∞ and (yn,ε(k))k≥1 ∈ ∥y∥n+ε
+2n
+U. Note that
+x =
+�
+yk̸=0
+sk
+xk
+yk
+ek +
+�xk
+yk
+yn,ε(k)
+�
+k≥1
+,
+�
+yk̸=0
+���sk
+xk
+yk
+��� ≤ �
+yk̸=0 |sk| ≤ 2n (∥y∥n + ε) and
+�
+xk
+yk yn,ε(k)
+�
+k≥1 ∈ ∥y∥n+ε
+2n
+U. Therefore, ∥x∥n ≤
+∥y∥n for every n ≥ 1, and x ∈ ℓF with with ∥x∥F ≤ ∥y∥F .
+To show that ℓF is symmetric, one only need to observe that for any permutation π and n ≥ 1,
+we have π(W) = W and π(U) = U. Therefore,
+���(y(k))k≥1
+���
+n =
+���(y (π(k)))k≥1
+���
+n for any y ∈ ℓF
+and any permutation π.
+□
+Corollary 3.2. Let ℓE be a symmetric sequence space which does not coincide with c0 and ℓ∞
+(up to equivalent norms). Then, there exists a reflexive symmetric sequence space ℓF such that
+ℓF ⊃ ℓE.
+4A nonempty subset A of a vector space is said to circled (or balanced) whenever x ∈ A and 0 ≤ λ ≤ 1 imply
+λx ∈ A) [4, p.134]. A convex circled hull of A is
+�
+n
+�
+i=1
+λixi : xi ∈ A for each i and
+n
+�
+i=1
+|λi| ≤ 1
+�
+.
+
+DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES
+7
+Proof. Since ℓE ̸= c0, ℓ∞, it follows that ℓ×
+E ̸= ℓ1 (see e.g. [46, Proposition 11] and [59]). By
+Lemma 3.1, we obtain that there exists a reflexive symmetric sequence space ℓG ⊂ ℓ×
+E. Therefore,
+letting ℓF := ℓ×
+G, which is a reflexive symmetric sequence space [32, Section 8.3] (see also [34,60,
+66]), we have
+ℓF = ℓ×
+G ⊃ ℓ××
+E
+⊃ ℓE,
+which completes the proof.
+□
+Now, we prove the main theorem of this section, which is an infinite measure version of [68,
+Corollary 3.2.3].
+Theorem 3.3. Let E(0, ∞) be a symmetric function space such that
+E(0, ∞) ∩ L∞(0, ∞) ⫋ C0(0, ∞).
+Then, there exists a symmetric KB-function space such that F(0, ∞) containing E(0, ∞). If, in
+addition, E(0, 1) :=
+�
+f ∈ L1(0, 1) : g(x) :=
+�
+f(x),
+if x ∈ (0, 1);
+0,
+otherwise
+∈ E(0, ∞), ∥f∥E(0,1) = ∥g∥E(0,∞)
+�
+satisfies that
+E(0, 1) ̸= L1(0, 1),
+then F(0, ∞) can be chosen as a reflexive symmetric space.
+Proof. Let ℓE be the symmetric sequence space generated by E(0, ∞) by setting
+ℓE :=
+
+
+(αn)n≥1 :
+�
+n≥1
+αnχ(n−1,n] ∈ E(0, ∞)
+
+
+ , ∥(αn)n≥1∥ℓE :=
+������
+�
+n≥1
+αnχ(n−1,n]
+������
+E(0,∞)
+.
+(4)
+By condition E(0, ∞) ∩ L∞(0, ∞) ⫋ C0(0, ∞), we obtain that ℓE ⫋ c0.
+Indeed, E(0, ∞) ∩
+L∞(0, ∞) ⫋ C0(0, ∞) implies that there exists z = µ(z) ∈ C0(0, ∞) such that µ(z) /∈ E(0, ∞) ∩
+L∞(0, ∞), and therefore, by (4), we obtain that (µ(n; z))n≥1 ∈ c0 but (µ(n; z))n≥1 /∈ ℓE.
+By Corollary 3.2, there exists a reflexive symmetric sequence space ℓG ⊃ ℓE. Observe that ℓG
+is fully symmetric [34,60,61].
+Now, we define a symmetric function space F1(0, ∞) by setting
+F1(0, ∞) :=
+�
+f ∈ (L1 + L∞)(0, ∞) :
+�� n
+n−1
+µ(t; f)dt
+�
+n≥1
+∈ ℓG
+�
+(5)
+equipped with the norm [62, Theorem 3.6.6]
+∥f∥F1(0,∞) :=
+�����
+�� n
+n−1
+µ(t; f)dt
+�
+n≥1
+�����
+ℓG
+, f ∈ F1(0, ∞),
+which also has the Fatou property and order continuous norm, i.e., F1(0, ∞) is a symmetric KB-
+space. Indeed, let {xβ} be an upwards directed net in F1(0, ∞)+ satisfying supβ ∥xβ∥F1(0,∞) < ∞.
+By the definition of F1(0, ∞),
+��� n
+n−1 µ(t; xβ)dt
+�
+n≥1
+�
+β
+is an upwards directed net in ℓ+
+G with
+supβ
+����
+�� n
+n−1 µ(t; xβ)dt
+�
+n≥1
+����
+ℓG
+< ∞. Let x ∈ S(0, ∞) be such that xβ ↑ x. Then, we have µ(xβ) ↑
+x and
+� n
+n−1 µ(t; xβ)dt ↑
+� n
+n−1 µ(t; x)dt for each n ≥ 1. By the Fatou property of ℓG, we obtain
+that
+�� n
+n−1 µ(t; x)dt
+�
+n≥1 ∈ ℓG with
+����
+�� n
+n−1 µ(t; x)dt
+�
+n≥1
+����
+ℓG
+= supβ
+����
+�� n
+n−1 µ(t; xβ)dt
+�
+n≥1
+����
+ℓG
+,
+i.e., x ∈ F1(0, ∞) with ∥x∥F1(0,∞) = supβ ∥xβ∥F1(0,∞) = supβ ∥xβ∥F1(0,∞). Therefore, F1(0, ∞)
+has the Fatou property. The order continuity of ∥·∥F1(0,∞) can be proved by a similar argument.
+Note that for any f = µ(f) ∈ E(0, ∞), we have
+�
+n≥1
+µ(n; f)χ(n−1,n] ≤ µ(f) and
+�
+n≥1
+µ(n; f)χ(n−1,n] ∈ E(0, ∞).
+
+8
+J. HUANG AND F. SUKOCHEV
+By (4), we have (µ(n; f))n≥1 ∈ ℓE. Therefore,
+�� n
+n−1
+µ(t; f)dt
+�
+n≥2
+=
+�� n+1
+n
+µ(t; f)dt
+�
+n≥1
+≤ (µ(n; f))n≥1 ∈ ℓE ⊂ ℓG.
+Therefore,
+�� n
+n−1 µ(t; f)dt
+�
+n≥1 ∈ ℓG, i.e.,
+f
+(5)
+∈ F1(0, ∞).
+That is, E(0, ∞) ⊂ F1(0, ∞). The proof for the first assertion is complete by defining F(0, ∞) :=
+F1(0, ∞).
+Assume that E(0, 1) ̸= L1(0, 1).
+If E(0, 1) = L∞(0, 1), then, defining G(0, 1) := L2(0, 1)
+(which is a reflexive symmetric function space), we have G(0, 1) ⊃ E(0, 1). If E(0, 1) ̸= L∞(0, 1),
+then, by [68, Corollary 3.2.3], there exists a reflexive symmetric function space G(0, 1) such that
+E(0, 1) ⊂ G(0, 1). We define
+G(0, ∞) :=
+�
+f ∈ (L1 + L∞)(0, ∞) : µ(f)χ(0,1) ∈ G(0, 1)
+�
+equipped with the norm5
+∥f∥G :=
+inf
+f=u+v, u∈G(0,∞),v∈L∞(0,∞)
+���µ(u)χ(0,1)
+��
+G(0,1) + ∥v∥∞
+�
+, f ∈ G(0, ∞).
+By the definition of a K¨othe dual (see e.g. [60, (4.31)]), it is readily verified that the K¨othe dual
+G(0, ∞)× of G(0, ∞) is given by
+�
+f ∈ (L1 + L∞)(0, ∞) :
+��µ(f)χ(0,1)
+��
+G(0,1)× < ∞
+�
+∩ L1(0, ∞)
+=
+�
+f ∈ (L1 + L∞)(0, ∞) : µ(f)χ(0,1) ∈ G(0, 1)× and µ(f)χ[1,∞) ∈ L1(0, ∞)
+�
+.
+equipped with the norm
+∥f∥G(0,∞)× := max
+���µ(f)χ(0,1)
+��
+G(0,1)× , ∥f∥1
+�
+, f ∈ G(0, ∞)×.
+For any decreasing net S0(0, ∞) ⊃ E(0, ∞) ∋ fi ↓ 0, we have µ(fi) ↓i 0 (see e.g [32, Proposition
+2(iv)]) and therefore, there exists a constant c(G) depending on G(0, 1) only [6, Chapter I, Theorem
+1.8] such that
+∥fi∥G(0,∞)× ≤ c(G)
+���µ(fi)χ(0,1)
+��
+G(0,1)× + ∥µ(fi)∥L1(0,∞)
+�
+↓i 0,
+(6)
+which shows that G(0, ∞)× has an order continuous norm.
+Now, we define
+F(0, ∞) := G(0, ∞) ∩ F1(0, ∞),
+equipped with ∥·∥F := max
+�
+∥·∥G , ∥·∥F1
+�
+.
+Since both G(0, ∞) and F1(0, ∞) have the Fatou
+property, it follows that F(0, ∞) has the Fatou property. The same reasoning as in (6) shows that
+∥·∥F is order continuous. Observe that [79, Section 2.3] (see also [64])
+F(0, ∞)× = G(0, ∞)× + F1(0, ∞)×
+is equipped with the norm
+∥f∥F (0,∞)× :=
+inf
+f=u+v, u∈G(0,∞)×,v∈F1(0,∞)×
+�
+∥u∥G× + ∥v∥F ×
+1
+�
+.
+We claim that ∥·∥F (0,∞)× is order continuous. Indeed, for any decreasing net F1(0, ∞) ∋ fi ↓ 0,
+we have µ(fi) ↓i 0. In particular, we have µ(n; fi) ↓i 0 for each n ≥ 1. By the triangle inequality,
+we have
+∥fi∥F (0,∞)× ≤
+��µ(fi)χ(0,1)
+��
+G(0,∞)× +
+��µ(fi)χ[1,∞)
+��
+F1(0,∞)× .
+5Let f, g ∈ G(0, ∞). By [6, Corollary 7.6], there exists A ⊂ (0, ∞) such that µ((f + g)χA) = µ(f + g)χ(0,1).
+Therefore, we have ��µ(f + g)χ(0,1)
+��
+G(0,1) = ∥(f + g)χA∥G(0,1) ≤ ∥fχA∥G(0,1) +∥gχA∥G(0,1) ≤ ��fχ(0,1)
+��
+G(0,1) +
+��gχ(0,1)
+��
+G(0,1). Hence, ∥·∥G(0,1) generates a symmetric norm on G(0, ∞). Since ∥·∥∞ is a symmetric norm, it
+follows ∥·∥G is a complete symmetric norm on G(0, ∞), see e.g. [60, Chapter I.3].
+
+DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES
+9
+By (6), it suffices to prove that
+��µ(fi)χ[1,∞)
+��
+F1(0,∞)× →i 0. By the definition of ∥·∥F1(0,∞), we
+have
+��µ(fi)χ[1,∞)
+��
+F1(0,∞)×
+[32, Prop. 26]
+=
+sup
+∥x∥F1(0,∞)=1
+� ∞
+0
+µ(t; µ(fi)χ[1,∞))µ(t; x)dt
+=
+sup
+∥x∥F1(0,∞)=1
+�
+n≥1
+� n
+n−1
+µ(t; µ(fi)χ[1,∞))µ(t; x)dt
+and
+�
+n≥1
+µ(n + 1; fi)
+� n
+n−1
+µ(t; x)dt =
+�
+n≥1
+µ(n; µ(fi)χ[1,∞))
+� n
+n−1
+µ(t; x)dt
+≤
+�
+n≥1
+� n
+n−1
+µ(t; µ(fi)χ[1,∞))µ(t; x)dt
+≤
+�
+n≥1
+µ(n − 1; µ(fi)χ[1,∞))
+� n
+n−1
+µ(t; x)dt =
+�
+n≥1
+µ(n; fi)
+� n
+n−1
+µ(t; x)dt.
+By the definition of F1(0, ∞), we have (µ(n; fi))n≥1 ∈ ℓ×
+G and
+��µ(fi)χ[1,∞)
+��
+F1(0,∞)× ≤ ∥{µ(n; fi)}n≥1∥ℓ×
+G ↓i 0.
+By [61, Theorem 1.c.5] (see also [66, Theorem 2.4.14] or [34,60]), F(0, ∞) is reflexive.
+□
+The following lemma is folklore, it provides a characterization for symmetric spaces whose ‘tail’
+is a proper subspace of C0(M, τ). For the sake of completeness, we include the full proof.
+Lemma 3.4. Assume that
+(1) E(M, τ) is a strongly symmetric space affiliated with a von Neumann algebra equipped
+with a semifinite infinite faithful normal trace τ, with cE = 1;
+(2) M is an atomless von Neumann algebra equipped with a semifinite infinite faithful normal
+trace τ or M is atomic equipped with a semifinite infinite faithful normal trace τ such that
+all minimal projections having equal trace.
+Then, E(M, τ)×× ⊂ S0(M, τ) if and only if E(M, τ) ∩ M ⫋ C0(M, τ).
+Proof. (⇒). Assume, contrapositively, that E(M, τ) ∩ M is not a proper subspace of C0(M, τ).
+Note that, if E(M, τ) contains some element x such that µ(∞; x) > 0, then E(M, τ) ⊃ M ⊃
+C0(M, τ). If for every x ∈ E(M, τ), we have µ(∞; x) = 0, then E(M, τ) ∩ M ⊂ C0(M, τ).
+Therefore, E(M, τ) ∩ M = C0(M, τ) or E(M, τ) ∩ M = M, i.e., E(M, τ) ⊃ E(M, τ) ∩ M ⊃
+C0(M, τ).
+By [29, Definition 5.1] (see also [34, Chapter IV, Proposition 3.12]), we have
+E(M, τ)× = {y ∈ S(M, τ) : xy ∈ L1(M, τ) for all x ∈ E(M, τ)}
+⊂ {y ∈ S(M, τ) : xy ∈ L1(M, τ) for all x ∈ C0(M, τ)}
+= C0(M, τ)× see e.g. [79, Lemma 8]
+=
+L1(M, τ),
+and
+E(M, τ)×× =
+�
+y ∈ S(M, τ) : xy ∈ L1(M, τ) for all x ∈ E(M, τ)×�
+⊃ {y ∈ S(M, τ) : xy ∈ L1(M, τ) for all x ∈ L1(M, τ)} = M,
+which is a contradiction with the assumption that E(M, τ)×× ⊂ S0(M, τ).
+(⇐). Assume by contradiction that E(M, τ)×× ⊃ M.
+By [32, Proposition 28] (see also [29,31]), there exists a fully symmetric function space G(0, ∞)
+having the Fatou property such that
+E(M, τ)×× = G(M, τ)
+
+10
+J. HUANG AND F. SUKOCHEV
+with ∥y∥E(M,τ)×× = ∥µ(y)∥G(0,∞), y ∈ E(M, τ)××.
+Since G(M, τ) ⊃ M, it follows that
+G(0, ∞) ⊃ L∞(0, ∞) and therefore, we have ∥·∥∞ ≥ c(G) ∥·∥G for some constant c(G) depending
+on G(0, ∞) only [6, Chapter I, Theorem 1.8]. Therefore, we have
+E(M, τ) ∩ M
+(3)
+⊃ (L1 ∩ L∞)(M, τ)
+∥·∥E ∩ M
+(2)
+= (L1 ∩ L∞)(M, τ)
+∥·∥G ∩ M
+⊃ (L1 ∩ L∞)(M, τ)
+∥·∥∞ = C0(M, τ),
+which is a contradiction with the assumption that E(M, τ) ∩ M ⫋ C0(M, τ).
+□
+Below, we establish a noncommutative version of Theorem 3.3.
+Observe that the result of
+Proposition 3.5 below holds for both finite and infinite traces.
+Proposition 3.5. Let M be a von Neumann algebra equipped with a semifinite faithful normal
+trace τ. Let E(M, τ) be a strongly symmetric space such that E(M, τ)×× ⊂ S0(M, τ)6. Then,
+there exists a symmetric KB-function space F(0, ∞) such that
+E(M, τ) ⊂ F(M, τ).
+Proof. Without loss of generality, we may assume that cE = 1. If τ(1) < ∞, then the assertion is
+trivial as E(M, τ)
+(3)
+⊂ L1(M, τ) and we can take F(0, ∞) := L1(0, ∞).
+Now, we assume that τ(1) = ∞.
+By [32, Proposition 28] (see also [29]), there exists a fully symmetric function space G(0, ∞)
+having the Fatou property such that
+E(M, τ)× = G(M, τ)
+with ∥y∥E×(M,τ) = ∥µ(y)∥G(0,∞), y ∈ E(M, τ)×.
+Let F1(0, ∞) := G(0, ∞)×. In particular, F1(0, ∞) has the Fatou property [32, Theorem 27].
+Observe that F1(M, τ) = E(M, τ)×× [32, Theorem 53]. We have [32, Section 5.3]
+E(M, τ) ⊂ E(M, τ)×× = F1(M, τ) ⊂ S0(M, τ).
+In particular, F1(0, ∞) ⊂ S0(0, ∞). Since both F1(0, ∞) and L∞(0, ∞) have the Fatou property,
+it follows that F1(0, ∞) ∩ L∞(0, ∞) has the Fatou property and therefore, F1(0, ∞) ∩ L∞(0, ∞) ̸=
+C0(0, ∞).
+By Theorem 3.3, there exists a symmetric KB-function space F(0, ∞) such that F1(0, ∞) ⊂
+F(0, ∞), which completes the proof.
+□
+4. Main results
+The starting point of this section is the following important result due to Akemann, Dodds and
+Gamlen [3, Theorem 4.2] (see also [2, Corollary II.9 and Theorem IV.3] and [75]). It should be
+viewed as a noncommutative version of Grothendieck’s theorem [41], which states that an arbitrary
+bounded operator from C(K) into a weakly sequentially complete Banach space is necessarily
+weakly compact.
+Theorem 4.1. A bounded linear map from a C∗-algebra into a weakly sequentially complete
+Banach space is weakly compact.
+Recall that a noncommutative strongly symmetric space is a KB-space if and only if it is weakly
+sequentially complete [33, Theorem 6.5], see also [28,31,32,34]. We have the following consequence
+of Theorem 4.1.
+Corollary 4.2. Let M be a von Neumann algebra equipped with a semifinite faithful normal trace
+τ, and let E(M, τ) be a strongly symmetric KB-space affiliated with M. Then every bounded map
+from a C∗-algebra into E(M, τ) is weakly compact.
+Proposition 4.3 below is a consequence of Corollary 4.2 and [28, Corollary 2.9] (see also [35,
+Theorem 5.4]). For a Banach space X, we denote by BX the unit ball of X.
+6Observe that if τ(1) < ∞, then S0(M, τ) = S(M, τ).
+
+DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES
+11
+Proposition 4.3. Let M be a von Neumann algebra equipped with a semifinite faithful normal
+trace τ. Assume that E(M, τ) is a strongly symmetric KB-space such that E(M, τ)× ⊂ S0(M, τ).
+Let A be a C∗-subalgebra M and let T be a bounded linear operator from A into E(M, τ). Then,
+the set
+BMT (BA)BM := {aT (x)b : a, b ∈ BM, x ∈ BA}
+is relatively weakly compact in E(M, τ).
+Proof. Without loss of generality, we may assume that cE = 1.
+Since every weakly compact operator sends bounded sets into weakly compact ones, we infer
+from Corollary 4.2 that the set
+T (BA)
+is relatively weakly compact in E(M, τ). Since E(M, τ) has the Fatou property, it follows that
+E(M, τ) = E(M, τ)×× ⊂ S0(M, τ) (see Section 2.2). Since E(M, τ)× ⊂ S0(M, τ), it follows
+from [28, Corollary 2.9] (or to [35, Theorem 5.4]) that
+�
+y∈BA
+Ω(T (y))
+is relatively σ(E××, E×)-compact (equivalently, relatively σ(E, E×)-compact, or weakly com-
+pact) in E(M, τ), where Ω(x) := {z ∈ (L1 + L∞)(M, τ) : z ≺≺ y}. Since BMT (BA)BM ⊂
+�
+y∈BA Ω(T (y)), it follows that
+BMT (BA)BM
+is weakly compact in E(M, τ)
+□
+Remark 4.4. Assume that E(M, τ) is a strongly symmetric space with cE = 1. Note that, if
+τ(1) < ∞, then the condition E(M, τ)× ⊂ S0(M, τ) holds for any symmetric space E(M, τ).
+If τ(1) = ∞, then E(M, τ)× ⊂ S0(M, τ) if and only if E(M, τ)∩M ⫌ L1(M, τ)∩M. Indeed,
+assume that E(M, τ) ∩ M ⫌ L1(M, τ) ∩ M. Then, there exists an element 0 ≤ z ∈ E(M, τ) ∩ M
+but z /∈ L1(M, τ). In particular, we have τ(z) = ∞. By the definition of K¨othe duals, we infer
+that 1 /∈ E(M, τ)×, which, in turn, implies that E(M, τ)× ⊂ S0(M, τ).
+On the other hand,
+assume by contradiction that L1(M, τ) ∩ M is not a proper subspace of E(M, τ) ∩ M. By (3), we
+have L1(M, τ) ∩ M ⊂ E(M, τ) ∩ M. Therefore, we obtain that E(M, τ) ∩ M = L1(M, τ) ∩ M.
+By the fact that E(M, τ)
+(3)
+⊂ (L1 + L∞)(M, τ), we obtain that for any element x ∈ E(M, τ),
+µ(x)χ(0,1) ∈ L1(0, 1). Hence, all elements in E(M, τ) belong to L1(M, τ). Therefore, E(M, τ)× ⊃
+L1(M, τ)× = M [29, Definition 5.1]. That is, E(M, τ)× ̸⊂ S0(M, τ), which completes the proof.
+Let U(A) denote the set of all unitary elements in a C∗-algebra A.
+Proposition 4.5. Let M be a von Neumann algebra equipped with a semifinite faithful normal
+trace τ and let A be a unital C∗-subalgebra M. Assume that E(M, τ) is a strongly symmetric
+KB-space such that E(M, τ)× ⊂ S0(M, τ). Let δ : A → E(M, τ) be a derivation. Then,
+{δ(u)u∗ | u ∈ U(A)}
+is relatively weakly compact in E(M, τ). Consequently, the closure conv{δ(u)u∗ | u ∈ U(A)}
+∥·∥E
+of the convex hull is weak compact.
+Proof. By Ringrose’s theorem [73, Theorem 2], δ is bounded from (A, ∥·∥∞) into (E(M, τ), ∥·∥E).
+By Proposition 4.3, {δ(u)u∗ | u ∈ U(A)} is relatively weakly compact in E(M, τ).
+The second assertion follows from the Krein–Smulian theorem (see e.g. [84]).
+□
+The following lemma shows that derivations into a “large” symmetric space are inner.
+Lemma 4.6. Let M be a von Neumann algebra with a faithful normal semifinite trace τ and
+let A be a unital C∗-subalgebra of M. Let E(M, τ) a strongly symmetric KB-space such that
+E(M, τ)× ⊂ S0(M, τ).
+For every derivation δ : A → E(M, τ), there exists an element a ∈
+conv{δ(u)u∗ | u ∈ U(A)}
+∥·∥E such that δ = δa on A. In particular, ∥a∥ ≤ ∥δ∥A→E.
+
+12
+J. HUANG AND F. SUKOCHEV
+Proof. Without loss of generality, we may assume that the carrier projection cE = 1.
+For every u ∈ U(A), we have δ(u) ∈ E(M, τ), and therefore we can define the mapping
+αu : E(M, τ) −→ E(M, τ), by setting
+αu(x) := uxu∗ + δ(u)u∗.
+For every u, v ∈ U(A), we have
+αu(αv(x)) = uvxv∗u∗ + uδ(v)v∗u∗ + δ(u)u∗
+= (uv)x(uv)∗ + uδ(v)v∗u∗ + δ(u)vv∗u∗
+= (uv)x(uv)∗ + δ(uv)(uv)∗ = αuv(x).
+In addition, the equality δ(1) = δ(12) = 2δ(1) implies that δ(1) = 0, and therefore α1(x) = x, x ∈
+E(M, τ). Thus, α is an action of the group U(A) on E(M, τ).
+We claim that the set
+K := conv {δ(u)u∗ | u ∈ U(A)}
+∥·∥E
+is invariant with respect to α. Since δ(u)u∗ = αu(0), it follows that k00 := {δ(u)u∗ | u ∈ U(A)} is
+an orbit of 0 with respect to α, and therefore, is an invariant subset with respect to α. In addition,
+for any positive scalars s and t with s + t = 1, we have
+αu(s · x + t · y) = s · uxu∗ + t · uyu∗ + (s + t) · δ(u)u∗ = s · αu(x) + t · αu(y),
+∀x, y ∈ E(M, τ).
+Hence, for every u ∈ U(A) the mapping αu is affine, which implies that conv(K00) is also an
+invariant subset with respect to α. Now, the equality αu(x) − αu(y) = u(x − y)u∗, x, y ∈ E(M, τ)
+implies that every αu, u ∈ U(A), is an isometry on E(M, τ). Hence, K is an invariant subset with
+respect to α.
+Furthermore, the fact that αU is an isometry combined with [25, Chapter V, Lemma 10.7]
+implies that the family {αu : u ∈ U(A)} is a noncontracting family of affine mappings. Clearly,
+αu is weakly continuous for every u ∈ U(A).
+On the other hand, by Proposition 4.5, K is weakly compact. Thus, the set K and the family
+{αu : u ∈ U(A)} satisfy the assumptions of the Ryll-Nardzewski fixed-point theorem [25, Chapter
+V, Theorem 10.8]. Hence, there exists a point a ∈ K fixed with respect to α, that is, we have
+a = αu(a) = uau∗ + δ(u)u∗
+for every u ∈ U(A) and ∥a∥E ≤ ∥δ∥A→E. Therefore au = ua + δ(u) for every u ∈ U(A). Thus,
+δ(u) = [a, u] for every u ∈ U(A). Since every element x ∈ A is a linear combination of four
+elements from U(A) [51, Theorem 4.1.7], we obtain that δ = δa on A.
+□
+A symmetric function space E(M, τ) is said to have the Levi property [1, Definition 7], or a
+monotone complete norm, or to satisfy property (B) [58, Chapter X.4], if for every upwards directed
+net {xβ} in E(M, τ)+, satisfying supβ ∥xβ∥E < ∞, there exists an element x ∈ E(M, τ)+ such
+that xβ ↑ x in E(M, τ). It is well known that if a norm is Levi, then necessarily it is also weak
+Fatou [1, p.89], i.e., there exists a constant K ≥ 1 such that
+0 ≤ xβ ↑ x ⇒ ∥x∥E ≤ K lim
+β ∥xβ∥E .
+Remark 4.7. For a symmetric function space E(0, ∞), it has the Levi property if and only if it
+has an equivalent symmetric norm having the Fatou property. The (⇐) implication is clear. For
+the (⇒) implication, one only need to observe that for any 0 ≤ x ∈ E(0, ∞), there exists a net
+(xβ)β ⊂ (L1 ∩ L∞)(0, ∞) such that xβ ↑ x [32, Proposition 1(vii)], and therefore,
+1
+K ∥x∥E ≤ sup
+β
+∥xβ∥E
+(2)
+= sup
+β
+∥xβ∥E×× = ∥x∥E××
+[32, Section 5.3]
+≤
+∥x∥E ,
+and (E(0, ∞)××, ∥·∥E××) has the Fatou property (see [30, Theorem 4.1] and [60]).
+Recall that E(0, ∞) has the Fatou property if and only E(0, ∞) is isometric to E(0, ∞)××. We
+obtain that E(0, ∞) has the Levi property if and only E(0, ∞) = E(0, ∞)×× (up to equivalent
+norms).
+
+DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES
+13
+The following theorem is the main result of the present paper, which resolves problems consid-
+ered in a numbers of papers, see e.g. [11,12,14,43,45,47,55,83] and references therein.
+Theorem 4.8. Let M be a von Neumann algebra with a faithful normal semifinite trace τ and let
+A be a C∗-subalgebra of M. If E(M, τ) is a strongly symmetric space of τ-compact operators (i.e.,
+E(M, τ) ⊂ S0(M, τ)7) having the Fatou property (resp., the Levi property), then every derivation
+δ : A → E(M, τ) is inner. That is, there exists an element a ∈ conv {δ(u)u∗ | u ∈ U(A)}
+tτ ⊂
+E(M, τ) with ∥a∥E ≤ ∥δ∥A→E (∥a∥E ≤ c ∥δ∥A→E for some constant c depending on E(0, ∞)
+only) such that δ = δa on A.
+Proof. Without loss of generality, we may assume that the carrier projection cE = 1 and assume
+that A is unital (see e.g. the proof of [11, Proposition 3.4]). By Remark 4.7, we may assume that
+E(M, τ) has the Fatou property.
+Since E(M, τ) has the Fatou property and E(M, τ) ⊂ S0(M, τ), it follows that E(M, τ)×× ⊂
+S0(M, τ). By Proposition 3.5, there exists a symmetric KB-function space F(0, ∞) ⫋ (L1 +
+C0)(0, ∞) such that
+E(M, τ) ⊂ F(M, τ) ⊂ S0(M, τ).
+Without loss of generality, we may assume that L2(0, ∞) ⊂ F(0, ∞) by replacing F(0, ∞) with
+L2(0, ∞) + F(0, ∞). In particular, F(0, ∞) ∩ L∞(0, ∞) ⫌ L1(0, ∞) ∩ L∞(0, ∞). By Remark 4.4,
+F(0, ∞)× ⊂ S0(0, ∞), and therefore, F(M, τ)× ⊂ S0(M, τ).
+Note that δ(A) ⊂ E(M, τ) ⊂ F(M, τ). By Lemma 4.6, there is an element
+a ∈ conv {δ(u)u∗ | u ∈ U(A)}
+∥·∥F
+such that δ = δa on A. Hence, there exists a sequence
+(xn)∞
+n=1 ⊂ conv {δ(u)u∗ | u ∈ U(A)}
+such that ∥xn − a∥F →n 0. Since F(M, τ) is a symmetric space, it follows from [32, Proposition
+20] that xn →tτ a as n → ∞.
+By Ringrose’s theorem [73], we have that δ : (A, ∥·∥∞) → (E(M, τ), ∥·∥E) is a bounded map-
+ping. Since E(M, τ) has the Fatou property, it follows that the closed ball (E(M, τ), ∥·∥E) with
+radius ∥δ∥A→E is closed in S(M, τ) with respect to the local measure topology (see Section 2.2).
+Noticing that every element xn, n ≥ 1 belongs to the ball of radius ∥δ∥A→E in E(M, τ) and
+xn → a in the local measure topology, we conclude that a ∈ E(M, τ) with ∥a∥E ≤ ∥δ∥A→E.
+□
+Recall that a symmetric function space having the Fatou/Levi property generates a noncommu-
+tative strongly symmetric space having the Fatou/Levi property (see e.g. [61, Proposition 2.a.8]
+and [32, Theorem 54]). We have the following consequence of Theorem 4.8.
+Corollary 4.9. Assume that E(0, ∞) ⊂ S0(0, ∞) is a symmetric function space having the Fatou
+property (resp., the Levi property). Then, a derivation
+δ : A → E(M, τ)
+is necessarily inner for any von Neumann algebra M with a semifinite faithful normal trace τ
+and a C∗-subalgebra A of M. That is, there exists an element a ∈ conv {δ(u)u∗ | u ∈ U(A)}
+tτ ⊂
+E(M, τ) with ∥a∥E ≤ ∥δ∥A→E (∥a∥E ≤ c ∥δ∥A→E for some constant c depending on E(0, ∞)
+only) such that δ = δa on A.
+Corollary 4.9 provides an alternative proof for the resolution of the derivation problem due
+to Bunce and Paschke [18], which differs from those in [5, 70] involving L-embeddedness of the
+predual of a von Neumann algebra. Moreover, we provide new information that implementing
+element a of the derivation can be found in the measure (or any symmetric norm topology which
+is weaker than the L1-topology) closure of the convex hull of {δ(u)u∗ | u ∈ U(A)}, which seems
+to be more natural in fixed point theory and derivation theory.
+7If τ(1) < ∞, then E(M, τ) ⊂ S0(M, τ) holds for any symmetric space E(M, τ) affiliated with M.
+
+14
+J. HUANG AND F. SUKOCHEV
+Corollary 4.10. Let M be a semifinite von Neumann algebra and let A be a C∗-subalgebra of M.
+Then, any derivation from A into M∗ is inner, and there exists an element a ∈ conv{δ(u)u∗ | u ∈ U(A)}
+tτ
+with ∥a∥L1 ≤ ∥δ∥A→L1 such that δ = δa on A.
+The following corollary significantly extends results from [11,43,55] for derivations into symmet-
+ric ideals. On the other hand, it shows that [12, Theorem 1.3] holds true even for C∗-subalgebras
+rather than von Neumann subalgebras.
+Corollary 4.11. Let M be a semifinite von Neumann algebra equipped with a semifinite faithful
+normal trace τ and let A be a C∗-subalgebra of M. Let E(M, τ) be an ideal of C0(M, τ) having
+strongly symmetric norm and the Levi property. Then, every derivation from A into E(M, τ) is
+inner.
+For a von Neumann algebra M equipped with a faithful normal tracial state, M ⊂ S0(M, τ)
+and M has the Fatou/Levi property. Therefore, we have the following consequence of Theorem 4.8.
+Corollary 4.12.
+[19, Section 5] [80] Let M be a von Neumann algebra equipped with a faithful
+normal tracial state and let A be a C∗-subalgebra of M. Then every derivation from A into M is
+inner.
+5. Outer derivations
+In this section, we present several examples of outer derivations with values in a symmetric
+space failing the Levi property.
+The study of derivations into ideals of a von Neumann algebra was initiated Johnson and
+Parrott [48] by showing that derivations from an abelian/properly infinite von Neumann subalgebra
+of B(H) into the algebra K(H) of all compact operators on H are inner. This result may be well
+seen as a precursor to Question 1 in the setting of symmetrically normed ideals in M = B(H).
+After that, the innerness of derivations into ideals of a von Neumann algebra have been studied
+extensively in [11,12,15,16,20,40,43,55,72].
+Recall that the Fatou/Levi property can not be dropped in Theorem 4.8, which has been
+demonstrated in [77, Example 4.1.8], [11, Theorem 3.8] and [44, Theorem 4.2.1]. The following
+theorem demonstrates that there exist outer derivations into E(M, τ) ∩ M whenever the ideal
+E(M, τ) ∩ M does not have the Levi property.
+Theorem 5.1. Let M be a non-finite factor equipped with a semifinite faithful normal trace τand
+let E(M, τ) be a symmetric space affiliated with M such that E(M, τ)∩M does not have the Levi
+property. Then, there exist outer derivations from the C∗-algebra C0(M, τ) into E(M, τ) ∩ M.
+Proof. Since M is a factor, it follows that it is either atomless or is atomic with all atoms having
+equal trace. Therefore,
+E(M, τ)××
+is a fully symmetric space having the Fatou property (see e.g. [29], [34, Chapter IV, Theorem
+5.4], [32, Remarks 5 and 6]). Hence,
+E(M, τ)×× ∩ M,
+equipped with the norm ∥x∥E(M,τ)××∩M = max {∥x∥E×× , ∥x∥∞} , x ∈ E(M, τ)×× ∩ M, has the
+Fatou property. On the other hand, the lack of the Levi property of E(M, τ) ∩ M implies that
+there exists a net (xi)i∈I of positive elements in E(M, τ) ∩ M increasing to 0 ≤ x ∈ M such
+that sup ∥xi∥E < ∞ but x /∈ E(M, τ). By the Fatou property of E(M, τ)×× and E(M, τ) ⊂
+E(M, τ)×× [32, Section 5.3], we have x ∈ E(M, τ)×× ∩ M.
+Since E(M, τ)∩M does not have the Levi property, it follows that E(M, τ)∩M ̸= M. Hence,
+E(M, τ) ⊂ S0(M, τ). There are two possible cases:
+(1) If E(M, τ)×× ⊂ C0(M, τ), then we consider δx.
+We claim that δx is a non-inner derivation from C0(M, τ) into E(M, τ) ∩ M. Recall that
+∥z∥E
+(2)
+= ∥z∥E×× , ∀z ∈ F(M, τ).
+
+DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES
+15
+For every y ∈ C0(M, τ), the projections e|y|(ε, ∞) and e|y|(ε1, ∞) are τ-finite for every ε > ε1 > 0.
+Thus, we have
+���xye|y|(ε, ∞) − xye|y|(ε1, ∞)
+���
+E ≤
+���xye|y|(ε1, ε]
+���
+E =
+���xye|y|(ε1, ε]
+���
+E×× ≤ ε ∥x∥E×× ,
+which shows that
+�
+xye|y|( 1
+n, ∞)
+�
+n≥1 is a Cauchy sequence in E(M, τ).
+On the other hand,
+xye|y|( 1
+n, ∞) → xy in measure, which implies that xy ∈ E(M, τ).
+Similarly, yx ∈ E(M, τ)
+and therefore δx(C0(M, τ)) ⊂ E(M, τ). Moreover, x ∈ M and C0(M, τ) ⊂ M imply that
+δx(C0(M, τ)) ⊂ E(M, τ) ∩ M.
+Finally, if there exists an operator x′ ∈ E(M, τ) ∩ M such that δx = δx′, then
+x − x′ commutes with all elements in C0(M, τ).
+Noticing that M is the closure of C0(M, τ) in the weak operator topology (see e.g. [62, Definition
+2.6.8]), we obtain that
+x − x′ commutes with all elements in M.
+However, M is a factor and therefore x − x′ ∈ C1, which is a contradiction with the assumption
+that E(M, τ)×× ⊂ C0(M, τ).
+(2) If E(M, τ)×× ⊃ M, then, by Lemma 3.4, we have E(M, τ) ∩ M ⊃ C0(M, τ). Recall that
+E(M, τ) ∩ M ̸= M. We have E(M, τ) ∩ M = C0(M, τ). Now, the existence of outer derivations
+follows from [11, Theorem 3.8] immediately.
+□
+Corollary 5.2. Let E(0, ∞) be a symmetric function space. If E(0, ∞)∩L∞(0, ∞) does not have
+the Levi property, then for any infinite factor M equipped with a semifinite faithful normal trace
+τ, there exists outer derivations from the C∗-algebra C0(M, τ) into E(M, τ) ∩ M.
+Proof. By Theorem 5.1, it suffices to prove that E(M, τ) ∩ M does not have the Levi property.
+Since E(0, ∞) ∩ L∞(0, ∞) has no Levi property, it follows that (see e.g. Remark 4.7)
+E(0, ∞) ∩ L∞(0, ∞) ̸= (E(0, ∞) ∩ L∞(0, ∞))×× = E(0, ∞)×× ∩ L∞(0, ∞).
+That is, there exists 0 ≤ x = µ(x) ∈ E(0, ∞)×× ∩ L∞(0, ∞) such that x /∈ E(0, ∞) ∩ L∞(0, ∞).
+Observe that
+�
+n≥1
+µ(n; x)χ(n−1,n] ≤ µ(x) ∈ E(0, ∞)×× ∩ L∞(0, ∞).
+There exists y ∈ M such that
+µ(y) =
+�
+n≥1
+µ(n; x)χ(n−1,n] ≤ µ(x) ∈ E(0, ∞)×× ∩ L∞(0, ∞).
+However, we have
+µ(y) ≥ µ
+�
+µ(x) − µ(x)χ(0,1]
+�
+.
+Since µ(x)χ(0,1] ∈ E(0, ∞)∩L∞(0, ∞) (see e.g. [32, p.245] or [60]), it follows that y /∈ E(M, τ)∩M.
+This shows that E(M, τ)∩M does not have the Levi property when M a semifinite infinite (atomic
+or atomless) factor, see Remark 4.7.
+□
+Recall that the dilation operator σs, s > 0, on S(0, ∞) is defined by [60–62]
+(σs)f(t) = f
+� t
+s
+�
+, ∀f ∈ S(0, ∞), s ∈ (0, ∞).
+For the case of unbounded operators, the example of an outer derivation is more involved. The
+result presented below is perhaps the first example of an outer derivation into a symmetric space
+affiliated with a finite von Neumann algebra.
+Example 5.3. Let M be the von Neumann algebra which is the closure of ⊕n≥1M2 in the weak
+operator topology equipped with the trace τ := ⊕n≥1
+1
+2n+1 Tr2, where Tr2 is the standard trace of
+M2. Let E(0, 1) be a symmetric function space without the Levi property. Then, there exist outer
+derivations from the C∗-algebra A := ⊕M2n
+∥·∥∞ into E(M, τ).
+
+16
+J. HUANG AND F. SUKOCHEV
+Proof. The lack of the Levi property of E(0, 1) shows that E(0, 1) ̸= E(0, 1)×× (see e.g. Re-
+mark 4.7), and there exists a positive function a ∈ E(0, 1)×× but a /∈ E(0, 1). Define
+b :=
+�
+n≥1
+µ
+�
+1
+2n−1 ; a
+�
+χ[ 1
+2n ,
+1
+2n−1 ).
+We have
+b ≤ a ≤
+�
+n≥1
+µ
+� 1
+2n ; a
+�
+χ[ 1
+2n ,
+1
+2n−1 ) = σ2(b).
+Therefore, b ∈ E(0, 1)×× but b /∈ E(0, 1).
+Now, we define a self-adjoint element y by setting
+z :=
+�
+n≥1
+an (pn1 − pn2) , an = µ
+� 1
+2n ; a
+�
+, n ≥ 1,
+where pn1 and pn2 are mutually orthogonal non-trivial projections in the n-th direct summand
+M2, and the series is taken in the measure topology. In particular, µ(z) = σ2(b) and therefore,
+z ∈ E(M, τ)×× but z /∈ E(M, τ).
+We claim that δz(A) ⊂ E(M, τ)b, the closure of M in E(M, τ). Indeed, since δz has image in
+E××(M, τ) (generated by E(0, 1)××), it follows from Ringrose’s theorem [73] that δz is bounded
+from (A, ∥·∥∞) into (E××(M, τ), ∥·∥E××). Let pn be the identity of the n-th direct summand M2.
+For any element y ∈ A, we have yn := y �n
+k=1 pk → y in the uniform norm topology. Note that
+ynz, zyn ∈ M. Hence, for any n > m, we have
+∥δz(yn) − δz(ym)∥E(M,τ) = ∥µ(δz(yn) − δz(ym))∥E(0,1)
+(2)
+= ∥µ(δz(yn) − δz(ym))∥E(0,1)××
+= ∥δz(yn) − δz(ym)∥E××(M,τ)
+≤ 2 ∥yn − ym∥∞ ∥z∥E××(M,τ) → 0 as n → ∞.
+Therefore, (δz(yn))n≥1 is a Cauchy sequence in E(M, τ), which implies that (δz(yn))n≥1 converges
+to some element in E(M, τ) (in particular, (δz(yn))n≥1 is tτ-convergent). On the other hand,
+δz(yn) →tτ δz(y) as n → ∞.
+Therefore, ∥δz(yn)n≥1 − δz(y)∥E(M,τ) →n 0.
+Hence, δz(y) ∈
+E(M, τ) for all y ∈ A.
+Assume that there exist z1 ∈ E(M, τ) such that δz1 = δz, i.e., z1 − z commutes with A. That
+is, z1 −z = �
+n≥1 bnpn for some sequence (bn)n≥1 of complex numbers, where the series converges
+in the measure topology. However,
+E(0, 1) ∋ µ(z1) = µ
+
+z +
+�
+n≥1
+bnpn
+
+ = µ
+
+�
+n≥1
+an (pn1 − pn2) +
+�
+n≥1
+bnpn
+
+
+= µ
+
+�
+n≥1
+(an + bn)pn1 − (an − bn)pn2
+
+
+≥ µ
+
+�
+n≥1
+max{|an + bn|, |an − bn|}pn1
+
+
+≥ µ
+
+�
+n≥1
+anpn1
+
+ = µ
+
+�
+n≥1
+anpn2
+
+ = σ1/2µ(z),
+which implies that z ∈ E(M, τ) [60, Theorem II.4.4]. This is a contradiction to the assumption.
+□
+The following result is an immediate consequence of Corollaries 4.9, 5.2 and Example 5.3.
+
+DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES
+17
+Corollary 5.4. For a given symmetric function space E(0, ∞) ⊂ S0(0, ∞), the following two
+statements are equivalent:
+(1) for any von Neumann algebra M equipped with a semifinite faithful normal trace τ and
+any C∗-subalgebra A of M, derivations δ : A → E(M, τ) are necessarily inner;
+(2) E(0, ∞) has the Levi property.
+The following result is an immediate consequence of Corollary 4.9 and Theorem 5.1.
+Corollary 5.5. Let CE be a symmetric ideal of compact operators in B(H). Then, the following
+two statements are equivalent:
+(1) for any C∗-subalgebra A of B(H), derivations δ : A → CE are necessarily inner;
+(2) the commutative core ℓE of CE has the Levi property.
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+DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES
+19
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+Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology, Harbin,
+150001, China
+Email address: jinghao.huang@hit.edu.cn
+School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, Australiaa
+Email address: f.sukochev@unsw.edu.au
+
diff --git a/cNE2T4oBgHgl3EQfaQf4/content/tmp_files/load_file.txt b/cNE2T4oBgHgl3EQfaQf4/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c6da61761266d520562f23b9d9f557702bdffa85
--- /dev/null
+++ b/cNE2T4oBgHgl3EQfaQf4/content/tmp_files/load_file.txt
@@ -0,0 +1,1254 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf,len=1253
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='03874v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='OA]  10 Jan 2023  INNERNESS OF DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC  SPACES IS DETERMINED COMMUTATIVELY  JINGHAO HUANG AND FEDOR SUKOCHEV  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let E = E(0, ∞) be a symmetric function space and E(M, τ) be a symmet-  ric operator space associated with a semifinite von Neumann algebra with a faithful normal  semifinite trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Our main result identifies the class of spaces E for which every derivation  δ : A → E(M, τ) is necessarily inner for each C∗-subalgebra A in the class of all semifinite von  Neumann algebras M as those with the Levi property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Introduction  Let A be a C∗-algebra and let J be an A-bimodule [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' A derivation δ : A → J is a linear  mapping satisfying δ(xy) = δ(x)y + xδ(y), x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In particular, if a ∈ J, then δa(x) := xa − ax  is a derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Such derivations implemented by elements in J are said to be inner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' One of the  classical problems in operator algebra theory is the question whether every derivation from A into  J is automatically inner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The celebrated Kadison–Sakai theorem [50,76] states that derivations are always inner when A  is a von Neumann algebra and the A-bimodule J coincides with the algebra A itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Further, it was  proved that every derivation from a von Neumann algebra into any of its ideal s is automatically  inner [15,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' However, when one considers more general C∗-algebras A and A-bimodules J, there  are examples of non-inner derivations for some specific A and J (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [8, 11, 36–38, 72, 77]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  We quote the following from the memoir by Johnson [47, Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='11]): “It would be desirable  to identify those spaces X with H1(G, X∗) = 0 for all G ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' A similar statement appeared in [43]  (see also [80, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='60]): “Here again it can be asked if such derivations are inner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' that is, are they  induced by an element of J as above?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In fancier language, the question asks if the cohomology  group H1(A, J) is trivial.” During the past decades, a number of important special cases have  been studied (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [5,13,21,22,26,47,48,53,54,63,69,71,74]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Due to the rapid development of noncommutative analysis and motivated by questions due  to Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=', there are a number of papers concerning various versions of the following  question [5,7,10,18,83]:  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Assume that M is a von Neumann algebra equipped with a faithful normal semifinite  trace τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let E(M, τ) be a symmetric space of τ-compact operators affiliated with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' How can  one identity those E(M, τ) such that derivations from an arbitrary C∗-subalgebra A of M into  E(M, τ) are necessarily inner?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Experts in the operator theory are probably more familiar with symmetrically normed ideals  in B(H), which are a special case of noncommutative symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Various versions of  Question 1 for derivations with values in ideals of a von Neumann algebra were asked and discussed  in [11,12,43,44,48,55,71,72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  It was a long-standing open question whether every derivation a C∗-subalgebra of a semifinite  von Neumann algebra M into M∗ must be inner (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='247]), which in our setting  is equivalent to the special version of Question 1 with E = L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  This question was resolved  completely by Bader, Gelander and Monod in 2012 [5] (see also [70] for a slightly different proof  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' 46L57, 47A56, 46L52, 46L10, 46E30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' derivation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' symmetric space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' measurable operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Huang was supported by the NNSF of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='12031004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Sukochev’s research was supported by the  Australian Research Council (FL170100052).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  1  2  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' HUANG AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' SUKOCHEV  due to Pfitzner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The method used in [5] (or [70]) does not have any chance to deliver the full  answer on Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' This fact was emphaiszed in [5, Section 3], where the following points were  raised:  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In marked contrast to the classical fixed point theorems, there is no hope to  find a fixed point inside a general bounded closed convex subset of L1 · · · the  weak compactness · · · seems almost unavoidable · · ·  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' · · · a canonical norm one projection V ∗∗ → V is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' It would be interesting to find a purely geometric version of the proposition  · ·  The fact that the “fixed point” obtained in [5] is not inside a general bounded closed convex subset  of L1 leads to extra difficulties in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  In this paper, we completely resolve Question 1 above, see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  We show that the Levi property1 of a symmetric space is a sufficient and necessary condition for  the Question 1 above having an affirmative answer for every semifinite von Neumann algebra  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Thus, the derivation theorem for preduals of von Neumann algebras in [5] (in the semifinite  setting) becomes a trivial corollary of our Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The new approach devised in this paper answers to points (a), (b) and (c) above raised  in [5], which provides an alternative proof for the resolution of the question raised by Bunce  and Paschke [18], without involving weak compactness of a subset in a L-embedded space as [5]  and [70] did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' This enables us to find a “fixed point” (implementing the derivation) from a not  necessarily weakly compact closed convex subset of a noncommutative symmetric space which is  1-complemented subspace of its bidual but not necessarily an L-embedded Banach space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=', a  symmetric space having the Levi property [30, 31]2), see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' On the other hand, the  Levi property of the space E(M, τ) means that E(M, τ) coincides with its second K¨othe dual  and this geometrical condition is the only one required in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8 thus delivering (at least  spiritually) an answer to the question suggested in [5, Comment c] above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' We believe that the  method developed in this work is of interest in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  It is important to emphasize that the Fatou/Levi property was hatched in the theory of Banach  lattices [1,17], and was even included into the original definition of Banach function spaces over  σ-finite measure spaces (see [6, 65]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The property is somewhat analogous to the so-called “dual  normal” property3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The importance of the Fatou/Levi property in the theory of Banach function  spaces and symmetric operator spaces is hard to overestimate [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' It seems appropriate to  recall here that every derivation from a hyperfinite von Neumann algebra A into a dual normal  A-bimodule is inner (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [80, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3], [23] and [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Recall also, that derivations from  a nuclear C∗-algebra A into a dual Banach A-module are inner [24, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' However, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8  below holds for arbitrary C∗-subalgebras A of M and for symmetric spaces which may not have  a predual space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The reflexive gate type result (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [68, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3]) is relatively unknown but plays  a significant role in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In Section 3, we establish a noncommutative version of this  result and lay the groundwork for its usage in the derivation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In Section 4, using the  weak compactness criteria for noncommutative symmetric spaces obtained in [28,35], we show that  the Ryll-Nardzewski fixed point theorem is applicable to any noncommutative strongly symmetric  KB-space (or a Kantorovich–Banach space, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2) whose bounded part does not coincides  with C1(M, τ) = L1(M, τ) ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' This is the key technical step in the proof of the main result  of the present paper, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In Section 5, we demonstrate why the Fatou/Levi property of  1A symmetric function space E(0, ∞) having the Levi property has an equivalent symmetric norm such that  E(0, ∞) has the Fatou property, see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The Soviet school on Banach lattices used the term monotone  complete norm or property (B) [58, Chapter X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4], see also [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In the theory of operator algebras, a similar  property is called ‘monotone closed’ [81, Chapter III, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  2Indeed, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8 holds for the case when the projection constant is not necessarily 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  3Let M be a von Neumann algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' An M-bimodule X is said to be a dual normal X-bimodule if X is a dual  space and the maps m �→ mx and m �→ xm are both ultraweak-weak∗ continuous from M into X for each fixed  element x ∈ X [80, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES  3  the space E(M, τ) is a necessary condition for an affirmative answer to Question 1 in the class of  all semifinite von Neumann algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Preliminaries  In this section, we recall some notions of the theory of noncommutative integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  In what follows, H is a Hilbert space and B(H) is the ∗-algebra of all bounded linear operators  on H equipped with the uniform norm ∥·∥∞, and 1 is the identity operator on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let M be a  von Neumann algebra on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' We denote by P(M) the collection of all projections in M, by M′  the commutant of M and by Z(M) the center of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For details on von Neumann algebra theory,  the reader is referred to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [51,52] or [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' General facts concerning measurable operators may  be found in [67], [78] (see also [82, Chapter IX] and the forthcoming book [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For convenience  of the reader, some of the basic definitions are recalled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' τ-measurable operators and generalized singular value functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' A closed, densely  defined operator x : D (x) → H with the domain D (x) is said to be affiliated with M if yx ⊆ xy for  all y ∈ M′, where M′ is the commutant of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' A closed, densely defined operator x : D (x) → H  affiliated with M is said to be measurable if there exists a sequence {pn}∞  n=1 ⊂ P (M), such that  pn ↑ 1, pn(H) ⊆ D (x) and 1−pn is a finite projection (with respect to M) for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The collection  of all measurable operators with respect to M is denoted by S (M), which is a unital ∗-algebra  with respect to strong sums and products (denoted simply by x + y and xy for all x, y ∈ S (M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Let x be a self-adjoint operator affiliated with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' We denote its spectral measure by {ex}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' It is  well known that if x is an operator affiliated with M with the polar decomposition x = u|x|, then  u ∈ M and e ∈ M for all projections e ∈ {e|x|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Moreover, x ∈ S(M) if and only if e|x|(λ, ∞) is a  finite projection for some λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' It follows immediately that in the case when M is a von Neumann  algebra of type III or a type I factor, we have S(M) = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For type II von Neumann algebras,  this is no longer true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' From now on, let M be a semifinite von Neumann algebra equipped with  a faithful normal semifinite trace τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  An operator x ∈ S (M) is called τ-measurable if there exists a sequence {pn}∞  n=1 in P (M)  such that pn ↑ 1, pn(H) ⊆ D (x) and τ(1 − pn) < ∞ for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The collection S (M, τ) of all  τ-measurable operators is a unital ∗-subalgebra of S (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' It is well known that a linear operator  x belongs to S (M, τ) if and only if x ∈ S(M) and there exists λ > 0 such that τ(e|x|(λ, ∞)) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Alternatively, an unbounded operator x affiliated with M is τ-measurable (see [39]) if and only if  τ  �  e|x|�  n, ∞  ��  → 0,  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  For convenience of the reader, we also recall the definition of the measure topology tτ on the  algebra S(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For every ε, δ > 0, we define the set  V (ε, δ) = {x ∈ S(M, τ) : ∃p ∈ P (M) such that ∥x(1 − p)∥∞ ≤ ε, τ(p) ≤ δ} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The topology generated by the sets V (ε, δ), ε, δ > 0, is called the measure topology tτ on S(M, τ)  [34, 39, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' It is well known that the algebra S(M, τ) equipped with the measure topology is a  complete metrizable topological algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' A sequence {xn}∞  n=1 ⊂ S(M, τ) converges to zero with  respect to measure topology tτ if and only if τ  �  e|xn|(ε, ∞)  �  → 0 as n → ∞ for all ε > 0 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Another important vector topology on S(M, τ) is the local measure topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For convenience  we denote by Pf(M) the collection of all τ-finite projections in M, that is, the set of all e ∈ P(M)  satisfying τ(e) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' A neighbourhood base for this topology is given by the sets V (ε, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' e), ε, δ > 0,  e ∈ Pf(M), where  V (ε, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' e) = {x ∈ S(M, τ) : exe ∈ V (ε, δ)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Obviously, local measure topology is weaker than measure topology [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' We note here, that the  local measure topology used in the present paper differs from the local measure topology defined  in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let M be a von Neumann algebra equipped with a faithful normal semi-finite  trace τ and let x ∈ S(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The generalized singular value function µ(x) : t �→ µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x) of the  4  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' HUANG AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' SUKOCHEV  operator x is defined by setting  µ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x) = inf{∥xp∥∞ : p ∈ P(M) with τ(1 − p) ≤ s}, ∀s ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  An equivalent definition in terms of the distribution function of the operator |x| is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  for every operator x ∈ S(M, τ), setting d|x|(t) = τ(e|x|(t, ∞)),  t > 0, we have (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [39])  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x) = inf{s ≥ 0 : d|x|(s) ≤ t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  (1)  Note that µ(x) is a function defined on (0, ∞) even if the trace τ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  In particular,  µ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x) = 0 when s ≥ τ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Consider the algebra M = L∞(0, ∞) of all Lebesgue measurable essentially bounded functions  on (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The algebra M can be seen as an abelian von Neumann algebra acting via multiplication  on the Hilbert space H = L2(0, ∞), with the trace given by integration with respect to Lebesgue  measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' It is easy to see that the algebra of all τ-measurable operators affiliated with M can be  identified with the subalgebra S(0, ∞) of the algebra of Lebesgue measurable functions L0(0, ∞)  which consists of all functions x such that m({|x| > s}) is finite for some s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  It should  also be pointed out that the generalized singular value function µ(x) is precisely the decreasing  rearrangement µ(x) of the function |x| (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [60]) defined by  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x) = inf{s ≥ 0 : m({|x| ≥ s}) ≤ t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  If M = B(H) (respectively, ℓ∞) and τ is the standard trace Tr (respectively, the counting  measure on N), then it is not difficult to see that S(M) = S(M, τ) = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In this case, for  x ∈ S(M, τ) we have  µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x) = µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x),  t ∈ [n, n + 1),  n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The sequence {µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)}n≥0 is just the sequence of singular values of the operator x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  If x, y ∈ S(M, τ), then x is said to be submajorized by y, denoted by x ≺≺ y, if  � t  0  µ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)ds ≤  � t  0  µ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' y)ds for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  In particular, for x, y ∈ S(0, ∞), x ≺≺ y if and only if  � t  0 µ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)ds ≤  � t  0 µ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' y)ds, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' A linear subspace E of S(M, τ) equipped with a complete norm ∥·∥E, is called a  symmetric space (of τ-measurable operators) if x ∈ S(M, τ), y ∈ E and µ(x) ≤ µ(y) imply that  x ∈ E and ∥x∥E ≤ ∥y∥E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  It is well-known that any symmetric space E is a normed M-bimodule, that is, axb ∈ E for any  x ∈ E, a, b ∈ M and ∥axb∥E ≤ ∥a∥∞ ∥b∥∞ ∥x∥E [32,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' A symmetric space E(M, τ) ⊂ S(M, τ)  is called strongly symmetric if its norm ∥·∥E has the additional property that ∥x∥E ≤ ∥y∥E  whenever x, y ∈ E(M, τ) satisfy x ≺≺ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In addition, if x ∈ S(M, τ), y ∈ E(M, τ) and x ≺≺ y  imply that x ∈ E(M, τ) and ∥x∥E ≤ ∥y∥E, then E(M, τ) is called fully symmetric space (of  τ-measurable operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  If E ⊂ S(M, τ) is a symmetric space, then the norm ∥·∥E is called order continuous if ∥xα∥E →  0 whenever {xα} is a downwards directed net in E+ satisfying xα ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' A symmetric space E(M, τ)  is said to have the Fatou property if for every upwards directed net {xβ} in E(M, τ)+, satisfying  supβ ∥xβ∥E < ∞, there exists an element x ∈ E(M, τ)+ such that xβ ↑ x in E(M, τ) and  ∥x∥E = supβ ∥xβ∥E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Examples such as Schatten–von Neumann operator ideals, Lorentz operator  ideals, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' all have symmetric norms which have the Fatou property [34,60,61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' If E has the Fatou  property and order continuous norm, then it is said to be a KB-space (or Kantorovich–Banach  space) [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  If E(M, τ) is a symmetric space, then the carrier projection cE ∈ P(M) is defined by setting  cE =  �  {p : p ∈ P(M), p ∈ E(M, τ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  We remark that, replacing the von Neumann algebra M by the reduced von Neumann algebra  McE (note that E(M, τ) = cEE(M, τ)cE, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [32, Corollary 6]), it is often assumed that the  carrier projection of E(M, τ) is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES  5  If E(M, τ) is a symmetric space, then the K¨othe dual E(M, τ)× of E(M, τ) is defined by  E(M, τ)× = {x ∈ S(M, τ) :  sup  ∥y∥E≤1,y∈E  τ(|xy|) < ∞},  and for every x ∈ E(M, τ)×, we set ∥x∥E× = sup {τ(|yx|) : y ∈ E(M, τ), ∥y∥E ≤ 1} (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [32,  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2], see also [28,62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' It is well-known that ∥·∥E× is a norm on E(M, τ)× if and only if  the carrier projection cE of E(M, τ) is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In this case, for a strongly symmetric space  E(M, τ), the following statements are equivalent [31,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  E(M, τ) has the Fatou property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  E(M, τ)×× = E(M, τ) and ∥x∥E = ∥x∥E×× for all X ∈ E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The norm closed unit ball BE of E(M, τ) is closed in S(M, τ) with respect to the local  measure topology .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  If E(M, τ) is a strongly symmetric space with cE = 1 (or a symmetric space affiliated with a  semifinite von Neumann algebra which is either atomless or atomic with all minimal projections  having equal trace), then [32, Lemma 30] (see also [34, Theorem IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='7] and [60])  ∥x∥E = ∥x∥E×× , x ∈ (L1 ∩ L∞)(M, τ),  (2)  and  L1 ∩ L∞(M, τ) ⊂ E(M, τ) ⊂ (L1 + L∞)(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  (3)  A wide class of symmetric operator spaces associated with the von Neumann algebra M can  be constructed from concrete symmetric function spaces studied extensively in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Let  (E(0, ∞), ∥·∥E(0,∞)) be a symmetric function space on the semi-axis (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The pair  E(M, τ) = {x ∈ S(M, τ) : µ(x) ∈ E(0, ∞)},  ∥x∥E(M,τ) := ∥µ(x)∥E(0,∞)  is a symmetric operator space affiliated with M with cE = 1 [56] (see also [62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For convenience,  we denote ∥·∥E(M,τ) by ∥·∥E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Let E(0, ∞) be a symmetric function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' We define a symmetric space [60, Chapter I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3]  (L1 + E)(0, ∞) := {f ∈ S(0, ∞) : ∥f∥L1+E :=  inf  x=u+v,u∈L1(0,∞),v∈E(0,∞) {∥u∥1 + ∥v∥E} < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The ideal of τ-compact operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For a self-adjoint operator x ∈ S(M, τ), we denote  by s(x) its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The two-sided ideal F(M, τ) in M consisting of all elements of τ-finite range  is defined by setting  F(M, τ) = {x ∈ M : τ(s(|x|)) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The C∗-algebra C0(M, τ) of all τ-compact bounded operators can be described as the closure  in the norm ∥ · ∥∞ of the linear span of all τ-finite projections [62, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Equivalently,  C0(M, τ) the set of all elements x ∈ M such that τ(E|x|(λ, ∞)) < ∞ for every λ > 0 (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [34, Chapter II, Section 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The space C0(M, τ) is associated to the ideal of essentially  bounded functions vanishing at infinity (see [62, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='9]), that is,  C0(M, τ) =  �  a ∈ S(M, τ) : µ(a) ∈ L∞(0, ∞), µ(∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' a) := lim  t→∞ µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' a) = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  In particular, if τ is finite, then M = C0(M, τ) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [62, Page 64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The space S0(M, τ) of  τ-compact operators is the space associated to the algebra of functions from S(0, ∞) vanishing at  infinity, that is,  S0(M, τ) = {x ∈ S(M, τ) : µ(∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  This is a two-sided ideal in S(M, τ) and, clearly, C0(M, τ) = S0(M, τ) ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  We denote  C0(0, ∞) := {f ∈ L∞(0, ∞) : µ(∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' f) = 0},  and  (L1 + C0)(0, ∞) := {f ∈ L1 + L∞(0, ∞) : µ(∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' f) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  6  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' HUANG AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' SUKOCHEV  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' A noncommutative reflexive gate type theorem  Recall the reflexive gate type result for symmetric function spaces E(0, 1) [68, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3],  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=', if E(0, 1) ̸= L1(0, 1) and E(0, 1) ̸= L∞(0, 1), then there exist two reflexive symmetric space  F1(0, 1) and F2(0, 1) such that F2(0, 1) ⊂ E(0, 1) ⊂ F1(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For a symmetric sequence space  ℓE such that ℓE ⫋ c0, by the well-known construction of Davis–Figiel–Johnson–Pe�lczy´nski [27]  (see [4, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='37], [25, Chapter VI, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3] or [57, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' 255] for detailed expositions),  there exists a reflexive sequence space ℓF ⫋ ℓE, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' the proof of [57, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In the  following lemma, we show that this reflexive sequence is actually symmetric, which is a discrete  analogue of the reflexive gate result [68, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3] for symmetric function spaces on the unit  interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let ℓE be a symmetric sequence space which does not coincide with ℓ1 (up to equiv-  alent norms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then, there exists a reflexive symmetric sequence space ℓF such that ℓF ⊂ ℓE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The proof relies on the construction of Davis–Figiel–Johnson–Pe�lczy´nski [27, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='313].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Let {en}n≥1 be the standard symmetric basis of ℓE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Since ℓE ̸= ℓ1, it follows that {en}n≥1  is weakly null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By the Krein–Smulian theorem [4, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='42], the convex circled4 hull W of  {en}n≥1 is weakly compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Let Un = 2nW +2−nU, where U stands for the unit ball of ℓE, and ∥·∥n denotes the Minkowski  functional of Un, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=',  ∥x∥n := inf {λ > 0 : x ∈ λUn} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Put ℓF :=  �  x ∈ ℓE : ∥x∥F :=  ��∞  n=1 ∥x∥2  n  �1/2  < ∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By [27, Lemma 1] (see also [4, Theorem  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='37]), ℓF is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The ideal property of ℓF was asserted in [57, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For the sake of completeness, we  include a short proof below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let y = (y(k))k≥1 ∈ ℓF and x = (x(k))k≥1 ∈ ℓ∞ such that |x| ≤ |y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  We have y ∈ (∥y∥n + ε)Un for any ε > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=',  y =  N  �  k=1  skek + yn,ε,  where �N  k=1 |sk| ≤ 2n (∥y∥n + ε) , 1 ≤ N < ∞ and (yn,ε(k))k≥1 ∈ ∥y∥n+ε  2n  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Note that  x =  �  yk̸=0  sk  xk  yk  ek +  �xk  yk  yn,ε(k)  �  k≥1  ,  �  yk̸=0  ���sk  xk  yk  ��� ≤ �  yk̸=0 |sk| ≤ 2n (∥y∥n + ε) and  �  xk  yk yn,ε(k)  �  k≥1 ∈ ∥y∥n+ε  2n  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Therefore, ∥x∥n ≤  ∥y∥n for every n ≥ 1, and x ∈ ℓF with with ∥x∥F ≤ ∥y∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  To show that ℓF is symmetric, one only need to observe that for any permutation π and n ≥ 1,  we have π(W) = W and π(U) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Therefore,  ���(y(k))k≥1  ���  n =  ���(y (π(k)))k≥1  ���  n for any y ∈ ℓF  and any permutation π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  □  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let ℓE be a symmetric sequence space which does not coincide with c0 and ℓ∞  (up to equivalent norms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then, there exists a reflexive symmetric sequence space ℓF such that  ℓF ⊃ ℓE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  4A nonempty subset A of a vector space is said to circled (or balanced) whenever x ∈ A and 0 ≤ λ ≤ 1 imply  λx ∈ A) [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' A convex circled hull of A is  �  n  �  i=1  λixi : xi ∈ A for each i and  n  �  i=1  |λi| ≤ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES  7  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Since ℓE ̸= c0, ℓ∞, it follows that ℓ×  E ̸= ℓ1 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [46, Proposition 11] and [59]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1, we obtain that there exists a reflexive symmetric sequence space ℓG ⊂ ℓ×  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Therefore,  letting ℓF := ℓ×  G, which is a reflexive symmetric sequence space [32, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3] (see also [34,60,  66]), we have  ℓF = ℓ×  G ⊃ ℓ××  E  ⊃ ℓE,  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  □  Now, we prove the main theorem of this section, which is an infinite measure version of [68,  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let E(0, ∞) be a symmetric function space such that  E(0, ∞) ∩ L∞(0, ∞) ⫋ C0(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Then, there exists a symmetric KB-function space such that F(0, ∞) containing E(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' If, in  addition, E(0, 1) :=  �  f ∈ L1(0, 1) : g(x) :=  �  f(x),  if x ∈ (0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  0,  otherwise  ∈ E(0, ∞), ∥f∥E(0,1) = ∥g∥E(0,∞)  �  satisfies that  E(0, 1) ̸= L1(0, 1),  then F(0, ∞) can be chosen as a reflexive symmetric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let ℓE be the symmetric sequence space generated by E(0, ∞) by setting  ℓE :=  \uf8f1  \uf8f2  \uf8f3(αn)n≥1 :  �  n≥1  αnχ(n−1,n] ∈ E(0, ∞)  \uf8fc  \uf8fd  \uf8fe , ∥(αn)n≥1∥ℓE :=  ������  �  n≥1  αnχ(n−1,n]  ������  E(0,∞)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  (4)  By condition E(0, ∞) ∩ L∞(0, ∞) ⫋ C0(0, ∞), we obtain that ℓE ⫋ c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Indeed, E(0, ∞) ∩  L∞(0, ∞) ⫋ C0(0, ∞) implies that there exists z = µ(z) ∈ C0(0, ∞) such that µ(z) /∈ E(0, ∞) ∩  L∞(0, ∞), and therefore, by (4), we obtain that (µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' z))n≥1 ∈ c0 but (µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' z))n≥1 /∈ ℓE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2, there exists a reflexive symmetric sequence space ℓG ⊃ ℓE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Observe that ℓG  is fully symmetric [34,60,61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Now, we define a symmetric function space F1(0, ∞) by setting  F1(0, ∞) :=  �  f ∈ (L1 + L∞)(0, ∞) :  �� n  n−1  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' f)dt  �  n≥1  ∈ ℓG  �  (5)  equipped with the norm [62, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='6]  ∥f∥F1(0,∞) :=  �����  �� n  n−1  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' f)dt  �  n≥1  �����  ℓG  , f ∈ F1(0, ∞),  which also has the Fatou property and order continuous norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=', F1(0, ∞) is a symmetric KB-  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Indeed, let {xβ} be an upwards directed net in F1(0, ∞)+ satisfying supβ ∥xβ∥F1(0,∞) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  By the definition of F1(0, ∞),  ��� n  n−1 µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' xβ)dt  �  n≥1  �  β  is an upwards directed net in ℓ+  G with  supβ  ����  �� n  n−1 µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' xβ)dt  �  n≥1  ����  ℓG  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let x ∈ S(0, ∞) be such that xβ ↑ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then, we have µ(xβ) ↑  x and  � n  n−1 µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' xβ)dt ↑  � n  n−1 µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)dt for each n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By the Fatou property of ℓG, we obtain  that  �� n  n−1 µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)dt  �  n≥1 ∈ ℓG with  ����  �� n  n−1 µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)dt  �  n≥1  ����  ℓG  = supβ  ����  �� n  n−1 µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' xβ)dt  �  n≥1  ����  ℓG  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=', x ∈ F1(0, ∞) with ∥x∥F1(0,∞) = supβ ∥xβ∥F1(0,∞) = supβ ∥xβ∥F1(0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Therefore, F1(0, ∞)  has the Fatou property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The order continuity of ∥·∥F1(0,∞) can be proved by a similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Note that for any f = µ(f) ∈ E(0, ∞), we have  �  n≥1  µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' f)χ(n−1,n] ≤ µ(f) and  �  n≥1  µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' f)χ(n−1,n] ∈ E(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  8  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' HUANG AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' SUKOCHEV  By (4), we have (µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' f))n≥1 ∈ ℓE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Therefore,  �� n  n−1  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' f)dt  �  n≥2  =  �� n+1  n  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' f)dt  �  n≥1  ≤ (µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' f))n≥1 ∈ ℓE ⊂ ℓG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Therefore,  �� n  n−1 µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' f)dt  �  n≥1 ∈ ℓG, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=',  f  (5)  ∈ F1(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  That is, E(0, ∞) ⊂ F1(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The proof for the first assertion is complete by defining F(0, ∞) :=  F1(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Assume that E(0, 1) ̸= L1(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  If E(0, 1) = L∞(0, 1), then, defining G(0, 1) := L2(0, 1)  (which is a reflexive symmetric function space), we have G(0, 1) ⊃ E(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' If E(0, 1) ̸= L∞(0, 1),  then, by [68, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3], there exists a reflexive symmetric function space G(0, 1) such that  E(0, 1) ⊂ G(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' We define  G(0, ∞) :=  �  f ∈ (L1 + L∞)(0, ∞) : µ(f)χ(0,1) ∈ G(0, 1)  �  equipped with the norm5  ∥f∥G :=  inf  f=u+v, u∈G(0,∞),v∈L∞(0,∞)  ���µ(u)χ(0,1)  ��  G(0,1) + ∥v∥∞  �  , f ∈ G(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  By the definition of a K¨othe dual (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [60, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='31)]), it is readily verified that the K¨othe dual  G(0, ∞)× of G(0, ∞) is given by  �  f ∈ (L1 + L∞)(0, ∞) :  ��µ(f)χ(0,1)  ��  G(0,1)× < ∞  �  ∩ L1(0, ∞)  =  �  f ∈ (L1 + L∞)(0, ∞) : µ(f)χ(0,1) ∈ G(0, 1)× and µ(f)χ[1,∞) ∈ L1(0, ∞)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  equipped with the norm  ∥f∥G(0,∞)× := max  ���µ(f)χ(0,1)  ��  G(0,1)× , ∥f∥1  �  , f ∈ G(0, ∞)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  For any decreasing net S0(0, ∞) ⊃ E(0, ∞) ∋ fi ↓ 0, we have µ(fi) ↓i 0 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g [32, Proposition  2(iv)]) and therefore, there exists a constant c(G) depending on G(0, 1) only [6, Chapter I, Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8] such that  ∥fi∥G(0,∞)× ≤ c(G)  ���µ(fi)χ(0,1)  ��  G(0,1)× + ∥µ(fi)∥L1(0,∞)  �  ↓i 0,  (6)  which shows that G(0, ∞)× has an order continuous norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Now, we define  F(0, ∞) := G(0, ∞) ∩ F1(0, ∞),  equipped with ∥·∥F := max  �  ∥·∥G , ∥·∥F1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Since both G(0, ∞) and F1(0, ∞) have the Fatou  property, it follows that F(0, ∞) has the Fatou property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The same reasoning as in (6) shows that  ∥·∥F is order continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Observe that [79, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3] (see also [64])  F(0, ∞)× = G(0, ∞)× + F1(0, ∞)×  is equipped with the norm  ∥f∥F (0,∞)× :=  inf  f=u+v, u∈G(0,∞)×,v∈F1(0,∞)×  �  ∥u∥G× + ∥v∥F ×  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  We claim that ∥·∥F (0,∞)× is order continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Indeed, for any decreasing net F1(0, ∞) ∋ fi ↓ 0,  we have µ(fi) ↓i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In particular, we have µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' fi) ↓i 0 for each n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By the triangle inequality,  we have  ∥fi∥F (0,∞)× ≤  ��µ(fi)χ(0,1)  ��  G(0,∞)× +  ��µ(fi)χ[1,∞)  ��  F1(0,∞)× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  5Let f, g ∈ G(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By [6, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='6], there exists A ⊂ (0, ∞) such that µ((f + g)χA) = µ(f + g)χ(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Therefore, we have ��µ(f + g)χ(0,1)  ��  G(0,1) = ∥(f + g)χA∥G(0,1) ≤ ∥fχA∥G(0,1) +∥gχA∥G(0,1) ≤ ��fχ(0,1)  ��  G(0,1) +  ��gχ(0,1)  ��  G(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Hence, ∥·∥G(0,1) generates a symmetric norm on G(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Since ∥·∥∞ is a symmetric norm, it  follows ∥·∥G is a complete symmetric norm on G(0, ∞), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [60, Chapter I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES  9  By (6), it suffices to prove that  ��µ(fi)χ[1,∞)  ��  F1(0,∞)× →i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By the definition of ∥·∥F1(0,∞), we  have  ��µ(fi)χ[1,∞)  ��  F1(0,∞)×  [32, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' 26]  =  sup  ∥x∥F1(0,∞)=1  � ∞  0  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' µ(fi)χ[1,∞))µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)dt  =  sup  ∥x∥F1(0,∞)=1  �  n≥1  � n  n−1  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' µ(fi)χ[1,∞))µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)dt  and  �  n≥1  µ(n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' fi)  � n  n−1  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)dt =  �  n≥1  µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' µ(fi)χ[1,∞))  � n  n−1  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)dt  ≤  �  n≥1  � n  n−1  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' µ(fi)χ[1,∞))µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)dt  ≤  �  n≥1  µ(n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' µ(fi)χ[1,∞))  � n  n−1  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)dt =  �  n≥1  µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' fi)  � n  n−1  µ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  By the definition of F1(0, ∞), we have (µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' fi))n≥1 ∈ ℓ×  G and  ��µ(fi)χ[1,∞)  ��  F1(0,∞)× ≤ ∥{µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' fi)}n≥1∥ℓ×  G ↓i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  By [61, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='5] (see also [66, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='14] or [34,60]), F(0, ∞) is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  □  The following lemma is folklore, it provides a characterization for symmetric spaces whose ‘tail’  is a proper subspace of C0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For the sake of completeness, we include the full proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Assume that  (1) E(M, τ) is a strongly symmetric space affiliated with a von Neumann algebra equipped  with a semifinite infinite faithful normal trace τ, with cE = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  (2) M is an atomless von Neumann algebra equipped with a semifinite infinite faithful normal  trace τ or M is atomic equipped with a semifinite infinite faithful normal trace τ such that  all minimal projections having equal trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Then, E(M, τ)×× ⊂ S0(M, τ) if and only if E(M, τ) ∩ M ⫋ C0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' (⇒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Assume, contrapositively, that E(M, τ) ∩ M is not a proper subspace of C0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Note that, if E(M, τ) contains some element x such that µ(∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x) > 0, then E(M, τ) ⊃ M ⊃  C0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' If for every x ∈ E(M, τ), we have µ(∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x) = 0, then E(M, τ) ∩ M ⊂ C0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Therefore, E(M, τ) ∩ M = C0(M, τ) or E(M, τ) ∩ M = M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=', E(M, τ) ⊃ E(M, τ) ∩ M ⊃  C0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  By [29, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1] (see also [34, Chapter IV, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='12]), we have  E(M, τ)× = {y ∈ S(M, τ) : xy ∈ L1(M, τ) for all x ∈ E(M, τ)}  ⊂ {y ∈ S(M, τ) : xy ∈ L1(M, τ) for all x ∈ C0(M, τ)}  = C0(M, τ)× see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [79, Lemma 8]  =  L1(M, τ),  and  E(M, τ)×× =  �  y ∈ S(M, τ) : xy ∈ L1(M, τ) for all x ∈ E(M, τ)×�  ⊃ {y ∈ S(M, τ) : xy ∈ L1(M, τ) for all x ∈ L1(M, τ)} = M,  which is a contradiction with the assumption that E(M, τ)×× ⊂ S0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  (⇐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Assume by contradiction that E(M, τ)×× ⊃ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  By [32, Proposition 28] (see also [29,31]), there exists a fully symmetric function space G(0, ∞)  having the Fatou property such that  E(M, τ)×× = G(M, τ)  10  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' HUANG AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' SUKOCHEV  with ∥y∥E(M,τ)×× = ∥µ(y)∥G(0,∞), y ∈ E(M, τ)××.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Since G(M, τ) ⊃ M, it follows that  G(0, ∞) ⊃ L∞(0, ∞) and therefore, we have ∥·∥∞ ≥ c(G) ∥·∥G for some constant c(G) depending  on G(0, ∞) only [6, Chapter I, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Therefore, we have  E(M, τ) ∩ M  (3)  ⊃ (L1 ∩ L∞)(M, τ)  ∥·∥E ∩ M  (2)  = (L1 ∩ L∞)(M, τ)  ∥·∥G ∩ M  ⊃ (L1 ∩ L∞)(M, τ)  ∥·∥∞ = C0(M, τ),  which is a contradiction with the assumption that E(M, τ) ∩ M ⫋ C0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  □  Below, we establish a noncommutative version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Observe that the result of  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='5 below holds for both finite and infinite traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let M be a von Neumann algebra equipped with a semifinite faithful normal  trace τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let E(M, τ) be a strongly symmetric space such that E(M, τ)×× ⊂ S0(M, τ)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then,  there exists a symmetric KB-function space F(0, ∞) such that  E(M, τ) ⊂ F(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Without loss of generality, we may assume that cE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' If τ(1) < ∞, then the assertion is  trivial as E(M, τ)  (3)  ⊂ L1(M, τ) and we can take F(0, ∞) := L1(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Now, we assume that τ(1) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  By [32, Proposition 28] (see also [29]), there exists a fully symmetric function space G(0, ∞)  having the Fatou property such that  E(M, τ)× = G(M, τ)  with ∥y∥E×(M,τ) = ∥µ(y)∥G(0,∞), y ∈ E(M, τ)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Let F1(0, ∞) := G(0, ∞)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In particular, F1(0, ∞) has the Fatou property [32, Theorem 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Observe that F1(M, τ) = E(M, τ)×× [32, Theorem 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' We have [32, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3]  E(M, τ) ⊂ E(M, τ)×× = F1(M, τ) ⊂ S0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  In particular, F1(0, ∞) ⊂ S0(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Since both F1(0, ∞) and L∞(0, ∞) have the Fatou property,  it follows that F1(0, ∞) ∩ L∞(0, ∞) has the Fatou property and therefore, F1(0, ∞) ∩ L∞(0, ∞) ̸=  C0(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3, there exists a symmetric KB-function space F(0, ∞) such that F1(0, ∞) ⊂  F(0, ∞), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Main results  The starting point of this section is the following important result due to Akemann, Dodds and  Gamlen [3, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2] (see also [2, Corollary II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='9 and Theorem IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3] and [75]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' It should be  viewed as a noncommutative version of Grothendieck’s theorem [41], which states that an arbitrary  bounded operator from C(K) into a weakly sequentially complete Banach space is necessarily  weakly compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' A bounded linear map from a C∗-algebra into a weakly sequentially complete  Banach space is weakly compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Recall that a noncommutative strongly symmetric space is a KB-space if and only if it is weakly  sequentially complete [33, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='5], see also [28,31,32,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' We have the following consequence  of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let M be a von Neumann algebra equipped with a semifinite faithful normal trace  τ, and let E(M, τ) be a strongly symmetric KB-space affiliated with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then every bounded map  from a C∗-algebra into E(M, τ) is weakly compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3 below is a consequence of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2 and [28, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='9] (see also [35,  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For a Banach space X, we denote by BX the unit ball of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  6Observe that if τ(1) < ∞, then S0(M, τ) = S(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES  11  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let M be a von Neumann algebra equipped with a semifinite faithful normal  trace τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Assume that E(M, τ) is a strongly symmetric KB-space such that E(M, τ)× ⊂ S0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Let A be a C∗-subalgebra M and let T be a bounded linear operator from A into E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then,  the set  BMT (BA)BM := {aT (x)b : a, b ∈ BM, x ∈ BA}  is relatively weakly compact in E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Without loss of generality, we may assume that cE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Since every weakly compact operator sends bounded sets into weakly compact ones, we infer  from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2 that the set  T (BA)  is relatively weakly compact in E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Since E(M, τ) has the Fatou property, it follows that  E(M, τ) = E(M, τ)×× ⊂ S0(M, τ) (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Since E(M, τ)× ⊂ S0(M, τ), it follows  from [28, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='9] (or to [35, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4]) that  �  y∈BA  Ω(T (y))  is relatively σ(E××, E×)-compact (equivalently, relatively σ(E, E×)-compact, or weakly com-  pact) in E(M, τ), where Ω(x) := {z ∈ (L1 + L∞)(M, τ) : z ≺≺ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Since BMT (BA)BM ⊂  �  y∈BA Ω(T (y)), it follows that  BMT (BA)BM  is weakly compact in E(M, τ)  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Assume that E(M, τ) is a strongly symmetric space with cE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Note that, if  τ(1) < ∞, then the condition E(M, τ)× ⊂ S0(M, τ) holds for any symmetric space E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  If τ(1) = ∞, then E(M, τ)× ⊂ S0(M, τ) if and only if E(M, τ)∩M ⫌ L1(M, τ)∩M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Indeed,  assume that E(M, τ) ∩ M ⫌ L1(M, τ) ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then, there exists an element 0 ≤ z ∈ E(M, τ) ∩ M  but z /∈ L1(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In particular, we have τ(z) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By the definition of K¨othe duals, we infer  that 1 /∈ E(M, τ)×, which, in turn, implies that E(M, τ)× ⊂ S0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  On the other hand,  assume by contradiction that L1(M, τ) ∩ M is not a proper subspace of E(M, τ) ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By (3), we  have L1(M, τ) ∩ M ⊂ E(M, τ) ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Therefore, we obtain that E(M, τ) ∩ M = L1(M, τ) ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  By the fact that E(M, τ)  (3)  ⊂ (L1 + L∞)(M, τ), we obtain that for any element x ∈ E(M, τ),  µ(x)χ(0,1) ∈ L1(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Hence, all elements in E(M, τ) belong to L1(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Therefore, E(M, τ)× ⊃  L1(M, τ)× = M [29, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' That is, E(M, τ)× ̸⊂ S0(M, τ), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Let U(A) denote the set of all unitary elements in a C∗-algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let M be a von Neumann algebra equipped with a semifinite faithful normal  trace τ and let A be a unital C∗-subalgebra M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Assume that E(M, τ) is a strongly symmetric  KB-space such that E(M, τ)× ⊂ S0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let δ : A → E(M, τ) be a derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then,  {δ(u)u∗ | u ∈ U(A)}  is relatively weakly compact in E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Consequently, the closure conv{δ(u)u∗ | u ∈ U(A)}  ∥·∥E  of the convex hull is weak compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By Ringrose’s theorem [73, Theorem 2], δ is bounded from (A, ∥·∥∞) into (E(M, τ), ∥·∥E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3, {δ(u)u∗ | u ∈ U(A)} is relatively weakly compact in E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The second assertion follows from the Krein–Smulian theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  □  The following lemma shows that derivations into a “large” symmetric space are inner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let M be a von Neumann algebra with a faithful normal semifinite trace τ and  let A be a unital C∗-subalgebra of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let E(M, τ) a strongly symmetric KB-space such that  E(M, τ)× ⊂ S0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  For every derivation δ : A → E(M, τ), there exists an element a ∈  conv{δ(u)u∗ | u ∈ U(A)}  ∥·∥E such that δ = δa on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In particular, ∥a∥ ≤ ∥δ∥A→E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  12  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' HUANG AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' SUKOCHEV  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Without loss of generality, we may assume that the carrier projection cE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  For every u ∈ U(A), we have δ(u) ∈ E(M, τ), and therefore we can define the mapping  αu : E(M, τ) −→ E(M, τ), by setting  αu(x) := uxu∗ + δ(u)u∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  For every u, v ∈ U(A), we have  αu(αv(x)) = uvxv∗u∗ + uδ(v)v∗u∗ + δ(u)u∗  = (uv)x(uv)∗ + uδ(v)v∗u∗ + δ(u)vv∗u∗  = (uv)x(uv)∗ + δ(uv)(uv)∗ = αuv(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  In addition, the equality δ(1) = δ(12) = 2δ(1) implies that δ(1) = 0, and therefore α1(x) = x, x ∈  E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Thus, α is an action of the group U(A) on E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  We claim that the set  K := conv {δ(u)u∗ | u ∈ U(A)}  ∥·∥E  is invariant with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Since δ(u)u∗ = αu(0), it follows that k00 := {δ(u)u∗ | u ∈ U(A)} is  an orbit of 0 with respect to α, and therefore, is an invariant subset with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In addition,  for any positive scalars s and t with s + t = 1, we have  αu(s · x + t · y) = s · uxu∗ + t · uyu∗ + (s + t) · δ(u)u∗ = s · αu(x) + t · αu(y),  ∀x, y ∈ E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Hence, for every u ∈ U(A) the mapping αu is affine, which implies that conv(K00) is also an  invariant subset with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Now, the equality αu(x) − αu(y) = u(x − y)u∗, x, y ∈ E(M, τ)  implies that every αu, u ∈ U(A), is an isometry on E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Hence, K is an invariant subset with  respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Furthermore, the fact that αU is an isometry combined with [25, Chapter V, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='7]  implies that the family {αu : u ∈ U(A)} is a noncontracting family of affine mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Clearly,  αu is weakly continuous for every u ∈ U(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  On the other hand, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='5, K is weakly compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Thus, the set K and the family  {αu : u ∈ U(A)} satisfy the assumptions of the Ryll-Nardzewski fixed-point theorem [25, Chapter  V, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Hence, there exists a point a ∈ K fixed with respect to α, that is, we have  a = αu(a) = uau∗ + δ(u)u∗  for every u ∈ U(A) and ∥a∥E ≤ ∥δ∥A→E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Therefore au = ua + δ(u) for every u ∈ U(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Thus,  δ(u) = [a, u] for every u ∈ U(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Since every element x ∈ A is a linear combination of four  elements from U(A) [51, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='7], we obtain that δ = δa on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  □  A symmetric function space E(M, τ) is said to have the Levi property [1, Definition 7], or a  monotone complete norm, or to satisfy property (B) [58, Chapter X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4], if for every upwards directed  net {xβ} in E(M, τ)+, satisfying supβ ∥xβ∥E < ∞, there exists an element x ∈ E(M, τ)+ such  that xβ ↑ x in E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' It is well known that if a norm is Levi, then necessarily it is also weak  Fatou [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='89], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=', there exists a constant K ≥ 1 such that  0 ≤ xβ ↑ x ⇒ ∥x∥E ≤ K lim  β ∥xβ∥E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For a symmetric function space E(0, ∞), it has the Levi property if and only if it  has an equivalent symmetric norm having the Fatou property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The (⇐) implication is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For  the (⇒) implication, one only need to observe that for any 0 ≤ x ∈ E(0, ∞), there exists a net  (xβ)β ⊂ (L1 ∩ L∞)(0, ∞) such that xβ ↑ x [32, Proposition 1(vii)], and therefore,  1  K ∥x∥E ≤ sup  β  ∥xβ∥E  (2)  = sup  β  ∥xβ∥E×× = ∥x∥E××  [32, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3]  ≤  ∥x∥E ,  and (E(0, ∞)××, ∥·∥E××) has the Fatou property (see [30, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1] and [60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Recall that E(0, ∞) has the Fatou property if and only E(0, ∞) is isometric to E(0, ∞)××.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' We  obtain that E(0, ∞) has the Levi property if and only E(0, ∞) = E(0, ∞)×× (up to equivalent  norms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES  13  The following theorem is the main result of the present paper, which resolves problems consid-  ered in a numbers of papers, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [11,12,14,43,45,47,55,83] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let M be a von Neumann algebra with a faithful normal semifinite trace τ and let  A be a C∗-subalgebra of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' If E(M, τ) is a strongly symmetric space of τ-compact operators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=',  E(M, τ) ⊂ S0(M, τ)7) having the Fatou property (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=', the Levi property), then every derivation  δ : A → E(M, τ) is inner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' That is, there exists an element a ∈ conv {δ(u)u∗ | u ∈ U(A)}  tτ ⊂  E(M, τ) with ∥a∥E ≤ ∥δ∥A→E (∥a∥E ≤ c ∥δ∥A→E for some constant c depending on E(0, ∞)  only) such that δ = δa on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Without loss of generality, we may assume that the carrier projection cE = 1 and assume  that A is unital (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' the proof of [11, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='7, we may assume that  E(M, τ) has the Fatou property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Since E(M, τ) has the Fatou property and E(M, τ) ⊂ S0(M, τ), it follows that E(M, τ)×× ⊂  S0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='5, there exists a symmetric KB-function space F(0, ∞) ⫋ (L1 +  C0)(0, ∞) such that  E(M, τ) ⊂ F(M, τ) ⊂ S0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Without loss of generality, we may assume that L2(0, ∞) ⊂ F(0, ∞) by replacing F(0, ∞) with  L2(0, ∞) + F(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In particular, F(0, ∞) ∩ L∞(0, ∞) ⫌ L1(0, ∞) ∩ L∞(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4,  F(0, ∞)× ⊂ S0(0, ∞), and therefore, F(M, τ)× ⊂ S0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Note that δ(A) ⊂ E(M, τ) ⊂ F(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='6, there is an element  a ∈ conv {δ(u)u∗ | u ∈ U(A)}  ∥·∥F  such that δ = δa on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Hence, there exists a sequence  (xn)∞  n=1 ⊂ conv {δ(u)u∗ | u ∈ U(A)}  such that ∥xn − a∥F →n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Since F(M, τ) is a symmetric space, it follows from [32, Proposition  20] that xn →tτ a as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  By Ringrose’s theorem [73], we have that δ : (A, ∥·∥∞) → (E(M, τ), ∥·∥E) is a bounded map-  ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Since E(M, τ) has the Fatou property, it follows that the closed ball (E(M, τ), ∥·∥E) with  radius ∥δ∥A→E is closed in S(M, τ) with respect to the local measure topology (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Noticing that every element xn, n ≥ 1 belongs to the ball of radius ∥δ∥A→E in E(M, τ) and  xn → a in the local measure topology, we conclude that a ∈ E(M, τ) with ∥a∥E ≤ ∥δ∥A→E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  □  Recall that a symmetric function space having the Fatou/Levi property generates a noncommu-  tative strongly symmetric space having the Fatou/Levi property (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [61, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8]  and [32, Theorem 54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' We have the following consequence of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Assume that E(0, ∞) ⊂ S0(0, ∞) is a symmetric function space having the Fatou  property (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=', the Levi property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then, a derivation  δ : A → E(M, τ)  is necessarily inner for any von Neumann algebra M with a semifinite faithful normal trace τ  and a C∗-subalgebra A of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' That is, there exists an element a ∈ conv {δ(u)u∗ | u ∈ U(A)}  tτ ⊂  E(M, τ) with ∥a∥E ≤ ∥δ∥A→E (∥a∥E ≤ c ∥δ∥A→E for some constant c depending on E(0, ∞)  only) such that δ = δa on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='9 provides an alternative proof for the resolution of the derivation problem due  to Bunce and Paschke [18], which differs from those in [5, 70] involving L-embeddedness of the  predual of a von Neumann algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Moreover, we provide new information that implementing  element a of the derivation can be found in the measure (or any symmetric norm topology which  is weaker than the L1-topology) closure of the convex hull of {δ(u)u∗ | u ∈ U(A)}, which seems  to be more natural in fixed point theory and derivation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  7If τ(1) < ∞, then E(M, τ) ⊂ S0(M, τ) holds for any symmetric space E(M, τ) affiliated with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  14  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' HUANG AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' SUKOCHEV  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let M be a semifinite von Neumann algebra and let A be a C∗-subalgebra of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Then, any derivation from A into M∗ is inner, and there exists an element a ∈ conv{δ(u)u∗ | u ∈ U(A)}  tτ  with ∥a∥L1 ≤ ∥δ∥A→L1 such that δ = δa on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The following corollary significantly extends results from [11,43,55] for derivations into symmet-  ric ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' On the other hand, it shows that [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3] holds true even for C∗-subalgebras  rather than von Neumann subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let M be a semifinite von Neumann algebra equipped with a semifinite faithful  normal trace τ and let A be a C∗-subalgebra of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let E(M, τ) be an ideal of C0(M, τ) having  strongly symmetric norm and the Levi property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then, every derivation from A into E(M, τ) is  inner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  For a von Neumann algebra M equipped with a faithful normal tracial state, M ⊂ S0(M, τ)  and M has the Fatou/Levi property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Therefore, we have the following consequence of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  [19, Section 5] [80] Let M be a von Neumann algebra equipped with a faithful  normal tracial state and let A be a C∗-subalgebra of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then every derivation from A into M is  inner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Outer derivations  In this section, we present several examples of outer derivations with values in a symmetric  space failing the Levi property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The study of derivations into ideals of a von Neumann algebra was initiated Johnson and  Parrott [48] by showing that derivations from an abelian/properly infinite von Neumann subalgebra  of B(H) into the algebra K(H) of all compact operators on H are inner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' This result may be well  seen as a precursor to Question 1 in the setting of symmetrically normed ideals in M = B(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  After that, the innerness of derivations into ideals of a von Neumann algebra have been studied  extensively in [11,12,15,16,20,40,43,55,72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Recall that the Fatou/Levi property can not be dropped in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8, which has been  demonstrated in [77, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8], [11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8] and [44, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The following  theorem demonstrates that there exist outer derivations into E(M, τ) ∩ M whenever the ideal  E(M, τ) ∩ M does not have the Levi property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let M be a non-finite factor equipped with a semifinite faithful normal trace τand  let E(M, τ) be a symmetric space affiliated with M such that E(M, τ)∩M does not have the Levi  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then, there exist outer derivations from the C∗-algebra C0(M, τ) into E(M, τ) ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Since M is a factor, it follows that it is either atomless or is atomic with all atoms having  equal trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Therefore,  E(M, τ)××  is a fully symmetric space having the Fatou property (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [29], [34, Chapter IV, Theorem  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4], [32, Remarks 5 and 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Hence,  E(M, τ)×× ∩ M,  equipped with the norm ∥x∥E(M,τ)××∩M = max {∥x∥E×× , ∥x∥∞} , x ∈ E(M, τ)×× ∩ M, has the  Fatou property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' On the other hand, the lack of the Levi property of E(M, τ) ∩ M implies that  there exists a net (xi)i∈I of positive elements in E(M, τ) ∩ M increasing to 0 ≤ x ∈ M such  that sup ∥xi∥E < ∞ but x /∈ E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By the Fatou property of E(M, τ)×× and E(M, τ) ⊂  E(M, τ)×× [32, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3], we have x ∈ E(M, τ)×× ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Since E(M, τ)∩M does not have the Levi property, it follows that E(M, τ)∩M ̸= M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Hence,  E(M, τ) ⊂ S0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' There are two possible cases:  (1) If E(M, τ)×× ⊂ C0(M, τ), then we consider δx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  We claim that δx is a non-inner derivation from C0(M, τ) into E(M, τ) ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Recall that  ∥z∥E  (2)  = ∥z∥E×× , ∀z ∈ F(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES  15  For every y ∈ C0(M, τ), the projections e|y|(ε, ∞) and e|y|(ε1, ∞) are τ-finite for every ε > ε1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Thus, we have  ���xye|y|(ε, ∞) − xye|y|(ε1, ∞)  ���  E ≤  ���xye|y|(ε1, ε]  ���  E =  ���xye|y|(ε1, ε]  ���  E×× ≤ ε ∥x∥E×× ,  which shows that  �  xye|y|( 1  n, ∞)  �  n≥1 is a Cauchy sequence in E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  On the other hand,  xye|y|( 1  n, ∞) → xy in measure, which implies that xy ∈ E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Similarly, yx ∈ E(M, τ)  and therefore δx(C0(M, τ)) ⊂ E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Moreover, x ∈ M and C0(M, τ) ⊂ M imply that  δx(C0(M, τ)) ⊂ E(M, τ) ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Finally, if there exists an operator x′ ∈ E(M, τ) ∩ M such that δx = δx′, then  x − x′ commutes with all elements in C0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Noticing that M is the closure of C0(M, τ) in the weak operator topology (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [62, Definition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8]), we obtain that  x − x′ commutes with all elements in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  However, M is a factor and therefore x − x′ ∈ C1, which is a contradiction with the assumption  that E(M, τ)×× ⊂ C0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  (2) If E(M, τ)×× ⊃ M, then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4, we have E(M, τ) ∩ M ⊃ C0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Recall that  E(M, τ) ∩ M ̸= M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' We have E(M, τ) ∩ M = C0(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Now, the existence of outer derivations  follows from [11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='8] immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  □  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let E(0, ∞) be a symmetric function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' If E(0, ∞)∩L∞(0, ∞) does not have  the Levi property, then for any infinite factor M equipped with a semifinite faithful normal trace  τ, there exists outer derivations from the C∗-algebra C0(M, τ) into E(M, τ) ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1, it suffices to prove that E(M, τ) ∩ M does not have the Levi property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Since E(0, ∞) ∩ L∞(0, ∞) has no Levi property, it follows that (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='7)  E(0, ∞) ∩ L∞(0, ∞) ̸= (E(0, ∞) ∩ L∞(0, ∞))×× = E(0, ∞)×× ∩ L∞(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  That is, there exists 0 ≤ x = µ(x) ∈ E(0, ∞)×× ∩ L∞(0, ∞) such that x /∈ E(0, ∞) ∩ L∞(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Observe that  �  n≥1  µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)χ(n−1,n] ≤ µ(x) ∈ E(0, ∞)×× ∩ L∞(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  There exists y ∈ M such that  µ(y) =  �  n≥1  µ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' x)χ(n−1,n] ≤ µ(x) ∈ E(0, ∞)×× ∩ L∞(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  However, we have  µ(y) ≥ µ  �  µ(x) − µ(x)χ(0,1]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Since µ(x)χ(0,1] ∈ E(0, ∞)∩L∞(0, ∞) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' [32, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='245] or [60]), it follows that y /∈ E(M, τ)∩M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  This shows that E(M, τ)∩M does not have the Levi property when M a semifinite infinite (atomic  or atomless) factor, see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  □  Recall that the dilation operator σs, s > 0, on S(0, ∞) is defined by [60–62]  (σs)f(t) = f  � t  s  �  , ∀f ∈ S(0, ∞), s ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  For the case of unbounded operators, the example of an outer derivation is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The  result presented below is perhaps the first example of an outer derivation into a symmetric space  affiliated with a finite von Neumann algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let M be the von Neumann algebra which is the closure of ⊕n≥1M2 in the weak  operator topology equipped with the trace τ := ⊕n≥1  1  2n+1 Tr2, where Tr2 is the standard trace of  M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let E(0, 1) be a symmetric function space without the Levi property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then, there exist outer  derivations from the C∗-algebra A := ⊕M2n  ∥·∥∞ into E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  16  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' HUANG AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' SUKOCHEV  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' The lack of the Levi property of E(0, 1) shows that E(0, 1) ̸= E(0, 1)×× (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Re-  mark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='7), and there exists a positive function a ∈ E(0, 1)×× but a /∈ E(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Define  b :=  �  n≥1  µ  �  1  2n−1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' a  �  χ[ 1  2n ,  1  2n−1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  We have  b ≤ a ≤  �  n≥1  µ  � 1  2n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' a  �  χ[ 1  2n ,  1  2n−1 ) = σ2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Therefore, b ∈ E(0, 1)×× but b /∈ E(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Now, we define a self-adjoint element y by setting  z :=  �  n≥1  an (pn1 − pn2) , an = µ  � 1  2n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' a  �  , n ≥ 1,  where pn1 and pn2 are mutually orthogonal non-trivial projections in the n-th direct summand  M2, and the series is taken in the measure topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' In particular, µ(z) = σ2(b) and therefore,  z ∈ E(M, τ)×× but z /∈ E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  We claim that δz(A) ⊂ E(M, τ)b, the closure of M in E(M, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Indeed, since δz has image in  E××(M, τ) (generated by E(0, 1)××), it follows from Ringrose’s theorem [73] that δz is bounded  from (A, ∥·∥∞) into (E××(M, τ), ∥·∥E××).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let pn be the identity of the n-th direct summand M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  For any element y ∈ A, we have yn := y �n  k=1 pk → y in the uniform norm topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Note that  ynz, zyn ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Hence, for any n > m, we have  ∥δz(yn) − δz(ym)∥E(M,τ) = ∥µ(δz(yn) − δz(ym))∥E(0,1)  (2)  = ∥µ(δz(yn) − δz(ym))∥E(0,1)××  = ∥δz(yn) − δz(ym)∥E××(M,τ)  ≤ 2 ∥yn − ym∥∞ ∥z∥E××(M,τ) → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Therefore, (δz(yn))n≥1 is a Cauchy sequence in E(M, τ), which implies that (δz(yn))n≥1 converges  to some element in E(M, τ) (in particular, (δz(yn))n≥1 is tτ-convergent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' On the other hand,  δz(yn) →tτ δz(y) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Therefore, ∥δz(yn)n≥1 − δz(y)∥E(M,τ) →n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Hence, δz(y) ∈  E(M, τ) for all y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Assume that there exist z1 ∈ E(M, τ) such that δz1 = δz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=', z1 − z commutes with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' That  is, z1 −z = �  n≥1 bnpn for some sequence (bn)n≥1 of complex numbers, where the series converges  in the measure topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' However,  E(0, 1) ∋ µ(z1) = µ  \uf8eb  \uf8edz +  �  n≥1  bnpn  \uf8f6  \uf8f8 = µ  \uf8eb  \uf8ed�  n≥1  an (pn1 − pn2) +  �  n≥1  bnpn  \uf8f6  \uf8f8  = µ  \uf8eb  \uf8ed�  n≥1  (an + bn)pn1 − (an − bn)pn2  \uf8f6  \uf8f8  ≥ µ  \uf8eb  \uf8ed�  n≥1  max{|an + bn|, |an − bn|}pn1  \uf8f6  \uf8f8  ≥ µ  \uf8eb  \uf8ed�  n≥1  anpn1  \uf8f6  \uf8f8 = µ  \uf8eb  \uf8ed�  n≥1  anpn2  \uf8f6  \uf8f8 = σ1/2µ(z),  which implies that z ∈ E(M, τ) [60, Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' This is a contradiction to the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  □  The following result is an immediate consequence of Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='9, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='2 and Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  DERIVATIONS INTO NONCOMMUTATIVE SYMMETRIC SPACES  17  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' For a given symmetric function space E(0, ∞) ⊂ S0(0, ∞), the following two  statements are equivalent:  (1) for any von Neumann algebra M equipped with a semifinite faithful normal trace τ and  any C∗-subalgebra A of M, derivations δ : A → E(M, τ) are necessarily inner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  (2) E(0, ∞) has the Levi property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  The following result is an immediate consequence of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='9 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Let CE be a symmetric ideal of compact operators in B(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Then, the following  two statements are equivalent:  (1) for any C∗-subalgebra A of B(H), derivations δ : A → CE are necessarily inner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  (2) the commutative core ℓE of CE has the Levi property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  References  [1] Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
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+page_content=' Abramovich, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
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+page_content=' Sukochev, Non-trivial derivations on commutative regular algebras, Extracta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
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+page_content=' Sukochev, Continuity of derivations of algebras of locally measurable operators, Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='  Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
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+page_content=' Sukochev, Continuous derivations on algebras of locally measurable operators are inner,  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
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+page_content='  Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology, Harbin,  150001, China  Email address: jinghao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='huang@hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='cn  School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, Australiaa  Email address: f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='sukochev@unsw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
+page_content='au' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfaQf4/content/2301.03874v1.pdf'}
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+Higher Order Dynamics in the Replicator Equation Produce a Limit Cycle in
+Rock-Paper-Scissors
+Christopher Griffin1, ∗ and Rongling Wu2, 3, †
+1Applied Research Laboratory, The Pennsylvania State University, University Park, PA 16802
+2Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China
+3Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
+Recent work has shown that pairwise interactions may not be sufficient to fully model ecological
+dynamics in the wild. In this letter, we consider a replicator dynamic that takes both pairwise
+and triadic interactions into consideration using a rank three tensor. We study this new nonlinear
+dynamics using a generalised rock-paper-scissors game whose dynamics are well understood in the
+classic replicator sense. We show that the addition of higher-order dynamics leads to the creation
+of a subcritical Hopf bifurcation and consequently an unstable limit cycle. It is known that this
+kind of behaviour cannot occur in the pairwise replicator in any three strategy games, showing the
+effect higher-order interactions can have on the resulting dynamics of the system. We numerically
+characterise parameter regimes in which limit cycles exist and discuss possible ways to generalise
+this approach to studying higher-order interactions.
+I.
+INTRODUCTION
+Pairwise interactions are frequently assumed in constructing dynamical systems models of ecological systems [1–4].
+This is a foundational assumption of classical evolutionary game theory in which the replicator dynamic is built
+from a matrix game [5–7]. In this case, pairwise interactions of players define the fitness function that governs the
+dynamics. This simplifying assumption is violated by the intrinsic existence of high-order interactions (HOI’s), for
+which there is growing evidence [8–18]. A higher-order interaction occurs when three or more species act together as a
+subgroup to shape community behaviour[2, 8–12, 19–21]. In the case of random interactions in ecological communities,
+the occurrence of HOIs can alter the established relationship between diversity and stability [10], leading to new
+evolutionary trajectories.
+In this letter we show how to modify the standard game matrix replicator dynamic by defining fitness in terms of
+both a game matrix (for pairwise interactions) and a game tensor (for triadic interactions). This work is related to the
+prior work of Gokhale and Traulsen [22] who studied evolutionary games with multiple (more than two) strategies and
+multiple players. In this work, they study the maximum number of mixed strategy equilibria that may emerge in the
+replicator dynamic. More recently, Zhang et al. [23] study multiplayer evolutionary games in the context of asymmetric
+payoffs, which we do not consider. In particular, we study the resulting dynamics on a generalised rock-paper-scissors
+game (RPS) [5]. We show that the resulting dynamics arising from triadic interactions are fundamentally different
+from those dynamics arising from RPS in the replicator equation with only pairwise interactions by showing that the
+HOI’s lead to the emergence of a limit cycle.
+Rock-paper-scissors (and its generalisations) has been studied in multiple different contexts [24–43] and there are
+at least two schools of partially compatible dynamics. Postlewaite and Rucklidge [26, 28–30, 33, 38] have extensively
+studied a dynamical systems model of RPS in both spatial and aspatial cases. Their dynamics are distinct [44] from
+the dynamics arising from the replicator equation as given in [5–7]. We do not consider their dynamics but instead
+focus on the replicator equation. In particular, Zeeman [45] showed that RPS dynamics under the replicator exhibit a
+degenerate Hopf bifurcation and cannot produce limit cycles. More generally, Zeeman showed that no three strategy
+game can produce a limit cycle under the replicator dynamics. However, since every dynamical system arising from
+the replicator equation is diffeomorphic to a generalised Lotka-Volterra system, limit cycles and chaos may emerge
+for games with more than three strategies.
+The main results of this letter are: (i) We show how to incorporate triadic interactions using a simple rank 3
+tensor and note that the replicator equation ansatz does not change as a result of these interactions. (ii) We show
+(numerically) that in generalised RPS with HOI’s a subcritical Hopf bifurcation occurs and a limit cycle emerges for
+appropriate parameter choices. This behaviour must be caused by the HOI’s, since such dynamics cannot emerge
+with only pairwise interactions. (iii) We use a statistical analysis to construct a three-dimensional bifurcation surface,
+∗ griffinch@psu.edu
+† ronglingwu@mail.tsinghua.edu.cn
+arXiv:2301.02518v1  [nlin.AO]  6 Jan 2023
+
+2
+showing parameter regions where the (unique) interior fixed point is stable and admits a limit cycle, is stable with no
+limit cycle, and is unstable.
+II.
+MODEL
+Let ∆n−1 be the n − 1 dimensional unit simplex embedded in Rn composed of vectors u = ⟨u1, . . . , un⟩ so that
+u1 + · · · + un = 1 and ui ≥ 0 for all i ∈ {1, . . . , n}. Consider a simple ecological model with n species where Ui is the
+size of the population of species i ∈ {1, . . . , n}. Let
+M =
+n
+�
+i=1
+Ui
+be the total number of organisms present. The following observation can be found in [46] and sets the stage for our
+analysis. Let ui = Ui/M be the proportion of biomass composed of species i. If fi : ∆n−1 → R is the fitness function
+so that:
+˙Ui = Uifi(u),
+then it is easy to show (with the quotient rule) that:
+˙ui = ui
+�
+f(u − ¯f(u)
+�
+,
+(1)
+where
+¯f(u) =
+n
+�
+i=1
+uifi(u).
+Eq. (1) is the standard replicator [5–7]. In classical evolutionary game theory, we assume there is a (payoff) matrix
+A ∈ Rn×n so that fi(u) = eT
+i Au, where ei is the standard basis vector in Euclidean space in direction i. This model
+assumes simple binary interactions among species with the payoff resulting from an interaction between species i
+and species j given by Aij and going to the first (row) player in an imaginary game. Consequently single matrix
+evolutionary game theory makes the most sense when the game described by A is symmetric. When A is skew-
+symmetric, the game is not only symmetric but also zero-sum [47] as in the case of rock-paper-scissors, whose
+interaction matrix is cyclic [48, 49] with
+A =
+�
+�
+0
+−1
+1
+1
+0
+−1
+−1
+1
+0
+�
+� .
+(2)
+In this matrix, index 1 corresponds to rock, index 2 to paper, and index 3 to scissors. We will maintain this convention
+when we analyse a generalisation of this game.
+Paik and Griffin [49] (as well as Itoh [50], Bogoyavlensky [51] and Vesselov & Shabbat[52] in different contexts)
+show that for generalisations of rock-paper-scissors to cyclic skew-symmetric game matrices with ±1 entries, the
+resulting dynamics are Liouville-Arnold integrable. This is an (almost) immediate consequence of the fact that all
+zero-sum replicator dynamics arise from a non-linear Poisson bracket as shown in [48]. In particular, this automatically
+precludes the emergence of more interesting dynamics in simple cyclic games with Zeeman [45] going so far as to show
+that in all three-strategy matrix games, the replicator dynamics cannot exhibit limit cycles.
+In the case of higher-order interactions, it suffices to redefine fi(u) to be
+fi(u) = eT
+i Au + uT Biu,
+(3)
+where Bi is a quadratic form (matrix) that takes two copies of the population proportion vector u and returns a payoff
+to species i that occurs when one member of species i meets two members of the population. In general, we could
+think of Bi as being a slice of a (0, 3) tensor B : ∆n−1 × ∆n−1 × ∆n−1 → R. Using Eq. (3), it is clear that Eq. (1)
+(the replicator equation) still holds for the dynamics of ui, but now incorporates payoffs from three-way interactions.
+In what follows, we will define
+Bi,j,k = pijkAij + qijkAik,
+
+3
+for constants of proportionality pijk and qijk. That is we assume that the three-way payoff is composed of payoffs from
+binary interactions that are scaled to model the ecological effects of the more complex interactions. In our analysis
+of generalised rock-paper-scissors, we choose pijk and qijk so that the Nash equilibria remain fixed points but the
+stability of the fixed points change, showing that the Folk theorem of evolutionary game theory (see [7]) need not hold
+in models of higher order interactions. It is left as a question for future work whether there are sufficient conditions
+on the tensor B that ensure the Nash equilibria of the game matrix A are preserved as fixed points in three-way
+dynamics.
+Before proceeding to the analysis of a specific case, we note that Eq. (3) could be generalised to include as many
+as n-way interactions by using higher rank tensors [22]. However, it is unlikely that such interactions are biologically
+meaningful. Statistical tests for these kinds of interactions are discussed in [53].
+III.
+HIGHER ORDER ROCK-PAPER-SCISSORS
+The remainder of this paper is dedicated to showing that higher-order-interactions in a generalised rock-paper-
+scissors game produce dynamics not seen when only pairwise interactions are modelled. For the remainder of this
+paper fix a parameterised payoff matrix
+A =
+�
+�
+0
+−b − 1
+a + 1
+a + 1
+0
+−b − 1
+−b − 1
+a + 1
+0
+�
+� ,
+(4)
+where we assume a, b ≥ 0. This matrix will govern the payoff from simple pairwise interactions. We now define the
+tensor B using its slices so that
+B1 =
+�
+�
+0
+1
+2(−b − 1)
+1
+2(a + 1)β
+1
+2(−b − 1)
+−b − 1
+0
+1
+2(a + 1)β
+0
+(a + 1)α
+�
+�
+(5)
+B2 =
+�
+�
+(a + 1)α
+1
+2(a + 1)β
+0
+1
+2(a + 1)β
+0
+1
+2(−b − 1)
+0
+1
+2(−b − 1)
+−b − 1
+�
+�
+(6)
+B3 =
+�
+�
+−b − 1
+0
+1
+2(−b − 1)
+0
+(a + 1)α
+1
+2(a + 1)β
+1
+2(−b − 1)
+1
+2(a + 1)β
+0
+�
+� .
+(7)
+Here we assume α ∈ (1, 2] and β ∈
+�
+0, 1
+2
+�
+for ecological sensibility. The reasoning behind defining B in this way is
+justified by considering the payoff associated to rock (index 1). If a rock plays against two other rocks, it receives no
+payoff – as expected from Eq. (4). If it meets two papers or a paper and a rock, then it receives the same net negative
+payoff −b − 1 as if it met one paper. On the other hand, if it meets two scissors, then its payoff increases by a factor
+of α ≥ 1. If a rock meets a rock and scissors, then the rocks split the payoff and each receives a payoff decreased by a
+factor β. Finally if a rock meets both a paper and a scissors, then mutual destruction leads to no net payoff for any
+player. The same logic is used for all other players.
+There are several ways to realise the logic described above in the tensor B. Eqs. (5) to (7) enforce symmetry in
+each slice and preserve a generalised cyclic property. Note that B1 can be recovered from B2 by rotating the rows
+and columns of B1 down and to the right (just as row i + 1 can be obtained from row i in A by rotating to the right).
+Thus B is a circulant (or Toeplitz) tensor.
+It is straightforward to see that ¯ui = ei (i ∈ {1, 2, 3}) are fixed points as is the symmetric Nash equilibrium
+¯u = ⟨ 1
+3, 1
+3, 1
+3⟩. The fact that A is a circulant matrix and B is a circulant tensor ensures that these are the only
+equilibria in the system occurring in ∆2 and ¯u is the only interior equilibrium.
+A.
+Fixed Point Analysis
+The eigenvalues of the Jacobian matrix at any fixed point ¯ui (i ∈ {1, 2, 3}) are
+Λ = {0, −2(b + 1), (a + 1)(α + 1)}.
+Based on the assumptions on the values of a, b and α, these are hyperbolic saddles, just as in the case of ordinary
+rock-paper-scissors in the replicator dynamic [7]. This is in contrast to the behaviour at the interior fixed point.
+
+4
+There are one real and two complex eigenvalues of the Jacobian matrix at the interior fixed point ¯u given by
+Λ =
+�
+r1
+9 , r2 ± i
+√
+3q
+18
+�
+,
+where
+r1 = 2 + 5b − α − β − a(α + β + 3)
+and
+r2 = 1 + a(β − 2α − 3) + 4b − 2α + β
+and
+q = 9 + 6b + 2α + β + a(3 + 2α + β).
+The first (real) eigenvalue is extraneous since the dynamics evolve on ∆2. Assuming α, β and b are free, r2 = 0 when
+a = a∗ = 1 + 4b + β − 2α
+3 + 2α − β
+.
+(8)
+When a = a∗ + ϵ, then
+r2 = ϵ(β − 2α − 3).
+If follows from our assumptions on α and β that if ϵ > 0, then a > a∗ and r2 < 0 and ¯u∗ is attracting. Otherwise, ¯u∗
+is repelling. For ϵ = 0, the system has two pure imaginary eigenvalues, which satisfies the first requirement of Hopf’s
+theorem (see [54], Page 152). Our assumptions that α ∈ (1, 2] and β ∈
+�
+0, 1
+2
+�
+ensures that
+r′
+2(ϵ) = (β − 2α − 3) ̸= 0.
+Thus the real part of the eigenvalues must cross the imaginary axis with non-zero speed, satisfying the second criterion
+of Hopf’s theorem. Thus the system exhibits a Hopf bifurcation.
+It is interesting to note that if we adhere to our ecological assumptions that α ∈ (1, 2] and β ∈
+�
+0, 1
+2
+�
+, then the
+ordinary rock-paper-scissors matrix has r2 < 0. Thus the mixed strategy equilibrium becomes stable for all values of
+α and β. To see this note that if b = 0 and we require a∗ = 0, then we must have α = (1 + β)/2, which takes values
+between 1
+2 and 3
+4, outside of the ecologically sensible range.
+B.
+Numerical Illustration of a Subcritical Limit Cycle
+We can show numerically that a subcritical limit cycle emerges for example parameters. For the remainder of this
+section, let b = 0. In our initial limit cycle construction we assume α = 2 and β = 1
+2. In this case, the interaction
+of (e.g.) a rock with two scissors doubles the payoff to rock, while two rocks interacting with a scissors will split
+the payoff associated to the interaction. We set ϵ =
+1
+100, implying the interior fixed point will be stable. In Fig. 1
+(left) we see an (approximated) limit cycle surrounding a stable interior fixed point. Outside the limit cycle, flow
+goes to the boundary. To complete the numeric proof, we apply the Poincar´e-Bendixson theorem. In Fig. 1 (right)
+we compute the distance from the three trajectories to the interior fixed point after a ternary transform. This is a
+true representation of the distance seen in the trajectories in Fig. 1 (left). We can see that the distance from the
+(approximated) limit cycle oscillates around a constant mean. The trajectory outside the limit cycle increases its
+distance to the interior fixed point, while the trajectory inside the limit cycle decreases its distance to the interior
+fixed point as expected. Thus the numerically identified limit cycle is subcritical.
+When ϵ < 0, the interior fixed point becomes unstable (as expected) and the limit cycle vanishes as shown in
+Fig. 2 (left). All trajectories approach the boundary as confirmed numerically in Fig. 2 (right), which again shows
+the normalised distance from the trajectory to the interior fixed point.
+When ϵ is increased beyond a certain value, the limit cycle disappears while the interior fixed point remains stable.
+This is illustrated in Fig. 3. In Fig. 3 (right) we show that the distance from any trajectory to the interior fixed point
+collapses to zero as time goes to infinity.
+We can numerically approximate the value of ϵ where the limit cycle disappears for arbitrarily values of α ∈ (1, 2]
+and β ∈ (0, 1
+2]. To do this, use the following steps:
+1. Input: α, β.
+
+5
+R
+P
+S
+0
+50
+100
+150
+200
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Time
+Normalized Distance
+IC=(0.7,0.15,0.15)
+IC=(0.4,0.117,0.483)
+IC=(0.4,0.2,0.4)
+FIG. 1. (Top) Example trajectories with a subcritical limit cycle are shown. (Bottom) The distance of the trajectories to the
+interior fixed point as a function of time. This proves numerically that there is a subcritical Hopf bifurcation in these dynamics
+via the Poincar´e-Bendixson theorem.
+2. Initialize: ϵ =
+1
+100. Compute a = a∗ + ϵ, where a∗ is given by Eq. (8).
+3. Numerically integrate the time-inverted dynamics:
+˙ui = −ui
+�
+f(u − ¯f(u)
+�
+,
+with u0 = ⟨0.99, 0.005, 0.005⟩. This accomplishes two things: (i) The modified dynamics invert the stability of
+the limit cycle and (ii) the initial condition starts the trajectory close to the boundary. Let u(t) be the resulting
+solution.
+4. Compute d(t) = ∥u(t) − ˆ¯u∥2.
+5. Fit d(t) ∼ γ0 + γ1t.
+6. If γ1 < 0 with p-value less than 0.001, then statistically the trajectory is decaying toward a limit cycle and we
+set ϵ := ϵ +
+1
+1000 and goto Step 3. Otherwise, stop and return ϵ.
+Using this technique, we can generate a surface showing the dependence of ϵ on the parameters α and β (see Fig. 4).
+Before continuing our analysis, we note that a similar procedure can be used to identify limit cycles within this
+dynamical system. Mathematica code to generate all figures is provided in the SI.
+There are five outlier points that are significantly above or below the trend surface. These are most likely due to
+numerical errors (from the integration) or sensitivity to the p-value test used in Line 6 of the procedure. The surface
+
+6
+R
+P
+S
+0
+50
+100
+150
+200
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Time
+Normalized Distance
+IC=(0.7,0.15,0.15)
+IC=(0.4,0.117,0.483)
+IC=(0.4,0.2,0.4)
+FIG. 2. (Top) Sett ϵ below zero destabilises the interior fixed point and destroys the limit cycle (as expected). All trajectories
+approach the boundary.
+(Bottom) The distances of the trajectories to the interior fixed point increase as the trajectories
+approach the boundary of ∆2.
+was fit using non-outlier points and is given by the expression:
+ˆϵmax(α, β) ≈ 0.063αβ − 0.075α − 0.191β + 0.217.
+The adjusted r2 of this fit is 0.98 showing good explanatory power. The parameter table for the model is
+Est.
+SE
+t-Stat p-val
+1
+0.217
+0.002 104.8
+1.09 × 10−170
+α
+−0.0751 0.001 −56.4 5.446 × 10−121
+β
+−0.191
+0.007 −28.7 3.098 × 10−71
+αβ 0.063
+0.004 14.7
+3.00 × 10−33
+This statistical analysis shows that the the maximum period of a limit cycle (correlated to ϵmax) is negatively propor-
+tional to both α and β. However, since interaction term is non-trivial the interaction between α and β can interact to
+increase the ϵmax before the limit cycle disappears. Fig. 4 illustrates a type of bifurcation diagram (for b = 0) with a
+triple (α, β, ϵ) generating a limit cycle if it falls below the surface and the interior fixed point being globally attracting
+if (α, β, ϵ) is above the surface.
+
+7
+R
+P
+S
+0
+50
+100
+150
+200
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Time
+Normalized Distance
+IC=(0.7,0.15,0.15)
+IC=(0.4,0.117,0.483)
+IC=(0.4,0.2,0.4)
+FIG. 3. (Top) If ϵ is increased beyond a critical value, the limit cycle is destroyed and all trajectories approach the interior
+fixed point. (Bottom) The distances of the trajectories to the interior fixed point decrease as expected.
+FIG. 4. Compute (α, β, ϵmax) points and the resulting fit surface showing the lifetime (ϵ) of the limit cycle as a function of α
+and β. Outlier points are most likely due to numerical errors or sensitivity to the p-value test.
+
+β
+Emax
+α8
+IV.
+CONCLUSION
+In this letter we used the replicator equation to model higher-order interactions between three (or more) species
+through the introduction of an interaction tensor. We showed that the dynamics that results in this case are fun-
+damentally different from the ordinary binary interactions modelled by the usual replicator dynamics with a payoff
+matrix. In particular, we studied a generalised rock-paper-scissors model and showed the existence of a non-degenerate
+Hopf bifurcation that allows a subcritical limit cycle to emerge when higher order interactions are allowed. This is in
+contrast to classical results on three strategy games in the ordinary replicator equation in which limit cycles cannot
+emerge as a result of the degeneracy of a similar Hopf bifurcation.
+While this letter provides a framework for modelling higher-order interactions within a replicator framework, there
+are several possible future directions. Generalising the interaction rules used to construct the three-way interaction
+tensor could lead to a generalisation of the Folk theorem in certain cases. It also would be interesting to introduce
+a spatial component as in [44, 55–58] and determine what spatial dynamics emerge as a result of these higher-order
+interactions.
+V.
+ACKNOWLEDGEMENTS
+C.G. was supported in part by the National Science Foundation under grant DMS-1814876. R.W. was supported
+by the Talents Grants at Beijing Forestry University.
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diff --git a/cdE0T4oBgHgl3EQfnwHq/content/tmp_files/load_file.txt b/cdE0T4oBgHgl3EQfnwHq/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf,len=576
+page_content='Higher Order Dynamics in the Replicator Equation Produce a Limit Cycle in  Rock-Paper-Scissors  Christopher Griffin1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' ∗ and Rongling Wu2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' †  1Applied Research Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' PA 16802  2Beijing Institute of Mathematical Sciences and Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Beijing 101408,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' China  3Yau Mathematical Sciences Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Beijing 100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' China  Recent work has shown that pairwise interactions may not be sufficient to fully model ecological  dynamics in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In this letter, we consider a replicator dynamic that takes both pairwise  and triadic interactions into consideration using a rank three tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' We study this new nonlinear  dynamics using a generalised rock-paper-scissors game whose dynamics are well understood in the  classic replicator sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' We show that the addition of higher-order dynamics leads to the creation  of a subcritical Hopf bifurcation and consequently an unstable limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' It is known that this  kind of behaviour cannot occur in the pairwise replicator in any three strategy games, showing the  effect higher-order interactions can have on the resulting dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' We numerically  characterise parameter regimes in which limit cycles exist and discuss possible ways to generalise  this approach to studying higher-order interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  INTRODUCTION  Pairwise interactions are frequently assumed in constructing dynamical systems models of ecological systems [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  This is a foundational assumption of classical evolutionary game theory in which the replicator dynamic is built  from a matrix game [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In this case, pairwise interactions of players define the fitness function that governs the  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' This simplifying assumption is violated by the intrinsic existence of high-order interactions (HOI’s), for  which there is growing evidence [8–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' A higher-order interaction occurs when three or more species act together as a  subgroup to shape community behaviour[2, 8–12, 19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In the case of random interactions in ecological communities,  the occurrence of HOIs can alter the established relationship between diversity and stability [10], leading to new  evolutionary trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  In this letter we show how to modify the standard game matrix replicator dynamic by defining fitness in terms of  both a game matrix (for pairwise interactions) and a game tensor (for triadic interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' This work is related to the  prior work of Gokhale and Traulsen [22] who studied evolutionary games with multiple (more than two) strategies and  multiple players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In this work, they study the maximum number of mixed strategy equilibria that may emerge in the  replicator dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' More recently, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' [23] study multiplayer evolutionary games in the context of asymmetric  payoffs, which we do not consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In particular, we study the resulting dynamics on a generalised rock-paper-scissors  game (RPS) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' We show that the resulting dynamics arising from triadic interactions are fundamentally different  from those dynamics arising from RPS in the replicator equation with only pairwise interactions by showing that the  HOI’s lead to the emergence of a limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  Rock-paper-scissors (and its generalisations) has been studied in multiple different contexts [24–43] and there are  at least two schools of partially compatible dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Postlewaite and Rucklidge [26, 28–30, 33, 38] have extensively  studied a dynamical systems model of RPS in both spatial and aspatial cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Their dynamics are distinct [44] from  the dynamics arising from the replicator equation as given in [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' We do not consider their dynamics but instead  focus on the replicator equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In particular, Zeeman [45] showed that RPS dynamics under the replicator exhibit a  degenerate Hopf bifurcation and cannot produce limit cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' More generally, Zeeman showed that no three strategy  game can produce a limit cycle under the replicator dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' However, since every dynamical system arising from  the replicator equation is diffeomorphic to a generalised Lotka-Volterra system, limit cycles and chaos may emerge  for games with more than three strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  The main results of this letter are: (i) We show how to incorporate triadic interactions using a simple rank 3  tensor and note that the replicator equation ansatz does not change as a result of these interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (ii) We show  (numerically) that in generalised RPS with HOI’s a subcritical Hopf bifurcation occurs and a limit cycle emerges for  appropriate parameter choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' This behaviour must be caused by the HOI’s, since such dynamics cannot emerge  with only pairwise interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (iii) We use a statistical analysis to construct a three-dimensional bifurcation surface,  ∗ griffinch@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='edu  † ronglingwu@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='cn  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='02518v1  [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='AO]  6 Jan 2023  2  showing parameter regions where the (unique) interior fixed point is stable and admits a limit cycle, is stable with no  limit cycle, and is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  MODEL  Let ∆n−1 be the n − 1 dimensional unit simplex embedded in Rn composed of vectors u = ⟨u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' , un⟩ so that  u1 + · · · + un = 1 and ui ≥ 0 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Consider a simple ecological model with n species where Ui is the  size of the population of species i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Let  M =  n  �  i=1  Ui  be the total number of organisms present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' The following observation can be found in [46] and sets the stage for our  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Let ui = Ui/M be the proportion of biomass composed of species i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' If fi : ∆n−1 → R is the fitness function  so that:  ˙Ui = Uifi(u),  then it is easy to show (with the quotient rule) that:  ˙ui = ui  �  f(u − ¯f(u)  �  ,  (1)  where  ¯f(u) =  n  �  i=1  uifi(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (1) is the standard replicator [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In classical evolutionary game theory, we assume there is a (payoff) matrix  A ∈ Rn×n so that fi(u) = eT  i Au, where ei is the standard basis vector in Euclidean space in direction i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' This model  assumes simple binary interactions among species with the payoff resulting from an interaction between species i  and species j given by Aij and going to the first (row) player in an imaginary game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Consequently single matrix  evolutionary game theory makes the most sense when the game described by A is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' When A is skew-  symmetric, the game is not only symmetric but also zero-sum [47] as in the case of rock-paper-scissors, whose  interaction matrix is cyclic [48, 49] with  A =  �  �  0  −1  1  1  0  −1  −1  1  0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  (2)  In this matrix, index 1 corresponds to rock, index 2 to paper, and index 3 to scissors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' We will maintain this convention  when we analyse a generalisation of this game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  Paik and Griffin [49] (as well as Itoh [50], Bogoyavlensky [51] and Vesselov & Shabbat[52] in different contexts)  show that for generalisations of rock-paper-scissors to cyclic skew-symmetric game matrices with ±1 entries, the  resulting dynamics are Liouville-Arnold integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' This is an (almost) immediate consequence of the fact that all  zero-sum replicator dynamics arise from a non-linear Poisson bracket as shown in [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In particular, this automatically  precludes the emergence of more interesting dynamics in simple cyclic games with Zeeman [45] going so far as to show  that in all three-strategy matrix games, the replicator dynamics cannot exhibit limit cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  In the case of higher-order interactions, it suffices to redefine fi(u) to be  fi(u) = eT  i Au + uT Biu,  (3)  where Bi is a quadratic form (matrix) that takes two copies of the population proportion vector u and returns a payoff  to species i that occurs when one member of species i meets two members of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In general, we could  think of Bi as being a slice of a (0, 3) tensor B : ∆n−1 × ∆n−1 × ∆n−1 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (3), it is clear that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (1)  (the replicator equation) still holds for the dynamics of ui, but now incorporates payoffs from three-way interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  In what follows, we will define  Bi,j,k = pijkAij + qijkAik,  3  for constants of proportionality pijk and qijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' That is we assume that the three-way payoff is composed of payoffs from  binary interactions that are scaled to model the ecological effects of the more complex interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In our analysis  of generalised rock-paper-scissors, we choose pijk and qijk so that the Nash equilibria remain fixed points but the  stability of the fixed points change, showing that the Folk theorem of evolutionary game theory (see [7]) need not hold  in models of higher order interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' It is left as a question for future work whether there are sufficient conditions  on the tensor B that ensure the Nash equilibria of the game matrix A are preserved as fixed points in three-way  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  Before proceeding to the analysis of a specific case, we note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (3) could be generalised to include as many  as n-way interactions by using higher rank tensors [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' However, it is unlikely that such interactions are biologically  meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Statistical tests for these kinds of interactions are discussed in [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  HIGHER ORDER ROCK-PAPER-SCISSORS  The remainder of this paper is dedicated to showing that higher-order-interactions in a generalised rock-paper-  scissors game produce dynamics not seen when only pairwise interactions are modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' For the remainder of this  paper fix a parameterised payoff matrix  A =  �  �  0  −b − 1  a + 1  a + 1  0  −b − 1  −b − 1  a + 1  0  �  � ,  (4)  where we assume a, b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' This matrix will govern the payoff from simple pairwise interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' We now define the  tensor B using its slices so that  B1 =  �  �  0  1  2(−b − 1)  1  2(a + 1)β  1  2(−b − 1)  −b − 1  0  1  2(a + 1)β  0  (a + 1)α  �  �  (5)  B2 =  �  �  (a + 1)α  1  2(a + 1)β  0  1  2(a + 1)β  0  1  2(−b − 1)  0  1  2(−b − 1)  −b − 1  �  �  (6)  B3 =  �  �  −b − 1  0  1  2(−b − 1)  0  (a + 1)α  1  2(a + 1)β  1  2(−b − 1)  1  2(a + 1)β  0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  (7)  Here we assume α ∈ (1, 2] and β ∈  �  0, 1  2  �  for ecological sensibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' The reasoning behind defining B in this way is  justified by considering the payoff associated to rock (index 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' If a rock plays against two other rocks, it receives no  payoff – as expected from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' If it meets two papers or a paper and a rock, then it receives the same net negative  payoff −b − 1 as if it met one paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' On the other hand, if it meets two scissors, then its payoff increases by a factor  of α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' If a rock meets a rock and scissors, then the rocks split the payoff and each receives a payoff decreased by a  factor β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Finally if a rock meets both a paper and a scissors, then mutual destruction leads to no net payoff for any  player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' The same logic is used for all other players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  There are several ways to realise the logic described above in the tensor B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (5) to (7) enforce symmetry in  each slice and preserve a generalised cyclic property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Note that B1 can be recovered from B2 by rotating the rows  and columns of B1 down and to the right (just as row i + 1 can be obtained from row i in A by rotating to the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  Thus B is a circulant (or Toeplitz) tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  It is straightforward to see that ¯ui = ei (i ∈ {1, 2, 3}) are fixed points as is the symmetric Nash equilibrium  ¯u = ⟨ 1  3, 1  3, 1  3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' The fact that A is a circulant matrix and B is a circulant tensor ensures that these are the only  equilibria in the system occurring in ∆2 and ¯u is the only interior equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  Fixed Point Analysis  The eigenvalues of the Jacobian matrix at any fixed point ¯ui (i ∈ {1, 2, 3}) are  Λ = {0, −2(b + 1), (a + 1)(α + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  Based on the assumptions on the values of a, b and α, these are hyperbolic saddles, just as in the case of ordinary  rock-paper-scissors in the replicator dynamic [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' This is in contrast to the behaviour at the interior fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  4  There are one real and two complex eigenvalues of the Jacobian matrix at the interior fixed point ¯u given by  Λ =  �  r1  9 , r2 ± i  √  3q  18  �  ,  where  r1 = 2 + 5b − α − β − a(α + β + 3)  and  r2 = 1 + a(β − 2α − 3) + 4b − 2α + β  and  q = 9 + 6b + 2α + β + a(3 + 2α + β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  The first (real) eigenvalue is extraneous since the dynamics evolve on ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Assuming α, β and b are free, r2 = 0 when  a = a∗ = 1 + 4b + β − 2α  3 + 2α − β  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  (8)  When a = a∗ + ϵ, then  r2 = ϵ(β − 2α − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  If follows from our assumptions on α and β that if ϵ > 0, then a > a∗ and r2 < 0 and ¯u∗ is attracting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Otherwise, ¯u∗  is repelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' For ϵ = 0, the system has two pure imaginary eigenvalues, which satisfies the first requirement of Hopf’s  theorem (see [54], Page 152).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Our assumptions that α ∈ (1, 2] and β ∈  �  0, 1  2  �  ensures that  r′  2(ϵ) = (β − 2α − 3) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  Thus the real part of the eigenvalues must cross the imaginary axis with non-zero speed, satisfying the second criterion  of Hopf’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Thus the system exhibits a Hopf bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  It is interesting to note that if we adhere to our ecological assumptions that α ∈ (1, 2] and β ∈  �  0, 1  2  �  , then the  ordinary rock-paper-scissors matrix has r2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Thus the mixed strategy equilibrium becomes stable for all values of  α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' To see this note that if b = 0 and we require a∗ = 0, then we must have α = (1 + β)/2, which takes values  between 1  2 and 3  4, outside of the ecologically sensible range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  Numerical Illustration of a Subcritical Limit Cycle  We can show numerically that a subcritical limit cycle emerges for example parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' For the remainder of this  section, let b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In our initial limit cycle construction we assume α = 2 and β = 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In this case, the interaction  of (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=') a rock with two scissors doubles the payoff to rock, while two rocks interacting with a scissors will split  the payoff associated to the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' We set ϵ =  1  100, implying the interior fixed point will be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 1  (left) we see an (approximated) limit cycle surrounding a stable interior fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Outside the limit cycle, flow  goes to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' To complete the numeric proof, we apply the Poincar´e-Bendixson theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 1 (right)  we compute the distance from the three trajectories to the interior fixed point after a ternary transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' This is a  true representation of the distance seen in the trajectories in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 1 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' We can see that the distance from the  (approximated) limit cycle oscillates around a constant mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' The trajectory outside the limit cycle increases its  distance to the interior fixed point, while the trajectory inside the limit cycle decreases its distance to the interior  fixed point as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Thus the numerically identified limit cycle is subcritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  When ϵ < 0, the interior fixed point becomes unstable (as expected) and the limit cycle vanishes as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 2 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' All trajectories approach the boundary as confirmed numerically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 2 (right), which again shows  the normalised distance from the trajectory to the interior fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  When ϵ is increased beyond a certain value, the limit cycle disappears while the interior fixed point remains stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 3 (right) we show that the distance from any trajectory to the interior fixed point  collapses to zero as time goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  We can numerically approximate the value of ϵ where the limit cycle disappears for arbitrarily values of α ∈ (1, 2]  and β ∈ (0, 1  2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' To do this, use the following steps:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Input: α, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  5  R  P  S  0  50  100  150  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='5  Time  Normalized Distance  IC=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='7,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='15,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='15)  IC=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='117,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='483)  IC=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='4)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (Top) Example trajectories with a subcritical limit cycle are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (Bottom) The distance of the trajectories to the  interior fixed point as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' This proves numerically that there is a subcritical Hopf bifurcation in these dynamics  via the Poincar´e-Bendixson theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Initialize: ϵ =  1  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Compute a = a∗ + ϵ, where a∗ is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Numerically integrate the time-inverted dynamics:  ˙ui = −ui  �  f(u − ¯f(u)  �  ,  with u0 = ⟨0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='99, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='005⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' This accomplishes two things: (i) The modified dynamics invert the stability of  the limit cycle and (ii) the initial condition starts the trajectory close to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Let u(t) be the resulting  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Compute d(t) = ∥u(t) − ˆ¯u∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Fit d(t) ∼ γ0 + γ1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' If γ1 < 0 with p-value less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='001, then statistically the trajectory is decaying toward a limit cycle and we  set ϵ := ϵ +  1  1000 and goto Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Otherwise, stop and return ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  Using this technique, we can generate a surface showing the dependence of ϵ on the parameters α and β (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  Before continuing our analysis, we note that a similar procedure can be used to identify limit cycles within this  dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Mathematica code to generate all figures is provided in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  There are five outlier points that are significantly above or below the trend surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' These are most likely due to  numerical errors (from the integration) or sensitivity to the p-value test used in Line 6 of the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' The surface  6  R  P  S  0  50  100  150  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='6  Time  Normalized Distance  IC=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='7,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='15,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='15)  IC=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='117,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='483)  IC=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='4)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (Top) Sett ϵ below zero destabilises the interior fixed point and destroys the limit cycle (as expected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' All trajectories  approach the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  (Bottom) The distances of the trajectories to the interior fixed point increase as the trajectories  approach the boundary of ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  was fit using non-outlier points and is given by the expression:  ˆϵmax(α, β) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='063αβ − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='075α − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='191β + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  The adjusted r2 of this fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='98 showing good explanatory power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' The parameter table for the model is  Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  SE  t-Stat p-val  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='217  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='002 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='09 × 10−170  α  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='0751 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='001 −56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='446 × 10−121  β  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='191  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='007 −28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='098 × 10−71  αβ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='063  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='004 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='00 × 10−33  This statistical analysis shows that the the maximum period of a limit cycle (correlated to ϵmax) is negatively propor-  tional to both α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' However, since interaction term is non-trivial the interaction between α and β can interact to  increase the ϵmax before the limit cycle disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 4 illustrates a type of bifurcation diagram (for b = 0) with a  triple (α, β, ϵ) generating a limit cycle if it falls below the surface and the interior fixed point being globally attracting  if (α, β, ϵ) is above the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  7  R  P  S  0  50  100  150  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='30  Time  Normalized Distance  IC=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='7,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='15,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='15)  IC=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='117,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='483)  IC=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='4)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (Top) If ϵ is increased beyond a critical value, the limit cycle is destroyed and all trajectories approach the interior  fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' (Bottom) The distances of the trajectories to the interior fixed point decrease as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Compute (α, β, ϵmax) points and the resulting fit surface showing the lifetime (ϵ) of the limit cycle as a function of α  and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Outlier points are most likely due to numerical errors or sensitivity to the p-value test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  β  Emax  α8  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  CONCLUSION  In this letter we used the replicator equation to model higher-order interactions between three (or more) species  through the introduction of an interaction tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' We showed that the dynamics that results in this case are fun-  damentally different from the ordinary binary interactions modelled by the usual replicator dynamics with a payoff  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' In particular, we studied a generalised rock-paper-scissors model and showed the existence of a non-degenerate  Hopf bifurcation that allows a subcritical limit cycle to emerge when higher order interactions are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' This is in  contrast to classical results on three strategy games in the ordinary replicator equation in which limit cycles cannot  emerge as a result of the degeneracy of a similar Hopf bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  While this letter provides a framework for modelling higher-order interactions within a replicator framework, there  are several possible future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' Generalising the interaction rules used to construct the three-way interaction  tensor could lead to a generalisation of the Folk theorem in certain cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' It also would be interesting to introduce  a spatial component as in [44, 55–58] and determine what spatial dynamics emerge as a result of these higher-order  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
+page_content=' was supported in part by the National Science Foundation under grant DMS-1814876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfnwHq/content/2301.02518v1.pdf'}
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diff --git a/eNE2T4oBgHgl3EQfxQh6/content/tmp_files/2301.04109v1.pdf.txt b/eNE2T4oBgHgl3EQfxQh6/content/tmp_files/2301.04109v1.pdf.txt
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@@ -0,0 +1,3329 @@
+Matching calipers and the precision of index
+estimation
+Ben B. Hansen∗
+January 11, 2023
+Abstract
+This paper characterizes the precision of index estimation as it car-
+ries over into precision of matching. In a model assuming Gaussian
+covariates and making best-case assumptions about matching quality,
+it sharply characterizes average and worst-case discrepancies between
+paired differences of true versus estimated index values. In this op-
+timistic setting, worst-case true and estimated index differences de-
+cline to zero if p = o[n/(log n)], the same restriction on model size
+that is needed for consistency of common index models.
+This re-
+mains so as the Gaussian assumption is relaxed to sub-gaussian, if
+in that case the characterization of paired index errors is less sharp.
+The formula derived under Gaussian assumptions is used as the basis
+for a matching caliper. Matching such that paired differences on the
+estimated index fall below this caliper brings the benefit that after
+matching, worst-case differences onan underlying index tend to 0 if
+p = o{[n/(log n)]2/3}. (With a linear index model, p = o[n/(log n)]
+suffices.) A proposed refinement of the caliper condition brings the
+same benefits without the sub-gaussian condition on covariates. When
+strong ignorability holds and the index is a well-specified propensity
+or prognostic score, ensuring in this way that worst-case matched dis-
+crepancies on it tend to 0 with increasing n also ensures the consistency
+of matched estimators of the treatment effect.
+∗This work has benefitted from comments of Jake Bowers, Joshua Errickson, Mark
+Fredrickson, Xuming He, Peter Schochet, Stilian Stoev and Lan Wang. Responsibility
+rests with the author for any shortcomings that remain.
+1
+arXiv:2301.04109v1  [stat.ME]  10 Jan 2023
+
+Key words and phrases:
+Matching, caliper, overlap, positivity, propensity
+score, prognostic score
+1
+Introduction
+In preparing a matched observational study, estimation of a treatment propensity
+briefly takes center stage, as covariates are chosen and a model specification is
+selected. These models quickly recede from view once propensity score estimates
+have been extracted from them, despite their carrying essential information about
+those estimates’ likely precision. The situation is little different in matching on
+prognostic or principal stratum scores: sampling variability of the model standing
+behind a matching index is rarely so much as even appraised. We seem to take it
+for granted that errors of estimation of a matching index can’t possibly be so large
+as to threaten the integrity of matching.
+Propensity matching is understood to be a large sample technique, as are
+logistic and other regression methods typically used for index model estimation,
+and classical asymptotics may seem to encourage inattention to index estimation
+error.
+As treatment/covariate samples (zi, ⃗xi) accumulate from any reasonable
+distribution of fixed dimension p+1, one expects errors of estimation of the index,
+{|⃗xi ˆβ −⃗xiβn| : i}, to be increasingly negligible, decreasing with or near n−1/2, just
+as |ˆβ − βn|2 = O(n−1/2). The problem is that the matching canon discourages
+parsimony in propensity modeling (Rubin and Thomas, 1996), and fixed-p large
+sample theory may describe non-parsimonious models poorly.
+Increasing-p asymptotics for logistic regression and similar techniques are avail-
+able (Portnoy, 1988; He and Shao, 2000), if less widely known. Given that p =
+o[(log n)/n], they deliver |ˆβ − βn|2 = OP [(p/n)1/2], not OP (n−1/2), as reviewed in
+§ 2 below. This suggests a still larger order, p/n1/2, for errors of form ⃗X(ˆβ − βn),
+�p
+j=1 X2
+j = O(p) corresponding to | ⃗X|2 =
+��p
+j=1 X2
+j
+�1/2
+= O(p1/2). While some-
+what of a simplification, the suggestion is correct in its implication that if p in-
+creases in proportion with n1/2, for example, then index estimation errors need not
+diminish even as coefficient estimation errors do. Outside of fixed-p asymptotics,
+consistency of the index model does not in itself make index errors asymptotically
+negligible: that calls for stronger assumptions, specialized matching techniques or
+a combination of the two.
+For control of index estimation error by way of stronger assumptions, note
+that if p is assumed to increase slowly enough, increasing dimension regression
+asymptotics resemble those with fixed p. It happens that p ∝ n1/2 is slightly too
+large for such correspondence to obtain, so it is unsurprising that fixed-p intuitions
+should fail in that regime.
+(Asymptotic normality of ˆβ, for example, calls for
+2
+
+p2 log(p) = o(n), not p = o(n1/2) [He and Shao, 2000].) But say the index model
+has sub-√n dimension, in the specific sense that p = o[(n/ log n)1/2]. Section 3.1
+of this paper shows that for consistently estimated index models with sub-gaussian
+covariates and sub-√n dimension, estimation errors of realized values of the index
+tend to zero.
+This convergence is in the strong, l∞, sense of maxi≤n | ⃗Xi(ˆβ −
+βn)| = oP (1), so it justifies the common practices of matching, subclassifying or
+simply trimming on the estimated propensity score, as analytic interventions to
+secure overlap assumptions of the stronger type, cl ≤ P(Z = 1| ⃗X) ≤ cu with
+[cl, cu] ⊂ (0, 1), as applied to the subset of available observations that remain after
+pruning.
+Section 3.3 goes on to study ordinary Cov(ˆβ) estimates’ adaptability to char-
+acterizing likely sizes of index estimation errors, ⃗xi(ˆβ −βn). A fitted index model’s
+Fisher information gives an estimate ˆC of Cˆβ = Cov(ˆβ), either directly or as part of
+an Eicker-Huber-White sandwich. In sub-n1/2 dimensional regimes, corresponding
+standard errors s. e.(⃗xˆβ) = [⃗x ˆC⃗x′]1/2 will be seen to estimate closely1 the sampling
+variabilities Var1/2[(⃗x − ¯⃗x)(ˆβ − βn)]. As the sub-√n condition is relaxed to sub-
+n (p = o[n/ log(n)]), information- and sandwich-based estimators underestimate
+Cov(ˆβ). This limits their utility for inference about βn, and according their behav-
+ior outside of sub-√n regimes has received less study. It does not follow, however,
+that they are are ill-suited to inform the selection of index-based matches. We
+explore conditions under which analytic Cov(ˆβ)-estimators continue to character-
+ize sampling variabilities of a linearization of ˆβ, offering a basis for estimators
+capturing the better part of Var[⃗x(ˆβ − βn)]. In both regimes, the largest values
+of {Var1/2[⃗xi(ˆβ − βn)] : i} may be well separated from the rest, in themselves ap-
+preciably increasing E[maxi≤n | ⃗Xi(ˆβ − βn)|]. Results presented in Section 4 are
+helpful for identifying the worst offenders, subjects i with large s. e.[⃗xi ˆβ]. Either
+or both of the sub-gaussian and sub-√n conditions can be relaxed, but then con-
+trol of index estimation errors necessitates that such subjects be pruned from the
+sample.
+In such circumstances, matching offers alternate practical remedies that can
+retain more of the sample.
+It helps first by shifting attention from particular
+{⃗xi ˆβ : i} or {⃗xiβn : i} to paired differences of indices, (⃗xi − ⃗xj)ˆβ or (⃗xi − ⃗xj)βn, as
+does § 4 below. While underestimates of Var1/2[(⃗xi − ⃗xj)ˆβ] may be unhelpful for
+inference about (⃗xi − ⃗xj)βn, but they can certainly inform matching procedures.
+Section 3.2 uses ˆCˆβ to associate deliberately reduced standard errors with paired
+contrasts such as (⃗xi − ⃗xj)ˆβ, so constructed that their average, the paired index
+contrast summary standard error (PIC SE), inexpensively approximates the root
+1For |⃗v|2 ̸= 0, |⃗v|−1
+2 {s. e.[⃗v(ˆβ − βn)] − Var1/2[⃗v(ˆβ − βn)]} is oP [(p/n)1/2], whereas
+|⃗v|−1
+2
+s. e.[⃗v(ˆβ − βn)] and |⃗v|−1
+2
+Var1/2[⃗v(ˆβ − βn)] are both OP [(p/n)1/2].
+3
+
+mean square of matched discrepancies on estimated scores, (⃗xi −⃗xj)ˆβ, across pairs
+{i, j} with little or no discrepancy on the true score, ⃗xiβ ≈ ⃗xjβ.
+The PIC SE tends to zero at the same rate as |ˆβ − βn|2, making it useful as
+a yardstick for matching. With sub-gaussian data and sub-√n model dimension,
+using it to set the width of a caliper on the index — permitting i’s pairing with j
+only if |(⃗xi − ⃗xj)ˆβ| ≤ cn pic se(ˆβ), with cn = 2, say — forces matched differences
+on the true index to vanish in the asymptotic limit:
+max
+1≤i∼j≤n |(⃗xi − ⃗xj)βn| P→ 0
+(where “i ∼ j” means “i is matched to j”),
+(1)
+in virtue of pic se(ˆβ) = OP [(p/n)1/2]. Indeed, under the p = o(n−1 log n) growth
+condition needed for consistency of common index models, (1) holds with non-
+constant cn, provided that cn = OP [(log n)1/2]. Such matching requirements are
+generally less likely to exclude subjects then comparable trimming rules, as they
+permit inclusion even of subjects far from the center of the distribution whenever
+the contrasting study arm has similarly situated subjects. If i is excluded from the
+matched sample for lack of counterparts j within caliper distance, it must have
+been separated from its comparison group by a distance exceeding the resolution
+of the estimate of the index. As will be seen in the data example, this resolution
+can be strikingly large, much larger than extant caliper width recommendations
+(Rosenbaum and Rubin, 1985; Rubin and Thomas, 2000; Austin, 2011; Wang et al.,
+2013); in many cases it will be much more sparing in its exclusions from the
+matched sample.
+This is fortunate, because the paper will recommend a nonstandard strength-
+ening of the requirement that |(⃗xi−⃗xj)ˆβ| be less than the designated caliper width,
+excluding potential pairs {i, j} either because |(⃗xi − ⃗xj)ˆβ| is too large or because
+s. e.[(⃗xi − ⃗xj)ˆβ] is. (Alternately put, because ⃗xi and ⃗xj are too separated on either
+the index itself or on a certain index estimator-dependent Mahalanobis distance.)
+The pairs E ⊆ {{i, j} : 1 ≤ i ̸= j ≤ n} that remain eligible by this criterion
+satisfy max{i,j}∈E |(⃗xi − ⃗xj)(ˆβ − βn)| = oP (1), even as maxi≤n | ⃗Xi(ˆβ − βn)| may no
+longer tend to 0. Thus (1) is maintained, even as the sub-gaussian assumption on
+⃗X is relaxed to a fourth moment condition, and p = o{[n/ log(n)]1/2} is relaxed
+to p = o{[n/ log(n)]2/3}. In matching on propensity scores, selecting pairs from
+within such an E ensures that the overlap criterion can be assessed in terms of
+estimated scores, because maxi∼j |(⃗xi − ⃗xj)ˆβ − (⃗xi − ⃗xj)βn| = oP (1). As the rec-
+ommended requirement can be viewed as a varying (by value of s. e.[(⃗xi − ⃗xj)ˆβ])
+limit on |(⃗xi − ⃗xj)ˆβ|, I continue to call it a caliper. In many cases, these PICSE
+calipers continue to be more inclusive than would Rosenbaum and Rubin’s (1985)
+canonical |(⃗xi − ⃗xj)ˆβ| ≤ 0.2 s. d.(xˆβ) requirement, and in all cases PICSE calipers
+4
+
+cause exclusion of a unit i only when our best estimate of the index distances from
+it to each potential counterpart j exceeds the resolution of the index’s estimation.
+2
+Context
+2.1
+Estimable index scores
+In an observational study with a treatment and a control condition, the propensity
+score is a function ⃗x �→ g−1[P(Z = 1 | ⃗X = ⃗x)], where g : ℜ → [0, 1] is continuous
+and increasing (Rosenbaum and Rubin, 1983).
+The z-on-⃗x regression is often
+assumed to follow a generalized linear model such as the logistic, P(Z = 1 | ⃗X =
+⃗x) = [1 + e−⃗xβ]−1. Taking g as that model’s link function identifies the propensity
+score with the index ⃗xβ. Rosenbaum and Rubin (1985) recommended matching on
+⃗xˆβ, not g−1(⃗xˆβ). Similarly prognostic scoring, confounder and risk scoring, and
+principal stratum scoring fit (parametric) regression models in order to extract
+indices ⃗xˆβ for use in matching, subclassification or sample trimming, if not also
+in weighting- or covariance adjustments to be applied once an analysis sample has
+been selected.
+Let R denote the dependent variable of the index model, e.g. Z for a propensity
+score or a response Y for a prognostic score. Let ˆβ be the solution of
+� n
+�
+i=1
+ψ(ri, ⃗xi; β)
+�
++ α(β) = 0
+(2)
+in β, where ψ(r, ⃗x; β) is the ℜp-valued gradient of a scalar-valued function ρ(r, ⃗x; β)
+that is convex in β. (In maximum likelihood estimation α(·) = 0, but Bayesian
+estimation [e.g., Gelman et al., 2008] and certain frequentist bias-reduction schemes
+[Firth, 1993; Kosmidis and Firth, 2009] minimize a penalized objective, in which
+cases α(·) is the penalty term’s gradient.) The index score (IS) is estimable if
+E
+�
+[
+n
+�
+i=1
+ψ(Ri, ⃗xi; β)] + α(β)
+�
+= 0
+(3)
+has a unique root βn, with supγ:|γ−βn|2≤1 |α(γ)|2 = oP (n1/2). Following He and
+Shao (2000), near-roots of (2) and/or (3) are acceptable when the equations do not
+have exact solutions, provided that there is a unique nearest root, but these exact
+or nearest roots are assumed to satisfy �p
+j=1
+��n
+i=1 ψj(ri, ⃗xi; ˆβ)
+�2
+= oP (n1/2) and
+�p
+j=1[E [�n
+i=1 ψj(Ri, ⃗xi; βn)]]2 = o(n1/2). Assume there are functions c0(·, ·, ·) and
+w(·) such that
+ψ(r, ⃗x, β) = c0(r, ⃗x, β0 + ⃗xβ)w(⃗x)(1, ⃗x)′.
+(4)
+5
+
+This structure accommodates robust (Cantoni and Ronchetti, 2001) and general-
+ized (Liang and Zeger, 1986) estimating equations as well as score functions such
+as logistic regression’s, ψ(r, ⃗x, β) = {r − [1 + exp(−β0 − ⃗xβ)]−1}(1, ⃗x)′. If there
+are pre-existing strata 1, . . . , L, with matches to be made within strata and any
+subclasses to further divide them, then (4) may be modified by replacement of β0
+with stratum-specific intercepts β1, . . . , βL. Sufficient conditions for consistency of
+an IS will be reviewed in § 2.3.
+2.2
+Sub-gaussian random variables
+A real-valued random variable V is sub-gaussian if its tails are no heavier than
+that of a centered Normal distribution: there is a finite constant sV such that for
+t > 0,
+P(V < t) ≤ exp[−t2/(2s2
+V )],
+P(V > t) ≤ exp[−t2/(2s2
+V )].
+When this holds, sV can be taken to be a constant multiple of ∥V ∥ψ2, the sub-
+gaussian norm of V . This norm is defined as the infimum of {t > 0 : E[exp(V 2/t2)] ≤
+2}, a nonempty set for sub-gaussian V , or as ∞ if V is not sub-gaussian. For vec-
+tor ⃗V , ∥⃗V ∥ψ2 = sup{∥⃗V γ∥ψ2 : |γ|2 = 1}, and ⃗V is said to be sub-gaussian if
+∥⃗V ∥ψ2 < ∞ (Vershynin, 2018, § 2.5, 3.4). It follows directly from these defini-
+tions that for fixed vectors γ, ∥V γ∥ψ2 = |γ|2∥V ∥ψ2, ∥⃗V γ∥ψ2 ≤ |γ|2∥⃗V ∥ψ2 and
+∥γ∥ψ2 = (log 2)−1/2|γ|2; and that for fixed matrices M, ∥⃗V M∥ψ2 ≤ |M|2∥⃗V ∥ψ2.
+Sums of sub-gaussian variables are sub-gaussian. Hoeffding’s inequality bounds
+tails of sums of independent sub-gaussians in terms of the sum of the summands’
+squared sub-gaussian norms. Maxima of sequences of sub-gaussian random vari-
+ables grow slowly: for an absolute constant k0,
+E max
+1≤i≤n Vi ≤ k0
+�
+max
+1≤i≤n ∥Vi∥ψ2
+�
+(log n)1/2,
+(5)
+with {Vi : i} independent or dependent. For {Gi : i} Normal with variance 1 or less,
+(5) holds with √2 in place of k0 maxi≤n ∥Vi∥ψ2; if {Gi : i} are independent N(0, 1)
+then this bound is sharp, in the sense of limn↑∞(log n)−1/2 E max1≤i≤n Gi = √2
+(Boucheron et al., 2013, § 2.5). Maxima of sub-gaussian vector sequences grow
+slowly as well: there are absolute constants k1 and k2 such that for any sub-
+gaussian {⃗Vi}
+with mean 0 and covariance I, and any deterministic matrices
+{Mi} with column dimension matching the extent of ⃗V ,
+E
+�
+max
+1≤i≤n |Mi⃗V ′
+i |2
+�
+≤ max
+i≤n ∥⃗Vi∥ψ2
+�
+k1 max
+i≤n |Mi|F + k2
+�
+max
+i≤n |Mi|2
+�
+(log n)1/2
+�
+(6)
+6
+
+where |M|F = tr(M′M)1/2 (Vershynin, 2018, Ex. 6.3.5). A consequence is that
+if for each n { ⃗Xni : 1 ≤ i ≤ n} are independent random vectors of length pn
+such that | Cov( ⃗Xni)|2 and ∥ ⃗Xni∥ψ2 are uniformly bounded, then max1≤i≤n | ⃗Xi|2 =
+OP [max(pn, log n)1/2]. This property of sub-gaussian covariates, A11 in Section 2.3
+below, will be assumed in Section 3 but then relaxed in Section 4. The scalar
+c0(R, ⃗x, ⃗xβ) in (4) above, on the other hand, will consistently be required to be
+sub-gaussian, via Section 2.3’s A5.
+2.3
+Consistently estimable index scores
+Let the data and model parameter be arranged in triangular arrays, with sample
+and model n having n observations of p independent variables xi, p (strictly, pn)
+increasing with n. Consistency of ˆβ for (βn : n) will mean that p = o(n) and
+|ˆβ−βn|2
+2 = OP (p/n). Conditions for such consistency are available in the literature.
+We orderto present them along with accompanying conditions characterizing the
+ˆβ’s relationship to its closest linear approximation.
+For estimable βn, define An = An(βn) and ˆAn = An(ˆβ), where
+An(γ) = 1
+n
+�
+n
+�
+i=1
+∇β E [ψ(Ri, ⃗xi; β)]|β=γ + α(γ)
+�
+,
+(7)
+with “∇β” interpreted in terms of weak differentiation if ordinary partial deriva-
+tives do not exist for some β values. For invertible An, the linearization of estimator
+ˆβ is given by
+˜βn = A−1
+n
+1
+n{[
+n
+�
+i=1
+ψ(ri, ⃗xi; βn)] + α(βn)}.
+(8)
+The random vector n−1 �n
+i=1 ψ(Ri, ⃗xi; βn) has covariance n−1Bn where
+Bn = Bn(βn) and Bn(γ) = n−1 E
+n
+�
+i=1
+ψ(Ri, ⃗xi; γ)ψ(Ri, ⃗xi; γ)′.
+(9)
+Proposition 1 (He and Shao, 2000). Under A2, A4, A6 and A9 as stated below:
+1. |ˆβ − βn|2
+P→ 0, with rate OP [(p/n)1/2];
+2. |ˆβ − ˜βn|2
+P→ 0, with rate OP [(p/n)(log n)1/2].
+Proposition 1 restates He and Shao’s Theorems 2.1 and 2.2 as applied to models
+in which only the coefficient parameter βn grows in dimension with n, with a
+slight strengthening of rate condition. (They assume p log(p) = o(n), while A9
+7
+
+says p log(n) = o(n).) Their Theorem 2.2 characterizes decline of the linearization
+error only for sub-√n dimensional models, but a straightforward adaptation of its
+proof gives part 2 of the proposition. See also He and Shao’s Example 3.
+Our regularity assumptions are as follows.
+A1. The columns of x are centered. There may be pre-existing strata 1, . . . , L,
+with L/n → 0. In this case the columns of x are also stratum-centered: for
+stratifying variable v, �
+i:vi=ℓ ⃗xi = 0, ℓ = 1, . . . , L.
+A2. The IS is estimable (as defined in § 2.1) and linear in x.
+A3. An is invertible. Furthermore there is δ > 0 such that An(γ) is invertible
+whenever |γ − βn|2 < δ, and supγ:|γ−βn|2<δ |An(γ)−1|2 is bounded.
+A4. ψ(r, x, β) is of form (4). The functions c0(r, ⃗x, ·) (i.e. η �→ c0(r, ⃗x, η)) are Lip-
+schitz continuous with a common Lipschitz constant, as are (∂/∂η) E c0(R, ⃗x, η).
+A5. The random variables c0(Ri, ⃗xi, ⃗xiβn) have bounded sub-gaussian norm.
+A6. For ℓ = 0, 2 and 4,
+max
+γ,δ:|γ|2=
+|δ|2=1
+n
+�
+i=1
+w(⃗xi)ℓ(⃗xiγ)2(⃗xiδ)2 = O(n).
+A7. p−1|βn|2
+2 = p−1 �
+k β2
+nk is bounded.
+A8. s2(xβn) = β′
+nS(x)βn tends to a limit in (0, ∞].
+A9. p = o[n/(log n)], i.e. (p log n)/n → 0 (sub-n model dimension).
+While the link function g is assumed the same for each n, the coefficient vector
+β grows in length, and βn needn’t converge. Indeed, according to A8 s(sβn) is
+permitted to diverge (but not tend to 0).
+It is less burdensome here to assume invertibility of An, as A3 does, than in
+other regression contexts, as for present applications one can freely change the basis
+of the design matrix, there being no interest in particular elements or contrasts of β.
+Via the Cauchy-Schwartz inequality, A6 entails that maxγ:|γ|2=1 n−1 �n
+i=1 w(⃗xi)ℓ(⃗xiγ)2 =
+O(1) for ℓ ∈ {0, 1, 2}, in turn giving |S(x)|2 = O(1). The appropriateness of these
+commitments can be evaluated in advance of IS estimation, whereas A3 calls for
+inspection of model-fitting artifacts after estimation of β. A simple measure to
+improve the fit of A3, as well as A12 below, is to trim explanatory variables that
+contribute relatively little to index model fit, as indicated by common model se-
+lection criteria.
+Certain results take stronger forms with one or more secondary conditions.
+8
+
+A10. p2 = o[n/(log n)] (sub-√n model dimension).
+A11. maxi≤n |⃗xi|2
+2 = O[max(p, log n)], and maxi≤n w(⃗xi)2 = O(log n) (sub-gaussian
+covariates).
+A12. Each S(x) = (n − L)−1x′x and Bn are of full rank, with |(S(x))−1|2 and
+|B−1
+n |2 bounded (full-rank covariance).
+If the ⃗xes (and w(⃗x)) are sub-gaussian in the sense of being realizations of ⃗X with
+E | ⃗X|ψ2 uniformly bounded, while also A6 holds in the sense of | Cov( ⃗X)|2 (and
+thus p−1 E | ⃗X|2
+2) being uniformly bounded as well, then A11 follows from (6), as
+discussed in § 2.2 following (6).
+According to Proposition 4 below, sandwich
+estimators of Cov(ˆβ) generally require sub-gaussian covariates, and A10, sub-√n
+model dimension; but the covariance estimator based on ˆAn but not ˆBn generally
+requires only sub-n model dimension, A9, and for present purposes will be simi-
+larly beneficial even when it lacks Fisher consistency as compared to the sandwich
+estimator. According to Proposition 6, full-rank covariance (A12) makes index
+sampling variabilities Var(⃗xˆβ) and Var
+�
+(⃗xi − ⃗xj)ˆβ
+�
+estimable without attention
+to size of ⃗x or ⃗xi − ⃗xj. However, our method for asymptotically exact matching
+does not require this, and is valid with S or Bn of less than full rank.
+3
+Asymptotically exact matching with Gaus-
+sian or sub-gaussian ⃗X
+For each n let Sn be a random partition of {1, . . . , n}. It is not presumed that
+Sn expands or extends any earlier partition S1, . . . , Sn−1.
+Denote by [i]Sn the
+unique Sn element containing i ≤ n, and write i Sn
+∼ j when there is s ∈ Sn such
+that i, j ∈ s. Absent ambiguity as to which n or partition sequence is intended,
+these symbols are given as “i∼j” or “[i],” respectively.
+The progression {Sn}
+constitutes an asymptotically exact index post-stratification if Sn-stratum width
+in the direction of the underlying index, max{|(⃗xi − ⃗xj)βn| : i Sn
+∼ j}, tends in
+probability to 0. This section presents a tolerance for Sn-stratum width in the
+direction of the estimated index, max{|(⃗xi − ⃗xj)ˆβ| : i Sn
+∼ j}, that is narrow enough
+to ensure asymptotic exactness in the special case of a sub-gaussian covariate
+(A11). It is also is sufficiently wide that, in a further special case to be described
+in Section 3.2, no i meriting placement in a poststratum with representation of
+the contrasting group can be excluded from such placement in virtue of ⃗xi ˆβ being
+isolated relative to {⃗xj ˆβ : zj ̸= zi}.
+9
+
+When βn is subject to estimation, the observable counterparts of differences
+(⃗xi − ⃗xj)βn, 1 ≤ i, j ≤ n, that is contrasts of form (⃗xi − ⃗xj)ˆβ, are termed paired
+index contrasts (PICs). The discrepancy between a PIC and the paired contrast
+it estimates, (⃗xi − ⃗xj)(ˆβ − βn), is a PIC error. Ensuring that a post-stratification
+is asymptotically exact calls for separate attention to PIC errors versus the PICs
+themselves. Our width tolerance will involve a novel estimate of PIC error size,
+the PIC SE.
+3.1
+PIC errors in the sub-gaussian case
+Recall that errors ˆβ − βn of index coefficient estimates decompose as (ˆβ − ˜β) +
+(˜β − βn), with ˜β the linearization defined in (8). Index and PIC errors decompose
+similarly.
+Corollary (of Proposition 1 part 2). If A2, A4, A6, A9, and A11, then maxi |⃗xi(ˆβ−
+˜βn)| = OP [p3/2(log n)1/2/n]. (Unless p = o(log n), in which case maxi |⃗xi(ˆβ −
+˜βn)| = OP [p(log n)/n]).
+Proposition 2. If A1, A2, A5, and A6, then ∥˜βn − βn∥ψ2 = O(n−1/2).
+Corollary (of Proposition 2). If A1, A2, A5, A6, and A11, then E maxi≤n |⃗xi(˜βn − βn)| =
+O
+�
+[(p log n)/n]1/2�
+. (Unless p = o(log n), in which case E maxi≤n |⃗xi(˜βn − βn)| =
+O
+�
+[(log n)/n1/2]
+�
+.)
+Proposition 1’s corollary is immediate from its part 2 in combination with A11.
+Prop. 2 is proved in the appendix; its corollary flows from A11 and (5). They follow
+D’Amour et al. (2021) in assuming sub-gaussian covariates, in the sense of A11,
+an assumption to be relaxed in Section 4 below.
+The corollaries characterize errors of estimation of index values rather than
+paired contrasts of them, but they have immediate extensions giving the same
+rates of decline for maxi,j≤n |(⃗xi−⃗xj)(ˆβ− ˜βn)| and E maxi,j≤n |(⃗xi − ⃗xj)(˜βn − βn)|,
+respectively. Because for any collection of W of of length-p row vectors,
+sup
+⃗w∈W
+|⃗w(ˆβ − βn)| ≤ sup
+⃗w∈W
+|⃗w(ˆβ − ˜β)| + sup
+⃗w∈W
+|⃗w(˜β − βn)|,
+(10)
+it follows that with sub-gaussian covariates the worst-case PIC or index error tends
+to 0 provided that (p3/2/n) log n does, i.e. if p = o{[n/(log n)]2/3}.
+When the covariate has sub-√n dimension, the corollaries indicate that in large
+samples the suprema of {|⃗xi(ˆβ− ˜β)| : i ≤ n} and {|(⃗xi−⃗xj)(ˆβ− ˜β)| : i, j ≤ n} will be
+smaller by an order of magnitude, p/n1/2 = oP (1), than those of {|⃗xi(˜β − βn)| : i}
+and {|(⃗xi − ⃗xj)(˜β − βn)| : i, j} (both of which are OP
+�
+((p/n)1/2 log n)
+�
+). Of the
+two errors at right of (10), the sup |⃗w(˜β − βn)| term ordinarily dominates; we turn
+attention to it.
+10
+
+3.2
+A thought experiment
+In a special case making both ⃗X and ˆβ Gaussian, sizes of PIC errors admit specific
+characterization in terms of readily estimable quantities. Ghosh and Cort´es (2019),
+among others, consider related issues under an assumption of Gaussian ⃗X. For
+vectors v ∈ ℜm let |v|2 and |v|∞ have their usual meanings, (m−1 �m
+i=1 v2
+i )1/2 and
+�
+i≤m |vi| respectively. For matrices M and N of like dimension, ⟨M, N⟩F denotes
+the Frobenius inner product tr(M′N).
+Proposition 3. Let S be a partition of {1, . . . , n} with an associated mapping ⃗µ :
+S → ℜp, and let ES(·) and CovS(·) denote expectations calculated with S and {µs :
+s ∈ S} held fixed. Given S and {µs : s ∈ S} let (˜βn − βn; ⃗Xi − ⃗µ([i]S); ⃗Xj − ⃗µ([j]S))
+be jointly multivariate Normal, for each i, j ≤ n, with mean zero, CovS(˜β) = C,
+CovS( ⃗Xi) = CovS( ⃗Xj) = Σ and, if i ̸= j, CovS( ⃗Xi, ⃗Xj) = 0. Then we have
+ES
+���[( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j, i < j]
+��2
+2
+�
+= ⟨2Σ, C⟩F
+and
+(11)
+ES
+���[( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j]
+��
+∞
+�
+≤ z∗
+nS⟨2Σ, C⟩1/2
+F ,
+(12)
+where nS := card({{i, j} : i S∼ j, i < j}) and z∗
+nS := (2 log 2nS)1/2 bounds E max1≤i≤n |Gi|,
+(Gi : i) independent N(0, 1), as described in Section 2.2.
+Given its strong assumptions on the covariate, Proposition 3 has limited prac-
+tical use for PIC error control; we shall arrive at methods for containment of
+[( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j] that relax those assumptions to the moment condition
+A6.
+But these methods call for a limit on sizes of PIC errors that are to be
+tolerated, and (12) will turn out to be helpfully specific in this regard.
+The statistician who sets out to select a matched sample has it as her oper-
+ating hypothesis that each member of the focal group has counterparts that are
+close enough, in terms of ⃗xβn, within the alternate group. Simplifying, so as to
+remove the question-begging “close enough,” let us suppose provisionally that for
+each focal group member i, the available sample contains within it at least one
+contrasting group member j that would be a perfect match on the underlying in-
+dex, ⃗xiβn = ⃗xjβn. Continue the thought experiment by supposing ˆβ and ⃗X to
+be as described in Proposition 3, and by taking the focal group to be the smaller
+of the treatment and control groups, implying no fewer than min(n0, n1) perfect
+pairs. Let S to be the collection of equivalence classes induced by the relation
+that i S∼ j if and only if ⃗xiβn = ⃗xjβn. The proposition then characterizes PICs
+|(⃗xi − ⃗xj)ˆβ| for which the contrast on the underlying index, |(⃗xi − ⃗xj)βn|, is 0.
+Each simplification made thus far in order to apply the proposition should err in
+the direction of understating the maximum PIC among pairs closely matched on
+11
+
+⃗xβn; but even if we continue to arrange our thought experiment so as to minimize
+this quantity we will find it to be almost unworkably large, in a sense to be given
+presently.
+As specified so far, our perfect pairing thought experiment permits no ⃗Xβn
+variation within strata of S. This means the stratified covariance CovS( ⃗X) must
+satisfy β′
+nΣβn = 0. Bending available covariate data to this constraint, take ⃗X
+to be distributed as in Proposition 3 with Σ = S⊥βn := (n − L)−1x⊥xβn′x⊥xβn,
+the observed covariates’ covariance as projected onto the orthocomplement of the
+index, x⊥xβn. (Here x is the n × p matrix of covariates as observed; x⊥v denotes
+the n×p matrix of residuals arising from the p regressions of x-columns on n-vector
+v; and L is the number of overt, preexisting strata, if such exist, and 1 otherwise.
+Following A1, x is assumed to be centered or stratum-centered, as appropriate, and
+x⊥xβn inherits this centering.) The natural estimate S⊥ˆβ = (n − L)−1x⊥xˆβ′x⊥xˆβ
+of S⊥βn is appropriately consistent for S⊥βn, as noted in Proposition 5 below.
+Observe that use of S⊥ˆβ (as opposed to S) again reduces (11) and (12), if in
+increasing-p regimes it leaves their order unchanged.
+The maximum PIC bound (12) is at its smallest, with nS = min(n0, n1),
+when each member of the smaller of the focal and comparison groups has just one
+perfectly matching counterpart. Among such configurations, the bound is sharpest
+when the pairs do not overlap, as in matching without replacement. (Conditionally
+given ˜β as well as S and {⃗µ(s) : s ∈ S}, ( ⃗Xi − ⃗Xj)(˜β − βn) is independent of
+( ⃗Xi′− ⃗Xj′)(˜β−βn), for i, j, i′, j′ with {i, j} ̸= {i′, j′}; by the discussion following (5)
+in Section 2.2, this causes inequality (31) in Appendix B.2 to be sharp.) Complete
+the specification of our perfect-pairing thought experiment by supposing its nS =
+min(n0, n1) pairs to be nonoverlapping.
+Then (12) more closely estimates the
+width in ⃗xˆβ of S.
+Across pairs S constituting the thought experiment, the maximum PIC error is
+expected to be z∗
+min(n0,n1)⟨2S⊥βn, C⟩1/2
+F , of order O{[(p log n)/n]1/2}. The accom-
+panying estimate is z∗
+min(n0,n1)⟨2S⊥ˆβ, ˆC⟩1/2
+F . These quantities are small enough to
+tend to zero, but only barely so: [(p log n)/n]1/2 is precisely the rate that A9 re-
+quires to decline to 0. Among pairs {i, j} that in actuality are perfectly matched,
+(⃗xi −⃗xj)βn = 0, even tame, Gaussian variation in other covariate directions engen-
+ders a maximum PIC |[( ⃗Xi − ⃗Xi)ˆβ : i S∼ j]|∞ of as large an order as can be tolerated
+of separations on the actual index, |[( ⃗Xi − ⃗Xi)βn : i S∼ j]|∞, if the matching is to
+be asymptotically exact. So we will recommend this number as a matching toler-
+ance, not only when making the restrictive assumptions of Proposition 3 but also
+when entertaining only the weaker A1–A9.
+12
+
+3.3
+A summary standard error for PICs
+In light of (8) and (9), the covariance of ˜βn is
+Cn = n−1A−1
+n Bn(A−1
+n )′.
+(13)
+This Cn approximates the covariance of ˆβ, particularly when the index model
+has sub-√n dimension.
+When p = o{[n/ log(n)]1/2}, Prop. 1 says |˜β − ˆβ|2 =
+oP (n−1/2), small enough to obviate distinctions between Cn = Cov(˜β) and Cov(ˆβ).
+For example, Lemma 1 below entails that s. d.(⃗x˜β) = [⃗xCn⃗x′]1/2 shares the or-
+der OP (n−1/2|⃗x|2) with ⃗x(ˆβ − βn), whereas part 2 of Prop. 1 says ⃗x(˜β − ˆβ) =
+oP (n−1/2|⃗x|2). The proposition following the lemma will show the larger order
+OP (n−1/2|⃗x|2) also to be shared by s. e.(⃗x˜β) = [⃗x ˆCn⃗x′]1/2.
+Lemma 1. Under A3, A5 and A6, |Bn|2 = O(1) and |Cn|2 = O(n−1).
+If ψ is the gradient a (well-specified) log-likelihood, then An = Bn, and n−1 ˆA−1
+n
+estimates Cn. More broadly, Cn is estimated by n−1 ˆA−1
+n
+ˆBn( ˆA−1
+n )′, where
+ˆBn = ˆBn(ˆβ),
+ˆBn(β) = n−1
+n
+�
+i=1
+ψ(ri, ⃗xi; β)ψ(ri, ⃗xi; β)′.
+The propositions that follow establish the consistency of natural covariance esti-
+mators and related quantities.
+Proposition 4. Under A1–A9, | ˆA−1
+n
+− A−1
+n |2
+P→ 0. If also A10 (sub-√n dimen-
+sion) and A11 (sub-gaussian covariates), then | ˆA−1
+n
+ˆBn( ˆA−1
+n )′ − nCn|2
+P→ 0.
+Proposition 5. Let A1–A9 hold, let ˆCn be a consistent estimate of Cn (| ˆCn −
+Cn|2 = oP (n−1)) and let S, S⊥βn and S⊥ˆβ be as defined in Section 3.2. Then
+|⟨S, ˆC⟩F − ⟨S, C⟩F | = oP (p/n)
+and
+|⟨S⊥ˆβ, ˆC⟩F − ⟨S⊥βn, C⟩F | = oP (p/n),
+whereas ⟨S, C⟩F = OP (p/n) and ⟨S⊥βn, C⟩F = OP (p/n).
+The quantity ⟨2S⊥ˆβ, ˆCˆβ⟩1/2
+F
+will be termed the PIC standard error (PIC SE);
+Proposition 5 says that it consistently estimates the analogous parameter appear-
+ing at right of (11) and (12) in Proposition 3. Proposition 4 is new as applied to
+increasing-p regimes; Proposition 5 is entirely new. Their proofs are given in Ap-
+pendix B.3, along with demonstrations of intermediate results including Lemma 1.
+13
+
+3.4
+PIC SE calipers
+We recommend matching within limits of z∗
+min(n0,n1) times the PIC SE ⟨S⊥ˆβ, ˆC⟩1/2
+F ,
+whether or not the Gaussian model of Proposition 3 applies. If it does apply, this
+ensures that the same multiple of the PIC SE characterizes matched discrepancies
+on the underlying index (Sections 3.2 and 3.3). If it does not apply but the covari-
+ate is sub-gaussian, (12) may no longer limit sizes of PIC errors, but they continue
+to tend to 0 as long as p = o{[n/(log n)]2/3} (Section 3.1). If neither the Gaussian
+nor sub-gaussian modeling assumptions apply, additional matching requirements
+to be described below will be necessary to force the PIC errors towards 0, but
+z∗
+min(n0,n1) times the PIC SE remains an appropriate tolerance for PICs. It tends
+to zero, so its use as a caliper width forces PICs toward zero; it tends to zero
+at the same [(p log n)/n]1/2 rate that Proposition 1 requires to tend to zero for
+index model consistency, making it no more restrictive than is necessary to force
+the PIC maximum to tend to 0. In the context of the idealized setting studied in
+Section 3.2 it was seen also to be minimal in a more quantitatively specific sense,
+in virtue of its sharp characterization of the notional experiment’s maximum PIC
+error: if such a paired experiment were to be lurking within the actual data, set-
+ting a tolerance for matching on xˆβ any smaller than z∗
+min(n0,n1) times the PIC SE
+would exclude pairs that are in fact perfectly matched on xβn.
+4
+Deconstructing Gaussian and sub-gaussian
+assumptions
+4.1
+PIC SEs with unrestricted X
+The PIC SE averages expected PIC errors in either of two ways.
+First, if the
+sample available for matching contains a subsample of perfectly matched subjects
+(as envisioned in Section 3.2) for which the within-pair covariance of covariates
+is S⊥βn, then the squared PIC SE estimates the expected mean of squared PIC
+errors, [(⃗xi − ⃗xj)(ˆβ − βn)]2, across perfectly matched pairs (i, j). Second, taking
+the entirety of the sample as-is but residualizing each subject’s covariate for xβn (as
+also discussed in Section 3.2), the squared PIC SE is approximately the expected
+mean square of reduced PIC errors, [(⃗x⊥xβn
+i
+− ⃗x⊥xβn
+j
+)(ˆβ − βn)]2, now across all
+pairs {i, j}, 1 ≤ i < j ≤ n.
+To see this, for 1 ≤ i, j ≤ n write ⃗d⊥βn
+ij
+:= ⃗x⊥xβn
+i
+− ⃗x⊥xβn
+j
+, noting that ⃗d⊥βn
+ij
+=
+14
+
+⃗xi − ⃗xj for pairs {i, j} that are perfectly matched for the index. Observe that
+[⃗d⊥βn
+ij
+(˜β − βn)]2 = [⃗d⊥βn
+ij
+(˜β − βn)]′[⃗d⊥βn
+ij
+(˜β − βn)]
+= (˜β − βn)′(⃗d⊥βn′
+ij
+⃗d⊥βn
+ij
+)(˜β − βn).
+Summing over perfectly matched pairs, for the first scenario, or all
+�n
+2
+�
+possible
+pairs, for the second, and in either case letting np denote the number of pairs
+contributing to the sum, we have
+1
+np
+�
+[⃗d⊥βn
+ij
+(˜β − βn)]2 = (˜β − βn)′�
+n−1
+m
+� ⃗d⊥βn′
+ij
+⃗d⊥βn
+ij
+�
+(˜β − βn)
+= (˜β − βn)′(2S⊥βn)(˜β − βn)
+(14)
+= ⟨2S⊥βn, (˜β − βn)(˜β − βn)′⟩F ,
+(15)
+where (14) invokes the U-statistic representation of covariance, (n−1)−1 �
+i=1(wi−
+¯w)(vi − ¯v) =
+�n
+2
+�−1 �n−1
+i=1
+�n
+j=i+1
+1
+2(wi − wj)(vi − vj), and (15) uses the sum of ele-
+mentwise products ⟨M, N⟩F of matrices M and N to re-express (14). Accordingly
+E
+� 1
+np
+�
+[⃗d⊥βn
+ij
+(˜β − βn)]2
+�
+= ⟨2S⊥βn, Cn⟩F .
+A similar argument reveals the PIC SE’s alternate interpretation as root mean
+square of pairwise distances |⃗d⊥ˆβ
+ij ˆC1/2|2 = (⃗d⊥ˆβ
+ij ˆC ⃗d⊥ˆβ′
+ij )1/2 over pairs {i, j}. Invok-
+ing in turn the cyclic property of the matrix trace, the definition of the Frobenius
+matrix product ⟨·, ·⟩F , the Frobenius product’s bilinearity, and the U-statistic rep-
+resentation of sample covariance:
+� ⃗d⊥ˆβ
+ij ˆCd⊥ˆβ′
+ij
+=
+�
+tr(⃗d⊥ˆβ
+ij ˆC ⃗d⊥ˆβ′
+ij ) =
+�
+tr(⃗d⊥ˆβ′
+ij
+⃗d⊥ˆβ
+ij ˆC)
+=
+�
+⟨⃗d⊥ˆβ′
+ij
+⃗d⊥ˆβ
+ij , ˆC⟩F = ⟨
+� ⃗d⊥ˆβ′
+ij
+⃗d⊥ˆβ
+ij , ˆC⟩F = np⟨2S⊥ˆβ, ˆC⟩F .
+When the average is only over {i, j} that are perfectly matched for the index,
+(⃗d⊥ˆβ
+ij ˆCd⊥ˆβ′
+ij )1/2 approximates s. d.[(⃗xi − ⃗xj)ˆβ], because ⃗xiβn = ⃗xjβn means that
+⃗d⊥ˆβ
+ij
+approximates ⃗xi − ⃗xj. The interpretation as a pairwise covariate distance,
+within the orthocomplement in x of xˆβ and after rescaling by ˆC1/2, is available
+both for the perfect pairing thought experiment and also when the mean is over
+all {i, j}, 1 ≤ i < j ≤ n.
+These arguments rely implicitly on A1–A9, via Proposition 5, but do not call for
+Gaussian covariates, nor for boundedness of covariates’ sub-gaussian norms. By
+the same token, none admit extensions offering maximum, rather than average,
+PIC error control, as is necessary for asymptotically exact matching.
+15
+
+4.2
+Caliper refinement with attention to index error
+distances
+It is intuitive that with covariates drawn from heavy tailed distributions there
+may be PICs exceeding the PIC SE by factors well above z∗
+np = √(2 log 2np),
+in contrast to the Gaussian situation described in Proposition 2.
+Writing δ[x]
+for the distribution placing point mass at x, heavier tails on the covariate mean
+heavier tails on the empirical distributions
+�n
+2
+�−1 �n−1
+i=1
+�n
+j=i+1 δ[s. d.[(⃗xi − ⃗xj)˜β]],
+�n
+2
+�−1 �n−1
+i=1
+�n
+j=i+1 δ[(⃗xi − ⃗xj)(˜β − βn)], and in turn
+�n
+2
+�−1 �n−1
+i=1
+�n
+j=i+1 δ[(⃗xi − ⃗xj)(ˆβ − βn)].
+Fortunately, estimates |(⃗xi−⃗xj) ˆC1/2|2 of standard deviations s. d.(i, j) = |(⃗xi−
+⃗xj) ˆC1/2|2 are available at the time of matching: one can simply avoid pairings {i, j}
+for which |(⃗xi−⃗xj) ˆC1/2|2 is too large. Call |(⃗xi−⃗xj) ˆC1/2|2 the index error distance
+separating i from j. Proposition 6 says index error distances can estimate pairwise
+index sampling variabilities uniformly well, even for models of sub-√n dimension
+if the estimator ˆC is appropriately chosen.
+Proposition 6. i. Let ψ(r, ⃗x, β) be the gradient of a likelihood function governing
+the conditional distribution of R given ⃗X, with ˆCn = n−1 ˆA−1
+n . Under A1–A9,
+�����
+�
+1 − |⃗xi ˆC1/2|2
+|⃗xiC1/2|2
+: i ≤ n
+������
+∞
+,
+�����
+�
+1 − |(⃗xi − ⃗xj) ˆC1/2|2
+|(⃗xi − ⃗xj)C1/2|2
+: i, j ≤ n
+������
+∞
+P→ 0.
+(16)
+(Here 0/0 is taken to be 1.)
+ii. Let {ϵn : n} satisfy ϵ−1
+n
+= O[n/ max(p, log n)]. Under A1–A11 with ˆCn =
+n−1 ˆA−1
+n
+ˆBn ˆA−1
+n ,
+�����
+�
+1 − max(ϵ1/2
+n , |⃗xi ˆC1/2|2)
+max(ϵ1/2
+n , |⃗xiC1/2|2)
+: i
+������
+∞
+P→ 0.
+If x and En ⊆ {{i, j} : 1 ≤ i, j ≤ n} satisfy |(|⃗xi − ⃗xj|2
+2 : {i, j} ∈ En)|∞ =
+OP (max(p, log n)), then under A1–A10
+�����
+�
+1 − max[ϵ1/2
+n , |(⃗xi − ⃗xj) ˆC1/2|2]
+max[ϵ1/2
+n , |(⃗xi − ⃗xj)C1/2|2]
+: {i, j} ∈ En
+������
+∞
+P→ 0.
+iii. Under A1–A12, (16) holds with ˆCn = n−1 ˆA−1
+n
+ˆBn ˆA−1
+n .
+We focus on situations conforming to the hypotheses of (i) or of (iii), warranting
+uniform convergence (16) of index error distances.
+Let us calibrate sizes of index error distances with reference to Section 3.2’s
+perfect pairing thought experiment. Proposition 7 adapts extant results about
+Gaussian chaos to characterize that setting’s maximum of |( ⃗Xi − ⃗Xj)C1/2|2.
+16
+
+Proposition 7. Let ⃗Xi, i ≤ n, be independent MVN(µ, Σ). Let C be a second
+positive semidefinite matrix of the same dimension as Σ, let E ⊆ {{i, j} : 1 ≤ i ̸=
+j ≤ n} and let nE := card(E). Then
+�
+E
+���{|( ⃗Xi − ⃗Xj)C1/2|2 : {i, j} ∈ E}
+��2
+∞
+��1/2
+≤
+⟨2Σ, C⟩1/2
+F
+�
+�1 +
+�
+log nE
+p[Σ1/2CΣ1/2]
+�1/2�
+�,
+(17)
+where p[M] denotes intrinsic dimension, tr(M)/|M|2, for positive semidefinite M.
+Proposition 7 is proved in Appendix B.2. With Σ = S⊥βn and C = Cn, it
+explicitly bounds the worst-case pairwise distance |( ⃗Xi − ⃗Xj)C1/2|2 within the
+perfect-pairing thought experiment of Section 3.2. To contain PIC errors of actual
+experiments to a similar level, I recommend the match-eligibility requirement that
+|(⃗xi − ⃗xj) ˆC1/2|2 ≤ ⟨2S⊥ˆβ, ˆC⟩1/2
+F
+�
+2 +
+�log min(n0, n1)
+p − 1
+�1/2�
+,
+(18)
+as a complement to the z∗
+min(n0,n1)⟨2S⊥ˆβ, ˆC⟩1/2
+F
+limit on PICs |(⃗xi − ⃗xj)ˆβ| that
+was recommended in Section 3.4. The heuristic by which Proposition 7 supports
+constraint (18), to be explained presently, more simply suggests the stricter cap
+on |(⃗xi − ⃗xj) ˆC1/2|2 of
+⟨2S⊥ˆβ, ˆC⟩1/2
+F
+�
+1 +
+�log min(n0, n1)
+p − 1
+�1/2�
+;
+(19)
+but it will subsequently be shown that (18) together with a softer penalty on
+index error distances respecting (18) while exceeding (19) is sufficient for present
+purposes.
+To relate (17) to (19), first recall that min(n0, n1) is the size of the perfect-
+pairing thought experiment (Sec. 3.2) and ⟨2S⊥ˆβ, ˆC⟩F is consistent for ⟨2S⊥βn, Cn⟩F
+(Prop. 5). In general 0 ≤ p[Σ1/2CΣ1/2] ≤ p, by definition (see also Tropp, 2015;
+Vershynin, 2018, §7.8). In the special cases that (R, ⃗X) has linear discriminant
+structure with Cor( ⃗X) known, or that R is linear in ⃗X, with ⃗X Gaussian and βn
+estimated accordingly in either case, intrinsic and extrinsic dimensions coincide:
+p[Σ1/2CnΣ1/2] = p or p − 1, depending as Σ = S(x) or S⊥βn. If supposed to contain
+min(n0, n1) distinct pairs {i, j} for which i ̸= j but ⃗xiβn = ⃗xjβn, then either of
+these Gaussian- ⃗X special cases closely models Section 3.2’s notional perfect pair-
+ing, with Σ = S⊥βn, and (19) estimates the expected maximum covariate distance
+17
+
+|(⃗xi − ⃗xj)C1/2|2 among perfect pairs. At the same time, an |(⃗xi − ⃗xj) ˆC1/2|2 limit
+of form (19) should rarely exclude perfect pairs because of separation in direc-
+tions orthogonal to the index, since (17) approximates such separations’ expected
+maximum from above.
+Regardless of what distribution the covariate may have been drawn from, limits
+(19) or (18) on covariate distances |(⃗xi − ⃗xj) ˆC1/2|2 also engender limits on PIC
+errors |(⃗xi −⃗xj)(ˆβ −βn)|. In part this is because (⃗xi −⃗xj)(˜β −βn) is sub-gaussian if
+˜β is, and is N[0, |(⃗xi − ⃗xj)C1/2
+n
+|2
+2] if ˜β ∼ MVN(βn, Cn). Proposition 8, stated here
+without proof, collects the relevant facts reviewed in Section 2.2.
+Proposition 8. Let E ⊆ {{i, j} : 1 ≤ i, j ≤ n} satisfy |(⃗xi − ⃗xj)C1/2
+n
+|2 > 0 for all
+{i, j} ∈ E, and let nE = card(E). If ˜β ∼ MVN(βn, Cn),
+E
+������
+�
+(⃗xi − ⃗xj)(˜β − βn)
+|(⃗xi − ⃗xj)C1/2
+n
+|2
+: {i, j} ∈ E
+������
+∞
+�
+≤ z∗
+nE = [2 log(2nE)]1/2
+and
+E
+�
+�
+�����
+�
+(⃗x⊥xβn
+i
+− ⃗x⊥xβn
+j
+)(˜β − βn)
+|(⃗x⊥xβn
+i
+− ⃗x⊥xβn
+j
+)C1/2
+n
+|2
+: {i, j} ∈ E
+������
+∞
+�
+� ≤ z∗
+nE.
+(20)
+If ˜β − βn is non-Normal but sub-gaussian with ∥˜β − βn∥ψ2 bounded, these expected
+maximums continue to be of order (log nE)1/2.
+Together with Prop. 6, Proposition 8 says requiring each of min(n0, n1) pairs
+{i, j} to have |(⃗xi −⃗xj) ˆC1/2
+n
+|2 below (19) puts the corresponding PIC errors below
+z∗
+min(n0,n1) times (19). If p increases faster than log n, then the ratio in (19) tends to
+0, and (19) is asymptotically equivalent to the PIC SE. That is, confining matching
+to pairs {i, j} for which |(⃗xi − ⃗xj) ˆC1/2
+n
+|2 falls left of (19) makes the supremum of
+matched errors |(⃗xi − ⃗xj)(˜β − βn)| asymptotically as it would be in the perfect-
+pairing thought experiment, given log n = o(p) but not special conditions on the
+distribution of the covariate. (If p increases no faster than log n, p = O(log n),
+these errors somewhat exceed those of the corresponding thought experiment, but
+they tend quickly to zero anyway, due to the p’s slow increase.)
+These considerations suggest (19) as a hard limit for pair distances |(⃗xi −
+⃗xj) ˆC1/2|2, but similar control of PIC errors can be had with a simple policy that
+encourages matches with |(⃗xi − ⃗xj) ˆC1/2|2 beneath (19) while only requiring (18).
+Make i is eligible for pairing to j if
+|(⃗xi − ⃗xj)ˆβ| ≤ z∗
+min(n0,n1)[⟨2S⊥ˆβ, ˆC⟩1/2
+F
+− ˆe(i, j)]
+(21)
+18
+
+where
+ˆe(i, j) :=
+�
+|(⃗xi − ⃗xj) ˆC1/2|2 − ⟨2S⊥ˆβ, ˆC⟩1/2
+F
+�
+1 +
+�log min(n0, n1)
+p − 1
+�1/2��
++
+, (22)
+v+ := max(0, v), represents excess in index error distance as compared to its nomi-
+nal supremum (19). If |(⃗xi −⃗xj) ˆC1/2|2 never exceeds this nominal supremum, (21)
+reduces to the requirement that |(⃗xi −⃗xj)ˆβ| ≤ z∗
+min(n0,n1)⟨2S⊥ˆβ, ˆC⟩1/2
+F , as proposed
+in Section 3.4. For potential pairings {i, j} with |(⃗xi − ⃗xj) ˆC1/2|2 exceeding (19),
+Section 3.4’s PIC allowance of z∗
+min(n0,n1)⟨2S⊥ˆβ, ˆC⟩1/2
+F
+is reduced in recognition of
+the pairing’s large standard error. When the index error distance |(⃗xi − ⃗xj) ˆC1/2|2
+is so large that its excess ˆe(i, j) exceeds the PIC SE — or equivalently, so large
+that (18) fails — (21) forbids i’s pairing with j.
+This selectively narrowed PIC SE caliper has important advantages over non-
+varying PIC SE calipers, alone or in combination with calipers of width (19) on
+the pairwise index error distance. Non-varying PIC SE calipers secure asymptotic
+exactness of a match only for sub-gaussian covariates, an assumption that selective
+narrowing of the caliper enables us to do without. Coupling a non-varying PIC
+SE caliper with a limit on the pairwise index error distance of (19) contains the
+sum
+|[(⃗xi − ⃗xj)ˆβ : i S∼ j]|∞ + |[(⃗xi − ⃗xj)(˜β − βn) : i S∼ j]|∞
+(23)
+at the product of z∗
+min(n0,n1) with the right hand side of (18), just as the selectively
+narrowed PIC SE caliper does, but at the cost of categorically disallowing pairwise
+index error distances in excess of (19). In contrast, the selectively narrowed caliper
+permits those pairings if their PICs are sufficient small. This additional tolerance
+is important because (19) systematically underestimates suprema of pairwise index
+error distances for some index models, even with Gaussian ⃗X, because of its use of
+p − 1 in lieu of Proposition 7’s p[(S⊥βn)1/2C(S⊥βn)1/2]. That minor embarrassment
+could be remedied by replacing p − 1 in (19) by p[(S⊥ ˆβ)1/2 ˆC((S⊥ ˆβ))1/2], but then as-
+sumption A12 would become necessary for asymptotic equivalence of (19) and the
+PIC SE — an equivalence needed even under A9, the weakest of the model dimen-
+sionality restrictions considered in this paper, to force (23) toward an asymptote
+of 0. Assumption A12 is discussed in the next section.
+4.3
+The contribution of linearization error
+Display (23) omits linearization error. Unless the estimator of the index regression
+is linear in its dependent variable R, to estimate |[(⃗xi − ⃗xj)βn : i ∼ j]| we must
+attend to |[(⃗xi − ⃗xj)(ˆβ − ˜β) : i ∼ j]| as well as |[(⃗xi − ⃗xj)ˆβ : i ∼ j]| and |[(⃗xi −
+19
+
+⃗xj)(˜β − βn) : i ∼ j]|.
+Index estimators are linear in R in the special cases of
+linear regression and linear discriminant modeling with fixed correlation, but not
+for indices estimated with probit or logistic regression.
+Recall from Section 3.1 that with sub-gaussian covariates (A11), |[(⃗xi−⃗xj)(ˆβ−
+˜β) : i, j ≤ n]|∞ tends to 0 provided that p = o{[n/(log n)]2/3}, and is smaller by an
+order of magnitude than |[(⃗xi − ⃗xj)(˜β − βn) : ⃗xiβn ≈ ⃗xjβn]|2 measured in the PIC
+SE, provided that p = o{[n/(log n)]1/2}. Matching within selectively narrowed
+PIC SE calipers secures these conclusions under conditions not including A11.
+However, depending on the specific side conditions and estimation routines that
+are employed, the matching procedure may need to observe additional caliper
+restrictions.
+First consider the case that A1–A9 hold, with Cn estimated by n−1 ˆA−1
+n . The
+matching requirement (18), a consequence of (21), ensures that |[|(⃗xi −⃗xj) ˆC1/2|2 :
+i ∼ j]|∞ = OP {[max(p, log n)/n]1/2}, since (by Proposition 5) the PIC SE is
+OP [(p/n)1/2] and since (p − 1)−1 log min(n0, n1) = O[max(1, p−1 log n)]. Proposi-
+tion 6 and ˆC = n−1 ˆA−1
+n in turn give |[|(⃗xi−⃗xj)A−1/2|2 : i ∼ j]|∞ = OP [max(p, log n)1/2].
+As A3 and Lemma 3 in Appendix B.3 entail |An|2 = O(1), |[|(⃗xi−⃗xj)|2 : i ∼ j]|∞ =
+OP [max(p, log n)1/2]. Whether or not ⃗X was drawn from a sub-gaussian distribu-
+tion, pairs {i, j} selected within selectively narrowed PIC SE calipers can be no
+more separated on ⃗x than they would have been under sub-gaussian sampling, and
+Proposition 1 entails
+|[(⃗xi − ⃗xj)(ˆβ − ˜β) : i ∼ j]|∞ = OP [pn−1(log n)1/2max(p, log n)1/2].
+(24)
+Within matched pairs, linearization error is as described in Section 3.1, even with-
+out A11 or additional matching restrictions.
+When Cn is instead estimated by n−1 ˆA−1
+n Bn ˆA−1
+n , A10 is needed in addition
+to A1–A9, for consistency of ˆCn (by Proposition 4).
+Then |[|(⃗xi − ⃗xj) ˆC1/2|2 :
+i ∼ j]|∞ = OP {[max(p, log n)/n]1/2}, by a similar argument as above. Case (ii) of
+Proposition 6 then gives that |[|(⃗xi − ⃗xj)C1/2|2 : i ∼ j]|∞ = OP {[max(p, log n)/n]1/2},
+provided that ⟨2S⊥ˆβ, ˆC⟩F
+�
+1 +
+�
+log min(n0,n1)
+p−1
+�1/2�
+is of the same order as max(p, log n)/n.
+(Proposition 5 gives that it is OP [max(p, log n)/n], but here we also require its
+reciprocal to be OP {[max(p, log n)/n]−1}.) At this point A12 also becomes nec-
+essary, to ensure |B−1/2
+n
+|2 = OP (1) and thus that |C−1/2
+n
+|2 = OP (n1/2). If so,
+Proposition 1 again gives (24).
+The full-rank covariance condition A12 merits careful consideration in prac-
+tice, however.
+It will poorly describe some otherwise unassailable index mod-
+els, as partitioners have long been encouraged to add covariates in such models
+without regard to their mutual correlations (Rubin and Thomas, 1996). Fortu-
+20
+
+nately A12 is straightforward to diagnose, by checking that neither S nor ˆBn is
+ill-conditioned. If sustainable, it delivers (in combination with Lemma 1 and A6)
+the needed assurance that the PIC SE declines no faster than (p/n)1/2. If A12
+cannot be sustained while an assumption that ⟨S⊥βn, Cn⟩F = O(p/n) can be, we
+can instead combine that weaker assumption with additional matching restrictions
+ensuring that maxi∼j |⃗xi −⃗xj| = OP [max(p, log n)1/2]. Matches can be required to
+fall within Euclidean distance calipers of width tr(S)1/2{1 + [(log n)/(p − 1)]1/2},
+or with calipers of width s(xj){1 + [(log n)/(p − 1)]1/2} on each dimension j =
+1, . . . p of ⃗x separately. Either way, the combination of additional assumptions and
+matching restrictions secures (24), and that the dominant part of the PIC error
+|[(⃗xi − ⃗xj)(ˆβ − βn)]|∞ is |[(⃗xi − ⃗xj)(˜β − βn)]|∞, not |[(⃗xi − ⃗xj)(ˆβ − ˜β)]|∞.
+5
+Asymptotically exact matching and consis-
+tency of impact estimation
+Matching within PIC SE calipers arranges that paired differences of the index tend
+uniformly to zero, given mild conditions on the index model. For propensity and
+certain other index models, this convergence is precisely what is needed to ensure
+that in the absence of unmeasured confounding, the matched structure enjoys the
+same consistency properties as would be enjoyed were paired differences on the
+index uniformly and identically zero. In this section we assume Z ∈ {0, 1}; the
+very weak overlap condition
+P[0 < P(Z = 1 | Xβn) < 1] = 1;
+(25)
+that E |YC|1+δ, E |YT |1+δ < ∞ for some δ > 0; and that the mapping v �→
+logit[P(Z = 1|Xβ = v)] is Lipschitz. If Xβ is a propensity score modeled on the
+logit scale, this mapping is the identity and (25) follows from Rosenbaum and Ru-
+bin’s (1983) overlap condition, 0 < P(Z = 1 | X) < 1; if Xβ is a risk or prognostic
+score, (25) is less restrictive than their already weak overlap requirement.
+A partition Sn is a finely stratified design (Fogarty, 2018) if it divides {1, . . . , n}
+into partition elements s ∈ Sn that satisfy �
+i∈s�zi = z� ≤ 1 for either or both of
+z = 0, 1. These can be 1 : m0 or m1 : 1 matched sets, for natural numbers m0, m1,
+if not m1 : m0 blocks with both m0, m1 ≥ 2; singleton elements, s of size ns = 1,
+represent unmatched units.
+Such Sn may emerge from pair matching, where each
+s ∈ Sn is either a 1:1 pair, �
+i∈s�zi = 1� = �
+i∈s�zi = 0� = 1, or an unmatched 0:1
+or 1 : 0 singleton; from matching with multiple controls, permitting 1 : m, m ≥ 1,
+matches as well as singletons; from 1-nearest neighbor matching, in which m:1 but
+not 1:m sets may arise; or from full matching (Rosenbaum, 1991), permitting both
+21
+
+m : 1 and 1 : m configurations for any m; or from full matching with symmetric
+restrictions (Stuart and Green, 2006; Fredrickson et al., 2020), permitting both
+m : 1 and 1 : m matched sets, but only for m falling below a designated m0. The
+notation [i]Sn for the partition element s ∈ Sn containing i is abbreviated to [i]
+when no partition other than Sn is under consideration.
+Consider estimates defined as roots of ψSn(·) = 0, where
+ψSn(η) :=
+�
+s∈Sn
+�
+i∈s ψsi(η)
+�
+s∈Sn wsns ¯Zs(1 − ¯Zs),
+ψsi(η) := ws[Yi − η(Zi − ¯Zs)](Zi − ¯Zs), (26)
+¯zs := ns−1�
+j∈s zj and ws is a nonnegative weight determined by zs and/or ns.
+For example, the z-coefficient in an ordinary regression of outcomes y on z and
+matched-set indicator variables is expressible as the solution ˆτ of ψSn(τ) = 0 for
+ws ≡ 1, since (zi − ¯z[i] : i) is the residual of z’s ordinary regression on matched-
+set indicator variables. As a second example, the effect of treatment-on-treated
+estimator
+n−1
+{i:Zi=1,n[i]>1}
+�
+{i:Zi=1,n[i]>1}
+Yi − avg(Yj : Zj = 0, j ∼ i)
+uniquely solves ψSn(·) = 0 with w[i]Sn = 0 for unmatched i and ws = (1 − ¯Zs)−1
+for s with ns > 1.
+Inferences will reflect Sn by conditioning on stratum-wise treatment alloca-
+tions, that is on a sigma field containing Fn := σ
+��
+i∈s Zi : s ∈ Sn
+�
+. Desite this
+notation, {Fn : n} is not a nested filtration: as a rule Sm ̸⊆ Sm, as strict contain-
+ment does not permit the maximum index discrepancy, |{⃗xiβn − ⃗xjβn : i Sn
+∼ j}|∞,
+to decline with increasing n. The statistic [�
+i∈s Zi : s ∈ Sn] that defines Fn is in
+itself uninformative, S-ancillary (Severini, 2000; Lehmann and Romano, 2022) to
+parameters defined as roots of η �→ E[ψSn(η)], ψSn as defined in (26).
+Proposition 9 says that under mild assumptions about the regularity of {Sn : n}
+and (YT , YC, Z), the solution of (26) tends to a probability limit. To state the
+regularity assumptions, write ¯Ys(z) = 0 if �
+i∈s�Zi = z� = 0, for z = 0 or 1,
+and (�
+i∈s�Zi = z�)−1�
+i∈s Yi�Zi = z� otherwise; and let V (n,s) := ¯Ys(1) − ¯Ys(0) −
+E( ¯Ys(1) − ¯Ys(0) | Fn).
+Proposition 9. Let {( ⃗Xi, YCi, YTi, Zi) : i} be i.i.d., let {Sn : n} be finely stratified
+designs and let Fn = σ((�
+i∈s Zi : s ∈ Sn)). Assume the moment condition that
+for some δ > 0 either: (i) E |YC|1+δ, E |YT |1+δ < ∞ and ns is bounded; or (ii)
+there is a V and δ > 0 with E |V |1+δ < ∞ such that for each n and s ∈ Sn, |V |
+22
+
+stochastically dominates |V (n,s)| given Fn. 2 Let {ws : s ∈ Sn} be nonnegative, Fn-
+measurable weights, and let ψSn(·) be as in (26). Assume that with probability one:
+mn → ∞, where mn := �
+s∈Sn�ws ¯Zs(1− ¯Zs) > 0� is the cardinality of Sn exclusive
+of unmatched singletons and strata receiving zero weight; wsns ¯Zs(1− ¯Zs) is bounded
+above; and m−1
+n
+�
+s∈Sn wsns ¯Zs(1− ¯Zs) is bounded away from 0. Conditionally given
+Fn we then have, for any sigma fields {Gn : n} with Fn ⊆ Gn:
+i. for each η, ψSn(η) − E[ψSn(η)|Gn] → 0 in probability and in L1; and
+ii. η �→ ψSn(η) and η �→ E[ψSn(η) | Gn] have unique roots ˆτn and τn.
+iii. In addition, if there is τ0 ∈ (−∞, ∞) such that τn
+P→ τ0, then ˆτn
+P→ τ0.
+As compared to the classical consistency principle for i.i.d. observations (Hu-
+ber, 1964; Serfling, 1980, Lemma A of § 7.2.1), Proposition 9 upgrades moment
+requirements from estimating equation contributions ψ(W; θ) being L1 to V (n,s)
+being L1+δ, some δ > 0. This enables conclusions in terms of L1 as well as in-
+probability convergence, which in turn accommodates refinement of Fn-conditioning
+to conditioning on finer sigma fields reflecting matched variation in index scores.
+Specifically, consider
+Gn := σ
+�
+( ⃗Xiβn : i ≤ n) ∪ Fn
+�
+.
+The generating statistic ( ⃗Xiβn, 1 ≤ i ≤ n; �
+j∈s Zj, s ∈ Sn) is again S-ancillary to
+matched treatment-control contrasts such as τn.
+Because remaining information about ( ⃗Xi, Zi), 1 ≤ i ≤ n, is barred, the trans-
+formed index scores θi := logit[P(Z = 1|Xβn = xiβn)] determine Gn-conditional
+assignment probabilities as follows. If s ∈ Sn and ζ : s → {0, 1} satisfies �
+i∈s ζi =
+�
+i∈s zi, then
+πs(ζ) := P(Zi = ζi all i ∈ s | Gn)
+=
+�
+�
+�
+exp(θi)
+�
+j∈s exp(θj),
+any i ∈ s s.t. ζi = 1, and ζj = 0 for all j ∈ s \ {i}
+exp(−θi)
+�
+j∈s exp(−θj),
+any i ∈ s s.t. ζi = 0, and ζj = 1 for all j ∈ s \ {i}.
+(27)
+(Because we assume (25), θi ∈ (−∞, ∞) for all i. When s = {i} is an unmatched
+singleton, πs(ζ) = 1 for the sole permissible ζ, {i �→ zi}. When s is a 1 : 1 matched
+pair {i1, i2}, one condition in (27) obtains with i = i1 while the other obtains
+2That is, the Fn-conditional distribution of |V (n,s)| falls at or below the unconditional
+distribution of |V | in the usual stochastic ordering where W ⪯ V iff P(W > a) ≤ P(V > a)
+for all a.
+23
+
+with i = i2, so that (27) presents two distinct expressions for πs(ζ). But these
+expressions then assign the same value to πs(ζ), for each ζ : {i1, i2} → {0, 1}.)
+Now define
+˜ψs(η) :=
+�
+i∈s ψsi(η)
+nsπs(Zs) ;
+˜ψSn(η) :=
+�
+s∈Sn ˜ψs(η)
+�
+s∈Sn wsns ¯Zs(1 − ¯Zs).
+(28)
+In contrast to ψs(η) := �
+i∈s ψsi(η), ˜ψs(η) cannot be calculated in practice, as
+its random denominator involves the unknown βn. Accordingly ˜ψSn(·) lacks direct
+application to effect estimation.
+However, it is useful for analysis of estimates
+based on ψSn(·).
+Proposition 10. i. The unique root of η �→ E[ ˜ψSn(η)|Gn] is
+� �
+s∈Sn
+˜wsns
+�−1 �
+s∈Sn
+˜ws
+�
+i∈s
+E(Y | Z = 1, ⃗Xβn = ⃗xiβn) − E(Y | Z = 0, ⃗Xβn = ⃗xiβn),
+(29)
+where ˜ws := ws¯zs(1 − ¯zs).
+ii. For all η and n,
+���E[ ˜ψSn(η) − ψSn(η) | Gn]
+��� ≤
+[exp(4|{θi − θj : i ∼ j}|∞) − 1] ·
+�
+s∈Sn ˜wsns E |V (n,s)|
+�
+s∈Sn ˜wsns
+,
+(30)
+where θi = logit[P(Z = 1|Xβn = xiβn)] and V (n,s) is as defined in § 5, above.
+iii. If v �→ logit[P(Z = 1|Xβ = v)] is Lipschitz and the conditions of Propo-
+sition 9 hold, |{(⃗xi − ⃗xj)βn : i ∼ j}|∞ → 0 entails that the difference
+of τn with (29) tends in probability to 0, where τn is the unique root of
+η �→ E[ψSn(η) | Gn].
+iv. If the averages (29) tends in probability to a finite limit τ0, then ˆτn
+P→ τ0.
+If the index deconfounds allocation of treatment, (YC, YT ) ⊥ Z|Xβn, then (29)
+coincides with the average causal effect
+�
+s∈Sn ˜ws
+�
+i∈s E(YTi − YCi | ⃗Xβn = ⃗xiβn)
+�
+s∈Sn ˜wsns
+.
+If the index is a propensity score, this deconfounding flows from strong ignora-
+bility in the sense of Rosenbaum and Rubin (1983); if Xβ is a prognostic score,
+strong ignorability entails index strong ignorability under a secondary “no effect
+modification” condition (Hansen, 2008, Prop. 3).
+24
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+Foundations and Trends in Machine Learning, 8, 1–230.
+van der Vaart, A. W. (1998), Asymptotic Statistics, Cambridge University Press.
+Vershynin, R. (2018), High-dimensional probability: An introduction with applica-
+tions in data science, vol. 47, Cambridge university press.
+Wang, L. (2011), “GEE analysis of clustered binary data with diverging number
+of covariates,” Annals of Statistics, 39, 389–417.
+Wang, Y., Cai, H., Li, C., Jiang, Z., Wang, L., Song, J., and Xia, J. (2013),
+“Optimal caliper width for propensity score matching of three treatment groups:
+a Monte Carlo study,” PloS one, 8, e81045.
+A
+Review of mathematical symbols
+The symbols | · |2 and | · |∞ indicate Euclidean and supremum norms as usual
+(§ 3.2). For scalar or vector random variables V , ∥V ∥ψ2 is the sub-gaussian norm
+of V ; for fixed matrices M, |M|2 and |M|F are M’s operator and Frobenius norms
+respectively (§ 2.2). For matrices M and N of like dimension, ⟨M, N⟩F is the
+Frobenius inner product tr(M′N) (§ 3.2).
+Symbols ˆβ, ψ(r, ⃗x, β) and c0(r, ⃗x, β0 + ⃗xβ) are defined in Section 2.1, while
+Section 2.3 defines βn, ˜β, An, and Bn. For partitions S of {1, . . . , n}, [i]S denotes
+the subset of {1, . . . , n} belonging to S that contains i and i S∼ j means there is
+s ∈ S with both i ∈ s and j ∈ s (§ 3). Section 3.2 defines nS (as card(S)) and
+z∗
+m, for positive integers m. �A� is the indicator of event A. Section 5 defines
+¯vs = n−1
+s
+�
+j∈s vj for s ⊆ {1, . . . , n}; associates estimating functions ψSn(·) and
+˜ψSn(·), and sigma fields Fn and Gn, with partitions Sn of {1, . . . , n}; and also
+defines ¯vs(z) for s ⊆ {1, . . . , n} and z ∈ {0, 1}.
+27
+
+B
+Proofs for Section 3
+B.1
+Section 3.1
+Proof of Prop. 2. In light of (8) and IS estimability (A2, A3), the difference be-
+tween (˜βn−βn) and A−1
+n
+1
+n[�n
+j=1 ψ(Rj, ⃗xj; βn)] has Euclidean norm of order smaller
+than n−1/2. Because it is nonrandom, the sub-gaussian norm of this difference is
+also o(n−1/2). So it suffices to show ∥A−1
+n
+1
+n[�n
+j=1 ψ(Rj, ⃗xj; βn)]∥ψ2 = O(n−1/2).
+Given A3, for this it suffices in turn to show that ∥ �n
+j=1 ψ(Rj, ⃗xj; βn)∥ψ2 =
+O(n1/2).
+By (4),
+∥
+n
+�
+i=1
+ψ(Ri, ⃗xi; βn)∥ψ2 =
+sup
+γ:|γ|2=1
+∥
+n
+�
+i=1
+γ′ψ(Ri, ⃗xi; βn)∥ψ2
+=
+sup
+γ:|γ|2=1
+∥
+n
+�
+1
+c0(Ri, ⃗xi, ⃗xiβn)w(⃗xi)⃗xiγ∥ψ2.
+Let k1 be a bound for ∥c0(Ri, ⃗xi, ⃗xiβn)∥ψ2, by A5. According to the general Ho-
+effding inequality (Vershynin, 2018, § 2.6), there is a universal k0 such that
+∥
+n
+�
+1
+c0(Ri, ⃗xi, ⃗xiβ)w(⃗xi)⃗xiγ∥2
+ψ2 ≤k0
+n
+�
+1
+∥c0(Ri, ⃗xi, ⃗xiβ)w(⃗xi)⃗xiγ∥2
+ψ2
+≤k0k1
+n
+�
+1
+|w(⃗xi)⃗xiγ|2
+2. So
+sup
+γ:|γ|2=1
+∥
+n
+�
+1
+c0(Ri, ⃗xi, ⃗xiβ)w(⃗xi)⃗xiγ∥2
+ψ2 ≤k0k1n
+�
+1
+n
+n
+�
+1
+w(⃗xi)2
+�
+·
+sup
+γ:|γ|2=1
+�
+1
+n
+n
+�
+1
+|⃗xiγ|2
+2
+�
+.
+The left-hand side equals the square of ∥ �n
+1 c0(Ri, ⃗xi, ⃗xiβ)⃗xi∥ψ2, whereas A6 says
+the product at right is O(n). The proof is complete.
+B.2
+Proofs for section 3.2
+The following lemma helps to prove Proposition 3.
+Lemma 2. Under the conditions of Proposition 3, for all 1 ≤ i < j < n we have
+( ⃗Xi − ⃗Xj) ⊥ ˜β − βn | S, {⃗µ(s) : s ∈ S}.
+28
+
+Proof. Recall that ˜β − βn = A−1
+n
+1
+n
+�n
+i=1 ψ(Ri, ⃗Xi; βn). Suppressing conditioning
+for S, {µs : s ∈ S} in the notation,
+Cov
+�
+A−1
+n
+1
+n
+n
+�
+i=1
+ψ(Ri, ⃗Xi; βn), ⃗X1 − ⃗X2
+�
+=
+1
+n Cov
+�
+A−1
+n ψ(R1, ⃗X1; βn), ⃗X1
+�
+−
+1
+n Cov
+�
+A−1
+n ψ(R2, ⃗X2; βn), ⃗X2
+�
+= 0.
+By joint Normality of {[ ⃗Xi; ψ′(Ri, ⃗Xi; βn)] : i}, 1
+nA−1
+n ψ(R1, ⃗X1; βn) and ( ⃗Xi − ⃗Xj)
+are jointly Normal, and the fact that they are uncorrelated means they are inde-
+pendent.
+Proof of Proposition 3. For fixed γ ∈ ℜp we have, after some algebra that I omit,
+E[( ⃗X1 − ⃗X2)γ]2 = 2γ′Σγ
+and
+E
+��{( ⃗Xi − ⃗Xj)γ : i S∼ j, i ̸= j}
+��2
+2 = 2γ′Σγ.
+(Throughout the proof I write ““E[·]” for “ES[·].”) By the conditional indepen-
+dence established in Lemma 2, it follows that
+E
+��{( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j, i ̸= j}
+��2
+2 = ⟨2Σ, C⟩F .
+Also for fixed γ, (5) as applied to Normal variables gives
+E
+��{( ⃗Xi − ⃗Xj)γ : i S∼ j, i ̸= j}
+��
+∞ ≤ [4γ′Σγ log 2nS]1/2.
+(31)
+Since ˜β is independent of ⃗Xi − ⃗Xj for each 1 ≤ i < j ≤ n (Lemma 2), (31)
+entails
+E
+�
+E
+���{( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j, i ̸= j}
+��
+∞ | ˜β
+�2�
+≤
+E
+�
+4(˜β − βn)′Σ(˜β − βn) log 2nS
+�
+= 4⟨Σ, Cn⟩F log 2nS.
+Combining this fact with Jensen’s inequality for conditional expectation,
+�
+E
+���{( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j, i ̸= j}
+��
+∞
+��2
+≤
+E
+�
+E
+���{( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j, i ̸= j}
+��
+∞ | ˜β
+�2�
+≤ 4⟨Σ, Cn⟩F log 2nS.
+29
+
+B.3
+Section 3.3
+Proof of Lemma 1. Letting k < ∞ denote the supremum (Condition A5) of sub-
+gaussian norms of {c0(Ri, ⃗xi, ⃗xiβn) : i}, E[c0(Ri, ⃗xi, ⃗xiβn)4] ≤ (2k)4 for all i (e.g.,
+Vershynin, 2018, § 2.5.2). Accordingly �
+i E
+�
+c0(Ri, ⃗xi, ⃗xiβn)4�
+= O(n) and in turn
+�
+i{E
+�
+c0(Ri, ⃗xi, ⃗xiβn)2�
+}2 = O(n). Combining this with supγ:|γ|2=1
+�
+i w(⃗xi)4(⃗xiγ)4 =
+O(n) (Condition A6), Cauchy-Schwartz gives supγ:|γ|2=1
+�
+i E
+�
+c0(Ri, ⃗xi, ⃗xiβn)2�
+w(⃗xi)2(⃗xiγ)2 =
+O(n). Rearranging terms in light of (4), this says supγ:|γ|2=1
+�
+i E
+�
+[γ′ψ(Ri, ⃗xi, β)]2�
+=
+O(n), or supγ:|γ|2=1 γ′nBnγ = O(n); thus |Bn|2 = O(1).
+Condition A3 gives
+|A−1
+n |2 = O(1), so also |C−1
+n |2 = O(n−1).
+Our demonstration of Proposition 4 relies on three supporting lemmas, as
+follows.
+Lemma 3. Under A4 and A6, supβ |An(β)|2 = O(1).
+Proof. Write c1(⃗x, η) := (∂/∂η) E[c0(R, ⃗x, η) | ⃗X = ⃗x] so that ∇β E[ψ(R, ⃗x, β) | ⃗X = ⃗x] =
+c1(⃗x, ⃗xβ)w(⃗x)⃗x′⃗x. By A4, there is K1 < ∞ such that |c1(⃗x, β0 + ⃗xβ)| < K1, for
+any β. By Cauchy-Schwartz, A6 gives supγ:|γ|2=1
+�n
+i=1 w(⃗xi)(⃗xiγ)2 = O(n). Since
+n|An(β)|2 =
+sup
+γ:|γ|2=1
+γ′
+��
+i
+c1(⃗xi, ⃗xiβ)w(⃗xi)⃗x′⃗x
+�
+γ ≤ K1
+sup
+γ:|γ|2=1
+�
+i
+w(⃗xi)(⃗xγ)2
+for any β, the result follows.
+Lemma 4. Under A1–A9, |An(ˆβ) − An(βn)|2
+P→ 0 and | ˆBn(ˆβ) − ˆBn(βn)|2
+P→ 0.
+Lemma 5. Under A1–A9 as well as A11 and sub-√n dimension, | ˆBn(βn) −
+Bn(βn)|2
+P→ 0.
+Proofs of Lemmas 4 and 5 are given following the proof of Proposition 4.
+Proof of Proposition 4. Since ˆAn ≡ An(ˆβ), | ˆAn −An|2
+P→ 0 follows from Lemma 4.
+Since ˆBn ≡ ˆBn(ˆβ), | ˆBn−Bn|2
+P→ 0 follows from | ˆBn(ˆβ)− ˆBn(βn)|2
+P→ 0 (Lemma 4)
+and | ˆBn(βn) − Bn|2
+P→ 0 (Lemma 5).
+Since |A−1
+n |2 = O(1) and | ˆA−1
+n |2 = OP (1) (Condition A3 and Prop. 1), it follows
+that | ˆA−1
+n
+− A−1
+n |2
+P→ 0, by applying sub-multiplicativity of the spectral norm to
+the right-hand side of ( ˆA−1
+n −A−1
+n ) = ˆA−1
+n ( ˆAn −An)A−1
+n . Since also |Bn|2 = OP (1)
+(Lemma 1), the 2-norms of the second and third summands in ˆA−1
+n
+ˆBn ˆA−1
+n
+=
+A−1
+n BnA−1
+n + ( ˆA−1
+n − A−1
+n )BnA−1
+n + ˆA−1
+n Bn( ˆA−1
+n − A−1
+n ) + ˆA−1
+n ( ˆBn − Bn) ˆA−1
+n
+30
+
+must tend in probability to 0. Thus the stochastic order of |A−1
+n BnA−1
+n − ˆA−1
+n
+ˆBn ˆA−1
+n |2
+can be no greater than that of | ˆA−1
+n ( ˆBn −Bn) ˆA−1
+n |2. But as A4 and A6 entail that
+| ˆAn|2 = OP (1), by Lemma 3, OP (| ˆA−1
+n ( ˆBn−Bn) ˆA−1
+n |2) = OP (| ˆBn−Bn|2) = oP (1);
+this means |Cn − ˆCn|2 = n−1|A−1
+n BnA−1
+n − ˆA−1
+n
+ˆBn ˆA−1
+n |2 = oP (n−1).
+This proof of Lemma 4 was based in part on Wang’s proof of a similar principle
+for generalized estimating equations (2011, Thm. 3.10).
+Proof of Lemma 4. To establish |An(ˆβ)−An(βn)|2 = supγ:|γ|2=1 γ′[An(ˆβ)−An(βn)]γ P→
+0, let K1 < ∞ be a Lipschitz constant for η �→ c1(r, ⃗x, η), each r and ⃗x, where c1(·)
+is as defined in the proof of Lemma 3, above. (By Condition A4.) Then
+|γ′{∇β E[ψ(R, ⃗xi, β) | ⃗X = ⃗xi]β=ˆβ − ∇β E[ψ(R, ⃗xi, β) | ⃗X = ⃗xi]β=βn}γ|
+≤ K1[⃗xi(ˆβ − βn)]w(⃗xi)(⃗xiγ)2.
+(32)
+Summing over i and applying Cauchy-Schwartz,
+(γ′[An(ˆβ) − An(βn)]γ)2 ≤K2
+1
+�
+1
+n
+n
+�
+i=1
+w(⃗xi)2(⃗xiγ)4
+� �
+� 1
+n
+n
+�
+i=1
+�
+⃗xi
+ˆβ − βn
+|ˆβ − βn|2
+�2�
+� × |ˆβ − βn|2
+2,
+interpreting “⃗x(δ/|δ|2)” as 0 when |δ|2 = 0. It follows that
+|An(ˆβ)−An(βn)|2
+2 ≤ K2
+1
+�
+sup
+γ:|γ|2=1
+1
+n
+n
+�
+i=1
+w(⃗xi)2(⃗xiγ)4
+��
+sup
+δ:|δ|2=1
+1
+n
+n
+�
+i=1
+(⃗xiδ)2
+�
+|ˆβ−βn|2
+2.
+Observe that the conditions of Proposition 1 follow from those of Proposition 4,
+so that we may assume |ˆβ−βn|2 = OP [(p/n)1/2]. This combines with Condition A6
+to give |An(ˆβ) − An(βn)|2 = O(1)O(1)OP [(p/n)1/2], which by A9 is oP (1).
+As to | ˆBn(ˆβ) − ˆBn(βn)|2,
+γ′{Ψ(Ri, ⃗xi, ˆβ)Ψ(Ri, ⃗xi, ˆβ)′ − Ψ(Ri, ⃗xi, βn)Ψ(Ri, ⃗xi, βn)′}γ
+(33)
+=(c2
+0(Ri, ⃗xi, ˆηi) − c2
+0(Ri, ⃗xi, ηi))w(⃗xi)2(⃗xiγ)2
+=(c0(Ri, ⃗xi, ˆηi) − c0(Ri, ⃗xi, ηi))(c0(Ri, ⃗xi, ˆηi) + c0(Ri, ⃗xi, ηi))w(⃗xi)2(⃗xiγ)2
+=(c0(Ri, ⃗xi, ˆηi) − c0(Ri, ⃗xi, ηi))2w(⃗xi)2(⃗xiγ)2
++ (c0(Ri, ⃗xi, ˆηi) − c0(Ri, ⃗xi, ηi)) · 2c0(Ri, ⃗xi, ηi)w(⃗xi)2(⃗xiγ)2
+=:Vi(γ) + Wi(γ).
+(34)
+31
+
+By the Lipschitz property (A4) of c0(r, ⃗x, ·),
+sup
+γ:|γ|2=1
+1
+n
+�
+i
+|Vi(γ)| ≤K2
+1
+sup
+γ:|γ|2=1
+1
+n
+�
+i
+(ˆηi − ηi)2w(⃗xi)2(⃗xiγ)2
+≤K2
+1|ˆβ − βn|2
+2
+sup
+δ,γ:|δ|2=|γ|2=1
+1
+n
+�
+i
+w(⃗xi)2(⃗xiδ)2(⃗xiγ)2
+= O(1)OP (p/n)O(1) = oP (1),
+(35)
+invoking Assumption A6 and consistency of ˆβ at (35). The Lipschitz property of
+c0(r, ⃗x, ·) also gives
+sup
+γ:|γ|2=1
+1
+n
+�
+i
+|Wi(γ)| ≤2K1
+sup
+γ:|γ|2=1
+1
+n
+�
+i
+|c0(Ri, ⃗xi, ηi)(ˆηi − ηi)|w(⃗xi)2(⃗xiγ)2
+≤2K1|ˆβ − βn|2
+sup
+γ,δ:|γ|2=|δ|2=1
+1
+n
+�
+i
+c0(Ri, ⃗xi, ηi)(⃗xiδ)w(⃗xi)2(⃗xiγ)2
+≤2K1|ˆβ − βn|2[ 1
+n
+�
+i
+c4
+0(Ri, ⃗xi, ηi)]1/4
+×
+sup
+δ:|δ|2=1
+[ 1
+n
+�
+i
+w(⃗xi)4(⃗xiδ)4]1/4
+sup
+γ:|γ|2=1
+[ 1
+n
+�
+i
+w(⃗xi)4(⃗xiγ)4]1/2
+(36)
+= OP (p/n)OP (1)O(1)O(1) = oP (1).
+Here we apply Cauchy-Schwartz (twice) at (36) and, to pass to the next line, con-
+sistency of ˆβ as per Proposition 1, Assumption A5 in combination with Markov’s
+inequality and Assumption A6. Since
+| ˆBn(ˆβ) − ˆBn(βn)|2 =
+sup
+γ:|γ|2=1
+γ′( ˆBn(ˆβ) − ˆBn(βn))γ
+≤
+sup
+γ:|γ|2=1
+1
+n
+�
+i
+|Vi(γ)| +
+sup
+γ:|γ|2=1
+1
+n
+�
+i
+|Wi(γ)|,
+the result follows.
+Proof of Lemma 5. To control |γ[ ˆBn(βn) − Bn(βn)]γ′|, fix γ with |γ|2 = 1 and
+consider
+γ[ ˆBn(βn) − Bn(βn)]γ′ =γ′�
+n−1 �
+i≤n
+{c2
+0(Ri, ⃗xi, ⃗xiβn) − E[c2
+0(R, ⃗xi, ⃗xiβn) | ⃗X = ⃗xi]}w2(⃗xi)⃗x′
+i⃗xi
+�
+γ
+=n−1 �
+i≤n
+{c2
+0i(Ri) − E[c2
+0i(R) | ⃗X = ⃗xi]}w2(⃗xi)(⃗xiγ)2.
+32
+
+Observe that A5 entails the random variables c2
+0i(Ri) − E[c2
+0i(R) | ⃗X = ⃗xi] to be
+sub-exponential with uniformly bounded sub-exponential norm (Vershynin, 2018,
+§ 2.7). Applying Bernstein’s inequality (Vershynin, 2018, § 2.8),
+P{|γ[ ˆBn(βn) − Bn(βn)]γ′| ≥ t} ≤
+2 exp
+�
+−k2 min
+�
+n2t2
+k2e
+�n
+i=1 w(⃗xi)4(⃗xiγ)4 ,
+nt
+ke maxi≤n w(⃗xi)2(⃗xiγ)2
+��
+,
+(37)
+where ke is a finite upper bound for the sub-exponential norms of c2
+0i(Ri) −
+E[c2
+0i(Ri)], i ≥ 1, and k2 is a universal constant.
+Now let N be a 1/4-net of the p-dimensional sphere, i.e. a finite subset of
+{γ : |γ|2 = 1} with the property that {γ : |γ|2 = 1} is covered by balls centered in
+N of radius 1/4, so that (Vershynin, 2018, § 4.4.1)
+sup
+γ:|γ|2=1
+γ[ ˆBn(βn) − Bn(βn)]γ′ ≤ 2 sup
+γ∈N
+γ[ ˆBn(βn) − Bn(βn)]γ′.
+(38)
+We may select this N to have cardinality no more than 9p (Vershynin, 2018,
+Corr. 4.2.13).
+Since (37) holds for arbitrary γ on the unit sphere, it follows that
+1
+2 P{ sup
+γ:|γ|2=1
+|γ[ ˆBn(βn) − Bn(βn)]γ′| ≥ t} ≤
+exp
+�
+p log(9) − nk2 min
+�
+t2
+k2en−1 �n
+i=1 w(⃗xi)4(⃗xiγ)4 ,
+t
+ke maxi≤n w(⃗xi)2(⃗xiγ)2
+��
+=
+exp
+�
+−k2 min
+�
+nt2
+k2en−1 �n
+i=1 w(⃗xi)4(⃗xiγ)4 − pk3,
+nt
+ke maxi≤n w(⃗xi)2(⃗xiγ)2 − pk3
+��
+,
+(39)
+where k3 = log(9)/k2.
+Recalling that n−1 �n
+i=1 w(⃗xi)4(⃗xiγ)4 = O(1) (by A6),
+t2/[k2
+en−1 �n
+i=1 w(⃗xi)4(⃗xiγ)4] has positive limit infimum and finite limit supremum
+(for any t); thus the first of the two quantities of which the minimum is taken tends
+to ∞ because p = o(n) (A9). Since maxi≤n w(⃗xi)2(⃗xiγ)2 = O{max[p log n, (log n)2]}
+(A11), p2 log n = o(n) (sub-√n dimension) entails that the second quantity also
+must tend to ∞. (If pn ≤ log n infinitely often, then on the subsequence for which
+this is true the term in question is bounded below by (log n){(t/ke)n/[(log n) maxi≤n w(⃗xi)2(⃗xiγ)2] − k3},
+which tends to ∞ with (log n)[n/(log n)3 − 1].
+Otherwise pn > log n so that
+max[pn log n, (log n)2] = pn log n. The term in question equals p{(t/ke)n/[p maxi≤n w(⃗xi)2(⃗xiγ)2] − k3},
+which tends to ∞ because p ↑ ∞, maxi≤n w(⃗xi)2(⃗xiγ)2 = O(p log n) and p2 log n =
+o(n).) So the minimum in (39) increases without bound, and (39) itself tends to
+0.
+33
+
+Proposition 5’s proof uses two supporting lemmas.
+Lemma 6. Under A1 and A6, s(xˆβ) − s(xβn) = OP (|ˆβ − βn|2).
+Proof. By A1,
+s2(xˆβ) − s2(xβn) = ˆβ′S ˆβ − β′
+nSβn
+= (ˆβ + βn)′S(ˆβ − βn)
+= (S1/2 ˆβ + S1/2βn)′S1/2(ˆβ − βn),
+where S1/2 denotes the matrix square root of S. Noting A6’s implication that
+|S1/2|2 = O(1),
+|s(xˆβ) − s(xβn)| =[s(xˆβ) + s(xβn)]−1|(S1/2 ˆβ + S1/2βn)′S1/2(ˆβ − βn)|
+≤ (|S1/2 ˆβ|2 + |S1/2βn|2)−1|S1/2 ˆβ + S1/2βn|2|S1/2|2|ˆβ − βn|2
+≤ |S1/2|2|ˆβ − βn|2 = O(1)OP (|ˆβ − βn|2).
+In the proof of Lemma 7 below, let (I, J) ⊆ {1, . . . , n} be a randomly ordered
+simple random sample of size 2, and let ⃗D (or ⃗D⊥w, w an n-vector) be a 1 × p
+random vector representing the paired difference ⃗xI − ⃗xJ (or ⃗x⊥w
+I
+− ⃗x⊥w
+J
+). Then
+s. e.2
+r(I, J) = ⃗D⊥xˆβ ˆC ⃗D⊥xˆβ′
+(40)
+Because of the U-statistic representation of covariance, Cov( ⃗D) = 2S(x). By
+symmetry of the distribution of (I, J), E
+�
+⃗D
+�
+= 0, so that E
+�
+⃗D′ ⃗D
+�
+= Cov(D) =
+2S(x). Selection of (I, J) pays no attention to the distinction between treatment
+and control, making ⃗D independent of {Zi}n
+i=1 and, by extension, of ˆβ and ˆC.
+Therefore its conditional and marginal moments coincide: E
+�
+⃗D|ˆβ, ˆC
+�
+= E
+�
+⃗D
+�
+=
+0; E
+�
+⃗D′ ⃗D|ˆβ, ˆC
+�
+= Cov( ⃗D|ˆβ, ˆC) = Cov( ⃗D) = 2S(x).
+Lemma 7. Let S = S(x) or S⊥v = (n − L)−1x⊥v′x⊥v, some categorical vari-
+able v with L categories. Then ⟨S⊥ˆβ, ˆC⟩F = ⟨S, ˆC⟩F − s−2(xˆβ)⟨S ˆβ ˆβ′S, ˆC⟩F and
+⟨S⊥βn, C⟩F = ⟨S, ˆC⟩F − s−2(xβn)⟨Sβnβn′S, C⟩F .
+Proof. If S = S(x), let ⃗D be as defined above. Otherwise, if S = S⊥v, then let
+(I, J) ⊆ {1, . . . , n} be a randomly ordered stratified simple random sample of size
+2 from one of the categories of v with at least two elements, after selecting one of
+these categories with probability proportional to size, and let ⃗D (or ⃗D⊥w, w an
+34
+
+n-vector) be a 1 × p random vector representing the paired difference ⃗xI − ⃗xJ (or
+⃗x⊥w
+I
+− ⃗x⊥w
+J
+).
+Evaluate E
+�
+⃗D⊥xˆβ′ ⃗D⊥xˆβ|ˆβ, ˆC
+�
+= Cov( ⃗D⊥xˆβ|ˆβ, ˆC) using the U-statistic rep-
+resentation of sample covariance to get Cov( ⃗D⊥xˆβ|ˆβ, ˆC) = 2(n − 1)−1x⊥xˆβ′x⊥xˆβ.
+Now compare to:
+1
+2 E
+�
+⃗D⊥xˆβ′ ⃗D⊥xˆβ|ˆβ, ˆC
+�
+=(I − ˆβ ˆβ′S(x)/s2(xˆβ))′S(x)(I − ˆβ ˆβ′S(x)/s2(xˆβ))
+=S(x) − 2s−2(xˆβ)S(x) ˆβ ˆβ′S(x)+
+s−4(xˆβ)S(x) ˆβ · ˆβ′S(x) ˆβ · ˆβ′S(x)
+=S(x) − s−2(xˆβ)S(x) ˆβ ˆβ′S(x) =: S⊥ˆβ.
+Proof of Prop. 5. It follows from A6 that ∥S∥F = tr(S′S)1/2 = O(p1/2), and from
+the assumed consistency of ˆCn that ∥ ˆCn − Cn∥F = oP (p1/2/n). So
+|⟨S, ˆCn⟩F − ⟨S, Cn⟩F | = |⟨S, ˆCn − Cn⟩F | = oP (p/n).
+By Lemma 7,
+⟨S⊥ˆβ, ˆCn⟩F − ⟨S⊥βn, Cn⟩F =⟨S, ˆCn − Cn⟩F
++ s−2(xˆβ)⟨S ˆβ ˆβ′S, ˆCn⟩F − s−2(xβn)⟨Sβnβn′S, Cn⟩F
+=oP
+� p
+n
+�
++ [s−2(xˆβ) − s−2(xβn)]⟨S ˆβ ˆβ′S, ˆCn⟩F
+(41)
++ s−2(xβn)
+�
+⟨S ˆβ ˆβ′S, ˆCn⟩F − ⟨Sβnβn′S, Cn⟩F
+�
+.
+(42)
+To analyze the rightmost summand of (41), first observe that s2(xβn) =
+βn′Sβn, so that |s(xβn)| ≤ |S|2|βn|2 = O(p1/2) . Together with Proposition 1,
+Lemma 6 entails that s(xˆβ) − s(xβn) = OP (p/n).
+So A9 says that s(xˆβ) =
+OP (p1/2) just as |s(xβn)| = OP (p1/2), whence s(xˆβ)+s(xβn) = OP (p1/2). Combin-
+ing these facts, s2(xˆβ)−s2(xβn) = [s(xˆβ) − s(xβn)][s(xˆβ) + s(xβn)] = OP (p/n1/2).
+In light of A8, it follows that |s−2(xˆβ) − s−2(xβn)| = OP {p/[n1/2s4(xβn)]}.
+As to the ⟨S ˆβ ˆβ′S, ˆCn⟩F factor of (41),
+|⟨S ˆβ ˆβ′S, ˆCn⟩F | = tr(S ˆβ ˆβ′S ˆCn)
+= | tr(ˆβ′S ˆCnS ˆβ)|
+= ˆβ′S ˆCnS ˆβ ≤ ∥ˆβ′S1/2∥2∥S1/2∥2∥ ˆCn∥2∥S1/2∥2∥S1/2 ˆβ∥2
+= OP (s2(xβn)
+n
+),
+35
+
+by A8, A6, consistency of ˆCn and Lemma 1. We now have
+|[s−2(xˆβ) − s−2(xβn)]⟨S ˆβ ˆβ′S, ˆCn⟩F | = OP
+�
+p
+n3/2s2(xβn)
+�
+= oP
+� p
+n
+�
+.
+To bound (42), in light of A8 we focus on the second factor:
+|⟨S ˆβ ˆβ′S, ˆCn⟩F − ⟨Sβnβn′S, Cn⟩F | ≤
+|⟨S ˆβ ˆβ′S − Sβnβn′S, ˆCn⟩F | + |⟨Sβnβn′S, ˆCn − Cn⟩F |.
+(43)
+We address the left term of (43) via Cauchy-Schwartz.
+By Lemma 1 and
+consistency of ˆCn, ∥ ˆCn∥2 = OP (n−1), so ∥ ˆCn∥F = OP (p1/2/n).
+As S ˆβ ˆβ′S −
+Sβnβn′S = S(ˆβ ˆβ′ − βnβn′)S, ∥S ˆβ ˆβ′S − Sβnβn′S∥F ≤ ∥S1/2∥2∥S1/2 ˆβ ˆβ′S1/2 −
+S1/2βnβn′S1/2∥F ∥S1/2∥2 = OP (1)OP
+�
+∥S1/2 ˆβ ˆβ′S1/2 − S1/2βnβn′S1/2∥F
+�
+OP (1). This
+decomposes as
+∥S1/2 ˆβ ˆβ′S1/2 − S1/2βnβn′S1/2∥F ≤ ∥S1/2(ˆβ − βn)(ˆβ + βn)′S1/2∥F +
+∥S1/2 ˆββ′
+nS1/2 − S1/2βn ˆβ′S1/2∥F
+where:
+∥S1/2(ˆβ − βn)(ˆβ + βn)′S1/2∥F ≤∥S1/2∥2∥(ˆβ − βn)(ˆβ + βn)′S1/2∥F
+= OP (1) tr[S1/2(ˆβ + βn)(ˆβ − βn)′(ˆβ − βn)(ˆβ + βn)′S1/2]1/2
+= OP (1) tr[(ˆβ + βn)′S(ˆβ + βn)(ˆβ − βn)′(ˆβ − βn)]1/2
+= OP (1)[(ˆβ + βn)′S(ˆβ + βn)∥ˆβ − βn∥2
+2]1/2
+= OP (1)[(ˆβ + βn)′S(ˆβ + βn)]1/2OP [( p
+n)1/2]
+= OP [( p
+n)1/2][(ˆβ − βn + 2βn)′S(ˆβ − βn + 2βn)]1/2
+= OP [( p
+n)1/2][∥S1/2(ˆβ − βn)∥2
+2 + 4(ˆβ − βn)′Sβn + 4∥S1/2βn∥2
+2]1/2
+= OP [( p
+n)1/2]
+�
+OP
+� p
+n
+�
++ OP
+�� p
+n
+�1/2
+s(xβn)
+�
++ OP [s2(xβn)]
+�1/2
+= OP
+�� p
+n
+�1/2�
+OP
+��� p
+n
+�1/2
++ s(xβn)
+�2·1/2�
+= OP
+�
+max
+� p
+n,
+� p
+n
+�1/2
+s(xβn)
+��
+;
+36
+
+and
+∥S1/2 ˆββ′
+nS1/2 − S1/2βn ˆβ′S1/2∥F =∥S1/2(ˆβ − βn)β′
+nS1/2 − S1/2βn(ˆβ − βn)′S1/2∥F
+≤∥S1/2(ˆβ − βn)β′
+nS1/2∥F + ∥S1/2βn(ˆβ − βn)′S1/2∥F
+= 2 tr[S1/2βn(ˆβ − βn)′S(ˆβ − βn)β′
+nS1/2]1/2
+= 2[(ˆβ − βn)′S(ˆβ − βn)β′
+nSβn]1/2
+= OP
+�� p
+n
+�1/2
+s(xβn)
+�
+.
+Thus the left term at right of (43) is OP {max[p/n, (p/n)1/2s(xβn)]}.
+The remaining term in (43) is bounded as follows, using the definition of ⟨·, ·⟩F
+and the cyclic property of the trace:
+⟨Sβnβn′S, ˆCn − Cn⟩F = tr[Sβnβ′
+nS( ˆCn − Cn)]
+= tr[β′
+nS( ˆCn − Cn)Sβn]
+=β′
+nS( ˆCn − Cn)Sβn
+≤|βn|2
+2∥S∥2
+2∥ ˆCn − Cn∥2
+= OP (p)OP (1)oP (n−1) = oP (p/n),
+using A7, A6 and consistency of ˆCn. This shows that (43) as a whole is OP {max[p/n, (p/n)1/2s(xβn)]},
+from which it follows that (42) is OP {max[(p/n)s−2(xβn), (p/n)1/2s−1(xβn)]} =
+oP (p/n), completing the proof of the proposition’s consistency claim. The remain-
+der of the proposition follows from A6 and Lemma 1.
+C
+Proofs for section 4
+C.1
+Proof for Section 4.2
+Proof of Proposition 6. For parts (i) and (iii) it suffices to show that supi |1 −
+|⃗xi ˆC1/2|2
+2/|⃗xiC1/2|2
+2| and supi,j |1 − |(⃗xi − ⃗xj) ˆC1/2|2
+2/|(⃗xi − ⃗xj)C1/2|2
+2| tend in prob-
+ability to 0 (from the definition of in-probability convergence), and to consider
+only (i, j) with ⃗xi ̸= 0 and ⃗xi ̸= ⃗xj.
+The latter ensures |⃗xiC1/2|2 > 0 and
+|(⃗xi − ⃗xj)C1/2|2 > 0, since Cn is of full rank. Indeed, Lemma 3 in the appendix
+gives |An|2 = O(1), covering case (i); for case (iii), A12 gives |B−1
+n |2 = O(1), and
+in turn |AnB−1
+n An|2 = O(1). Note that this shows not only that Cn has full rank
+but also that |C−1
+n |2 = O(1).
+37
+
+For arbitrary nonzero p-vectors v we have
+1 − v′ ˆCnv
+v′Cnv = v′(Cn − ˆCn)v
+v′v
+v′v
+v′Cnv
+so that
+�����1 − v′ ˆCnv
+v′Cnv
+����� ≤
+�����
+v′(Cn − ˆCn)v
+v′v
+v′C−1
+n v
+v′v
+����� ≤ |Cn − ˆCn|2|C−1
+n |2,
+using general inequalities for positive definite matrices (e.g., Gentle, 2007, § 8.4).
+Proposition 4 gives | ˆCn − Cn|2 = oP (n−1), completing the proof of parts (i) and
+(iii).
+For part (ii), write a ∨ b := max(a, b) and a ∧ b := min(a, b). For arbitrary
+nonzero p-vectors v
+�����
+v′Cnv ∨ ϵn − v′ ˆCnv ∨ ϵn
+v′v
+����� ≤
+�����
+v′Cnv − v′ ˆCnv
+v′v
+����� , since
+���v′Cnv ∨ ϵn − v′ ˆCnv ∨ ϵn
+��� ≤
+���v′Cnv − v′ ˆCnv
+��� .
+(The latter is true because the expressions on either side of the inequality are equal
+if v′Cnv and v′ ˆCnv belong to the same half-interval, (0, ϵn] or [ϵn, ∞), whereas if
+they are separated by ϵn then the left side expression is smaller.) So
+�����
+v′Cnv ∨ ϵn − v′ ˆCnv ∨ ϵn
+v′v
+����� ≤ |Cn − ˆCn|2 = oP (n−1).
+(44)
+According to A11, supi ⃗xi⃗x′
+i = O[max(p, log n)]. So
+sup
+i
+⃗xi⃗x′
+i
+ϵn
+,
+sup
+{i,j}∈En
+(⃗xi − ⃗xj)(⃗xi − ⃗xj)′
+ϵn
+= OP (n).
+Thus
+sup
+i
+⃗xi⃗x′
+i
+⃗xiCn⃗x′
+i ∨ ϵn
+,
+sup
+{i,j}∈En
+(⃗xi − ⃗xj)(⃗xi − ⃗xj)′
+(⃗xi − ⃗xj)Cn(⃗xi − ⃗xj)′ ∨ ϵn
+= OP (n).
+By (44),
+sup
+i:|⃗xi|2̸=0
+⃗xiCn⃗x′
+i ∨ ϵn − ⃗xi ˆCn⃗x′
+i ∨ ϵn
+⃗xi⃗x′
+i
+⃗xi⃗x′
+i
+⃗xiCn⃗x′
+i ∨ ϵn
+= oP (1),
+38
+
+and likewise
+sup
+{i,j}∈En:
+⃗xi̸=⃗xj
+(⃗xi − ⃗xj)Cn(⃗xi − ⃗xj)′ ∨ ϵn − (⃗xi − ⃗xj) ˆCn(⃗xi − ⃗xj)′ ∨ ϵn
+(⃗xi − ⃗xj)(⃗xi − ⃗xj)′
+×
+(⃗xi − ⃗xj)(⃗xi − ⃗xj)′
+(⃗xi − ⃗xj)Cn(⃗xi − ⃗xj)′ ∨ ϵn
+= oP (1).
+Proof of Proposition 7. The assumption on { ⃗Xi : i ≤ n} entails that ( ⃗Xi − ⃗Xj)
+is MVN(0, 2Σ) and ( ⃗Xi − ⃗Xj)C1/2 is MVN(0, 2C1/2ΣC1/2).
+Let 2C1/2ΣC1/2
+have eigendecomposition Q′ΛQ, with Λ nonnegative real diagonal and Q an or-
+thogonal matrix. Then ( ⃗Xi − ⃗Xj)C1/2Q′ ∼ MVN(0, Λ). Writing Wij1, . . . , Wijp
+for the p coordinates of ( ⃗Xi − ⃗Xj)C1/2Q′ =: ⃗Wij, we see that Wij1, . . . , Wijp
+are mutually independent mean-0 Gaussians with variances v1, . . . , vp, the diago-
+nal entries of Λ and eigenvalues of 2C1/2ΣC1/2, while ( ⃗Xi − ⃗Xj)C( ⃗Xi − ⃗Xj)′ =
+( ⃗Xi − ⃗Xj)C1/2Q′QC1/2( ⃗Xi − ⃗Xj)′ = ⃗Wij ⃗W ′
+ij.
+Straightforwardly, for any i ̸= j E ⃗Wij ⃗W ′
+ij = �
+i vi, or tr(2C1/2ΣC1/2) =
+2⟨C, Σ⟩F .
+We proceed to characterize the moment generating function (MGF)
+of ⃗Wij ⃗W ′
+ij − E ⃗Wij ⃗W ′
+ij, or �p
+k=1(W 2
+ijk − E W 2
+ijk).
+The centered χ2
+1 distribution
+having MGF e−t(1 − 2t)−1/2, valid for t < 1/2, W 2
+ijk − E W 2
+ijk has log MGF
+(1/2)[−2vkt − log(1 − 2vkt)], valid for t < 1/(2vk). Applying the relation − log(1−
+x)−x ≤ x2/[2(1 − x)], 0 < x < 1 (a transformation of log(1−x)’s degree 2 Taylor
+expansion), we now have
+log E{exp t[W 2
+ijk − E W 2
+ijk]} ≤
+(vkt)2
+1−2vkt
+�
+for t <
+1
+2vk
+�
+≤
+(vkt)2
+1−2(maxk vk)t
+�
+t <
+1
+2 maxk vk
+�
+; so
+log E exp{t
+p
+�
+k=1
+[W 2
+ijk − E W 2
+ijk)]}
+≤
+t2 �
+k v2
+k
+1−2(maxk vk)t,
+t <
+1
+2 maxk vk
+due to independence of uncorrelated Gaussians. That is, �p
+k=1 W 2
+ijk − E W 2
+ijk is
+sub-gamma on its right tail with variance factor �
+k v2
+k and scale factor maxk vk;
+in symbols, �p
+k=1 W 2
+ijk − E W 2
+ijk ∈ Γ+(�
+k v2
+k, maxk vk) (Boucheron et al., 2013,
+§ 2.4). Since �
+k v2
+k = tr(Λ2), the cyclic property of the trace combines with defi-
+nitions above to reduce this variance factor to 4|C1/2ΣC1/2|2
+F , as tr(Q′ΛQQ′ΛQ) =
+4 tr(C1/2ΣC1/2C1/2ΣC1/2); and the scale factor reduces to 2|C1/2ΣC1/2|2, as maxk vk =
+|Λ|2 and |Λ|2 = |Q′ΛQ|2.
+39
+
+For an MGF characterization in terms of more familiar quantities, note that
+|2C1/2ΣC1/2|2
+F = �p
+i=1 v2
+i ≤ (maxk vk)(�
+k vk) = 2|C1/2ΣC1/2|2(�
+k vk), while
+�
+k vk = tr(D2) = tr(2C1/2ΣC1/2) = 2⟨Σ, C⟩F . So the variance factor can be
+taken as 4|C1/2ΣC1/2|2|S1/2C1/2|2
+F , or 4|C1/2ΣC1/2|2⟨Σ, C⟩F .
+These MGF characterizations give control of the supremum of { ⃗Wij ⃗W ′
+ij −
+E ⃗Wij ⃗W ′
+ij : i ̸= j ≤ n}. This class containing
+�n
+2
+�
+distinct Γ+
+�
+4|C1/2ΣC1/2|2⟨Σ, C⟩F ,
+2|C1/2ΣC1/2|2
+�
+random variables, Corollary 2.6 of Boucheron et al. (2013) yields
+E max
+i̸=j≤n[ ⃗Wij ⃗W ′
+ij − E( ⃗Wij ⃗W ′
+ij)] ≤
+2
+�
+2|C1/2ΣC1/2|2⟨Σ, C⟩F log
+�n
+2
+��1/2
++ 2|C1/2ΣC1/2|2 log
+�n
+2
+�
+.
+Simplifying via E ⃗Wij ⃗W ′
+ij = 2⟨Σ, C⟩F , i ̸= j, and 2a2+2ab+b2 = a2[1 + (1 + b/a)2],
+E max
+i̸=j≤n
+⃗Wij ⃗W ′
+ij ≤⟨Σ, C⟩F
+�
+�
+�1 +
+�
+�1 +
+�
+2|C1/2ΣC1/2|2 log
+�n
+2
+�
+⟨Σ, C⟩F
+�1/2�
+�
+2�
+�
+�
+= ⟨Σ, C⟩F
+�
+�
+�1 +
+�
+�1 +
+�
+2 log
+�n
+2
+�
+p[C1/2ΣC1/2]
+�1/2�
+�
+2�
+�
+�.
+(45)
+Observe next that on the positive real line x �→ √2+x dominates x �→ [1 + (1 + x)2]1/2.
+(The functions coincide at x = 0; otherwise the latter has derivative equal to the
+square root of (1+x)2/[1 + (1 + x)2], which is nowhere greater than 1.) Thus (45)
+gives
+�
+E max
+i̸=j≤n
+⃗Wij ⃗W ′
+ij
+�1/2
+≤⟨Σ, C⟩1/2
+F
+�
+�√2 +
+�
+2 log
+�n
+2
+�
+p[C1/2ΣC1/2]
+�1/2�
+�
+= √2⟨Σ, C⟩1/2
+F
+�
+�1 +
+�
+log
+�n
+2
+�
+p[C1/2ΣC1/2]
+�1/2�
+�.
+D
+Section 5
+Proof of Prop. 9. Throughout the proof, expected value is interpreted to be con-
+ditional on Fn.
+(Because each Fn is the sigma field of finitely many discrete
+40
+
+random variables, this introduces no measure-theoretic considerations that were
+not already present.)
+Also assume version (ii) of the moment condition, not-
+ing that it is entailed by version (i) and 0 < P(Z = 1) < 1 (which follows
+from 0 < P(Z = 1 | ⃗Xβn) < 1): E |YT |p < ∞ means E(|YT |p | Z) < ∞ also,
+which entails E |V (n,s)|p < ∞.
+If ns is bounded, then there are finitely many
+(�
+i∈s�Zi = 0�, �
+i∈s�Zi = 1�) achievable configurations for fine strata s, each with
+a characteristic distribution function x �→ P(V (n,s) ≤ x) such that E
+�
+|V (n,s)|p�
+=
+p
+� ∞
+0 yp−1 P(|V (n,s)| > y)dy < ∞ (e.g., Durrett, 2019, Lemma 2.2.13). The dis-
+tribution for |V | on ℜ+ given by setting P(|V | > y) to the maximum over these
+distributions of P(V (n,s) > y) stochastically dominates the relevant V (n,s) distri-
+butions. It also satisfies p
+� ∞
+0 yp−1 P(|V | > y)dy < ∞, because the sum of a finite
+collection of functions y �→ P(|V (n,s)| > y) dominates their maximum; so V ∈ Lp.
+Part i of proposition.
+From (26),
+�
+i∈s
+ψsi(η) = ws[Yi − η(Zi − ¯Zs)](Zi − ¯Zs)
+= ws{
+�
+i∈s:
+Zi=1
+[Yi(1 − ¯Zs) − η(1 − ¯Zs)2] +
+�
+i∈s:
+Zi=0
+[−Yi ¯Zs − η ¯Z2
+s ]}
+= ws[(1 − ¯Zs)
+�
+i∈s:
+Zi=1
+Yi − ηns ¯Zs(1 − ¯Zs)2 − ¯Zs
+�
+i∈s:
+Zi=0
+Yi − ηns(1 − ¯Zs) ¯Z2
+s ]
+= wsns ¯Zs(1 − ¯Zs)
+� ¯Ys(1) − η(1 − ¯Zs) − ¯Ys(0) − η ¯Zs
+�
+= wsns ¯Zs(1 − ¯Zs)( ¯Ys(1) − ¯Ys(0) − η),
+(46)
+so that E[�
+i∈s ψsi(η)] = wsns¯zs(1−¯zs) E( ¯Ys(1) − ¯Ys(0) − η). Recalling that V (n,s) =
+¯Ys(1) − ¯Ys(0) − E( ¯Ys(1) − ¯Ys(0)), we have
+ψSn(η) − E[ψSn(η)] =
+�
+s∈Sn ˜wn,snsV (n,s)
+�
+s∈Sn ˜wn,sns
+(47)
+where ˜wn,s := ws¯zs(1 − ¯zs) ∈ Fn. (When there is no risk of ambiguity, “ ˜wn,s” is
+abbreviated to “ ˜ws.”) Convergence of ψSn(η)−E[ψSn(η)|Gn] will be seen to follow
+from suitable convergence of (47), i.e. ψSn(η) − E[ψSn(η)|Fn].
+In-probability convergence of ψSn(η) − E[ψSn(η)].
+We adapt to the
+independent non-identically distributed case an argument for the L1-weak law of
+large numbers by truncation of increments.
+Following Durrett (2019, § 2.2.3),
+set ¯V (s)
+n
+:= V (n,s)�wn,s|V (n,s)| ≤ mn� and ¯Vn := V �|V | ≤ mn�. (Recall mn =
+41
+
+�
+s∈Sn�ws¯zs(1 − ¯zs) > 0�.) Whereas E V (n,s) = 0 by definition, E ¯V (s)
+n
+may differ
+from 0. Our first task is to show that (�
+s∈Sn ˜wn,s)−1 E �
+s∈Sn ˜wn,s ¯V (s)
+n
+→ 0.
+We have
+(
+�
+s∈Sn
+˜wsns)−1 E
+�
+s∈Sn
+˜wsns ¯V (s)
+n
+= (
+�
+s∈Sn
+˜wsns)−1
+�
+E
+�
+s∈Sn
+˜wsns ¯V (s)
+n+ + E
+�
+s∈Sn
+˜wsns ¯V (s)
+n−
+�
+where a+ and a− denote positive and negative parts of a, max(a, 0) and min(a, 0).
+Now
+(
+�
+s∈Sn
+˜wsns)−1 E
+�
+s∈Sn
+˜wsns ¯V (s)
+n+ = (
+�
+s∈Sn
+˜wsns)−1 �
+s∈Sn
+˜wsns E ¯V (s)
+n+
+= (
+�
+s∈Sn
+˜wsns)−1 �
+s∈Sn
+˜wsns
+� ∞
+0
+P( ¯V (s)
+n
+> x)dx
+=
+� ∞
+0
+�
+s∈Sn ˜wsns P( ¯V (s)
+n
+> x)
+�
+s∈Sn ˜wsns
+dx.
+(48)
+By the stochastic ordering assumption, P(V (n,s) > x) is dominated by P(V > x);
+but P( ¯V (s)
+n
+> x) ≤ P(V (n,s) > x), so the integrand in (48) is dominated by
+P(V > x) as well. As
+� ∞
+0 P(V > x)dx = E V+ < ∞, dominated convergence gives
+(�
+s∈Sn ˜wsns)−1 E �
+s∈Sn ˜wsns ¯V (s)
+n+ − (�
+s∈Sn ˜wsns)−1 E �
+s∈Sn ˜wsnsV (n,s)
++
+= o(1).
+Similarly (�
+s∈Sn ˜wsns)−1 E �
+s∈Sn ˜wsns ¯V (s)
+n− −(�
+s∈Sn ˜wsns)−1 E �
+s∈Sn ˜wsnsV (n,s)
+−
+=
+o(1), and
+� �
+s∈Sn
+˜wsns
+�−1�
+E
+�
+s∈Sn
+˜wsns ¯V (s)
+n
+− E
+�
+s∈Sn
+˜wsnsV (n,s)
+�
+= o(1).
+Since E V (n,s) = 0, this means (�
+s∈Sn ˜wsns)−1 E �
+s∈Sn ˜wsns ¯V (s)
+n
+→ 0.
+For in-probability convergence of (47) it now suffices to show
+�
+s∈Sn ˜wsns(V (n,s) − E ¯V (s)
+n )
+�
+s∈Sn ˜wsns
+= oP (1),
+(49)
+the conclusion of the weak law for triangular arrays. We now verify the premises
+of that principle, as it is given in Durrett’s (2019) Theorem 2.2.11. For each n
+{V (n,s) : s ∈ Sn} are independent because {(YTi, YCi, Zi) : i} are unconditionally
+independent and conditioning on Fn induces dependence within but not across
+strata s. Premise (i) of the theorem, �
+s∈Sn P( ˜wsnsV (n,s) > �
+s∈Sn ˜wsns) → 0,
+42
+
+will follow, by the assumptions of stochastic dominance and boundedness of ˜wsns,
+from convergence to 0 of
+�
+s∈Sn: ˜wsns>0
+P(u ˜wV >
+�
+s∈Sn
+˜wsns) = mn P(u ˜wV >
+�
+s∈Sn
+˜wsns),
+(50)
+where u ˜w is an upper bound for { ˜wsns : n; s ∈ Sn}. By hypothesis mn/(�
+s∈Sn ˜wsns) =
+OP (1), so (50)=oP (1) will follow from (�
+s∈Sn ˜wsns) P(u ˜wV > �
+s∈Sn ˜wsns) → 0.
+As we also hypothesize that mn → ∞ as n ↑ ∞, we also have �
+s∈Sn ˜wsns → ∞ as
+n increases, and convergence to 0 of (50) follows if x P(u ˜wV > x) → 0 as x ↑ ∞.
+This is true by dominated convergence, since x P(u ˜wV > x) ≤ E(u ˜wV �u ˜wV > x�),
+u ˜wV �u ˜wV > x� → 0 a.s. as x ↑ ∞, and E (u ˜wV ) < ∞.
+Premise (ii) of Durrett’s (2019) Theorem 2.2.11 is that
+(
+�
+s∈Sn
+˜wsns)−2 �
+n2
+s ˜wsn2
+s E
+�
+( ¯V (s)
+n )2�
+→ 0.
+(51)
+To verify this, observe first that
+E
+�
+(ws ¯V (s)
+n )2�
+=2
+� ∞
+0
+y P(|ws ¯V (s)
+n | > y)dy = 2
+� mn
+0
+y P(|ws ¯V (s)
+n | > y)dy
+≤ 2
+� mn
+0
+y P(|wsV (n,s)| > y)dy
+(52)
+≤ 2
+� mn
+0
+y P(|u ˜wV | > y)dy,
+(53)
+with (53) following from (52) by the stochastic dominance assumption. In conse-
+quence,
+(
+�
+s∈Sn
+˜wsns)−2 �
+n2
+s ˜w2
+s E
+�
+( ¯V (s)
+n )2�
+≤ (
+�
+s∈Sn
+˜wsns)−22mn
+� mn
+0
+y P(|u ˜wV | > y)dy.
+As we assume mn/(�
+s∈Sn ˜wsns) = OP (1), for (51) it suffices to show m−1
+n
+� mn
+0
+y P(|u ˜wV | >
+y) → 0 as n ↑ ∞. By the hypothesis that mn → ∞, this flows from x−1 � x
+0 y P(|u ˜wV | >
+y) → 0, which is a consequence of E |u ˜wV | < ∞, as shown by Durrett (2019) in
+the proofs of Theorems 2.1.12 and 2.1.14. This completes the verification that (47)
+converges in probability to 0.
+L1 convergence of ψSn(η)−E[ψSn(η)].
+By hypothesis there is p > 1 such
+that ∥V (n,s)∥Lp ≤ ∥V ∥Lp for all n and s ∈ Sn. Accordingly ∥(�
+s∈Sn ˜wn,s)−1�
+s∈Sn ˜wn,sV (n,s)∥Lp ≤
+∥V ∥Lp also. By the dominated convergence principle for random variables as in
+Theorem 1.6.8 and Exercise 2.3.5 of Durrett (2019), therefore, (47) converges to 0
+in L1 as well as in probability.
+43
+
+L1- and in-probability convergence of ψSn(η) − E[ψSn(η) | Gn].
+Ob-
+serve that
+∥ E[ψSn(η)|Gn] − E[ψSn(η)]∥L1 =∥ E{ψSn(η) − E[ψSn(η)] | Gn}∥L1
+≤ ∥ψSn(η) − E[ψSn(η)]∥L1,
+Fn being the smaller of the sigma fields {Fn, Gn}, E[ψSn(η)] being the same as
+E[ψSn(η) | Fn] and the conditional expectation operator being a contraction in L1
+(Durrett, 2019, Thm. 4.1.11). In tandem with
+∥ψSn(η)−E[ψSn(η)|Gn]∥L1 ≤ ∥ψSn(η)−E[ψSn(η)]∥L1+∥ E[ψSn(η)|Gn]−E[ψSn(η)]∥L1
+this means L1 convergence of ψSn(η)−E[ψSn(η)|Fn] entails that ψSn(η)−E[ψSn(η)|Gn]
+also converges to zero in L1. Finally, L1 convergence entails convergence in prob-
+ability.
+Part ii of proposition.
+Provided that �
+s ˜wsns is positive, both η �→ ψSn(η)
+and η �→ E[ψSn(η) | Gn] are everywhere differentiable with slope −1, and can be
+seen to tend to ±∞ as η tends to ∓∞. It follows that they have unique roots.
+Part iii of proposition.
+If the solutions τn of E[ψSn(η) | Gn] = 0 tend to a
+limit τ0 ∈ (−∞, ∞), then the following adaptation of the Huber argument for
+consistency of scalar solutions of monotone estimating equations (Huber, 1964;
+van der Vaart, 1998, Lemma 5.10) shows that ˆτn → τ0 in probability. Fix ϵ > 0.
+Then
+P[ψSn(τ0 − ϵ) > ϵ/2, ψSn(τ0 + ϵ) < −ϵ/2] ≤ P(τ0 − ϵ < ˆτn < τ0 + ϵ).
+The left side tends to 1 because ψSn(τ0 ± ϵ) − E[ψSn(τ0 ± ϵ) | Gn] = oP (1), P(|τn −
+τ0| < ϵ/2) → 1, E[ψSn(η) | Gn] > ϵ/2 if η < τn − ϵ/2 and E[ψSn(η) | Gn] < −ϵ/2 if
+η > τn + ϵ/2. Therefore the right hand side tends to 1 as well.
+Lemma 8. Let θi = logit[P(Z = 1|Xβn = xiβn)]. If |θi − θj| < δ whenever i Sn
+∼ j,
+then for all s ∈ Sn and z : s → {0, 1} such that �
+i∈s ζi = �
+i∈s zi ∈ {1, ns − 1},
+����
+πs(ζ)
+n−1
+s
+− 1
+���� ≤ (1 − n−1
+s )(e2δ − 1)
+(54)
+and
+����
+n−1
+s
+πs(ζ) − 1
+���� ≤ (1 − n−1
+s )(e4δ − 1).
+(55)
+44
+
+Proof of Lemma 8. For s an 1 : m matched set, some nonnegative integer m, by
+(27) we have
+πs(ζ)
+n−1
+s
+− 1 = 1
+n−1
+s
+·
+exp θi
+�
+j∈[i] exp(θj) − 1
+=
+ns − �
+j∈[i] exp(θj − θi)
+�
+j∈[i] exp(θj − θi)
+(56)
+=
+m − �
+j∈[i]\{i} exp(θj − θi)
+�
+j∈[i] exp(θj − θi)
+;
+so
+m − m exp(δ)
+(m + 1) exp(−δ) ≤ πs(ζ)
+n−1
+s
+− 1 ≤ m − m exp(−δ)
+(m + 1) exp(−δ),
+(57)
+−
+m
+m + 1eδ(eδ − 1) ≤ πs(ζ)
+n−1
+s
+− 1 ≤
+m
+m + 1(eδ − 1)
+and
+����
+πs(ζ)
+n−1
+s
+− 1
+���� ≤ ns − 1
+ns
+eδ(eδ − 1).
+(58)
+Now observe that e2δ − eδ < e2δ − 1 for positive δ; (54) follows.
+If s is an m : 1 matched set, m ≥ 0, the argument culminating in (58) again
+applies after substitution of −θi and −θj for θi and θj. Again (54) follows.
+With (54) established in all cases, (55) follows by (54)’s consequence that
+����
+πs(ζ)
+n−1
+s
+���� ≤ e2δ;
+the identity |x−1−1| ≤ |x−1|·|x−1|, valid for x ̸= 0; and e2δ(e2δ −1) ≤ e4δ −1.
+Proof of Prop. 10. Claim (i) follows from
+E[ ˜ψs(η) | Gn] = ˜ws
+�
+i∈s
+[E(Y | Z = 1, ⃗Xβn = ⃗xiβn) − E(Y | Z = 0, ⃗Xβn = ⃗xiβn) − η].
+(59)
+To show this, first observe that (46) and (28) combine to give
+˜ψs(η) =
+˜ws
+πs(Zs)( ¯Ys(1) − ¯Ys(0) − η).
+(60)
+As Sn is assumed to be a fine stratification, one or both of ¯Ys(1) = avg[(Yi : i ∈ s, Zi = 1)]
+and ¯Ys(0) = avg[(Yi : i ∈ s, Zi = 0)] is in actuality a single observation. (60) is taken
+to be zero, as is ψs(η) = ˜ws
+�
+i∈s ψsi(η) = ˜ws( ¯Ys(1) − ¯Ys(0) − η), when ˜ws = 0 be-
+cause ¯Zs = 0 or 1.
+45
+
+For s with �
+i∈s Zi = 1, by (27) the expectation of (60) evaluates to E[ ˜ψs(η) | Gn] =
+�
+i∈s
+{E(Y | Z = 1, ⃗Xβn = ⃗xiβn) − avg
+j∈s\{i}
+[E(Y | Z = 0, ⃗Xβn = ⃗xjβn)] − η}
+·
+˜ws
+πs(0(s)
++i)
+πs(0(s)
++i)
+= ˜ws
+�
+i∈s
+{E(Y | Z = 1, ⃗Xβn = ⃗xiβn) − avg
+j∈s\{i}
+[E(Y | Z = 0, ⃗Xβn = ⃗xjβn)] − η},
+(61)
+where we use 0(s)
++i to denote the mapping on s taking i to 1 and remaining elements
+to 0; (59) follows. For s with �
+i∈s�zi = 0� = 1, this argument gives E[ ˜ψs(η) | Gn] =
+˜ws
+�
+i∈s
+{ avg
+j∈s\{i}
+[E(Y | Z = 1, ⃗Xβn = ⃗xjβn)] − E(Y | Z = 0, ⃗Xβn = ⃗xiβn) − η} (62)
+where 1(s)
+−i denotes the mapping of s that takes i to 0 and remaining elements to
+1. Again (59) follows.
+Part (ii).
+Given s ∈ Sn, write µ(s)
+0
+= E( ¯Ys(0) | Fn) and µ(s)
+1
+= E( ¯Ys(1) | Fn).
+By symmetry, µ(s)
+z
+= E(Y | Z1 = z, �ns
+i=1 Zi = �
+i∈s zi), z = 0, 1. For s ∈ Sn with
+�
+i∈s�zi = 1� = 1, one has ˜w−1
+s
+E[ψs(η) | Gn] =
+�
+i∈s
+{[E(Y | Z = 1, ⃗Xβn = ⃗xiβn) − avg
+j∈s\{i}
+[E(Y | Z = 0, ⃗Xβn = ⃗xjβn)]] − η}πs(0(s)
++i)
+=
+�
+i∈s
+{[E(Y − µ(s)
+1
+| Z = 1, ⃗Xβn = ⃗xiβn) − avg
+j∈s\{i}
+[E(Y − µ(s)
+0
+| Z = 0, ⃗Xβn = ⃗xjβn)]]}πs(0(s)
++i)
++ µ(s)
+1
+− µ(s)
+0
+− η,
+whereas if �
+i∈s�zi = 0� = 1 then ˜w−1
+s
+E[ψs(η) | Gn] =
+�
+i∈s
+{[ avg
+j∈s\{i}
+[E(Y | Z = 1, ⃗Xβn = ⃗xjβn)] − E(Y | Z = 0, ⃗Xβn = ⃗xiβn)] − η}πs(1(s)
+−i)
+=
+�
+i∈s
+{ avg
+j∈s\{i}
+[E(Y − µ(s)
+1
+| Z = 1, ⃗Xβn = ⃗xjβn)] − E(Y − µ(s)
+0
+| Z = 0, ⃗Xβn = ⃗xiβn)}πs(1(s)
+−i)
++ µ(s)
+1
+− µ(s)
+0
+− η.
+46
+
+At the same time, (61) and (62) give
+˜w−1
+s
+E[ψs(η) | Gn] = µ(s)
+1
+− µ(s)
+0
+− η+
+�
+i∈s
+{[E(Y − µ(s)
+1
+| Z = 1, ⃗Xβn = ⃗xiβn) − avg
+j∈s\{i}
+[E(Y − µ(s)
+0
+| Z = 0, ⃗Xβn = ⃗xjβn)]]}n−1
+s
+or
+˜w−1
+s
+E[ψs(η) | Gn] = µ(s)
+1
+− µ(s)
+0
+− η+
+�
+i∈s
+{ avg
+j∈s\{i}
+[E(Y − µ(s)
+1
+| Z = 1, ⃗Xβn = ⃗xjβn)] − E(Y − µ(s)
+0
+| Z = 0, ⃗Xβn = ⃗xiβn)}n−1
+s ,
+depending as �
+i∈s�zi = 1� = 1 or �
+i∈s�zi = 0� = 1, respectively. Differencing
+these expressions,
+˜w−1
+s
+E[ ˜ψs(η) − ψs(η) | Gn] =
+�
+i∈s
+�
+n−1
+s
+πs(0s
++i) − 1
+�
+·{[E(Y − µ(s)
+1
+| Z = 1, ⃗Xβn = ⃗xiβn) − avg
+j∈s\{i}
+[E(Y − µ(s)
+0
+| Z = 0, ⃗Xβn = ⃗xjβn)]]}πs(0s
++i)
+(63)
+if �
+i∈s�zi = 1� = 1, and if �
+i∈s�zi = 0� = 1 then
+˜w−1
+s
+E[ ˜ψs(η) − ψs(η) | Gn] =
+�
+i∈s
+�
+n−1
+s
+πs(1s
+−i) − 1
+�
+·{ avg
+j∈s\{i}
+[E(Y − µ(s)
+1
+| Z = 1, ⃗Xβn = ⃗xjβn)] − E(Y − µ(s)
+0
+| Z = 0, ⃗Xβn = ⃗xiβn)}πs(1(s)
+−i).
+(64)
+Observing that ns E(|V (n,s)| | Gn) ≥ | E(nsV (n,s) | Gn)| =
+�����
+�
+i∈s
+{[E(Y − µ(s)
+1
+| Z = 1, ⃗Xβn = ⃗xiβn) − avg
+j∈s\{i}
+[E(Y − µ(s)
+0
+| Z = 0, ⃗Xβn = ⃗xjβn)]]}πs(0s
++i)
+�����
+or
+�����
+�
+i∈s
+{ avg
+j∈s\{i}
+[E(Y − µ(s)
+1
+| Z = 1, ⃗Xβn = ⃗xjβn)] − E(Y − µ(s)
+0
+| Z = 0, ⃗Xβn = ⃗xiβn)}πs(1(s)
+−i)
+�����
+depending as �
+i∈s�zi = 1� = 1 or �
+i∈s�zi = 0� = 1, and that under the same
+respective conditions
+����
+n−1
+s
+πs(0s
++i) − 1
+���� ≤ exp(2 sup
+i,j∈s
+|θi − θj|)−1 or
+����
+n−1
+s
+πs(1s
+−i) − 1
+���� ≤ exp(2 sup
+i,j∈s
+|θi − θj|)−1
+47
+
+by Lemma 8, we have:
+| E[ ˜ψs(η) − ψs(η) | Gn]| ≤ ˜wsns[exp(2 sup
+i,j∈s
+|θi − θj|) − 1] E(|V (n,s)| | Gn).
+(65)
+This establishes (30).
+Part (iii).
+Because the conditional expectation operator is a contraction in
+Lp,
+E
+��
+s∈Sn ˜wsns E(|V (n,s)| | Gn)
+�
+s∈Sn ˜wsns
+�
+≤
+�
+s∈Sn ˜wsns E |V (n,s)|
+�
+s∈Sn ˜wsns
+≤
+�
+s∈Sn ˜wsns E |V |
+�
+s∈Sn ˜wsns
+= E |V |.
+Markov’s inequality now gives that �
+s∈Sn ˜wsns
+−1�
+s∈Sn ˜wsns E(|V (n,s)| | Gn) =
+OP (1). Accordingly, |{θi − θj : i ∼ j}|∞ = oP (1) combines with (30) to entail
+sup
+η | E[ ˜ψSn(η) − ψSn(η) | Gn]| = oP (1).
+(66)
+By Prop. 9, η �→ E(ψSn(η) | Gn) has the unique root τn, and by part (i) of this
+proposition, η �→ E( ˜ψSn(η) | Gn) has a unique root given by (29); as either of these
+functions’ slopes are bounded away from zero, (66) entails that these roots must
+converge together.
+Part (iv).
+Part (iv) of the proposition now follows from conclusion iii of
+Proposition 9.
+48
+
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@@ -0,0 +1,1150 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf,len=1149
+page_content='Matching calipers and the precision of index  estimation  Ben B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Hansen∗  January 11, 2023  Abstract  This paper characterizes the precision of index estimation as it car-  ries over into precision of matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In a model assuming Gaussian  covariates and making best-case assumptions about matching quality,  it sharply characterizes average and worst-case discrepancies between  paired differences of true versus estimated index values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In this op-  timistic setting, worst-case true and estimated index differences de-  cline to zero if p = o[n/(log n)], the same restriction on model size  that is needed for consistency of common index models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  This re-  mains so as the Gaussian assumption is relaxed to sub-gaussian, if  in that case the characterization of paired index errors is less sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The formula derived under Gaussian assumptions is used as the basis  for a matching caliper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Matching such that paired differences on the  estimated index fall below this caliper brings the benefit that after  matching, worst-case differences onan underlying index tend to 0 if  p = o{[n/(log n)]2/3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (With a linear index model, p = o[n/(log n)]  suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=') A proposed refinement of the caliper condition brings the  same benefits without the sub-gaussian condition on covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' When  strong ignorability holds and the index is a well-specified propensity  or prognostic score, ensuring in this way that worst-case matched dis-  crepancies on it tend to 0 with increasing n also ensures the consistency  of matched estimators of the treatment effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  ∗This work has benefitted from comments of Jake Bowers, Joshua Errickson, Mark  Fredrickson, Xuming He, Peter Schochet, Stilian Stoev and Lan Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Responsibility  rests with the author for any shortcomings that remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='04109v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='ME]  10 Jan 2023  Key words and phrases:  Matching, caliper, overlap, positivity, propensity  score, prognostic score  1  Introduction  In preparing a matched observational study, estimation of a treatment propensity  briefly takes center stage, as covariates are chosen and a model specification is  selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' These models quickly recede from view once propensity score estimates  have been extracted from them, despite their carrying essential information about  those estimates’ likely precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The situation is little different in matching on  prognostic or principal stratum scores: sampling variability of the model standing  behind a matching index is rarely so much as even appraised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' We seem to take it  for granted that errors of estimation of a matching index can’t possibly be so large  as to threaten the integrity of matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Propensity matching is understood to be a large sample technique, as are  logistic and other regression methods typically used for index model estimation,  and classical asymptotics may seem to encourage inattention to index estimation  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  As treatment/covariate samples (zi, ⃗xi) accumulate from any reasonable  distribution of fixed dimension p+1, one expects errors of estimation of the index,  {|⃗xi ˆβ −⃗xiβn| : i}, to be increasingly negligible, decreasing with or near n−1/2, just  as |ˆβ − βn|2 = O(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The problem is that the matching canon discourages  parsimony in propensity modeling (Rubin and Thomas, 1996), and fixed-p large  sample theory may describe non-parsimonious models poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Increasing-p asymptotics for logistic regression and similar techniques are avail-  able (Portnoy, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' He and Shao, 2000), if less widely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Given that p =  o[(log n)/n], they deliver |ˆβ − βn|2 = OP [(p/n)1/2], not OP (n−1/2), as reviewed in  § 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This suggests a still larger order, p/n1/2, for errors of form ⃗X(ˆβ − βn),  �p  j=1 X2  j = O(p) corresponding to | ⃗X|2 =  ��p  j=1 X2  j  �1/2  = O(p1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' While some-  what of a simplification, the suggestion is correct in its implication that if p in-  creases in proportion with n1/2, for example, then index estimation errors need not  diminish even as coefficient estimation errors do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Outside of fixed-p asymptotics,  consistency of the index model does not in itself make index errors asymptotically  negligible: that calls for stronger assumptions, specialized matching techniques or  a combination of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  For control of index estimation error by way of stronger assumptions, note  that if p is assumed to increase slowly enough, increasing dimension regression  asymptotics resemble those with fixed p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' It happens that p ∝ n1/2 is slightly too  large for such correspondence to obtain, so it is unsurprising that fixed-p intuitions  should fail in that regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (Asymptotic normality of ˆβ, for example, calls for  2  p2 log(p) = o(n), not p = o(n1/2) [He and Shao, 2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=') But say the index model  has sub-√n dimension, in the specific sense that p = o[(n/ log n)1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1  of this paper shows that for consistently estimated index models with sub-gaussian  covariates and sub-√n dimension, estimation errors of realized values of the index  tend to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  This convergence is in the strong, l∞, sense of maxi≤n | ⃗Xi(ˆβ −  βn)| = oP (1), so it justifies the common practices of matching, subclassifying or  simply trimming on the estimated propensity score, as analytic interventions to  secure overlap assumptions of the stronger type, cl ≤ P(Z = 1| ⃗X) ≤ cu with  [cl, cu] ⊂ (0, 1), as applied to the subset of available observations that remain after  pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3 goes on to study ordinary Cov(ˆβ) estimates’ adaptability to char-  acterizing likely sizes of index estimation errors, ⃗xi(ˆβ −βn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' A fitted index model’s  Fisher information gives an estimate ˆC of Cˆβ = Cov(ˆβ), either directly or as part of  an Eicker-Huber-White sandwich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In sub-n1/2 dimensional regimes, corresponding  standard errors s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='(⃗xˆβ) = [⃗x ˆC⃗x′]1/2 will be seen to estimate closely1 the sampling  variabilities Var1/2[(⃗x − ¯⃗x)(ˆβ − βn)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' As the sub-√n condition is relaxed to sub-  n (p = o[n/ log(n)]), information- and sandwich-based estimators underestimate  Cov(ˆβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This limits their utility for inference about βn, and according their behav-  ior outside of sub-√n regimes has received less study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' It does not follow, however,  that they are are ill-suited to inform the selection of index-based matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' We  explore conditions under which analytic Cov(ˆβ)-estimators continue to character-  ize sampling variabilities of a linearization of ˆβ, offering a basis for estimators  capturing the better part of Var[⃗x(ˆβ − βn)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In both regimes, the largest values  of {Var1/2[⃗xi(ˆβ − βn)] : i} may be well separated from the rest, in themselves ap-  preciably increasing E[maxi≤n | ⃗Xi(ˆβ − βn)|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Results presented in Section 4 are  helpful for identifying the worst offenders, subjects i with large s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='[⃗xi ˆβ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Either  or both of the sub-gaussian and sub-√n conditions can be relaxed, but then con-  trol of index estimation errors necessitates that such subjects be pruned from the  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  In such circumstances, matching offers alternate practical remedies that can  retain more of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  It helps first by shifting attention from particular  {⃗xi ˆβ : i} or {⃗xiβn : i} to paired differences of indices, (⃗xi − ⃗xj)ˆβ or (⃗xi − ⃗xj)βn, as  does § 4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' While underestimates of Var1/2[(⃗xi − ⃗xj)ˆβ] may be unhelpful for  inference about (⃗xi − ⃗xj)βn, but they can certainly inform matching procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2 uses ˆCˆβ to associate deliberately reduced standard errors with paired  contrasts such as (⃗xi − ⃗xj)ˆβ, so constructed that their average, the paired index  contrast summary standard error (PIC SE), inexpensively approximates the root  1For |⃗v|2 ̸= 0, |⃗v|−1  2 {s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='[⃗v(ˆβ − βn)] − Var1/2[⃗v(ˆβ − βn)]} is oP [(p/n)1/2], whereas  |⃗v|−1  2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='[⃗v(ˆβ − βn)] and |⃗v|−1  2  Var1/2[⃗v(ˆβ − βn)] are both OP [(p/n)1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  3  mean square of matched discrepancies on estimated scores, (⃗xi −⃗xj)ˆβ, across pairs  {i, j} with little or no discrepancy on the true score, ⃗xiβ ≈ ⃗xjβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The PIC SE tends to zero at the same rate as |ˆβ − βn|2, making it useful as  a yardstick for matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' With sub-gaussian data and sub-√n model dimension,  using it to set the width of a caliper on the index — permitting i’s pairing with j  only if |(⃗xi − ⃗xj)ˆβ| ≤ cn pic se(ˆβ), with cn = 2, say — forces matched differences  on the true index to vanish in the asymptotic limit:  max  1≤i∼j≤n |(⃗xi − ⃗xj)βn| P→ 0  (where “i ∼ j” means “i is matched to j”),  (1)  in virtue of pic se(ˆβ) = OP [(p/n)1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Indeed, under the p = o(n−1 log n) growth  condition needed for consistency of common index models, (1) holds with non-  constant cn, provided that cn = OP [(log n)1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Such matching requirements are  generally less likely to exclude subjects then comparable trimming rules, as they  permit inclusion even of subjects far from the center of the distribution whenever  the contrasting study arm has similarly situated subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If i is excluded from the  matched sample for lack of counterparts j within caliper distance, it must have  been separated from its comparison group by a distance exceeding the resolution  of the estimate of the index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' As will be seen in the data example, this resolution  can be strikingly large, much larger than extant caliper width recommendations  (Rosenbaum and Rubin, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Rubin and Thomas, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Austin, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=',  2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' in many cases it will be much more sparing in its exclusions from the  matched sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  This is fortunate, because the paper will recommend a nonstandard strength-  ening of the requirement that |(⃗xi−⃗xj)ˆβ| be less than the designated caliper width,  excluding potential pairs {i, j} either because |(⃗xi − ⃗xj)ˆβ| is too large or because  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='[(⃗xi − ⃗xj)ˆβ] is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (Alternately put, because ⃗xi and ⃗xj are too separated on either  the index itself or on a certain index estimator-dependent Mahalanobis distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=')  The pairs E ⊆ {{i, j} : 1 ≤ i ̸= j ≤ n} that remain eligible by this criterion  satisfy max{i,j}∈E |(⃗xi − ⃗xj)(ˆβ − βn)| = oP (1), even as maxi≤n | ⃗Xi(ˆβ − βn)| may no  longer tend to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Thus (1) is maintained, even as the sub-gaussian assumption on  ⃗X is relaxed to a fourth moment condition, and p = o{[n/ log(n)]1/2} is relaxed  to p = o{[n/ log(n)]2/3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In matching on propensity scores, selecting pairs from  within such an E ensures that the overlap criterion can be assessed in terms of  estimated scores, because maxi∼j |(⃗xi − ⃗xj)ˆβ − (⃗xi − ⃗xj)βn| = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' As the rec-  ommended requirement can be viewed as a varying (by value of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='[(⃗xi − ⃗xj)ˆβ])  limit on |(⃗xi − ⃗xj)ˆβ|, I continue to call it a caliper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In many cases, these PICSE  calipers continue to be more inclusive than would Rosenbaum and Rubin’s (1985)  canonical |(⃗xi − ⃗xj)ˆβ| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='(xˆβ) requirement, and in all cases PICSE calipers  4  cause exclusion of a unit i only when our best estimate of the index distances from  it to each potential counterpart j exceeds the resolution of the index’s estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  2  Context  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1  Estimable index scores  In an observational study with a treatment and a control condition, the propensity  score is a function ⃗x �→ g−1[P(Z = 1 | ⃗X = ⃗x)], where g : ℜ → [0, 1] is continuous  and increasing (Rosenbaum and Rubin, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The z-on-⃗x regression is often  assumed to follow a generalized linear model such as the logistic, P(Z = 1 | ⃗X =  ⃗x) = [1 + e−⃗xβ]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Taking g as that model’s link function identifies the propensity  score with the index ⃗xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Rosenbaum and Rubin (1985) recommended matching on  ⃗xˆβ, not g−1(⃗xˆβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Similarly prognostic scoring, confounder and risk scoring, and  principal stratum scoring fit (parametric) regression models in order to extract  indices ⃗xˆβ for use in matching, subclassification or sample trimming, if not also  in weighting- or covariance adjustments to be applied once an analysis sample has  been selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Let R denote the dependent variable of the index model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Z for a propensity  score or a response Y for a prognostic score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Let ˆβ be the solution of  � n  �  i=1  ψ(ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' β)  �  + α(β) = 0  (2)  in β, where ψ(r, ⃗x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' β) is the ℜp-valued gradient of a scalar-valued function ρ(r, ⃗x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' β)  that is convex in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (In maximum likelihood estimation α(·) = 0, but Bayesian  estimation [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', 2008] and certain frequentist bias-reduction schemes  [Firth, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Kosmidis and Firth, 2009] minimize a penalized objective, in which  cases α(·) is the penalty term’s gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=') The index score (IS) is estimable if  E  �  [  n  �  i=1  ψ(Ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' β)] + α(β)  �  = 0  (3)  has a unique root βn, with supγ:|γ−βn|2≤1 |α(γ)|2 = oP (n1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Following He and  Shao (2000), near-roots of (2) and/or (3) are acceptable when the equations do not  have exact solutions, provided that there is a unique nearest root, but these exact  or nearest roots are assumed to satisfy �p  j=1  ��n  i=1 ψj(ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ˆβ)  �2  = oP (n1/2) and  �p  j=1[E [�n  i=1 ψj(Ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn)]]2 = o(n1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Assume there are functions c0(·, ·, ·) and  w(·) such that  ψ(r, ⃗x, β) = c0(r, ⃗x, β0 + ⃗xβ)w(⃗x)(1, ⃗x)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (4)  5  This structure accommodates robust (Cantoni and Ronchetti, 2001) and general-  ized (Liang and Zeger, 1986) estimating equations as well as score functions such  as logistic regression’s, ψ(r, ⃗x, β) = {r − [1 + exp(−β0 − ⃗xβ)]−1}(1, ⃗x)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If there  are pre-existing strata 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , L, with matches to be made within strata and any  subclasses to further divide them, then (4) may be modified by replacement of β0  with stratum-specific intercepts β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , βL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Sufficient conditions for consistency of  an IS will be reviewed in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2  Sub-gaussian random variables  A real-valued random variable V is sub-gaussian if its tails are no heavier than  that of a centered Normal distribution: there is a finite constant sV such that for  t > 0,  P(V < t) ≤ exp[−t2/(2s2  V )],  P(V > t) ≤ exp[−t2/(2s2  V )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  When this holds, sV can be taken to be a constant multiple of ∥V ∥ψ2, the sub-  gaussian norm of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This norm is defined as the infimum of {t > 0 : E[exp(V 2/t2)] ≤  2}, a nonempty set for sub-gaussian V , or as ∞ if V is not sub-gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For vec-  tor ⃗V , ∥⃗V ∥ψ2 = sup{∥⃗V γ∥ψ2 : |γ|2 = 1}, and ⃗V is said to be sub-gaussian if  ∥⃗V ∥ψ2 < ∞ (Vershynin, 2018, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' It follows directly from these defini-  tions that for fixed vectors γ, ∥V γ∥ψ2 = |γ|2∥V ∥ψ2, ∥⃗V γ∥ψ2 ≤ |γ|2∥⃗V ∥ψ2 and  ∥γ∥ψ2 = (log 2)−1/2|γ|2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and that for fixed matrices M, ∥⃗V M∥ψ2 ≤ |M|2∥⃗V ∥ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Sums of sub-gaussian variables are sub-gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Hoeffding’s inequality bounds  tails of sums of independent sub-gaussians in terms of the sum of the summands’  squared sub-gaussian norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Maxima of sequences of sub-gaussian random vari-  ables grow slowly: for an absolute constant k0,  E max  1≤i≤n Vi ≤ k0  �  max  1≤i≤n ∥Vi∥ψ2  �  (log n)1/2,  (5)  with {Vi : i} independent or dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For {Gi : i} Normal with variance 1 or less,  (5) holds with √2 in place of k0 maxi≤n ∥Vi∥ψ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' if {Gi : i} are independent N(0, 1)  then this bound is sharp, in the sense of limn↑∞(log n)−1/2 E max1≤i≤n Gi = √2  (Boucheron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', 2013, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Maxima of sub-gaussian vector sequences grow  slowly as well: there are absolute constants k1 and k2 such that for any sub-  gaussian {⃗Vi}  with mean 0 and covariance I, and any deterministic matrices  {Mi} with column dimension matching the extent of ⃗V ,  E  �  max  1≤i≤n |Mi⃗V ′  i |2  �  ≤ max  i≤n ∥⃗Vi∥ψ2  �  k1 max  i≤n |Mi|F + k2  �  max  i≤n |Mi|2  �  (log n)1/2  �  (6)  6  where |M|F = tr(M′M)1/2 (Vershynin, 2018, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' A consequence is that  if for each n { ⃗Xni : 1 ≤ i ≤ n} are independent random vectors of length pn  such that | Cov( ⃗Xni)|2 and ∥ ⃗Xni∥ψ2 are uniformly bounded, then max1≤i≤n | ⃗Xi|2 =  OP [max(pn, log n)1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This property of sub-gaussian covariates, A11 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3  below, will be assumed in Section 3 but then relaxed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The scalar  c0(R, ⃗x, ⃗xβ) in (4) above, on the other hand, will consistently be required to be  sub-gaussian, via Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3’s A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3  Consistently estimable index scores  Let the data and model parameter be arranged in triangular arrays, with sample  and model n having n observations of p independent variables xi, p (strictly, pn)  increasing with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Consistency of ˆβ for (βn : n) will mean that p = o(n) and  |ˆβ−βn|2  2 = OP (p/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Conditions for such consistency are available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  We orderto present them along with accompanying conditions characterizing the  ˆβ’s relationship to its closest linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  For estimable βn, define An = An(βn) and ˆAn = An(ˆβ), where  An(γ) = 1  n  �  n  �  i=1  ∇β E [ψ(Ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' β)]|β=γ + α(γ)  �  ,  (7)  with “∇β” interpreted in terms of weak differentiation if ordinary partial deriva-  tives do not exist for some β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For invertible An, the linearization of estimator  ˆβ is given by  ˜βn = A−1  n  1  n{[  n  �  i=1  ψ(ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn)] + α(βn)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (8)  The random vector n−1 �n  i=1 ψ(Ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn) has covariance n−1Bn where  Bn = Bn(βn) and Bn(γ) = n−1 E  n  �  i=1  ψ(Ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' γ)ψ(Ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' γ)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (9)  Proposition 1 (He and Shao, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Under A2, A4, A6 and A9 as stated below:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' |ˆβ − βn|2  P→ 0, with rate OP [(p/n)1/2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' |ˆβ − ˜βn|2  P→ 0, with rate OP [(p/n)(log n)1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 1 restates He and Shao’s Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2 as applied to models  in which only the coefficient parameter βn grows in dimension with n, with a  slight strengthening of rate condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (They assume p log(p) = o(n), while A9  7  says p log(n) = o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=') Their Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2 characterizes decline of the linearization  error only for sub-√n dimensional models, but a straightforward adaptation of its  proof gives part 2 of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' See also He and Shao’s Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Our regularity assumptions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The columns of x are centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' There may be pre-existing strata 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , L,  with L/n → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In this case the columns of x are also stratum-centered: for  stratifying variable v, �  i:vi=ℓ ⃗xi = 0, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The IS is estimable (as defined in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1) and linear in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' An is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Furthermore there is δ > 0 such that An(γ) is invertible  whenever |γ − βn|2 < δ, and supγ:|γ−βn|2<δ |An(γ)−1|2 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ψ(r, x, β) is of form (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The functions c0(r, ⃗x, ·) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' η �→ c0(r, ⃗x, η)) are Lip-  schitz continuous with a common Lipschitz constant, as are (∂/∂η) E c0(R, ⃗x, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The random variables c0(Ri, ⃗xi, ⃗xiβn) have bounded sub-gaussian norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For ℓ = 0, 2 and 4,  max  γ,δ:|γ|2=  |δ|2=1  n  �  i=1  w(⃗xi)ℓ(⃗xiγ)2(⃗xiδ)2 = O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' p−1|βn|2  2 = p−1 �  k β2  nk is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' s2(xβn) = β′  nS(x)βn tends to a limit in (0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' p = o[n/(log n)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (p log n)/n → 0 (sub-n model dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  While the link function g is assumed the same for each n, the coefficient vector  β grows in length, and βn needn’t converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Indeed, according to A8 s(sβn) is  permitted to diverge (but not tend to 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  It is less burdensome here to assume invertibility of An, as A3 does, than in  other regression contexts, as for present applications one can freely change the basis  of the design matrix, there being no interest in particular elements or contrasts of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Via the Cauchy-Schwartz inequality, A6 entails that maxγ:|γ|2=1 n−1 �n  i=1 w(⃗xi)ℓ(⃗xiγ)2 =  O(1) for ℓ ∈ {0, 1, 2}, in turn giving |S(x)|2 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The appropriateness of these  commitments can be evaluated in advance of IS estimation, whereas A3 calls for  inspection of model-fitting artifacts after estimation of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' A simple measure to  improve the fit of A3, as well as A12 below, is to trim explanatory variables that  contribute relatively little to index model fit, as indicated by common model se-  lection criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Certain results take stronger forms with one or more secondary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  8  A10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' p2 = o[n/(log n)] (sub-√n model dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' maxi≤n |⃗xi|2  2 = O[max(p, log n)], and maxi≤n w(⃗xi)2 = O(log n) (sub-gaussian  covariates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Each S(x) = (n − L)−1x′x and Bn are of full rank, with |(S(x))−1|2 and  |B−1  n |2 bounded (full-rank covariance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  If the ⃗xes (and w(⃗x)) are sub-gaussian in the sense of being realizations of ⃗X with  E | ⃗X|ψ2 uniformly bounded, while also A6 holds in the sense of | Cov( ⃗X)|2 (and  thus p−1 E | ⃗X|2  2) being uniformly bounded as well, then A11 follows from (6), as  discussed in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2 following (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  According to Proposition 4 below, sandwich  estimators of Cov(ˆβ) generally require sub-gaussian covariates, and A10, sub-√n  model dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' but the covariance estimator based on ˆAn but not ˆBn generally  requires only sub-n model dimension, A9, and for present purposes will be simi-  larly beneficial even when it lacks Fisher consistency as compared to the sandwich  estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' According to Proposition 6, full-rank covariance (A12) makes index  sampling variabilities Var(⃗xˆβ) and Var  �  (⃗xi − ⃗xj)ˆβ  �  estimable without attention  to size of ⃗x or ⃗xi − ⃗xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' However, our method for asymptotically exact matching  does not require this, and is valid with S or Bn of less than full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  3  Asymptotically exact matching with Gaus-  sian or sub-gaussian ⃗X  For each n let Sn be a random partition of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' It is not presumed that  Sn expands or extends any earlier partition S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Denote by [i]Sn the  unique Sn element containing i ≤ n, and write i Sn  ∼ j when there is s ∈ Sn such  that i, j ∈ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Absent ambiguity as to which n or partition sequence is intended,  these symbols are given as “i∼j” or “[i],” respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The progression {Sn}  constitutes an asymptotically exact index post-stratification if Sn-stratum width  in the direction of the underlying index, max{|(⃗xi − ⃗xj)βn| : i Sn  ∼ j}, tends in  probability to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This section presents a tolerance for Sn-stratum width in the  direction of the estimated index, max{|(⃗xi − ⃗xj)ˆβ| : i Sn  ∼ j}, that is narrow enough  to ensure asymptotic exactness in the special case of a sub-gaussian covariate  (A11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' It is also is sufficiently wide that, in a further special case to be described  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2, no i meriting placement in a poststratum with representation of  the contrasting group can be excluded from such placement in virtue of ⃗xi ˆβ being  isolated relative to {⃗xj ˆβ : zj ̸= zi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  9  When βn is subject to estimation, the observable counterparts of differences  (⃗xi − ⃗xj)βn, 1 ≤ i, j ≤ n, that is contrasts of form (⃗xi − ⃗xj)ˆβ, are termed paired  index contrasts (PICs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The discrepancy between a PIC and the paired contrast  it estimates, (⃗xi − ⃗xj)(ˆβ − βn), is a PIC error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Ensuring that a post-stratification  is asymptotically exact calls for separate attention to PIC errors versus the PICs  themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Our width tolerance will involve a novel estimate of PIC error size,  the PIC SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1  PIC errors in the sub-gaussian case  Recall that errors ˆβ − βn of index coefficient estimates decompose as (ˆβ − ˜β) +  (˜β − βn), with ˜β the linearization defined in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Index and PIC errors decompose  similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Corollary (of Proposition 1 part 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If A2, A4, A6, A9, and A11, then maxi |⃗xi(ˆβ−  ˜βn)| = OP [p3/2(log n)1/2/n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (Unless p = o(log n), in which case maxi |⃗xi(ˆβ −  ˜βn)| = OP [p(log n)/n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If A1, A2, A5, and A6, then ∥˜βn − βn∥ψ2 = O(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Corollary (of Proposition 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If A1, A2, A5, A6, and A11, then E maxi≤n |⃗xi(˜βn − βn)| =  O  �  [(p log n)/n]1/2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (Unless p = o(log n), in which case E maxi≤n |⃗xi(˜βn − βn)| =  O  �  [(log n)/n1/2]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=')  Proposition 1’s corollary is immediate from its part 2 in combination with A11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 2 is proved in the appendix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' its corollary flows from A11 and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' They follow  D’Amour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (2021) in assuming sub-gaussian covariates, in the sense of A11,  an assumption to be relaxed in Section 4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The corollaries characterize errors of estimation of index values rather than  paired contrasts of them, but they have immediate extensions giving the same  rates of decline for maxi,j≤n |(⃗xi−⃗xj)(ˆβ− ˜βn)| and E maxi,j≤n |(⃗xi − ⃗xj)(˜βn − βn)|,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Because for any collection of W of of length-p row vectors,  sup  ⃗w∈W  |⃗w(ˆβ − βn)| ≤ sup  ⃗w∈W  |⃗w(ˆβ − ˜β)| + sup  ⃗w∈W  |⃗w(˜β − βn)|,  (10)  it follows that with sub-gaussian covariates the worst-case PIC or index error tends  to 0 provided that (p3/2/n) log n does, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' if p = o{[n/(log n)]2/3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  When the covariate has sub-√n dimension, the corollaries indicate that in large  samples the suprema of {|⃗xi(ˆβ− ˜β)| : i ≤ n} and {|(⃗xi−⃗xj)(ˆβ− ˜β)| : i, j ≤ n} will be  smaller by an order of magnitude, p/n1/2 = oP (1), than those of {|⃗xi(˜β − βn)| : i}  and {|(⃗xi − ⃗xj)(˜β − βn)| : i, j} (both of which are OP  �  ((p/n)1/2 log n)  �  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Of the  two errors at right of (10), the sup |⃗w(˜β − βn)| term ordinarily dominates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' we turn  attention to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2  A thought experiment  In a special case making both ⃗X and ˆβ Gaussian, sizes of PIC errors admit specific  characterization in terms of readily estimable quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Ghosh and Cort´es (2019),  among others, consider related issues under an assumption of Gaussian ⃗X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For  vectors v ∈ ℜm let |v|2 and |v|∞ have their usual meanings, (m−1 �m  i=1 v2  i )1/2 and  �  i≤m |vi| respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For matrices M and N of like dimension, ⟨M, N⟩F denotes  the Frobenius inner product tr(M′N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Let S be a partition of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , n} with an associated mapping ⃗µ :  S → ℜp, and let ES(·) and CovS(·) denote expectations calculated with S and {µs :  s ∈ S} held fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Given S and {µs : s ∈ S} let (˜βn − βn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xi − ⃗µ([i]S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xj − ⃗µ([j]S))  be jointly multivariate Normal, for each i, j ≤ n, with mean zero, CovS(˜β) = C,  CovS( ⃗Xi) = CovS( ⃗Xj) = Σ and, if i ̸= j, CovS( ⃗Xi, ⃗Xj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Then we have  ES  ���[( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j, i < j]  ��2  2  �  = ⟨2Σ, C⟩F  and  (11)  ES  ���[( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j]  ��  ∞  �  ≤ z∗  nS⟨2Σ, C⟩1/2  F ,  (12)  where nS := card({{i, j} : i S∼ j, i < j}) and z∗  nS := (2 log 2nS)1/2 bounds E max1≤i≤n |Gi|,  (Gi : i) independent N(0, 1), as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Given its strong assumptions on the covariate, Proposition 3 has limited prac-  tical use for PIC error control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' we shall arrive at methods for containment of  [( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j] that relax those assumptions to the moment condition  A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  But these methods call for a limit on sizes of PIC errors that are to be  tolerated, and (12) will turn out to be helpfully specific in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The statistician who sets out to select a matched sample has it as her oper-  ating hypothesis that each member of the focal group has counterparts that are  close enough, in terms of ⃗xβn, within the alternate group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Simplifying, so as to  remove the question-begging “close enough,” let us suppose provisionally that for  each focal group member i, the available sample contains within it at least one  contrasting group member j that would be a perfect match on the underlying in-  dex, ⃗xiβn = ⃗xjβn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Continue the thought experiment by supposing ˆβ and ⃗X to  be as described in Proposition 3, and by taking the focal group to be the smaller  of the treatment and control groups, implying no fewer than min(n0, n1) perfect  pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Let S to be the collection of equivalence classes induced by the relation  that i S∼ j if and only if ⃗xiβn = ⃗xjβn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The proposition then characterizes PICs  |(⃗xi − ⃗xj)ˆβ| for which the contrast on the underlying index, |(⃗xi − ⃗xj)βn|, is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Each simplification made thus far in order to apply the proposition should err in  the direction of understating the maximum PIC among pairs closely matched on  11  ⃗xβn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' but even if we continue to arrange our thought experiment so as to minimize  this quantity we will find it to be almost unworkably large, in a sense to be given  presently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  As specified so far, our perfect pairing thought experiment permits no ⃗Xβn  variation within strata of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This means the stratified covariance CovS( ⃗X) must  satisfy β′  nΣβn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Bending available covariate data to this constraint, take ⃗X  to be distributed as in Proposition 3 with Σ = S⊥βn := (n − L)−1x⊥xβn′x⊥xβn,  the observed covariates’ covariance as projected onto the orthocomplement of the  index, x⊥xβn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (Here x is the n × p matrix of covariates as observed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' x⊥v denotes  the n×p matrix of residuals arising from the p regressions of x-columns on n-vector  v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and L is the number of overt, preexisting strata, if such exist, and 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Following A1, x is assumed to be centered or stratum-centered, as appropriate, and  x⊥xβn inherits this centering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=') The natural estimate S⊥ˆβ = (n − L)−1x⊥xˆβ′x⊥xˆβ  of S⊥βn is appropriately consistent for S⊥βn, as noted in Proposition 5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Observe that use of S⊥ˆβ (as opposed to S) again reduces (11) and (12), if in  increasing-p regimes it leaves their order unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The maximum PIC bound (12) is at its smallest, with nS = min(n0, n1),  when each member of the smaller of the focal and comparison groups has just one  perfectly matching counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Among such configurations, the bound is sharpest  when the pairs do not overlap, as in matching without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (Conditionally  given ˜β as well as S and {⃗µ(s) : s ∈ S}, ( ⃗Xi − ⃗Xj)(˜β − βn) is independent of  ( ⃗Xi′− ⃗Xj′)(˜β−βn), for i, j, i′, j′ with {i, j} ̸= {i′, j′};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' by the discussion following (5)  in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2, this causes inequality (31) in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2 to be sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=') Complete  the specification of our perfect-pairing thought experiment by supposing its nS =  min(n0, n1) pairs to be nonoverlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Then (12) more closely estimates the  width in ⃗xˆβ of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Across pairs S constituting the thought experiment, the maximum PIC error is  expected to be z∗  min(n0,n1)⟨2S⊥βn, C⟩1/2  F , of order O{[(p log n)/n]1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The accom-  panying estimate is z∗  min(n0,n1)⟨2S⊥ˆβ, ˆC⟩1/2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' These quantities are small enough to  tend to zero, but only barely so: [(p log n)/n]1/2 is precisely the rate that A9 re-  quires to decline to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Among pairs {i, j} that in actuality are perfectly matched,  (⃗xi −⃗xj)βn = 0, even tame, Gaussian variation in other covariate directions engen-  ders a maximum PIC |[( ⃗Xi − ⃗Xi)ˆβ : i S∼ j]|∞ of as large an order as can be tolerated  of separations on the actual index, |[( ⃗Xi − ⃗Xi)βn : i S∼ j]|∞, if the matching is to  be asymptotically exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' So we will recommend this number as a matching toler-  ance, not only when making the restrictive assumptions of Proposition 3 but also  when entertaining only the weaker A1–A9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3  A summary standard error for PICs  In light of (8) and (9), the covariance of ˜βn is  Cn = n−1A−1  n Bn(A−1  n )′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (13)  This Cn approximates the covariance of ˆβ, particularly when the index model  has sub-√n dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  When p = o{[n/ log(n)]1/2}, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 1 says |˜β − ˆβ|2 =  oP (n−1/2), small enough to obviate distinctions between Cn = Cov(˜β) and Cov(ˆβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  For example, Lemma 1 below entails that s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='(⃗x˜β) = [⃗xCn⃗x′]1/2 shares the or-  der OP (n−1/2|⃗x|2) with ⃗x(ˆβ − βn), whereas part 2 of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 1 says ⃗x(˜β − ˆβ) =  oP (n−1/2|⃗x|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The proposition following the lemma will show the larger order  OP (n−1/2|⃗x|2) also to be shared by s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='(⃗x˜β) = [⃗x ˆCn⃗x′]1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Under A3, A5 and A6, |Bn|2 = O(1) and |Cn|2 = O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  If ψ is the gradient a (well-specified) log-likelihood, then An = Bn, and n−1 ˆA−1  n  estimates Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' More broadly, Cn is estimated by n−1 ˆA−1  n  ˆBn( ˆA−1  n )′, where  ˆBn = ˆBn(ˆβ),  ˆBn(β) = n−1  n  �  i=1  ψ(ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' β)ψ(ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' β)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The propositions that follow establish the consistency of natural covariance esti-  mators and related quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Under A1–A9, | ˆA−1  n  − A−1  n |2  P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If also A10 (sub-√n dimen-  sion) and A11 (sub-gaussian covariates), then | ˆA−1  n  ˆBn( ˆA−1  n )′ − nCn|2  P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Let A1–A9 hold, let ˆCn be a consistent estimate of Cn (| ˆCn −  Cn|2 = oP (n−1)) and let S, S⊥βn and S⊥ˆβ be as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Then  |⟨S, ˆC⟩F − ⟨S, C⟩F | = oP (p/n)  and  |⟨S⊥ˆβ, ˆC⟩F − ⟨S⊥βn, C⟩F | = oP (p/n),  whereas ⟨S, C⟩F = OP (p/n) and ⟨S⊥βn, C⟩F = OP (p/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The quantity ⟨2S⊥ˆβ, ˆCˆβ⟩1/2  F  will be termed the PIC standard error (PIC SE);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 5 says that it consistently estimates the analogous parameter appear-  ing at right of (11) and (12) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Proposition 4 is new as applied to  increasing-p regimes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Proposition 5 is entirely new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Their proofs are given in Ap-  pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3, along with demonstrations of intermediate results including Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='4  PIC SE calipers  We recommend matching within limits of z∗  min(n0,n1) times the PIC SE ⟨S⊥ˆβ, ˆC⟩1/2  F ,  whether or not the Gaussian model of Proposition 3 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If it does apply, this  ensures that the same multiple of the PIC SE characterizes matched discrepancies  on the underlying index (Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If it does not apply but the covari-  ate is sub-gaussian, (12) may no longer limit sizes of PIC errors, but they continue  to tend to 0 as long as p = o{[n/(log n)]2/3} (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If neither the Gaussian  nor sub-gaussian modeling assumptions apply, additional matching requirements  to be described below will be necessary to force the PIC errors towards 0, but  z∗  min(n0,n1) times the PIC SE remains an appropriate tolerance for PICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' It tends  to zero, so its use as a caliper width forces PICs toward zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' it tends to zero  at the same [(p log n)/n]1/2 rate that Proposition 1 requires to tend to zero for  index model consistency, making it no more restrictive than is necessary to force  the PIC maximum to tend to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In the context of the idealized setting studied in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2 it was seen also to be minimal in a more quantitatively specific sense,  in virtue of its sharp characterization of the notional experiment’s maximum PIC  error: if such a paired experiment were to be lurking within the actual data, set-  ting a tolerance for matching on xˆβ any smaller than z∗  min(n0,n1) times the PIC SE  would exclude pairs that are in fact perfectly matched on xβn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  4  Deconstructing Gaussian and sub-gaussian  assumptions  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1  PIC SEs with unrestricted X  The PIC SE averages expected PIC errors in either of two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  First, if the  sample available for matching contains a subsample of perfectly matched subjects  (as envisioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2) for which the within-pair covariance of covariates  is S⊥βn, then the squared PIC SE estimates the expected mean of squared PIC  errors, [(⃗xi − ⃗xj)(ˆβ − βn)]2, across perfectly matched pairs (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Second, taking  the entirety of the sample as-is but residualizing each subject’s covariate for xβn (as  also discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2), the squared PIC SE is approximately the expected  mean square of reduced PIC errors, [(⃗x⊥xβn  i  − ⃗x⊥xβn  j  )(ˆβ − βn)]2, now across all  pairs {i, j}, 1 ≤ i < j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  To see this, for 1 ≤ i, j ≤ n write ⃗d⊥βn  ij  := ⃗x⊥xβn  i  − ⃗x⊥xβn  j  , noting that ⃗d⊥βn  ij  =  14  ⃗xi − ⃗xj for pairs {i, j} that are perfectly matched for the index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Observe that  [⃗d⊥βn  ij  (˜β − βn)]2 = [⃗d⊥βn  ij  (˜β − βn)]′[⃗d⊥βn  ij  (˜β − βn)]  = (˜β − βn)′(⃗d⊥βn′  ij  ⃗d⊥βn  ij  )(˜β − βn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Summing over perfectly matched pairs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' for the first scenario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' or all  �n  2  �  possible  pairs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' for the second,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and in either case letting np denote the number of pairs  contributing to the sum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' we have  1  np  �  [⃗d⊥βn  ij  (˜β − βn)]2 = (˜β − βn)′�  n−1  m  � ⃗d⊥βn′  ij  ⃗d⊥βn  ij  �  (˜β − βn)  = (˜β − βn)′(2S⊥βn)(˜β − βn)  (14)  = ⟨2S⊥βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (˜β − βn)(˜β − βn)′⟩F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (15)  where (14) invokes the U-statistic representation of covariance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (n−1)−1 �  i=1(wi−  ¯w)(vi − ¯v) =  �n  2  �−1 �n−1  i=1  �n  j=i+1  1  2(wi − wj)(vi − vj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and (15) uses the sum of ele-  mentwise products ⟨M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' N⟩F of matrices M and N to re-express (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Accordingly  E  � 1  np  �  [⃗d⊥βn  ij  (˜β − βn)]2  �  = ⟨2S⊥βn, Cn⟩F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A similar argument reveals the PIC SE’s alternate interpretation as root mean  square of pairwise distances |⃗d⊥ˆβ  ij ˆC1/2|2 = (⃗d⊥ˆβ  ij ˆC ⃗d⊥ˆβ′  ij )1/2 over pairs {i, j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Invok-  ing in turn the cyclic property of the matrix trace, the definition of the Frobenius  matrix product ⟨·, ·⟩F , the Frobenius product’s bilinearity, and the U-statistic rep-  resentation of sample covariance:  � ⃗d⊥ˆβ  ij ˆCd⊥ˆβ′  ij  =  �  tr(⃗d⊥ˆβ  ij ˆC ⃗d⊥ˆβ′  ij ) =  �  tr(⃗d⊥ˆβ′  ij  ⃗d⊥ˆβ  ij ˆC)  =  �  ⟨⃗d⊥ˆβ′  ij  ⃗d⊥ˆβ  ij , ˆC⟩F = ⟨  � ⃗d⊥ˆβ′  ij  ⃗d⊥ˆβ  ij , ˆC⟩F = np⟨2S⊥ˆβ, ˆC⟩F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  When the average is only over {i, j} that are perfectly matched for the index,  (⃗d⊥ˆβ  ij ˆCd⊥ˆβ′  ij )1/2 approximates s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='[(⃗xi − ⃗xj)ˆβ], because ⃗xiβn = ⃗xjβn means that  ⃗d⊥ˆβ  ij  approximates ⃗xi − ⃗xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The interpretation as a pairwise covariate distance,  within the orthocomplement in x of xˆβ and after rescaling by ˆC1/2, is available  both for the perfect pairing thought experiment and also when the mean is over  all {i, j}, 1 ≤ i < j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  These arguments rely implicitly on A1–A9, via Proposition 5, but do not call for  Gaussian covariates, nor for boundedness of covariates’ sub-gaussian norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' By  the same token, none admit extensions offering maximum, rather than average,  PIC error control, as is necessary for asymptotically exact matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2  Caliper refinement with attention to index error  distances  It is intuitive that with covariates drawn from heavy tailed distributions there  may be PICs exceeding the PIC SE by factors well above z∗  np = √(2 log 2np),  in contrast to the Gaussian situation described in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Writing δ[x]  for the distribution placing point mass at x, heavier tails on the covariate mean  heavier tails on the empirical distributions  �n  2  �−1 �n−1  i=1  �n  j=i+1 δ[s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='[(⃗xi − ⃗xj)˜β]],  �n  2  �−1 �n−1  i=1  �n  j=i+1 δ[(⃗xi − ⃗xj)(˜β − βn)], and in turn  �n  2  �−1 �n−1  i=1  �n  j=i+1 δ[(⃗xi − ⃗xj)(ˆβ − βn)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Fortunately, estimates |(⃗xi−⃗xj) ˆC1/2|2 of standard deviations s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='(i, j) = |(⃗xi−  ⃗xj) ˆC1/2|2 are available at the time of matching: one can simply avoid pairings {i, j}  for which |(⃗xi−⃗xj) ˆC1/2|2 is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Call |(⃗xi−⃗xj) ˆC1/2|2 the index error distance  separating i from j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Proposition 6 says index error distances can estimate pairwise  index sampling variabilities uniformly well, even for models of sub-√n dimension  if the estimator ˆC is appropriately chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Let ψ(r, ⃗x, β) be the gradient of a likelihood function governing  the conditional distribution of R given ⃗X, with ˆCn = n−1 ˆA−1  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Under A1–A9,  �����  �  1 − |⃗xi ˆC1/2|2  |⃗xiC1/2|2  : i ≤ n  ������  ∞  ,  �����  �  1 − |(⃗xi − ⃗xj) ˆC1/2|2  |(⃗xi − ⃗xj)C1/2|2  : i, j ≤ n  ������  ∞  P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (16)  (Here 0/0 is taken to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=')  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Let {ϵn : n} satisfy ϵ−1  n  = O[n/ max(p, log n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Under A1–A11 with ˆCn =  n−1 ˆA−1  n  ˆBn ˆA−1  n ,  �����  �  1 − max(ϵ1/2  n , |⃗xi ˆC1/2|2)  max(ϵ1/2  n , |⃗xiC1/2|2)  : i  ������  ∞  P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  If x and En ⊆ {{i, j} : 1 ≤ i, j ≤ n} satisfy |(|⃗xi − ⃗xj|2  2 : {i, j} ∈ En)|∞ =  OP (max(p, log n)), then under A1–A10  �����  �  1 − max[ϵ1/2  n , |(⃗xi − ⃗xj) ˆC1/2|2]  max[ϵ1/2  n , |(⃗xi − ⃗xj)C1/2|2]  : {i, j} ∈ En  ������  ∞  P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Under A1–A12, (16) holds with ˆCn = n−1 ˆA−1  n  ˆBn ˆA−1  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  We focus on situations conforming to the hypotheses of (i) or of (iii), warranting  uniform convergence (16) of index error distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Let us calibrate sizes of index error distances with reference to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2’s  perfect pairing thought experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Proposition 7 adapts extant results about  Gaussian chaos to characterize that setting’s maximum of |( ⃗Xi − ⃗Xj)C1/2|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  16  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Let ⃗Xi, i ≤ n, be independent MVN(µ, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Let C be a second  positive semidefinite matrix of the same dimension as Σ, let E ⊆ {{i, j} : 1 ≤ i ̸=  j ≤ n} and let nE := card(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Then  �  E  ���{|( ⃗Xi − ⃗Xj)C1/2|2 : {i, j} ∈ E}  ��2  ∞  ��1/2  ≤  ⟨2Σ, C⟩1/2  F  �  �1 +  �  log nE  p[Σ1/2CΣ1/2]  �1/2�  �,  (17)  where p[M] denotes intrinsic dimension, tr(M)/|M|2, for positive semidefinite M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 7 is proved in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' With Σ = S⊥βn and C = Cn, it  explicitly bounds the worst-case pairwise distance |( ⃗Xi − ⃗Xj)C1/2|2 within the  perfect-pairing thought experiment of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' To contain PIC errors of actual  experiments to a similar level, I recommend the match-eligibility requirement that  |(⃗xi − ⃗xj) ˆC1/2|2 ≤ ⟨2S⊥ˆβ, ˆC⟩1/2  F  �  2 +  �log min(n0, n1)  p − 1  �1/2�  ,  (18)  as a complement to the z∗  min(n0,n1)⟨2S⊥ˆβ, ˆC⟩1/2  F  limit on PICs |(⃗xi − ⃗xj)ˆβ| that  was recommended in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The heuristic by which Proposition 7 supports  constraint (18), to be explained presently, more simply suggests the stricter cap  on |(⃗xi − ⃗xj) ˆC1/2|2 of  ⟨2S⊥ˆβ, ˆC⟩1/2  F  �  1 +  �log min(n0, n1)  p − 1  �1/2�  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (19)  but it will subsequently be shown that (18) together with a softer penalty on  index error distances respecting (18) while exceeding (19) is sufficient for present  purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  To relate (17) to (19), first recall that min(n0, n1) is the size of the perfect-  pairing thought experiment (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2) and ⟨2S⊥ˆβ, ˆC⟩F is consistent for ⟨2S⊥βn, Cn⟩F  (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In general 0 ≤ p[Σ1/2CΣ1/2] ≤ p, by definition (see also Tropp, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Vershynin, 2018, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In the special cases that (R, ⃗X) has linear discriminant  structure with Cor( ⃗X) known, or that R is linear in ⃗X, with ⃗X Gaussian and βn  estimated accordingly in either case, intrinsic and extrinsic dimensions coincide:  p[Σ1/2CnΣ1/2] = p or p − 1, depending as Σ = S(x) or S⊥βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If supposed to contain  min(n0, n1) distinct pairs {i, j} for which i ̸= j but ⃗xiβn = ⃗xjβn, then either of  these Gaussian- ⃗X special cases closely models Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2’s notional perfect pair-  ing, with Σ = S⊥βn, and (19) estimates the expected maximum covariate distance  17  |(⃗xi − ⃗xj)C1/2|2 among perfect pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' At the same time, an |(⃗xi − ⃗xj) ˆC1/2|2 limit  of form (19) should rarely exclude perfect pairs because of separation in direc-  tions orthogonal to the index, since (17) approximates such separations’ expected  maximum from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Regardless of what distribution the covariate may have been drawn from, limits  (19) or (18) on covariate distances |(⃗xi − ⃗xj) ˆC1/2|2 also engender limits on PIC  errors |(⃗xi −⃗xj)(ˆβ −βn)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In part this is because (⃗xi −⃗xj)(˜β −βn) is sub-gaussian if  ˜β is, and is N[0, |(⃗xi − ⃗xj)C1/2  n  |2  2] if ˜β ∼ MVN(βn, Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Proposition 8, stated here  without proof, collects the relevant facts reviewed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Let E ⊆ {{i, j} : 1 ≤ i, j ≤ n} satisfy |(⃗xi − ⃗xj)C1/2  n  |2 > 0 for all  {i, j} ∈ E, and let nE = card(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If ˜β ∼ MVN(βn, Cn),  E  ������  �  (⃗xi − ⃗xj)(˜β − βn)  |(⃗xi − ⃗xj)C1/2  n  |2  : {i, j} ∈ E  ������  ∞  �  ≤ z∗  nE = [2 log(2nE)]1/2  and  E  �  �  �����  �  (⃗x⊥xβn  i  − ⃗x⊥xβn  j  )(˜β − βn)  |(⃗x⊥xβn  i  − ⃗x⊥xβn  j  )C1/2  n  |2  : {i, j} ∈ E  ������  ∞  �  � ≤ z∗  nE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (20)  If ˜β − βn is non-Normal but sub-gaussian with ∥˜β − βn∥ψ2 bounded, these expected  maximums continue to be of order (log nE)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Together with Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 6, Proposition 8 says requiring each of min(n0, n1) pairs  {i, j} to have |(⃗xi −⃗xj) ˆC1/2  n  |2 below (19) puts the corresponding PIC errors below  z∗  min(n0,n1) times (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If p increases faster than log n, then the ratio in (19) tends to  0, and (19) is asymptotically equivalent to the PIC SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' That is, confining matching  to pairs {i, j} for which |(⃗xi − ⃗xj) ˆC1/2  n  |2 falls left of (19) makes the supremum of  matched errors |(⃗xi − ⃗xj)(˜β − βn)| asymptotically as it would be in the perfect-  pairing thought experiment, given log n = o(p) but not special conditions on the  distribution of the covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (If p increases no faster than log n, p = O(log n),  these errors somewhat exceed those of the corresponding thought experiment, but  they tend quickly to zero anyway, due to the p’s slow increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=')  These considerations suggest (19) as a hard limit for pair distances |(⃗xi −  ⃗xj) ˆC1/2|2, but similar control of PIC errors can be had with a simple policy that  encourages matches with |(⃗xi − ⃗xj) ˆC1/2|2 beneath (19) while only requiring (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Make i is eligible for pairing to j if  |(⃗xi − ⃗xj)ˆβ| ≤ z∗  min(n0,n1)[⟨2S⊥ˆβ, ˆC⟩1/2  F  − ˆe(i, j)]  (21)  18  where  ˆe(i, j) :=  �  |(⃗xi − ⃗xj) ˆC1/2|2 − ⟨2S⊥ˆβ, ˆC⟩1/2  F  �  1 +  �log min(n0, n1)  p − 1  �1/2��  +  , (22)  v+ := max(0, v), represents excess in index error distance as compared to its nomi-  nal supremum (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If |(⃗xi −⃗xj) ˆC1/2|2 never exceeds this nominal supremum, (21)  reduces to the requirement that |(⃗xi −⃗xj)ˆβ| ≤ z∗  min(n0,n1)⟨2S⊥ˆβ, ˆC⟩1/2  F , as proposed  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For potential pairings {i, j} with |(⃗xi − ⃗xj) ˆC1/2|2 exceeding (19),  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='4’s PIC allowance of z∗  min(n0,n1)⟨2S⊥ˆβ, ˆC⟩1/2  F  is reduced in recognition of  the pairing’s large standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' When the index error distance |(⃗xi − ⃗xj) ˆC1/2|2  is so large that its excess ˆe(i, j) exceeds the PIC SE — or equivalently, so large  that (18) fails — (21) forbids i’s pairing with j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  This selectively narrowed PIC SE caliper has important advantages over non-  varying PIC SE calipers, alone or in combination with calipers of width (19) on  the pairwise index error distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Non-varying PIC SE calipers secure asymptotic  exactness of a match only for sub-gaussian covariates, an assumption that selective  narrowing of the caliper enables us to do without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Coupling a non-varying PIC  SE caliper with a limit on the pairwise index error distance of (19) contains the  sum  |[(⃗xi − ⃗xj)ˆβ : i S∼ j]|∞ + |[(⃗xi − ⃗xj)(˜β − βn) : i S∼ j]|∞  (23)  at the product of z∗  min(n0,n1) with the right hand side of (18), just as the selectively  narrowed PIC SE caliper does, but at the cost of categorically disallowing pairwise  index error distances in excess of (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In contrast, the selectively narrowed caliper  permits those pairings if their PICs are sufficient small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This additional tolerance  is important because (19) systematically underestimates suprema of pairwise index  error distances for some index models, even with Gaussian ⃗X, because of its use of  p − 1 in lieu of Proposition 7’s p[(S⊥βn)1/2C(S⊥βn)1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' That minor embarrassment  could be remedied by replacing p − 1 in (19) by p[(S⊥ ˆβ)1/2 ˆC((S⊥ ˆβ))1/2], but then as-  sumption A12 would become necessary for asymptotic equivalence of (19) and the  PIC SE — an equivalence needed even under A9, the weakest of the model dimen-  sionality restrictions considered in this paper, to force (23) toward an asymptote  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Assumption A12 is discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3  The contribution of linearization error  Display (23) omits linearization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Unless the estimator of the index regression  is linear in its dependent variable R, to estimate |[(⃗xi − ⃗xj)βn : i ∼ j]| we must  attend to |[(⃗xi − ⃗xj)(ˆβ − ˜β) : i ∼ j]| as well as |[(⃗xi − ⃗xj)ˆβ : i ∼ j]| and |[(⃗xi −  19  ⃗xj)(˜β − βn) : i ∼ j]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Index estimators are linear in R in the special cases of  linear regression and linear discriminant modeling with fixed correlation, but not  for indices estimated with probit or logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Recall from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1 that with sub-gaussian covariates (A11), |[(⃗xi−⃗xj)(ˆβ−  ˜β) : i, j ≤ n]|∞ tends to 0 provided that p = o{[n/(log n)]2/3}, and is smaller by an  order of magnitude than |[(⃗xi − ⃗xj)(˜β − βn) : ⃗xiβn ≈ ⃗xjβn]|2 measured in the PIC  SE, provided that p = o{[n/(log n)]1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Matching within selectively narrowed  PIC SE calipers secures these conclusions under conditions not including A11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  However, depending on the specific side conditions and estimation routines that  are employed, the matching procedure may need to observe additional caliper  restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  First consider the case that A1–A9 hold, with Cn estimated by n−1 ˆA−1  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The  matching requirement (18), a consequence of (21), ensures that |[|(⃗xi −⃗xj) ˆC1/2|2 :  i ∼ j]|∞ = OP {[max(p, log n)/n]1/2}, since (by Proposition 5) the PIC SE is  OP [(p/n)1/2] and since (p − 1)−1 log min(n0, n1) = O[max(1, p−1 log n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Proposi-  tion 6 and ˆC = n−1 ˆA−1  n in turn give |[|(⃗xi−⃗xj)A−1/2|2 : i ∼ j]|∞ = OP [max(p, log n)1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  As A3 and Lemma 3 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3 entail |An|2 = O(1), |[|(⃗xi−⃗xj)|2 : i ∼ j]|∞ =  OP [max(p, log n)1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Whether or not ⃗X was drawn from a sub-gaussian distribu-  tion, pairs {i, j} selected within selectively narrowed PIC SE calipers can be no  more separated on ⃗x than they would have been under sub-gaussian sampling, and  Proposition 1 entails  |[(⃗xi − ⃗xj)(ˆβ − ˜β) : i ∼ j]|∞ = OP [pn−1(log n)1/2max(p, log n)1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (24)  Within matched pairs, linearization error is as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1, even with-  out A11 or additional matching restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  When Cn is instead estimated by n−1 ˆA−1  n Bn ˆA−1  n , A10 is needed in addition  to A1–A9, for consistency of ˆCn (by Proposition 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Then |[|(⃗xi − ⃗xj) ˆC1/2|2 :  i ∼ j]|∞ = OP {[max(p, log n)/n]1/2}, by a similar argument as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Case (ii) of  Proposition 6 then gives that |[|(⃗xi − ⃗xj)C1/2|2 : i ∼ j]|∞ = OP {[max(p, log n)/n]1/2},  provided that ⟨2S⊥ˆβ, ˆC⟩F  �  1 +  �  log min(n0,n1)  p−1  �1/2�  is of the same order as max(p, log n)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (Proposition 5 gives that it is OP [max(p, log n)/n], but here we also require its  reciprocal to be OP {[max(p, log n)/n]−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=') At this point A12 also becomes nec-  essary, to ensure |B−1/2  n  |2 = OP (1) and thus that |C−1/2  n  |2 = OP (n1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If so,  Proposition 1 again gives (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The full-rank covariance condition A12 merits careful consideration in prac-  tice, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  It will poorly describe some otherwise unassailable index mod-  els, as partitioners have long been encouraged to add covariates in such models  without regard to their mutual correlations (Rubin and Thomas, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Fortu-  20  nately A12 is straightforward to diagnose, by checking that neither S nor ˆBn is  ill-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If sustainable, it delivers (in combination with Lemma 1 and A6)  the needed assurance that the PIC SE declines no faster than (p/n)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If A12  cannot be sustained while an assumption that ⟨S⊥βn, Cn⟩F = O(p/n) can be, we  can instead combine that weaker assumption with additional matching restrictions  ensuring that maxi∼j |⃗xi −⃗xj| = OP [max(p, log n)1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Matches can be required to  fall within Euclidean distance calipers of width tr(S)1/2{1 + [(log n)/(p − 1)]1/2},  or with calipers of width s(xj){1 + [(log n)/(p − 1)]1/2} on each dimension j =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' p of ⃗x separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Either way, the combination of additional assumptions and  matching restrictions secures (24), and that the dominant part of the PIC error  |[(⃗xi − ⃗xj)(ˆβ − βn)]|∞ is |[(⃗xi − ⃗xj)(˜β − βn)]|∞, not |[(⃗xi − ⃗xj)(ˆβ − ˜β)]|∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  5  Asymptotically exact matching and consis-  tency of impact estimation  Matching within PIC SE calipers arranges that paired differences of the index tend  uniformly to zero, given mild conditions on the index model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For propensity and  certain other index models, this convergence is precisely what is needed to ensure  that in the absence of unmeasured confounding, the matched structure enjoys the  same consistency properties as would be enjoyed were paired differences on the  index uniformly and identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In this section we assume Z ∈ {0, 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' the  very weak overlap condition  P[0 < P(Z = 1 | Xβn) < 1] = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (25)  that E |YC|1+δ, E |YT |1+δ < ∞ for some δ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and that the mapping v �→  logit[P(Z = 1|Xβ = v)] is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If Xβ is a propensity score modeled on the  logit scale, this mapping is the identity and (25) follows from Rosenbaum and Ru-  bin’s (1983) overlap condition, 0 < P(Z = 1 | X) < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' if Xβ is a risk or prognostic  score, (25) is less restrictive than their already weak overlap requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A partition Sn is a finely stratified design (Fogarty, 2018) if it divides {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , n}  into partition elements s ∈ Sn that satisfy �  i∈s�zi = z� ≤ 1 for either or both of  z = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' These can be 1 : m0 or m1 : 1 matched sets, for natural numbers m0, m1,  if not m1 : m0 blocks with both m0, m1 ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' singleton elements, s of size ns = 1,  represent unmatched units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Such Sn may emerge from pair matching, where each  s ∈ Sn is either a 1:1 pair, �  i∈s�zi = 1� = �  i∈s�zi = 0� = 1, or an unmatched 0:1  or 1 : 0 singleton;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' from matching with multiple controls, permitting 1 : m, m ≥ 1,  matches as well as singletons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' from 1-nearest neighbor matching, in which m:1 but  not 1:m sets may arise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' or from full matching (Rosenbaum, 1991), permitting both  21  m : 1 and 1 : m configurations for any m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' or from full matching with symmetric  restrictions (Stuart and Green, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Fredrickson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', 2020), permitting both  m : 1 and 1 : m matched sets, but only for m falling below a designated m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The  notation [i]Sn for the partition element s ∈ Sn containing i is abbreviated to [i]  when no partition other than Sn is under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Consider estimates defined as roots of ψSn(·) = 0, where  ψSn(η) :=  �  s∈Sn  �  i∈s ψsi(η)  �  s∈Sn wsns ¯Zs(1 − ¯Zs),  ψsi(η) := ws[Yi − η(Zi − ¯Zs)](Zi − ¯Zs), (26)  ¯zs := ns−1�  j∈s zj and ws is a nonnegative weight determined by zs and/or ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  For example, the z-coefficient in an ordinary regression of outcomes y on z and  matched-set indicator variables is expressible as the solution ˆτ of ψSn(τ) = 0 for  ws ≡ 1, since (zi − ¯z[i] : i) is the residual of z’s ordinary regression on matched-  set indicator variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' As a second example, the effect of treatment-on-treated  estimator  n−1  {i:Zi=1,n[i]>1}  �  {i:Zi=1,n[i]>1}  Yi − avg(Yj : Zj = 0, j ∼ i)  uniquely solves ψSn(·) = 0 with w[i]Sn = 0 for unmatched i and ws = (1 − ¯Zs)−1  for s with ns > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Inferences will reflect Sn by conditioning on stratum-wise treatment alloca-  tions, that is on a sigma field containing Fn := σ  ��  i∈s Zi : s ∈ Sn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Desite this  notation, {Fn : n} is not a nested filtration: as a rule Sm ̸⊆ Sm, as strict contain-  ment does not permit the maximum index discrepancy, |{⃗xiβn − ⃗xjβn : i Sn  ∼ j}|∞,  to decline with increasing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The statistic [�  i∈s Zi : s ∈ Sn] that defines Fn is in  itself uninformative, S-ancillary (Severini, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Lehmann and Romano, 2022) to  parameters defined as roots of η �→ E[ψSn(η)], ψSn as defined in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 9 says that under mild assumptions about the regularity of {Sn : n}  and (YT , YC, Z), the solution of (26) tends to a probability limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' To state the  regularity assumptions, write ¯Ys(z) = 0 if �  i∈s�Zi = z� = 0, for z = 0 or 1,  and (�  i∈s�Zi = z�)−1�  i∈s Yi�Zi = z� otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and let V (n,s) := ¯Ys(1) − ¯Ys(0) −  E( ¯Ys(1) − ¯Ys(0) | Fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Let {( ⃗Xi, YCi, YTi, Zi) : i} be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', let {Sn : n} be finely stratified  designs and let Fn = σ((�  i∈s Zi : s ∈ Sn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Assume the moment condition that  for some δ > 0 either: (i) E |YC|1+δ, E |YT |1+δ < ∞ and ns is bounded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' or (ii)  there is a V and δ > 0 with E |V |1+δ < ∞ such that for each n and s ∈ Sn, |V |  22  stochastically dominates |V (n,s)| given Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 2 Let {ws : s ∈ Sn} be nonnegative, Fn-  measurable weights, and let ψSn(·) be as in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Assume that with probability one:  mn → ∞, where mn := �  s∈Sn�ws ¯Zs(1− ¯Zs) > 0� is the cardinality of Sn exclusive  of unmatched singletons and strata receiving zero weight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' wsns ¯Zs(1− ¯Zs) is bounded  above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and m−1  n  �  s∈Sn wsns ¯Zs(1− ¯Zs) is bounded away from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Conditionally given  Fn we then have, for any sigma fields {Gn : n} with Fn ⊆ Gn:  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' for each η, ψSn(η) − E[ψSn(η)|Gn] → 0 in probability and in L1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' η �→ ψSn(η) and η �→ E[ψSn(η) | Gn] have unique roots ˆτn and τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In addition, if there is τ0 ∈ (−∞, ∞) such that τn  P→ τ0, then ˆτn  P→ τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  As compared to the classical consistency principle for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' observations (Hu-  ber, 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Serfling, 1980, Lemma A of § 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1), Proposition 9 upgrades moment  requirements from estimating equation contributions ψ(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' θ) being L1 to V (n,s)  being L1+δ, some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This enables conclusions in terms of L1 as well as in-  probability convergence, which in turn accommodates refinement of Fn-conditioning  to conditioning on finer sigma fields reflecting matched variation in index scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Specifically, consider  Gn := σ  �  ( ⃗Xiβn : i ≤ n) ∪ Fn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The generating statistic ( ⃗Xiβn, 1 ≤ i ≤ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' �  j∈s Zj, s ∈ Sn) is again S-ancillary to  matched treatment-control contrasts such as τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Because remaining information about ( ⃗Xi, Zi), 1 ≤ i ≤ n, is barred, the trans-  formed index scores θi := logit[P(Z = 1|Xβn = xiβn)] determine Gn-conditional  assignment probabilities as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If s ∈ Sn and ζ : s → {0, 1} satisfies �  i∈s ζi =  �  i∈s zi, then  πs(ζ) := P(Zi = ζi all i ∈ s | Gn)  =  �  �  �  exp(θi)  �  j∈s exp(θj),  any i ∈ s s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ζi = 1, and ζj = 0 for all j ∈ s \\ {i}  exp(−θi)  �  j∈s exp(−θj),  any i ∈ s s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ζi = 0, and ζj = 1 for all j ∈ s \\ {i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (27)  (Because we assume (25), θi ∈ (−∞, ∞) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' When s = {i} is an unmatched  singleton, πs(ζ) = 1 for the sole permissible ζ, {i �→ zi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' When s is a 1 : 1 matched  pair {i1, i2}, one condition in (27) obtains with i = i1 while the other obtains  2That is, the Fn-conditional distribution of |V (n,s)| falls at or below the unconditional  distribution of |V | in the usual stochastic ordering where W ⪯ V iff P(W > a) ≤ P(V > a)  for all a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  23  with i = i2, so that (27) presents two distinct expressions for πs(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' But these  expressions then assign the same value to πs(ζ), for each ζ : {i1, i2} → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=')  Now define  ˜ψs(η) :=  �  i∈s ψsi(η)  nsπs(Zs) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  ˜ψSn(η) :=  �  s∈Sn ˜ψs(η)  �  s∈Sn wsns ¯Zs(1 − ¯Zs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (28)  In contrast to ψs(η) := �  i∈s ψsi(η), ˜ψs(η) cannot be calculated in practice, as  its random denominator involves the unknown βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Accordingly ˜ψSn(·) lacks direct  application to effect estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  However, it is useful for analysis of estimates  based on ψSn(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The unique root of η �→ E[ ˜ψSn(η)|Gn] is  � �  s∈Sn  ˜wsns  �−1 �  s∈Sn  ˜ws  �  i∈s  E(Y | Z = 1, ⃗Xβn = ⃗xiβn) − E(Y | Z = 0, ⃗Xβn = ⃗xiβn),  (29)  where ˜ws := ws¯zs(1 − ¯zs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For all η and n,  ���E[ ˜ψSn(η) − ψSn(η) | Gn]  ��� ≤  [exp(4|{θi − θj : i ∼ j}|∞) − 1] ·  �  s∈Sn ˜wsns E |V (n,s)|  �  s∈Sn ˜wsns  ,  (30)  where θi = logit[P(Z = 1|Xβn = xiβn)] and V (n,s) is as defined in § 5, above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If v �→ logit[P(Z = 1|Xβ = v)] is Lipschitz and the conditions of Propo-  sition 9 hold, |{(⃗xi − ⃗xj)βn : i ∼ j}|∞ → 0 entails that the difference  of τn with (29) tends in probability to 0, where τn is the unique root of  η �→ E[ψSn(η) | Gn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If the averages (29) tends in probability to a finite limit τ0, then ˆτn  P→ τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  If the index deconfounds allocation of treatment, (YC, YT ) ⊥ Z|Xβn, then (29)  coincides with the average causal effect  �  s∈Sn ˜ws  �  i∈s E(YTi − YCi | ⃗Xβn = ⃗xiβn)  �  s∈Sn ˜wsns  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  If the index is a propensity score, this deconfounding flows from strong ignora-  bility in the sense of Rosenbaum and Rubin (1983);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' if Xβ is a prognostic score,  strong ignorability entails index strong ignorability under a secondary “no effect  modification” condition (Hansen, 2008, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
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+page_content='  Severini, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (2000), Likelihood methods in statistics, Oxford University Press, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  26  Stuart, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and Green, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (2006), “Using Full Matching to Estimate Causal  Effects in Non-Experimental Studies: Examining the Relationship between Ado-  lescent Marijuana Use and Adult Outcomes,” Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', Johns Hopkins Uni-  versity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Tropp, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (2015), “An Introduction to Matrix Concentration Inequalities,”  Foundations and Trends in Machine Learning, 8, 1–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  van der Vaart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (1998), Asymptotic Statistics, Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Vershynin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (2018), High-dimensional probability: An introduction with applica-  tions in data science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 47, Cambridge university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (2011), “GEE analysis of clustered binary data with diverging number  of covariates,” Annals of Statistics, 39, 389–417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', Cai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', Jiang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', Song, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', and Xia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (2013),  “Optimal caliper width for propensity score matching of three treatment groups:  a Monte Carlo study,” PloS one, 8, e81045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  A  Review of mathematical symbols  The symbols | · |2 and | · |∞ indicate Euclidean and supremum norms as usual  (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For scalar or vector random variables V , ∥V ∥ψ2 is the sub-gaussian norm  of V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' for fixed matrices M, |M|2 and |M|F are M’s operator and Frobenius norms  respectively (§ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For matrices M and N of like dimension, ⟨M, N⟩F is the  Frobenius inner product tr(M′N) (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Symbols ˆβ, ψ(r, ⃗x, β) and c0(r, ⃗x, β0 + ⃗xβ) are defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1, while  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3 defines βn, ˜β, An, and Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For partitions S of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , n}, [i]S denotes  the subset of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , n} belonging to S that contains i and i S∼ j means there is  s ∈ S with both i ∈ s and j ∈ s (§ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2 defines nS (as card(S)) and  z∗  m, for positive integers m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' �A� is the indicator of event A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Section 5 defines  ¯vs = n−1  s  �  j∈s vj for s ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' associates estimating functions ψSn(·) and  ˜ψSn(·), and sigma fields Fn and Gn, with partitions Sn of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and also  defines ¯vs(z) for s ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , n} and z ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  27  B  Proofs for Section 3  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1  Proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In light of (8) and IS estimability (A2, A3), the difference be-  tween (˜βn−βn) and A−1  n  1  n[�n  j=1 ψ(Rj, ⃗xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn)] has Euclidean norm of order smaller  than n−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Because it is nonrandom, the sub-gaussian norm of this difference is  also o(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' So it suffices to show ∥A−1  n  1  n[�n  j=1 ψ(Rj, ⃗xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn)]∥ψ2 = O(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Given A3, for this it suffices in turn to show that ∥ �n  j=1 ψ(Rj, ⃗xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn)∥ψ2 =  O(n1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  By (4),  ∥  n  �  i=1  ψ(Ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn)∥ψ2 =  sup  γ:|γ|2=1  ∥  n  �  i=1  γ′ψ(Ri, ⃗xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn)∥ψ2  =  sup  γ:|γ|2=1  ∥  n  �  1  c0(Ri, ⃗xi, ⃗xiβn)w(⃗xi)⃗xiγ∥ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Let k1 be a bound for ∥c0(Ri, ⃗xi, ⃗xiβn)∥ψ2, by A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' According to the general Ho-  effding inequality (Vershynin, 2018, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='6), there is a universal k0 such that  ∥  n  �  1  c0(Ri, ⃗xi, ⃗xiβ)w(⃗xi)⃗xiγ∥2  ψ2 ≤k0  n  �  1  ∥c0(Ri, ⃗xi, ⃗xiβ)w(⃗xi)⃗xiγ∥2  ψ2  ≤k0k1  n  �  1  |w(⃗xi)⃗xiγ|2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' So  sup  γ:|γ|2=1  ∥  n  �  1  c0(Ri, ⃗xi, ⃗xiβ)w(⃗xi)⃗xiγ∥2  ψ2 ≤k0k1n  �  1  n  n  �  1  w(⃗xi)2  � sup  γ:|γ|2=1  �  1  n  n  �  1  |⃗xiγ|2  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The left-hand side equals the square of ∥ �n  1 c0(Ri, ⃗xi, ⃗xiβ)⃗xi∥ψ2, whereas A6 says  the product at right is O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2  Proofs for section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2  The following lemma helps to prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Under the conditions of Proposition 3, for all 1 ≤ i < j < n we have  ( ⃗Xi − ⃗Xj) ⊥ ˜β − βn | S, {⃗µ(s) : s ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  28  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Recall that ˜β − βn = A−1  n  1  n  �n  i=1 ψ(Ri, ⃗Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Suppressing conditioning  for S, {µs : s ∈ S} in the notation,  Cov  �  A−1  n  1  n  n  �  i=1  ψ(Ri, ⃗Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn), ⃗X1 − ⃗X2  �  =  1  n Cov  �  A−1  n ψ(R1, ⃗X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn), ⃗X1  �  −  1  n Cov  �  A−1  n ψ(R2, ⃗X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn), ⃗X2  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  By joint Normality of {[ ⃗Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ψ′(Ri, ⃗Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn)] : i}, 1  nA−1  n ψ(R1, ⃗X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' βn) and ( ⃗Xi − ⃗Xj)  are jointly Normal, and the fact that they are uncorrelated means they are inde-  pendent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For fixed γ ∈ ℜp we have, after some algebra that I omit,  E[( ⃗X1 − ⃗X2)γ]2 = 2γ′Σγ  and  E  ��{( ⃗Xi − ⃗Xj)γ : i S∼ j, i ̸= j}  ��2  2 = 2γ′Σγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (Throughout the proof I write ““E[·]” for “ES[·].”) By the conditional indepen-  dence established in Lemma 2, it follows that  E  ��{( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j, i ̸= j}  ��2  2 = ⟨2Σ, C⟩F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Also for fixed γ, (5) as applied to Normal variables gives  E  ��{( ⃗Xi − ⃗Xj)γ : i S∼ j, i ̸= j}  ��  ∞ ≤ [4γ′Σγ log 2nS]1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (31)  Since ˜β is independent of ⃗Xi − ⃗Xj for each 1 ≤ i < j ≤ n (Lemma 2), (31)  entails  E  �  E  ���{( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j, i ̸= j}  ��  ∞ | ˜β  �2�  ≤  E  �  4(˜β − βn)′Σ(˜β − βn) log 2nS  �  = 4⟨Σ, Cn⟩F log 2nS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Combining this fact with Jensen’s inequality for conditional expectation,  �  E  ���{( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j, i ̸= j}  ��  ∞  ��2  ≤  E  �  E  ���{( ⃗Xi − ⃗Xj)(˜β − βn) : i S∼ j, i ̸= j}  ��  ∞ | ˜β  �2�  ≤ 4⟨Σ, Cn⟩F log 2nS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  29  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3  Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Letting k < ∞ denote the supremum (Condition A5) of sub-  gaussian norms of {c0(Ri, ⃗xi, ⃗xiβn) : i}, E[c0(Ri, ⃗xi, ⃗xiβn)4] ≤ (2k)4 for all i (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=',  Vershynin, 2018, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Accordingly �  i E  �  c0(Ri, ⃗xi, ⃗xiβn)4�  = O(n) and in turn  �  i{E  �  c0(Ri, ⃗xi, ⃗xiβn)2�  }2 = O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Combining this with supγ:|γ|2=1  �  i w(⃗xi)4(⃗xiγ)4 =  O(n) (Condition A6), Cauchy-Schwartz gives supγ:|γ|2=1  �  i E  �  c0(Ri, ⃗xi, ⃗xiβn)2�  w(⃗xi)2(⃗xiγ)2 =  O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Rearranging terms in light of (4), this says supγ:|γ|2=1  �  i E  �  [γ′ψ(Ri, ⃗xi, β)]2�  =  O(n), or supγ:|γ|2=1 γ′nBnγ = O(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' thus |Bn|2 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Condition A3 gives  |A−1  n |2 = O(1), so also |C−1  n |2 = O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Our demonstration of Proposition 4 relies on three supporting lemmas, as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Under A4 and A6, supβ |An(β)|2 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Write c1(⃗x, η) := (∂/∂η) E[c0(R, ⃗x, η) | ⃗X = ⃗x] so that ∇β E[ψ(R, ⃗x, β) | ⃗X = ⃗x] =  c1(⃗x, ⃗xβ)w(⃗x)⃗x′⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' By A4, there is K1 < ∞ such that |c1(⃗x, β0 + ⃗xβ)| < K1, for  any β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' By Cauchy-Schwartz, A6 gives supγ:|γ|2=1  �n  i=1 w(⃗xi)(⃗xiγ)2 = O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Since  n|An(β)|2 =  sup  γ:|γ|2=1  γ′  ��  i  c1(⃗xi, ⃗xiβ)w(⃗xi)⃗x′⃗x  �  γ ≤ K1  sup  γ:|γ|2=1  �  i  w(⃗xi)(⃗xγ)2  for any β, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Under A1–A9, |An(ˆβ) − An(βn)|2  P→ 0 and | ˆBn(ˆβ) − ˆBn(βn)|2  P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Under A1–A9 as well as A11 and sub-√n dimension, | ˆBn(βn) −  Bn(βn)|2  P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proofs of Lemmas 4 and 5 are given following the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Since ˆAn ≡ An(ˆβ), | ˆAn −An|2  P→ 0 follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Since ˆBn ≡ ˆBn(ˆβ), | ˆBn−Bn|2  P→ 0 follows from | ˆBn(ˆβ)− ˆBn(βn)|2  P→ 0 (Lemma 4)  and | ˆBn(βn) − Bn|2  P→ 0 (Lemma 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Since |A−1  n |2 = O(1) and | ˆA−1  n |2 = OP (1) (Condition A3 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 1), it follows  that | ˆA−1  n  − A−1  n |2  P→ 0, by applying sub-multiplicativity of the spectral norm to  the right-hand side of ( ˆA−1  n −A−1  n ) = ˆA−1  n ( ˆAn −An)A−1  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Since also |Bn|2 = OP (1)  (Lemma 1), the 2-norms of the second and third summands in ˆA−1  n  ˆBn ˆA−1  n  =  A−1  n BnA−1  n + ( ˆA−1  n − A−1  n )BnA−1  n + ˆA−1  n Bn( ˆA−1  n − A−1  n ) + ˆA−1  n ( ˆBn − Bn) ˆA−1  n  30  must tend in probability to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Thus the stochastic order of |A−1  n BnA−1  n − ˆA−1  n  ˆBn ˆA−1  n |2  can be no greater than that of | ˆA−1  n ( ˆBn −Bn) ˆA−1  n |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' But as A4 and A6 entail that  | ˆAn|2 = OP (1), by Lemma 3, OP (| ˆA−1  n ( ˆBn−Bn) ˆA−1  n |2) = OP (| ˆBn−Bn|2) = oP (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  this means |Cn − ˆCn|2 = n−1|A−1  n BnA−1  n − ˆA−1  n  ˆBn ˆA−1  n |2 = oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  This proof of Lemma 4 was based in part on Wang’s proof of a similar principle  for generalized estimating equations (2011, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' To establish |An(ˆβ)−An(βn)|2 = supγ:|γ|2=1 γ′[An(ˆβ)−An(βn)]γ P→  0, let K1 < ∞ be a Lipschitz constant for η �→ c1(r, ⃗x, η), each r and ⃗x, where c1(·)  is as defined in the proof of Lemma 3, above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (By Condition A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=') Then  |γ′{∇β E[ψ(R, ⃗xi, β) | ⃗X = ⃗xi]β=ˆβ − ∇β E[ψ(R, ⃗xi, β) | ⃗X = ⃗xi]β=βn}γ|  ≤ K1[⃗xi(ˆβ − βn)]w(⃗xi)(⃗xiγ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (32)  Summing over i and applying Cauchy-Schwartz,  (γ′[An(ˆβ) − An(βn)]γ)2 ≤K2  1  �  1  n  n  �  i=1  w(⃗xi)2(⃗xiγ)4  � �  � 1  n  n  �  i=1  �  ⃗xi  ˆβ − βn  |ˆβ − βn|2  �2�  � × |ˆβ − βn|2  2,  interpreting “⃗x(δ/|δ|2)” as 0 when |δ|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' It follows that  |An(ˆβ)−An(βn)|2  2 ≤ K2  1  �  sup  γ:|γ|2=1  1  n  n  �  i=1  w(⃗xi)2(⃗xiγ)4  ��  sup  δ:|δ|2=1  1  n  n  �  i=1  (⃗xiδ)2  �  |ˆβ−βn|2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Observe that the conditions of Proposition 1 follow from those of Proposition 4,  so that we may assume |ˆβ−βn|2 = OP [(p/n)1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This combines with Condition A6  to give |An(ˆβ) − An(βn)|2 = O(1)O(1)OP [(p/n)1/2], which by A9 is oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  As to | ˆBn(ˆβ) − ˆBn(βn)|2,  γ′{Ψ(Ri, ⃗xi, ˆβ)Ψ(Ri, ⃗xi, ˆβ)′ − Ψ(Ri, ⃗xi, βn)Ψ(Ri, ⃗xi, βn)′}γ  (33)  =(c2  0(Ri, ⃗xi, ˆηi) − c2  0(Ri, ⃗xi, ηi))w(⃗xi)2(⃗xiγ)2  =(c0(Ri, ⃗xi, ˆηi) − c0(Ri, ⃗xi, ηi))(c0(Ri, ⃗xi, ˆηi) + c0(Ri, ⃗xi, ηi))w(⃗xi)2(⃗xiγ)2  =(c0(Ri, ⃗xi, ˆηi) − c0(Ri, ⃗xi, ηi))2w(⃗xi)2(⃗xiγ)2  + (c0(Ri, ⃗xi, ˆηi) − c0(Ri, ⃗xi, ηi)) · 2c0(Ri, ⃗xi, ηi)w(⃗xi)2(⃗xiγ)2  =:Vi(γ) + Wi(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (34)  31  By the Lipschitz property (A4) of c0(r, ⃗x, ·),  sup  γ:|γ|2=1  1  n  �  i  |Vi(γ)| ≤K2  1  sup  γ:|γ|2=1  1  n  �  i  (ˆηi − ηi)2w(⃗xi)2(⃗xiγ)2  ≤K2  1|ˆβ − βn|2  2  sup  δ,γ:|δ|2=|γ|2=1  1  n  �  i  w(⃗xi)2(⃗xiδ)2(⃗xiγ)2  = O(1)OP (p/n)O(1) = oP (1),  (35)  invoking Assumption A6 and consistency of ˆβ at (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The Lipschitz property of  c0(r, ⃗x, ·) also gives  sup  γ:|γ|2=1  1  n  �  i  |Wi(γ)| ≤2K1  sup  γ:|γ|2=1  1  n  �  i  |c0(Ri, ⃗xi, ηi)(ˆηi − ηi)|w(⃗xi)2(⃗xiγ)2  ≤2K1|ˆβ − βn|2  sup  γ,δ:|γ|2=|δ|2=1  1  n  �  i  c0(Ri, ⃗xi, ηi)(⃗xiδ)w(⃗xi)2(⃗xiγ)2  ≤2K1|ˆβ − βn|2[ 1  n  �  i  c4  0(Ri, ⃗xi, ηi)]1/4  ×  sup  δ:|δ|2=1  [ 1  n  �  i  w(⃗xi)4(⃗xiδ)4]1/4  sup  γ:|γ|2=1  [ 1  n  �  i  w(⃗xi)4(⃗xiγ)4]1/2  (36)  = OP (p/n)OP (1)O(1)O(1) = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Here we apply Cauchy-Schwartz (twice) at (36) and, to pass to the next line, con-  sistency of ˆβ as per Proposition 1, Assumption A5 in combination with Markov’s  inequality and Assumption A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Since  | ˆBn(ˆβ) − ˆBn(βn)|2 =  sup  γ:|γ|2=1  γ′( ˆBn(ˆβ) − ˆBn(βn))γ  ≤  sup  γ:|γ|2=1  1  n  �  i  |Vi(γ)| +  sup  γ:|γ|2=1  1  n  �  i  |Wi(γ)|,  the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' To control |γ[ ˆBn(βn) − Bn(βn)]γ′|, fix γ with |γ|2 = 1 and  consider  γ[ ˆBn(βn) − Bn(βn)]γ′ =γ′�  n−1 �  i≤n  {c2  0(Ri, ⃗xi, ⃗xiβn) − E[c2  0(R, ⃗xi, ⃗xiβn) | ⃗X = ⃗xi]}w2(⃗xi)⃗x′  i⃗xi  �  γ  =n−1 �  i≤n  {c2  0i(Ri) − E[c2  0i(R) | ⃗X = ⃗xi]}w2(⃗xi)(⃗xiγ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  32  Observe that A5 entails the random variables c2  0i(Ri) − E[c2  0i(R) | ⃗X = ⃗xi] to be  sub-exponential with uniformly bounded sub-exponential norm (Vershynin, 2018,  § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Applying Bernstein’s inequality (Vershynin, 2018, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='8),  P{|γ[ ˆBn(βn) − Bn(βn)]γ′| ≥ t} ≤  2 exp  �  −k2 min  �  n2t2  k2e  �n  i=1 w(⃗xi)4(⃗xiγ)4 ,  nt  ke maxi≤n w(⃗xi)2(⃗xiγ)2  ��  ,  (37)  where ke is a finite upper bound for the sub-exponential norms of c2  0i(Ri) −  E[c2  0i(Ri)], i ≥ 1, and k2 is a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Now let N be a 1/4-net of the p-dimensional sphere, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' a finite subset of  {γ : |γ|2 = 1} with the property that {γ : |γ|2 = 1} is covered by balls centered in  N of radius 1/4, so that (Vershynin, 2018, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1)  sup  γ:|γ|2=1  γ[ ˆBn(βn) − Bn(βn)]γ′ ≤ 2 sup  γ∈N  γ[ ˆBn(βn) − Bn(βn)]γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (38)  We may select this N to have cardinality no more than 9p (Vershynin, 2018,  Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Since (37) holds for arbitrary γ on the unit sphere, it follows that  1  2 P{ sup  γ:|γ|2=1  |γ[ ˆBn(βn) − Bn(βn)]γ′| ≥ t} ≤  exp  �  p log(9) − nk2 min  �  t2  k2en−1 �n  i=1 w(⃗xi)4(⃗xiγ)4 ,  t  ke maxi≤n w(⃗xi)2(⃗xiγ)2  ��  =  exp  �  −k2 min  �  nt2  k2en−1 �n  i=1 w(⃗xi)4(⃗xiγ)4 − pk3,  nt  ke maxi≤n w(⃗xi)2(⃗xiγ)2 − pk3  ��  ,  (39)  where k3 = log(9)/k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Recalling that n−1 �n  i=1 w(⃗xi)4(⃗xiγ)4 = O(1) (by A6),  t2/[k2  en−1 �n  i=1 w(⃗xi)4(⃗xiγ)4] has positive limit infimum and finite limit supremum  (for any t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' thus the first of the two quantities of which the minimum is taken tends  to ∞ because p = o(n) (A9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Since maxi≤n w(⃗xi)2(⃗xiγ)2 = O{max[p log n, (log n)2]}  (A11), p2 log n = o(n) (sub-√n dimension) entails that the second quantity also  must tend to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (If pn ≤ log n infinitely often, then on the subsequence for which  this is true the term in question is bounded below by (log n){(t/ke)n/[(log n) maxi≤n w(⃗xi)2(⃗xiγ)2] − k3},  which tends to ∞ with (log n)[n/(log n)3 − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Otherwise pn > log n so that  max[pn log n, (log n)2] = pn log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The term in question equals p{(t/ke)n/[p maxi≤n w(⃗xi)2(⃗xiγ)2] − k3},  which tends to ∞ because p ↑ ∞, maxi≤n w(⃗xi)2(⃗xiγ)2 = O(p log n) and p2 log n =  o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=') So the minimum in (39) increases without bound, and (39) itself tends to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  33  Proposition 5’s proof uses two supporting lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Under A1 and A6, s(xˆβ) − s(xβn) = OP (|ˆβ − βn|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' By A1,  s2(xˆβ) − s2(xβn) = ˆβ′S ˆβ − β′  nSβn  = (ˆβ + βn)′S(ˆβ − βn)  = (S1/2 ˆβ + S1/2βn)′S1/2(ˆβ − βn),  where S1/2 denotes the matrix square root of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Noting A6’s implication that  |S1/2|2 = O(1),  |s(xˆβ) − s(xβn)| =[s(xˆβ) + s(xβn)]−1|(S1/2 ˆβ + S1/2βn)′S1/2(ˆβ − βn)|  ≤ (|S1/2 ˆβ|2 + |S1/2βn|2)−1|S1/2 ˆβ + S1/2βn|2|S1/2|2|ˆβ − βn|2  ≤ |S1/2|2|ˆβ − βn|2 = O(1)OP (|ˆβ − βn|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  In the proof of Lemma 7 below, let (I, J) ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , n} be a randomly ordered  simple random sample of size 2, and let ⃗D (or ⃗D⊥w, w an n-vector) be a 1 × p  random vector representing the paired difference ⃗xI − ⃗xJ (or ⃗x⊥w  I  − ⃗x⊥w  J  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Then  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2  r(I, J) = ⃗D⊥xˆβ ˆC ⃗D⊥xˆβ′  (40)  Because of the U-statistic representation of covariance, Cov( ⃗D) = 2S(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' By  symmetry of the distribution of (I, J), E  �  ⃗D  �  = 0, so that E  �  ⃗D′ ⃗D  �  = Cov(D) =  2S(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Selection of (I, J) pays no attention to the distinction between treatment  and control, making ⃗D independent of {Zi}n  i=1 and, by extension, of ˆβ and ˆC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Therefore its conditional and marginal moments coincide: E  �  ⃗D|ˆβ, ˆC  �  = E  �  ⃗D  �  =  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' E  �  ⃗D′ ⃗D|ˆβ, ˆC  �  = Cov( ⃗D|ˆβ, ˆC) = Cov( ⃗D) = 2S(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Let S = S(x) or S⊥v = (n − L)−1x⊥v′x⊥v, some categorical vari-  able v with L categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Then ⟨S⊥ˆβ, ˆC⟩F = ⟨S, ˆC⟩F − s−2(xˆβ)⟨S ˆβ ˆβ′S, ˆC⟩F and  ⟨S⊥βn, C⟩F = ⟨S, ˆC⟩F − s−2(xβn)⟨Sβnβn′S, C⟩F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If S = S(x), let ⃗D be as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Otherwise, if S = S⊥v, then let  (I, J) ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , n} be a randomly ordered stratified simple random sample of size  2 from one of the categories of v with at least two elements, after selecting one of  these categories with probability proportional to size, and let ⃗D (or ⃗D⊥w, w an  34  n-vector) be a 1 × p random vector representing the paired difference ⃗xI − ⃗xJ (or  ⃗x⊥w  I  − ⃗x⊥w  J  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Evaluate E  �  ⃗D⊥xˆβ′ ⃗D⊥xˆβ|ˆβ, ˆC  �  = Cov( ⃗D⊥xˆβ|ˆβ, ˆC) using the U-statistic rep-  resentation of sample covariance to get Cov( ⃗D⊥xˆβ|ˆβ, ˆC) = 2(n − 1)−1x⊥xˆβ′x⊥xˆβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Now compare to:  1  2 E  �  ⃗D⊥xˆβ′ ⃗D⊥xˆβ|ˆβ, ˆC  �  =(I − ˆβ ˆβ′S(x)/s2(xˆβ))′S(x)(I − ˆβ ˆβ′S(x)/s2(xˆβ))  =S(x) − 2s−2(xˆβ)S(x) ˆβ ˆβ′S(x)+  s−4(xˆβ)S(x) ˆβ · ˆβ′S(x) ˆβ · ˆβ′S(x)  =S(x) − s−2(xˆβ)S(x) ˆβ ˆβ′S(x) =: S⊥ˆβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' It follows from A6 that ∥S∥F = tr(S′S)1/2 = O(p1/2), and from  the assumed consistency of ˆCn that ∥ ˆCn − Cn∥F = oP (p1/2/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' So  |⟨S, ˆCn⟩F − ⟨S, Cn⟩F | = |⟨S, ˆCn − Cn⟩F | = oP (p/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  By Lemma 7,  ⟨S⊥ˆβ, ˆCn⟩F − ⟨S⊥βn, Cn⟩F =⟨S, ˆCn − Cn⟩F  + s−2(xˆβ)⟨S ˆβ ˆβ′S, ˆCn⟩F − s−2(xβn)⟨Sβnβn′S, Cn⟩F  =oP  � p  n  �  + [s−2(xˆβ) − s−2(xβn)]⟨S ˆβ ˆβ′S, ˆCn⟩F  (41)  + s−2(xβn)  �  ⟨S ˆβ ˆβ′S, ˆCn⟩F − ⟨Sβnβn′S, Cn⟩F  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (42)  To analyze the rightmost summand of (41), first observe that s2(xβn) =  βn′Sβn, so that |s(xβn)| ≤ |S|2|βn|2 = O(p1/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Together with Proposition 1,  Lemma 6 entails that s(xˆβ) − s(xβn) = OP (p/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  So A9 says that s(xˆβ) =  OP (p1/2) just as |s(xβn)| = OP (p1/2), whence s(xˆβ)+s(xβn) = OP (p1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Combin-  ing these facts, s2(xˆβ)−s2(xβn) = [s(xˆβ) − s(xβn)][s(xˆβ) + s(xβn)] = OP (p/n1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  In light of A8, it follows that |s−2(xˆβ) − s−2(xβn)| = OP {p/[n1/2s4(xβn)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  As to the ⟨S ˆβ ˆβ′S, ˆCn⟩F factor of (41),  |⟨S ˆβ ˆβ′S, ˆCn⟩F | = tr(S ˆβ ˆβ′S ˆCn)  = | tr(ˆβ′S ˆCnS ˆβ)|  = ˆβ′S ˆCnS ˆβ ≤ ∥ˆβ′S1/2∥2∥S1/2∥2∥ ˆCn∥2∥S1/2∥2∥S1/2 ˆβ∥2  = OP (s2(xβn)  n  ),  35  by A8, A6, consistency of ˆCn and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' We now have  |[s−2(xˆβ) − s−2(xβn)]⟨S ˆβ ˆβ′S, ˆCn⟩F | = OP  �  p  n3/2s2(xβn)  �  = oP  � p  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  To bound (42), in light of A8 we focus on the second factor:  |⟨S ˆβ ˆβ′S, ˆCn⟩F − ⟨Sβnβn′S, Cn⟩F | ≤  |⟨S ˆβ ˆβ′S − Sβnβn′S, ˆCn⟩F | + |⟨Sβnβn′S, ˆCn − Cn⟩F |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (43)  We address the left term of (43) via Cauchy-Schwartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  By Lemma 1 and  consistency of ˆCn, ∥ ˆCn∥2 = OP (n−1), so ∥ ˆCn∥F = OP (p1/2/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  As S ˆβ ˆβ′S −  Sβnβn′S = S(ˆβ ˆβ′ − βnβn′)S, ∥S ˆβ ˆβ′S − Sβnβn′S∥F ≤ ∥S1/2∥2∥S1/2 ˆβ ˆβ′S1/2 −  S1/2βnβn′S1/2∥F ∥S1/2∥2 = OP (1)OP  �  ∥S1/2 ˆβ ˆβ′S1/2 − S1/2βnβn′S1/2∥F  �  OP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='decomposes as  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='∥S1/2 ˆβ ˆβ′S1/2 − S1/2βnβn′S1/2∥F ≤ ∥S1/2(ˆβ − βn)(ˆβ + βn)′S1/2∥F +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='∥S1/2 ˆββ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='nS1/2 − S1/2βn ˆβ′S1/2∥F  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='where:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='∥S1/2(ˆβ − βn)(ˆβ + βn)′S1/2∥F ≤∥S1/2∥2∥(ˆβ − βn)(ˆβ + βn)′S1/2∥F  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='= OP (1) tr[S1/2(ˆβ + βn)(ˆβ − βn)′(ˆβ − βn)(ˆβ + βn)′S1/2]1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='= OP (1) tr[(ˆβ + βn)′S(ˆβ + βn)(ˆβ − βn)′(ˆβ − βn)]1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='= OP (1)[(ˆβ + βn)′S(ˆβ + βn)∥ˆβ − βn∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2]1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='= OP (1)[(ˆβ + βn)′S(ˆβ + βn)]1/2OP [( p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='n)1/2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='= OP [( p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='n)1/2][(ˆβ − βn + 2βn)′S(ˆβ − βn + 2βn)]1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='= OP [( p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='n)1/2][∥S1/2(ˆβ − βn)∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2 + 4(ˆβ − βn)′Sβn + 4∥S1/2βn∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2]1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='= OP [( p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='n)1/2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='OP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='� p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='+ OP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='�� p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='�1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='s(xβn)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='+ OP [s2(xβn)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='�1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='= OP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='�� p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='�1/2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='OP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='��� p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='�1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='+ s(xβn)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='�2·1/2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='= OP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='max  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='� p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  � p  n  �1/2  s(xβn)  ��  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  36  and  ∥S1/2 ˆββ′  nS1/2 − S1/2βn ˆβ′S1/2∥F =∥S1/2(ˆβ − βn)β′  nS1/2 − S1/2βn(ˆβ − βn)′S1/2∥F  ≤∥S1/2(ˆβ − βn)β′  nS1/2∥F + ∥S1/2βn(ˆβ − βn)′S1/2∥F  = 2 tr[S1/2βn(ˆβ − βn)′S(ˆβ − βn)β′  nS1/2]1/2  = 2[(ˆβ − βn)′S(ˆβ − βn)β′  nSβn]1/2  = OP  �� p  n  �1/2  s(xβn)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Thus the left term at right of (43) is OP {max[p/n, (p/n)1/2s(xβn)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The remaining term in (43) is bounded as follows, using the definition of ⟨·, ·⟩F  and the cyclic property of the trace:  ⟨Sβnβn′S, ˆCn − Cn⟩F = tr[Sβnβ′  nS( ˆCn − Cn)]  = tr[β′  nS( ˆCn − Cn)Sβn]  =β′  nS( ˆCn − Cn)Sβn  ≤|βn|2  2∥S∥2  2∥ ˆCn − Cn∥2  = OP (p)OP (1)oP (n−1) = oP (p/n),  using A7, A6 and consistency of ˆCn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This shows that (43) as a whole is OP {max[p/n, (p/n)1/2s(xβn)]},  from which it follows that (42) is OP {max[(p/n)s−2(xβn), (p/n)1/2s−1(xβn)]} =  oP (p/n), completing the proof of the proposition’s consistency claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The remain-  der of the proposition follows from A6 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  C  Proofs for section 4  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1  Proof for Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2  Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For parts (i) and (iii) it suffices to show that supi |1 −  |⃗xi ˆC1/2|2  2/|⃗xiC1/2|2  2| and supi,j |1 − |(⃗xi − ⃗xj) ˆC1/2|2  2/|(⃗xi − ⃗xj)C1/2|2  2| tend in prob-  ability to 0 (from the definition of in-probability convergence), and to consider  only (i, j) with ⃗xi ̸= 0 and ⃗xi ̸= ⃗xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The latter ensures |⃗xiC1/2|2 > 0 and  |(⃗xi − ⃗xj)C1/2|2 > 0, since Cn is of full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Indeed, Lemma 3 in the appendix  gives |An|2 = O(1), covering case (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' for case (iii), A12 gives |B−1  n |2 = O(1), and  in turn |AnB−1  n An|2 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Note that this shows not only that Cn has full rank  but also that |C−1  n |2 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  37  For arbitrary nonzero p-vectors v we have  1 − v′ ˆCnv  v′Cnv = v′(Cn − ˆCn)v  v′v  v′v  v′Cnv  so that  �����1 − v′ ˆCnv  v′Cnv  ����� ≤  �����  v′(Cn − ˆCn)v  v′v  v′C−1  n v  v′v  ����� ≤ |Cn − ˆCn|2|C−1  n |2,  using general inequalities for positive definite matrices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', Gentle, 2007, § 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proposition 4 gives | ˆCn − Cn|2 = oP (n−1), completing the proof of parts (i) and  (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  For part (ii), write a ∨ b := max(a, b) and a ∧ b := min(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For arbitrary  nonzero p-vectors v  �����  v′Cnv ∨ ϵn − v′ ˆCnv ∨ ϵn  v′v  ����� ≤  �����  v′Cnv − v′ ˆCnv  v′v  ����� , since  ���v′Cnv ∨ ϵn − v′ ˆCnv ∨ ϵn  ��� ≤  ���v′Cnv − v′ ˆCnv  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (The latter is true because the expressions on either side of the inequality are equal  if v′Cnv and v′ ˆCnv belong to the same half-interval, (0, ϵn] or [ϵn, ∞), whereas if  they are separated by ϵn then the left side expression is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=') So  �����  v′Cnv ∨ ϵn − v′ ˆCnv ∨ ϵn  v′v  ����� ≤ |Cn − ˆCn|2 = oP (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (44)  According to A11, supi ⃗xi⃗x′  i = O[max(p, log n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' So  sup  i  ⃗xi⃗x′  i  ϵn  ,  sup  {i,j}∈En  (⃗xi − ⃗xj)(⃗xi − ⃗xj)′  ϵn  = OP (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Thus  sup  i  ⃗xi⃗x′  i  ⃗xiCn⃗x′  i ∨ ϵn  ,  sup  {i,j}∈En  (⃗xi − ⃗xj)(⃗xi − ⃗xj)′  (⃗xi − ⃗xj)Cn(⃗xi − ⃗xj)′ ∨ ϵn  = OP (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  By (44),  sup  i:|⃗xi|2̸=0  ⃗xiCn⃗x′  i ∨ ϵn − ⃗xi ˆCn⃗x′  i ∨ ϵn  ⃗xi⃗x′  i  ⃗xi⃗x′  i  ⃗xiCn⃗x′  i ∨ ϵn  = oP (1),  38  and likewise  sup  {i,j}∈En:  ⃗xi̸=⃗xj  (⃗xi − ⃗xj)Cn(⃗xi − ⃗xj)′ ∨ ϵn − (⃗xi − ⃗xj) ˆCn(⃗xi − ⃗xj)′ ∨ ϵn  (⃗xi − ⃗xj)(⃗xi − ⃗xj)′  ×  (⃗xi − ⃗xj)(⃗xi − ⃗xj)′  (⃗xi − ⃗xj)Cn(⃗xi − ⃗xj)′ ∨ ϵn  = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The assumption on { ⃗Xi : i ≤ n} entails that ( ⃗Xi − ⃗Xj)  is MVN(0, 2Σ) and ( ⃗Xi − ⃗Xj)C1/2 is MVN(0, 2C1/2ΣC1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Let 2C1/2ΣC1/2  have eigendecomposition Q′ΛQ, with Λ nonnegative real diagonal and Q an or-  thogonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Then ( ⃗Xi − ⃗Xj)C1/2Q′ ∼ MVN(0, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Writing Wij1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , Wijp  for the p coordinates of ( ⃗Xi − ⃗Xj)C1/2Q′ =: ⃗Wij, we see that Wij1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , Wijp  are mutually independent mean-0 Gaussians with variances v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' , vp, the diago-  nal entries of Λ and eigenvalues of 2C1/2ΣC1/2, while ( ⃗Xi − ⃗Xj)C( ⃗Xi − ⃗Xj)′ =  ( ⃗Xi − ⃗Xj)C1/2Q′QC1/2( ⃗Xi − ⃗Xj)′ = ⃗Wij ⃗W ′  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Straightforwardly, for any i ̸= j E ⃗Wij ⃗W ′  ij = �  i vi, or tr(2C1/2ΣC1/2) =  2⟨C, Σ⟩F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  We proceed to characterize the moment generating function (MGF)  of ⃗Wij ⃗W ′  ij − E ⃗Wij ⃗W ′  ij, or �p  k=1(W 2  ijk − E W 2  ijk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The centered χ2  1 distribution  having MGF e−t(1 − 2t)−1/2, valid for t < 1/2, W 2  ijk − E W 2  ijk has log MGF  (1/2)[−2vkt − log(1 − 2vkt)], valid for t < 1/(2vk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Applying the relation − log(1−  x)−x ≤ x2/[2(1 − x)], 0 < x < 1 (a transformation of log(1−x)’s degree 2 Taylor  expansion), we now have  log E{exp t[W 2  ijk − E W 2  ijk]} ≤  (vkt)2  1−2vkt  �  for t <  1  2vk  �  ≤  (vkt)2  1−2(maxk vk)t  �  t <  1  2 maxk vk  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' so  log E exp{t  p  �  k=1  [W 2  ijk − E W 2  ijk)]}  ≤  t2 �  k v2  k  1−2(maxk vk)t,  t <  1  2 maxk vk  due to independence of uncorrelated Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' That is, �p  k=1 W 2  ijk − E W 2  ijk is  sub-gamma on its right tail with variance factor �  k v2  k and scale factor maxk vk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  in symbols, �p  k=1 W 2  ijk − E W 2  ijk ∈ Γ+(�  k v2  k, maxk vk) (Boucheron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', 2013,  § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Since �  k v2  k = tr(Λ2), the cyclic property of the trace combines with defi-  nitions above to reduce this variance factor to 4|C1/2ΣC1/2|2  F , as tr(Q′ΛQQ′ΛQ) =  4 tr(C1/2ΣC1/2C1/2ΣC1/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and the scale factor reduces to 2|C1/2ΣC1/2|2, as maxk vk =  |Λ|2 and |Λ|2 = |Q′ΛQ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  39  For an MGF characterization in terms of more familiar quantities, note that  |2C1/2ΣC1/2|2  F = �p  i=1 v2  i ≤ (maxk vk)(�  k vk) = 2|C1/2ΣC1/2|2(�  k vk), while  �  k vk = tr(D2) = tr(2C1/2ΣC1/2) = 2⟨Σ, C⟩F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' So the variance factor can be  taken as 4|C1/2ΣC1/2|2|S1/2C1/2|2  F , or 4|C1/2ΣC1/2|2⟨Σ, C⟩F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  These MGF characterizations give control of the supremum of { ⃗Wij ⃗W ′  ij −  E ⃗Wij ⃗W ′  ij : i ̸= j ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This class containing  �n  2  �  distinct Γ+  �  4|C1/2ΣC1/2|2⟨Σ, C⟩F ,  2|C1/2ΣC1/2|2  �  random variables, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='6 of Boucheron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (2013) yields  E max  i̸=j≤n[ ⃗Wij ⃗W ′  ij − E( ⃗Wij ⃗W ′  ij)] ≤  2  �  2|C1/2ΣC1/2|2⟨Σ, C⟩F log  �n  2  ��1/2  + 2|C1/2ΣC1/2|2 log  �n  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Simplifying via E ⃗Wij ⃗W ′  ij = 2⟨Σ, C⟩F , i ̸= j, and 2a2+2ab+b2 = a2[1 + (1 + b/a)2],  E max  i̸=j≤n  ⃗Wij ⃗W ′  ij ≤⟨Σ, C⟩F  �  �  �1 +  �  �1 +  �  2|C1/2ΣC1/2|2 log  �n  2  �  ⟨Σ, C⟩F  �1/2�  �  2�  �  �  = ⟨Σ, C⟩F  �  �  �1 +  �  �1 +  �  2 log  �n  2  �  p[C1/2ΣC1/2]  �1/2�  �  2�  �  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (45)  Observe next that on the positive real line x �→ √2+x dominates x �→ [1 + (1 + x)2]1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (The functions coincide at x = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' otherwise the latter has derivative equal to the  square root of (1+x)2/[1 + (1 + x)2], which is nowhere greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=') Thus (45)  gives  �  E max  i̸=j≤n  ⃗Wij ⃗W ′  ij  �1/2  ≤⟨Σ, C⟩1/2  F  �  �√2 +  �  2 log  �n  2  �  p[C1/2ΣC1/2]  �1/2�  �  = √2⟨Σ, C⟩1/2  F  �  �1 +  �  log  �n  2  �  p[C1/2ΣC1/2]  �1/2�  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  D  Section 5  Proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Throughout the proof, expected value is interpreted to be con-  ditional on Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (Because each Fn is the sigma field of finitely many discrete  40  random variables, this introduces no measure-theoretic considerations that were  not already present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=')  Also assume version (ii) of the moment condition, not-  ing that it is entailed by version (i) and 0 < P(Z = 1) < 1 (which follows  from 0 < P(Z = 1 | ⃗Xβn) < 1): E |YT |p < ∞ means E(|YT |p | Z) < ∞ also,  which entails E |V (n,s)|p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  If ns is bounded, then there are finitely many  (�  i∈s�Zi = 0�, �  i∈s�Zi = 1�) achievable configurations for fine strata s, each with  a characteristic distribution function x �→ P(V (n,s) ≤ x) such that E  �  |V (n,s)|p�  =  p  � ∞  0 yp−1 P(|V (n,s)| > y)dy < ∞ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=', Durrett, 2019, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' The dis-  tribution for |V | on ℜ+ given by setting P(|V | > y) to the maximum over these  distributions of P(V (n,s) > y) stochastically dominates the relevant V (n,s) distri-  butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' It also satisfies p  � ∞  0 yp−1 P(|V | > y)dy < ∞, because the sum of a finite  collection of functions y �→ P(|V (n,s)| > y) dominates their maximum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' so V ∈ Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Part i of proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  From (26),  �  i∈s  ψsi(η) = ws[Yi − η(Zi − ¯Zs)](Zi − ¯Zs)  = ws{  �  i∈s:  Zi=1  [Yi(1 − ¯Zs) − η(1 − ¯Zs)2] +  �  i∈s:  Zi=0  [−Yi ¯Zs − η ¯Z2  s ]}  = ws[(1 − ¯Zs)  �  i∈s:  Zi=1  Yi − ηns ¯Zs(1 − ¯Zs)2 − ¯Zs  �  i∈s:  Zi=0  Yi − ηns(1 − ¯Zs) ¯Z2  s ]  = wsns ¯Zs(1 − ¯Zs)  � ¯Ys(1) − η(1 − ¯Zs) − ¯Ys(0) − η ¯Zs  �  = wsns ¯Zs(1 − ¯Zs)( ¯Ys(1) − ¯Ys(0) − η),  (46)  so that E[�  i∈s ψsi(η)] = wsns¯zs(1−¯zs) E( ¯Ys(1) − ¯Ys(0) − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Recalling that V (n,s) =  ¯Ys(1) − ¯Ys(0) − E( ¯Ys(1) − ¯Ys(0)), we have  ψSn(η) − E[ψSn(η)] =  �  s∈Sn ˜wn,snsV (n,s)  �  s∈Sn ˜wn,sns  (47)  where ˜wn,s := ws¯zs(1 − ¯zs) ∈ Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (When there is no risk of ambiguity, “ ˜wn,s” is  abbreviated to “ ˜ws.”) Convergence of ψSn(η)−E[ψSn(η)|Gn] will be seen to follow  from suitable convergence of (47), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ψSn(η) − E[ψSn(η)|Fn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  In-probability convergence of ψSn(η) − E[ψSn(η)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  We adapt to the  independent non-identically distributed case an argument for the L1-weak law of  large numbers by truncation of increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Following Durrett (2019, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3),  set ¯V (s)  n  := V (n,s)�wn,s|V (n,s)| ≤ mn� and ¯Vn := V �|V | ≤ mn�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (Recall mn =  41  �  s∈Sn�ws¯zs(1 − ¯zs) > 0�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=') Whereas E V (n,s) = 0 by definition, E ¯V (s)  n  may differ  from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Our first task is to show that (�  s∈Sn ˜wn,s)−1 E �  s∈Sn ˜wn,s ¯V (s)  n  → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  We have  (  �  s∈Sn  ˜wsns)−1 E  �  s∈Sn  ˜wsns ¯V (s)  n  = (  �  s∈Sn  ˜wsns)−1  �  E  �  s∈Sn  ˜wsns ¯V (s)  n+ + E  �  s∈Sn  ˜wsns ¯V (s)  n−  �  where a+ and a− denote positive and negative parts of a, max(a, 0) and min(a, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Now  (  �  s∈Sn  ˜wsns)−1 E  �  s∈Sn  ˜wsns ¯V (s)  n+ = (  �  s∈Sn  ˜wsns)−1 �  s∈Sn  ˜wsns E ¯V (s)  n+  = (  �  s∈Sn  ˜wsns)−1 �  s∈Sn  ˜wsns  � ∞  0  P( ¯V (s)  n  > x)dx  =  � ∞  0  �  s∈Sn ˜wsns P( ¯V (s)  n  > x)  �  s∈Sn ˜wsns  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (48)  By the stochastic ordering assumption, P(V (n,s) > x) is dominated by P(V > x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  but P( ¯V (s)  n  > x) ≤ P(V (n,s) > x), so the integrand in (48) is dominated by  P(V > x) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' As  � ∞  0 P(V > x)dx = E V+ < ∞, dominated convergence gives  (�  s∈Sn ˜wsns)−1 E �  s∈Sn ˜wsns ¯V (s)  n+ − (�  s∈Sn ˜wsns)−1 E �  s∈Sn ˜wsnsV (n,s)  +  = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Similarly (�  s∈Sn ˜wsns)−1 E �  s∈Sn ˜wsns ¯V (s)  n− −(�  s∈Sn ˜wsns)−1 E �  s∈Sn ˜wsnsV (n,s)  −  =  o(1), and  � �  s∈Sn  ˜wsns  �−1�  E  �  s∈Sn  ˜wsns ¯V (s)  n  − E  �  s∈Sn  ˜wsnsV (n,s)  �  = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Since E V (n,s) = 0, this means (�  s∈Sn ˜wsns)−1 E �  s∈Sn ˜wsns ¯V (s)  n  → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  For in-probability convergence of (47) it now suffices to show  �  s∈Sn ˜wsns(V (n,s) − E ¯V (s)  n )  �  s∈Sn ˜wsns  = oP (1),  (49)  the conclusion of the weak law for triangular arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' We now verify the premises  of that principle, as it is given in Durrett’s (2019) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For each n  {V (n,s) : s ∈ Sn} are independent because {(YTi, YCi, Zi) : i} are unconditionally  independent and conditioning on Fn induces dependence within but not across  strata s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Premise (i) of the theorem, �  s∈Sn P( ˜wsnsV (n,s) > �  s∈Sn ˜wsns) → 0,  42  will follow, by the assumptions of stochastic dominance and boundedness of ˜wsns,  from convergence to 0 of  �  s∈Sn: ˜wsns>0  P(u ˜wV >  �  s∈Sn  ˜wsns) = mn P(u ˜wV >  �  s∈Sn  ˜wsns),  (50)  where u ˜w is an upper bound for { ˜wsns : n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' s ∈ Sn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' By hypothesis mn/(�  s∈Sn ˜wsns) =  OP (1), so (50)=oP (1) will follow from (�  s∈Sn ˜wsns) P(u ˜wV > �  s∈Sn ˜wsns) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  As we also hypothesize that mn → ∞ as n ↑ ∞, we also have �  s∈Sn ˜wsns → ∞ as  n increases, and convergence to 0 of (50) follows if x P(u ˜wV > x) → 0 as x ↑ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  This is true by dominated convergence, since x P(u ˜wV > x) ≤ E(u ˜wV �u ˜wV > x�),  u ˜wV �u ˜wV > x� → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' as x ↑ ∞, and E (u ˜wV ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Premise (ii) of Durrett’s (2019) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='11 is that  (  �  s∈Sn  ˜wsns)−2 �  n2  s ˜wsn2  s E  �  ( ¯V (s)  n )2�  → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (51)  To verify this, observe first that  E  �  (ws ¯V (s)  n )2�  =2  � ∞  0  y P(|ws ¯V (s)  n | > y)dy = 2  � mn  0  y P(|ws ¯V (s)  n | > y)dy  ≤ 2  � mn  0  y P(|wsV (n,s)| > y)dy  (52)  ≤ 2  � mn  0  y P(|u ˜wV | > y)dy,  (53)  with (53) following from (52) by the stochastic dominance assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In conse-  quence,  (  �  s∈Sn  ˜wsns)−2 �  n2  s ˜w2  s E  �  ( ¯V (s)  n )2�  ≤ (  �  s∈Sn  ˜wsns)−22mn  � mn  0  y P(|u ˜wV | > y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  As we assume mn/(�  s∈Sn ˜wsns) = OP (1), for (51) it suffices to show m−1  n  � mn  0  y P(|u ˜wV | >  y) → 0 as n ↑ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' By the hypothesis that mn → ∞, this flows from x−1 � x  0 y P(|u ˜wV | >  y) → 0, which is a consequence of E |u ˜wV | < ∞, as shown by Durrett (2019) in  the proofs of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='12 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' This completes the verification that (47)  converges in probability to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  L1 convergence of ψSn(η)−E[ψSn(η)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  By hypothesis there is p > 1 such  that ∥V (n,s)∥Lp ≤ ∥V ∥Lp for all n and s ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Accordingly ∥(�  s∈Sn ˜wn,s)−1�  s∈Sn ˜wn,sV (n,s)∥Lp ≤  ∥V ∥Lp also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' By the dominated convergence principle for random variables as in  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='8 and Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='5 of Durrett (2019), therefore, (47) converges to 0  in L1 as well as in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  43  L1- and in-probability convergence of ψSn(η) − E[ψSn(η) | Gn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Ob-  serve that  ∥ E[ψSn(η)|Gn] − E[ψSn(η)]∥L1 =∥ E{ψSn(η) − E[ψSn(η)] | Gn}∥L1  ≤ ∥ψSn(η) − E[ψSn(η)]∥L1,  Fn being the smaller of the sigma fields {Fn, Gn}, E[ψSn(η)] being the same as  E[ψSn(η) | Fn] and the conditional expectation operator being a contraction in L1  (Durrett, 2019, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' In tandem with  ∥ψSn(η)−E[ψSn(η)|Gn]∥L1 ≤ ∥ψSn(η)−E[ψSn(η)]∥L1+∥ E[ψSn(η)|Gn]−E[ψSn(η)]∥L1  this means L1 convergence of ψSn(η)−E[ψSn(η)|Fn] entails that ψSn(η)−E[ψSn(η)|Gn]  also converges to zero in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Finally, L1 convergence entails convergence in prob-  ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Part ii of proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Provided that �  s ˜wsns is positive, both η �→ ψSn(η)  and η �→ E[ψSn(η) | Gn] are everywhere differentiable with slope −1, and can be  seen to tend to ±∞ as η tends to ∓∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' It follows that they have unique roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Part iii of proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  If the solutions τn of E[ψSn(η) | Gn] = 0 tend to a  limit τ0 ∈ (−∞, ∞), then the following adaptation of the Huber argument for  consistency of scalar solutions of monotone estimating equations (Huber, 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  van der Vaart, 1998, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='10) shows that ˆτn → τ0 in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Fix ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Then  P[ψSn(τ0 − ϵ) > ϵ/2, ψSn(τ0 + ϵ) < −ϵ/2] ≤ P(τ0 − ϵ < ˆτn < τ0 + ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  The left side tends to 1 because ψSn(τ0 ± ϵ) − E[ψSn(τ0 ± ϵ) | Gn] = oP (1), P(|τn −  τ0| < ϵ/2) → 1, E[ψSn(η) | Gn] > ϵ/2 if η < τn − ϵ/2 and E[ψSn(η) | Gn] < −ϵ/2 if  η > τn + ϵ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Therefore the right hand side tends to 1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Let θi = logit[P(Z = 1|Xβn = xiβn)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' If |θi − θj| < δ whenever i Sn  ∼ j,  then for all s ∈ Sn and z : s → {0, 1} such that �  i∈s ζi = �  i∈s zi ∈ {1, ns − 1},  ����  πs(ζ)  n−1  s  − 1  ���� ≤ (1 − n−1  s )(e2δ − 1)  (54)  and  ����  n−1  s  πs(ζ) − 1  ���� ≤ (1 − n−1  s )(e4δ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (55)  44  Proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For s an 1 : m matched set, some nonnegative integer m, by  (27) we have  πs(ζ)  n−1  s  − 1 = 1  n−1  s exp θi  �  j∈[i] exp(θj) − 1  =  ns − �  j∈[i] exp(θj − θi)  �  j∈[i] exp(θj − θi)  (56)  =  m − �  j∈[i]\\{i} exp(θj − θi)  �  j∈[i] exp(θj − θi)  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  so  m − m exp(δ)  (m + 1) exp(−δ) ≤ πs(ζ)  n−1  s  − 1 ≤ m − m exp(−δ)  (m + 1) exp(−δ),  (57)  −  m  m + 1eδ(eδ − 1) ≤ πs(ζ)  n−1  s  − 1 ≤  m  m + 1(eδ − 1)  and  ����  πs(ζ)  n−1  s  − 1  ���� ≤ ns − 1  ns  eδ(eδ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (58)  Now observe that e2δ − eδ < e2δ − 1 for positive δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (54) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  If s is an m : 1 matched set, m ≥ 0, the argument culminating in (58) again  applies after substitution of −θi and −θj for θi and θj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Again (54) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  With (54) established in all cases, (55) follows by (54)’s consequence that  ����  πs(ζ)  n−1  s  ���� ≤ e2δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  the identity |x−1−1| ≤ |x−1|·|x−1|, valid for x ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and e2δ(e2δ −1) ≤ e4δ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Claim (i) follows from  E[ ˜ψs(η) | Gn] = ˜ws  �  i∈s  [E(Y | Z = 1, ⃗Xβn = ⃗xiβn) − E(Y | Z = 0, ⃗Xβn = ⃗xiβn) − η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (59)  To show this, first observe that (46) and (28) combine to give  ˜ψs(η) =  ˜ws  πs(Zs)( ¯Ys(1) − ¯Ys(0) − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (60)  As Sn is assumed to be a fine stratification, one or both of ¯Ys(1) = avg[(Yi : i ∈ s, Zi = 1)]  and ¯Ys(0) = avg[(Yi : i ∈ s, Zi = 0)] is in actuality a single observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (60) is taken  to be zero, as is ψs(η) = ˜ws  �  i∈s ψsi(η) = ˜ws( ¯Ys(1) − ¯Ys(0) − η), when ˜ws = 0 be-  cause ¯Zs = 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  45  For s with �  i∈s Zi = 1, by (27) the expectation of (60) evaluates to E[ ˜ψs(η) | Gn] =  �  i∈s  {E(Y | Z = 1, ⃗Xβn = ⃗xiβn) − avg  j∈s\\{i}  [E(Y | Z = 0, ⃗Xβn = ⃗xjβn)] − η} ˜ws  πs(0(s)  +i)  πs(0(s)  +i)  = ˜ws  �  i∈s  {E(Y | Z = 1, ⃗Xβn = ⃗xiβn) − avg  j∈s\\{i}  [E(Y | Z = 0, ⃗Xβn = ⃗xjβn)] − η},  (61)  where we use 0(s)  +i to denote the mapping on s taking i to 1 and remaining elements  to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' (59) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For s with �  i∈s�zi = 0� = 1, this argument gives E[ ˜ψs(η) | Gn] =  ˜ws  �  i∈s  { avg  j∈s\\{i}  [E(Y | Z = 1, ⃗Xβn = ⃗xjβn)] − E(Y | Z = 0, ⃗Xβn = ⃗xiβn) − η} (62)  where 1(s)  −i denotes the mapping of s that takes i to 0 and remaining elements to  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Again (59) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Part (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Given s ∈ Sn, write µ(s)  0  = E( ¯Ys(0) | Fn) and µ(s)  1  = E( ¯Ys(1) | Fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  By symmetry, µ(s)  z  = E(Y | Z1 = z, �ns  i=1 Zi = �  i∈s zi), z = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' For s ∈ Sn with  �  i∈s�zi = 1� = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' one has ˜w−1  s  E[ψs(η) | Gn] =  �  i∈s  {[E(Y | Z = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xβn = ⃗xiβn) − avg  j∈s\\{i}  [E(Y | Z = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xβn = ⃗xjβn)]] − η}πs(0(s)  +i)  =  �  i∈s  {[E(Y − µ(s)  1  | Z = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xβn = ⃗xiβn) − avg  j∈s\\{i}  [E(Y − µ(s)  0  | Z = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xβn = ⃗xjβn)]]}πs(0(s)  +i)  + µ(s)  1  − µ(s)  0  − η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  whereas if �  i∈s�zi = 0� = 1 then ˜w−1  s  E[ψs(η) | Gn] =  �  i∈s  {[ avg  j∈s\\{i}  [E(Y | Z = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xβn = ⃗xjβn)] − E(Y | Z = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xβn = ⃗xiβn)] − η}πs(1(s)  −i)  =  �  i∈s  { avg  j∈s\\{i}  [E(Y − µ(s)  1  | Z = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xβn = ⃗xjβn)] − E(Y − µ(s)  0  | Z = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xβn = ⃗xiβn)}πs(1(s)  −i)  + µ(s)  1  − µ(s)  0  − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  46  At the same time, (61) and (62) give  ˜w−1  s  E[ψs(η) | Gn] = µ(s)  1  − µ(s)  0  − η+  �  i∈s  {[E(Y − µ(s)  1  | Z = 1, ⃗Xβn = ⃗xiβn) − avg  j∈s\\{i}  [E(Y − µ(s)  0  | Z = 0, ⃗Xβn = ⃗xjβn)]]}n−1  s  or  ˜w−1  s  E[ψs(η) | Gn] = µ(s)  1  − µ(s)  0  − η+  �  i∈s  { avg  j∈s\\{i}  [E(Y − µ(s)  1  | Z = 1, ⃗Xβn = ⃗xjβn)] − E(Y − µ(s)  0  | Z = 0, ⃗Xβn = ⃗xiβn)}n−1  s ,  depending as �  i∈s�zi = 1� = 1 or �  i∈s�zi = 0� = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Differencing  these expressions,  ˜w−1  s  E[ ˜ψs(η) − ψs(η) | Gn] =  �  i∈s  �  n−1  s  πs(0s  +i) − 1  �  {[E(Y − µ(s)  1  | Z = 1, ⃗Xβn = ⃗xiβn) − avg  j∈s\\{i}  [E(Y − µ(s)  0  | Z = 0, ⃗Xβn = ⃗xjβn)]]}πs(0s  +i)  (63)  if �  i∈s�zi = 1� = 1, and if �  i∈s�zi = 0� = 1 then  ˜w−1  s  E[ ˜ψs(η) − ψs(η) | Gn] =  �  i∈s  �  n−1  s  πs(1s  −i) − 1  �  { avg  j∈s\\{i}  [E(Y − µ(s)  1  | Z = 1, ⃗Xβn = ⃗xjβn)] − E(Y − µ(s)  0  | Z = 0, ⃗Xβn = ⃗xiβn)}πs(1(s)  −i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (64)  Observing that ns E(|V (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='s)| | Gn) ≥ | E(nsV (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='s) | Gn)| =  �����  �  i∈s  {[E(Y − µ(s)  1  | Z = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xβn = ⃗xiβn) − avg  j∈s\\{i}  [E(Y − µ(s)  0  | Z = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xβn = ⃗xjβn)]]}πs(0s  +i)  �����  or  �����  �  i∈s  { avg  j∈s\\{i}  [E(Y − µ(s)  1  | Z = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xβn = ⃗xjβn)] − E(Y − µ(s)  0  | Z = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' ⃗Xβn = ⃗xiβn)}πs(1(s)  −i)  �����  depending as �  i∈s�zi = 1� = 1 or �  i∈s�zi = 0� = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' and that under the same  respective conditions  ����  n−1  s  πs(0s  +i) − 1  ���� ≤ exp(2 sup  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='j∈s  |θi − θj|)−1 or  ����  n−1  s  πs(1s  −i) − 1  ���� ≤ exp(2 sup  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='j∈s  |θi − θj|)−1  47  by Lemma 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' we have:  | E[ ˜ψs(η) − ψs(η) | Gn]| ≤ ˜wsns[exp(2 sup  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='j∈s  |θi − θj|) − 1] E(|V (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='s)| | Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (65)  This establishes (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Part (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Because the conditional expectation operator is a contraction in  Lp,  E  ��  s∈Sn ˜wsns E(|V (n,s)| | Gn)  �  s∈Sn ˜wsns  �  ≤  �  s∈Sn ˜wsns E |V (n,s)|  �  s∈Sn ˜wsns  ≤  �  s∈Sn ˜wsns E |V |  �  s∈Sn ˜wsns  = E |V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Markov’s inequality now gives that �  s∈Sn ˜wsns  −1�  s∈Sn ˜wsns E(|V (n,s)| | Gn) =  OP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' Accordingly, |{θi − θj : i ∼ j}|∞ = oP (1) combines with (30) to entail  sup  η | E[ ˜ψSn(η) − ψSn(η) | Gn]| = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  (66)  By Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' 9, η �→ E(ψSn(η) | Gn) has the unique root τn, and by part (i) of this  proposition, η �→ E( ˜ψSn(η) | Gn) has a unique root given by (29);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content=' as either of these  functions’ slopes are bounded away from zero, (66) entails that these roots must  converge together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Part (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  Part (iv) of the proposition now follows from conclusion iii of  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
+page_content='  48' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE2T4oBgHgl3EQfxQh6/content/2301.04109v1.pdf'}
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+Sensitivity of CP Violation of Λ decay in J/ψ → Λ¯Λ at STCF
+Yue Xu∗1, Xiaorong Zhou∗2,3, Xiaodong Shi2,3, Yongxin Guo1, Kuiyong Liu1, Li Gong1,
+and Xiaoshen Kang†1
+1Department of Physics, Liaoning University, Shenyang 110036, People’s Republic of China
+2Department of Modern Physics, University of Science and Technology of China, Hefei
+230026, People’s Republic of China
+3State Key Laboratory of Particle Detection and Electronics, Hefei 230026, People’s
+Republic of China
+Abstract
+The process of J/ψ → Λ¯Λ is studied using 1.0 × 1012
+J/ψ Monte Carlo (MC) events at √s=3.097 GeV
+with a fast simulation software at future Super Tau
+Charm Facility (STCF). The statistical sensitivity
+for CP violation is determined to be the order of
+O (10−4) by measuring the asymmetric parameters
+of the Λ decay. Furthermore, the decay of J/ψ → Λ¯Λ
+also serves as a benchmark process to optimize the de-
+tector responses using the interface provided by the
+fast simulation software.
+1
+Introduction
+The electromagnetic force, weak nuclear force, and
+strong nuclear force are addressed with the Standard
+Model (SM), which is established as a well-tested
+physics theory. Although SM is so successful, there
+are still some unresolved issues including the source
+of CP violation [1]. In SM, CP violation can be in-
+cluded by introducing a complex phase in the quark
+mixing matrix, which is named Cabibbo-Kobayashi-
+Maskawa (CKM) matrix. Experimentally, starting in
+1964, people subsequently observed CP violation in
+the weak decay process of the K, B, and D meson
+∗These authors contributed equally to the work.
+†Corresponding author.
+systems [2, 3, 4, 5, 6, 7]. The CKM quark mixing
+matrix can give a wonderful explanation of the ob-
+served CP violation in the meson systems. However,
+the magnitude of CP violation predicted by the SM
+cannot explain the matter-antimatter asymmetry in
+the universe [8, 9, 10]. Moreover, many extensions
+of the SM imply that the CKM matrix may not be
+the only source of CP violation [11, 12].
+So more
+experimental studies are required to further test the
+CP violation mechanism in SM and search for other
+sources of CP violation.
+In 1956, Lee and Yang first proposed the vio-
+lation of parity (P) conservation in the weak de-
+cays of baryons [13].
+The degree of violation can
+be expressed in terms of the asymmetry parameters,
+α = 2Re (s∗p)/ (|s|2 +|p|2), where s and p stand for
+the parity-violating s-wave and parity-conserving p-
+wave amplitudes in the weak decay. In 1986, theoret-
+ical physicist Pakvasa proposed that the observable
+quantity of CP violation could be constructed us-
+ing asymmetric parameters in the decay of baryons,
+and predicted that the CP violation of baryons in
+the SM is O (10−5) [14, 15]. The processes of pionic
+decays of hyperons provide a good place to explore
+CP violation as they have a large branch ratio close
+to 1 [16, 17]. The CP asymmetry can be described
+as ACP = α+¯α
+α−¯α, and the asymmetric parameters are
+CP-odd for the charge conjugate decay of B/ ¯B (B
+is a spin-1/2 baryon). Therefore, if CP is conserved,
+1
+arXiv:2301.04337v1  [hep-ph]  11 Jan 2023
+
+α = −¯α, ACP is equal to 0 [16, 17].
+The
+Fermilab
+has
+specially
+designed
+Hy-
+perCP (E871) experiment to study CP violation of
+baryons in charged-Ξ and Λ hyperon decays. They
+have analyzed 11.7×107 Ξ− → Λπ− → pπ−π−
+and 4.1×107 Ξ+
+→
+Λπ+
+→
+pπ+π+ events to
+determine
+the
+products
+αΞαΛ
+and
+¯αΞ ¯αΛ
+[18].
+The
+sum
+AΛ
+CP +AΞ
+CP
+was
+estimated
+to
+be (0.0 ± 5.1 ± 4.4) × 10−4 [18]. In 2019, by studying
+the quantum entanglement of baryon pairs in the
+J/ψ → Λ¯Λ process and using a multi-dimensional
+fitting method, the BESIII experiment obtained an
+independent measurement of AΛ
+CP with matching
+precision:
+AΛ
+CP = −0.006 ± 0.012 ± 0.007, under
+the statistics of 0.4×106 J/ψ → Λ¯Λ → pπ−¯pπ+
+events [19].
+Recently, the asymmetries from the
+direct and subsequent J/ψ → Ξ−¯Ξ+ decays were
+measured for the first time at BESIII and found
+to be AΞ
+CP
+=
+−0.0029 ± 0.0133 ± 0.0057 and
+∆φΞ = −0.0075 ± 0.0137 ± 0.0037 rad [20]. Despite
+these, the CP violation measurement accuracy of the
+current experiment still does not meet the prediction
+of the SM and is mainly dominated by statistics
+uncertainty [14, 15].
+To test for the existence of new sources of CP
+violation other than SM, a hyperon sample with
+larger statistics is required. The STCF is a futural
+high-luminosity collider and also one of major op-
+tions for the accelerator-based high-energy project
+in China in the post-BEPCII era.
+The center-of-
+mass energy (√s) of the STCF collision will cover
+2 ∼ 7 GeV, which has been doubled compared to
+BEPCII. The peaking luminosity is expected to be
+over 0.5 × 1035 cm−2 s−1 or higher at √s = 4 GeV.
+It is expected to provide more than 1.0 × 1012 J/ψ
+events per year and has great potential for improv-
+ing luminosity and realizing beam polarization. So
+STCF will be an ideal place to study CP violation of
+Λ decay.
+In this analysis, we performed the sensitivity study
+of decay asymmetries of Λ decay and the decay chan-
+nel is e+e− → J/ψ → Λ (→ pπ−)¯Λ (→ ¯nπ0) with
+the statistics of 1.0×1012J/ψ MC events. The ampli-
+tude of the signal process follows the helicity ampli-
+tude method which is described explicitly, as shown
+in Eq. 6. Furthermore, the final states of Λ → pπ−
+decay have one low-momentum π− particle, which
+plays a key role in limiting the overall reconstruction
+efficiency. Therefore, it is essential to improve the re-
+construction efficiency of the low-momentum π− to
+get better sensitivity, so the decay of J/ψ → Λ¯Λ is
+also used as a benchmark process in this analysis to
+perform optimization of detector performance design.
+2
+Formalism
+The production process e+e− → J/ψ → Λ¯Λ is de-
+scribed in the c.m. system of J/ψ. The scattering
+angle θ of Λ is defined by
+cos θ = ˆp · ˆk,
+(1)
+where p and k are the three momenta of outgoing
+Λ and initial positron, respectively. The scattering
+plane with the vector p and k is used to form the xz-
+plane, and the corresponding y-axis is perpendicular
+to the scattering plane. The right-handed coordinate
+system is defined as follows:
+ex =
+1
+sin θ (ˆk × ˆp) × ˆp,
+ey =
+1
+sin θ (ˆk × ˆp),
+ez = ˆp.
+(2)
+The spin density matrix for a two spin 1/2 particle
+state can be expressed in terms of a set of 4 × 4
+matrices obtained from the outer product, ⊗, of σµ
+and σ¯ν [21]:
+ρ = 1
+4
+�
+µ¯ν
+Cµ¯νσΛ
+µ ⊗ σ
+¯Λ
+¯ν ,
+(3)
+where σµ,¯ν with µ, ¯ν= 0, 1, 2, 3, represent spin-1/2
+base matrices for baryon Λ/¯Λ in the rest frame. The
+2 × 2 matrices are σ0 = 12, σ1 = σx, σ2 = σy, and
+σ3 = σz. In particular, the spin matrices σµ and σ¯ν
+are given in the helicity frames of the baryons Λ and
+¯Λ, respectively. We define the coordinate system for
+Λ¯Λ decay, as shown in Fig. 1. The real coefficients
+Cµ¯ν for e+e− → J/ψ → Λ¯Λ with non-polarized inject
+2
+
+beams are given by Eq. 4,
+Cµ¯ν =
+�
+�
+�
+�
+1 + α cos2 θ
+0
+β sin θ cos θ
+0
+0
+sin2 θ
+0
+γ sin θ cos θ
+−β sin θ cos θ
+0
+α sin2 θ
+0
+0
+−γ sin θ cos θ
+0
+−α − cos2 θ
+�
+�
+�
+�
+(4)
+,
+where β
+=
+�
+(1 − α2) sin
+(∆Φ) and γ
+=
+�
+(1 − α2) cos (∆Φ), are functions of the scatter-
+ing angle θ of Λ. In the real coefficients Cµ¯ν of Eq. 4,
+there are two parameters related to the production
+process of e+e− → J/ψ → Λ¯Λ, the ratio of two he-
+licity amplitudes α, and the relative phase of the two
+helicity amplitudes ∆Φ.
+After considering the subsequent two-body weak
+decays into pπ−/¯nπ0, the joint angular distribution
+of the p/¯n pair is given within the present formalism
+as [21]:
+Trρp¯n ∝
+3
+�
+µ,¯ν=0
+Cµ¯ν (θ)aΛ
+µ0a
+¯Λ
+¯ν0,
+(5)
+where the aΛ
+µ0 (θ1, φ1; α1) and a¯Λ
+¯ν0 (θ2, φ2; α2) repre-
+sent the correlation of the spin density matrices in
+the sequential decays and the full expressions can be
+found in Ref. [21]. α1/α2 are the decay asymmetries
+for Λ → pπ−/¯Λ → ¯nπ0.
+The variables θ1 and φ1
+are the proton spherical coordinates in the Λ helicity
+frame with the axes x1, y1, z1 defined in Fig. 1. The
+variables θ2 and φ2 are the anti-neutron spherical an-
+gles in the ¯Λ helicity frame with the axes x2, y2, z2.
+!
+"
+#"
+$%
+$&
+'()
+*+)
+*,)
+'(-
+*+-
+*,-
+'(
+Figure 1: The reaction system with the defined he-
+licity angles in Λ¯Λ decay.
+An event of the reaction e+e− → J/ψ → Λ (→
+pπ−)¯Λ (→ ¯nπ0) is specified by the five-dimensional
+vector ξ = (θ, Ω1 (θ1, φ1), Ω2 (θ2, φ2)), and the joint
+angular distribution W (ξ) can be expressed as:
+W (ξ) = F0 (ξ) + αF5 (ξ)
++ α1α2 (F1 (ξ) +
+�
+1 − α2 cos (∆Φ)F2 (ξ) + αF6 (ξ))
++
+�
+1 − α2 sin (∆Φ) (−α1F3 (ξ) + α2F4 (ξ))
+(6)
+with a set of angular functions Fi (ξ) defined as:
+F0 (ξ) = 1
+F1 (ξ) = sin2 θ sin θ1 sin θ2 cos φ1 cos φ2 − cos2 θ cos θ1 cos θ2
+F2 (ξ) = sin θ cos θ (sin θ1 cos θ2 cos φ1 − cos θ1 sin θ2 cos φ2)
+F3 (ξ) = sin θ cos θ sin θ1 sin φ1
+F4 (ξ) = sin θ cos θ sin θ2 sin φ2
+F5 (ξ) = cos2 θ
+F6 (ξ) = sin2 θ sin θ1 sin θ2 sin φ1 sin φ2 − cos θ1 cos θ2.
+(7)
+There are four terms in Eq. 6: the first two (F0 +
+αF5) describe the production angular distribution,
+and the third and fourth terms give the spin correla-
+tion and polarization, respectively. The polarization
+is in the ey direction and is related to the phase ∆Φ
+via [22]
+Py = −
+√
+1 − α2 sin θ cos θ
+1 + α cos2 θ
+sin (∆Φ).
+(8)
+The polarization can only occur when ∆Φ is not equal
+to 0. As a consequence, the decay asymmetries can be
+determined with nonzero ∆Φ. Using this conclusion,
+the BESIII experiment used the angular distribution
+analysis method to observe the nonzero relative phase
+∆Φ of Λ in the baryon system for the first time, and
+then measured the decay asymmetry of Λ decay [19].
+3
+Detector and MC simulations
+The design structure of the STCF detector from the
+interaction point to the outside mainly includes a
+tracking system, a particle identification (PID) sys-
+tem, an electromagnetic calorimeter (EMC), a super-
+conducting solenoid and a muon detector (MUD).
+3
+
+The detailed conceptual design of each sub-detector
+can be found in [23, 24].
+The STCF detector and offline software system are
+under research and development at present. In order
+to study the physical potential of STCF and further
+optimize the detector design, a fast simulation soft-
+ware package dedicated to STCF detectors has been
+developed [23, 24] and it has proven to be a useful
+tool for analysis in STCF. The fast simulation is sim-
+ple to use and can simulate the response of objects
+in each sub-detector without Geant4, including vari-
+ables such as efficiency, and resolution (space, mo-
+mentum, energy, time, etc.). By default, all the pa-
+rameterized parameters for each sub-detector perfor-
+mance are based on the BESIII performance [25], but
+can be adjusted flexibly by scaling a factor according
+to the expected performance of the STCF detector,
+or by implementing a special interface to model any
+performance described with an external histogram,
+an input curve, or a series of discrete data [23]. In
+this analysis, the default scale factor is set to 1.0,
+which can be used to optimize the detector design
+according to physical requirements.
+4
+Analysis of J/ψ → Λ¯Λ with
+fast simulation
+The J/ψ → Λ¯Λ reaction is identified with the Λ sub-
+sequently decaying into pπ− and ¯Λ decay into ¯nπ0 re-
+sulting in a final state of pπ−¯nγγ. So, the candidate
+events are required to have at least two oppositely
+charged tracks and at least three showers.
+The combination of positive and negative charged
+tracks closest to the PDG mass of Λ was chosen as
+the Λ candidate [26]. In addition, the two daughter
+tracks are constrained to originate from a common
+decay vertex.
+The most energetic shower with en-
+ergy deposition greater than 350 MeV is selected as
+¯n. The two showers except the ¯n candidate are con-
+sistent with photons and are used to reconstruct the
+π0 candidates. At least, one good π0 is required. In
+order to select the J/ψ → Λ (pπ−)¯Λ (¯nπ0) candidate
+events, a two-constrained (2C) kinematic fit was per-
+formed, where ¯n is treated as a missed particle with
+mass fixed to 0.938 GeV [26], and the constraints in-
+cluding the four-momentum conservation of J/ψ and
+an additional constraint of photon pair to have an in-
+variant mass equal to π0. Furthermore, θ¯n is required
+to be less than 5◦, where θ¯n is defined as the angle
+between the ¯n direction obtained from kinematic fit
+and the most energetic shower. To further suppress
+the background, Λ and ¯Λ candidates are required to
+be within 1.110 GeV/c2 < Mpπ− < 1.120 GeV/c2
+and 1.098 GeV/c2 < M¯nπ0 < 1.127 GeV/c2.
+The 1.0 × 106 events of the J/ψ → Λ¯Λ → pπ−¯nπ0
+process were generated to optimize the selection cri-
+teria and evaluate the selection efficiencies for the
+baryon pair production. Based on the above selection
+conditions, with the help of fast simulation software,
+129575 candidate events of J/ψ → Λ¯Λ → pπ−¯nπ0
+were selected. The step-by-step selection efficiency is
+shown in Table 1.
+Furthermore, these MC samples also are used to
+optimize the detector response and 1.0 × 1012 events
+of signal process were generated to test the sensi-
+tivity of CP violation.
+To analyze the potential
+background process, 1.0 × 106 events of J/ψ →
+anything were generated as the inclusive MC. Af-
+ter the above event selection criteria were applied
+on the inclusive MC and by topology analysis, the
+J/ψ → Λ¯Σ0 → pπ−¯nπ0γ process has be shown to be
+the dominant background. So, 1.0 × 1012 events of
+the J/ψ → Λ¯Σ0 → pπ−¯nπ0γ process were generated
+to do the background test in the next chapter. Fur-
+thermore, 0.7×109 events of the signal process were
+generated using the phase space (PHSP) generator to
+estimate the normalization coefficient in Maximum
+Likelihood (MLL) fit.
+5
+Optimization of detector per-
+formance
+After the above event selection, the final selection
+efficiency is about 12.96%. The performance of the
+detector can be optimized from the following aspects:
+the selection efficiency of the charged tracks, the mo-
+mentum resolution of the charged tracks, and the po-
+sition resolution of the photons. Utilizing the signal
+4
+
+MC sample and with the help of fast simulation soft-
+ware tools, the optimized results of the detector re-
+sponse are as follows:
+a.Tracking efficiency
+The charged particles in the final state that can be
+identified by the detector include electrons, muons,
+pions, kaons, and protons.
+These charged parti-
+cles have a wide range of momentum, some can be
+as high as 3.5 GeV/c, and some can be less than
+1 GeV/c. This situation requires the detector to have
+the ability to cover a large momentum range and
+high-reconstruction efficiency.
+In the part of track
+system design of STCF, different materials or ad-
+vanced tracking algorithms can be used to further
+improve the ability of low-momentum track recon-
+struction. The J/ψ → Λ¯Λ → pπ−¯nπ0 decay has low-
+momentum final state particle π−, which is a good
+choice for optimizing the detector response, improv-
+ing the resolution of low-momentum particles.
+In this analysis, we gradually adjusted the scale
+factor of tracking efficiency from 1.0 to 2.0. It can be
+seen from Fig. 2 that the final selection efficiency has
+increased significantly in the range from 1.0 to 1.1
+of the scale factor, and the selection efficiency will
+increase from 12.96% to 13.67%.
+Charged track efficiency scale
+1
+1.2
+1.4
+1.6
+1.8
+2
+ Efficiency(%)
+13
+13.2
+13.4
+13.6
+13.8
+14
+14.2
+Figure 2: Charged track efficiency scale versus the
+selection efficiency.
+b.Momentum resolution of the charged tracks
+The momentum resolution of the charged tracks
+can also be optimized by the fast simulation.
+σxy
+and σz are the spatial resolutions of tracks in the
+xy-plane and z-direction. By default, σxy = 130 µm
+and σz=2480 µm.
+Optimizing σxy from 52 µm to
+130 µm, and the corresponding σz is optimized from
+992 µm to 2480 µm. There is no significant change
+in efficiency, as shown in Fig. 3.
+m)
+µ
+ Momentum resolution of charged tracks(
+50
+60
+70
+80
+90
+100
+110
+120
+130
+ Efficiency(%)
+12.7
+12.8
+12.9
+13
+13.1
+13.2
+13.3
+Figure 3: Momentum resolution of charged tracks
+versus the selection efficiency.
+In addition, the transverse momentum PT and po-
+lar angle cos θ are two characteristic quantities of
+track reconstruction in MDC. They are related to the
+level of track bending and hit positions of tracks in
+the MDC. The optimization curve of the transverse
+momentum of low-momentum π− is shown in Fig. 4,
+where the black and red points represent the ratio of
+signal efficiency to MC truth before and after all the
+above optimization, respectively.
+c.Position resolution of photon
+The decay of J/ψ → Λ¯Λ → pπ−¯nπ0 has a final
+state particle π0, π0 is reconstructed by two photons,
+so this process is also very sensitive to the EMC per-
+formance. With the increase in the resolution of the
+π0, there will be a better signal-to-background ra-
+tio and higher detection efficiency. Optimizing the
+signal-to-background ratio can provide a reference for
+the EMC design. In this analysis, the signal process
+5
+
+) GeV
+-π
+(
+T
+P
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+Trk. efficiency (%)
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Before optimization
+After optimization
+Figure 4: The Optimization curve of the transverse
+momentum of π.
+J/ψ → Λ¯Λ → pπ−¯nπ0 and the main background
+process J/ψ → Λ¯Σ0 → pπ−¯nπ0γ were studied. By
+fitting the distribution of the invariant mass of M¯nπ0,
+a 3σ mass interval of M¯nπ0 is obtained to further re-
+duce the impact of the background process. Figure 5
+shows the signal selection efficiency and background
+rejection under the change of photon position reso-
+lution.
+The scale factor of the position resolution
+of photon varies from 0.4 to 1.0. The red and blue
+points represent the case of using the nominal ¯Λ mass
+window and the optimized ¯Λ mass window, respec-
+tively.
+Although this will lose some signal events,
+it can reduce more background and make the signal
+cleaner. It is appropriate to set the scale factor to
+0.7, which corresponds to the position resolution of
+4 mm. The signal selection efficiency will get increase
+from 12.96% to 15.11%, while the main background
+will be reduced from 3.27% to 3.17%.
+After all the optimization of detector responds, the
+events of signal MC will increase from 12.96% to
+15.97%, while the events of main background (J/ψ →
+Λ¯Σ0 → pπ−¯nπ0γ) will reduce from 3.27% to 3.09%.
+The selection efficiency is as shown in Table 1.
+Signal efficiency(%) 
+13
+14
+15
+16
+17
+18
+ Background rejection
+0.95
+0.955
+0.96
+0.965
+0.97
+0.975
+0.98
+ mass window
+Λ
+Nominal 
+ mass window
+Λ
+Optimized 
+Figure 5: The change of signal selection efficiency and
+background rejection with position resolution of the
+photon.
+6
+Extraction of the parameters
+In this analysis, the parameters can be extracted by
+applying an unbinned MLL fit. The probability den-
+sity function of the ith event can be expressed by
+P (ξi; pars) = W (ξi; pars)ϵ (ξi)/N (pars)
+(9)
+, where ϵ (ξi) is the efficiency of each event, ξi and
+pars are a set of angular vectors and parameters:
+ξi = (θ, Ω1, Ω2), pars= (α, α1, α2, ∆Φ), as described
+in Sec. 2.
+The joint probability density for observing N
+events in the data sample is [27]:
+P (ξ1, ξ2, ..., ξN; pars) =
+N
+�
+i=1
+P (ξi; pars)
+=
+N
+�
+i=1
+W (ξi; pars)ϵ (ξi)
+N (pars)
+.
+(10)
+By taking the natural logarithm of the joint probabil-
+ity density, the efficiency function can be separated
+lnP (ξ1..., ξN; pars) =
+N
+�
+i
+lnW (ξi; pars)
+N (pars) +
+N
+�
+i
+lnϵ (ξi).
+(11)
+6
+
+Table 1: Events selection efficiency.
+No optimized eff. (%)
+Optimized eff. (%)
+Increased efficiency after
+optimization in step (%)
+Charged tracks
+74.21
+79.38
+5.17
+Λ reconstruction
+66.27
+70.88
+4.61
+Good showers
+31.73
+33.57
+1.84
+π0 1C fit (Nγ ≥2)
+28.76
+29.93
+1.17
+Kinematic 2-C fit
+25.33
+27.31
+1.98
+Energy deposition of ¯n >0.35 GeV
+21.18
+22.88
+1.70
+θ¯n < 5◦
+14.54
+18.34
+3.80
+Λ and ¯Λ mass window
+12.96
+15.97
+3.01
+Usually, the minimization of -lnL is performed by
+using MINUIT [28]
+− lnL = −
+N
+�
+i
+lnW (ξi; pars)ϵ (ξi)
+N (pars)
+(12)
+, where N is the normalization factor, given by
+N =
+�
+W (ξ)ϵ (ξi)d cos θdΩ1dΩ2.
+(13)
+For a certain set of pars, N (pars) can be rewritten as
+the integration on each Fi term according to Eq. 6.
+To test the statistical sensitivity, the fitting was ap-
+plied on J/ψ samples with different statistics. The
+precision for the decay parameters is shown in Fig. 6.
+It is found that the precision of the parameters is pro-
+portional to the square root of the J/ψ sample. The
+correlation matrix among the parameters is shown in
+Table 2.
+Table 2: Correlation Matrix for the parameters, ob-
+tained with MINUIT.
+pars
+α
+α1
+α2
+∆Φ
+α
+1.000
+-0.089
+0.104
+0.339
+α1
+-0.089
+1.000
+0.853
+-0.120
+α2
+0.104
+0.853
+1.000
+0.058
+∆Φ
+0.339
+-0.120
+0.058
+1.000
+According to Eq. 6, the moment of sin θ1 sin φ1 is
+ samples(T)
+ψ
+J/
+0.1 0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Error of MLL fitting
+0.0005
+0.001
+0.0015
+0.002
+0.0025
+0.003
+α
+1
+α
+2
+α
+Φ
+∆
+Figure 6: The statistical sensitivity of J/ψ samples
+with different statistics.
+7
+
+given by
+⟨sin θ1 sin φ1⟩ =
+1
+Nnorm
+�
+W (ξ) sin θ1 sin φ1dΩ1dΩ2
+≈ −
+√
+1 − α2α1 sin (∆Φ)
+3 + α
+sin θ cos θ.
+(14)
+In the analysis of experimental data, ⟨sin θ1 sin φ1⟩
+can be calculated by the average of sin θ1 sin φ1 in
+each cos θ bin. The moment of sin θ1 sin φ1 can be
+connected with the polarization according to Eq. 8,
+⟨sin θ1 sin φ1⟩ ≈ (1 + α cos2 θ)α1
+3 + α
+Py.
+(15)
+c_signal
+θ
+cos
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+Polarization
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+c_signal
+Signal MC
+PHSP MC
+c_signal
+Figure 7:
+Polarization as a function of cos θ for
+J/ψ → Λ¯Λ → pπ−¯nπ0. The points with error bars
+are the signal MC, and the blue dashed histogram is
+the no-polarization scenario of PHSP MC.
+The distribution of polarization versus cos θ as
+shown in Fig. 7 and the events are not corrected with
+detection efficiency.
+7
+Prospect of CP sensitivity at
+STCF
+The asymmetry parameters used to observe CP vi-
+olation are affected by statistics and proportional to
+the �NJ/ψ, where NJ/ψ is the number of J/ψ events.
+By generating a 1.0 × 1012 MC sample, after event
+selection and detector optimization, the statistical ac-
+curacy of CP violation is 10−4. The STCF has great
+potential in improving luminosity and realizing beam
+polarization. It is expected that more than 1 ab−1
+experimental data and 3.4×1013 J/ψ events will be
+obtained per year, with the substantial increase in
+statistics, larger data samples will be generated on
+STCF, and in the future, it will hopefully reach a
+level of accuracy and theoretical prediction compati-
+bility.
+8
+Summary and prospect
+With the fast simulation software package, the MC
+samples of J/ψ → Λ¯Λ → pπ−¯nπ0 process were gen-
+erated.
+After the optimization of detector perfor-
+mance, the events selection efficiency of the signal
+process is increased by 23.22% compared to the unop-
+timized and the main background process is reduced
+by 5.5%. Furthermore, the 1.0 × 1012 J/ψ MC was
+used to pre-studied the sensitivity of CP violation
+of the J/ψ → Λ¯Λ process at the future STCF. The
+statistical accuracy of CP violation of Λ hyperon is
+10−4, which is close to the prediction of SM of CP
+violation in Λ hyperon decay [29].
+Acknowledgments
+The authors would like to thank the USTC Super-
+computing Center and the Hefei Comprehensive Na-
+tional Science Center for their strong support. This
+work is supported in part by National Natural Science
+Foundation of China (NSFC) under Contracts Nos.
+11872030, 11905092, 11972177, 12122509, 11625523,
+12105132, 11705078. The international partnership
+program of the Chinese Academy of Sciences un-
+der Grant No. 211134KYSB20200057 and by USTC
+Research Funds of the Double First-Class Initiative
+and the Fundamental Research Funds for the Cen-
+tral Universities.
+The Doctoral Scientific Research
+Foundation of Liaoning Province No. 2019-BS-113,
+the Foundation of Liaoning Educational Committee
+8
+
+No.
+LQN201902, the Natural Science Foundation
+of Liaoning Provincial Department of Education No.
+LCJ202003. China Postdoctoral Science Foundation
+under Contracts Nos. 2021M693181. The PhD Start-
+up Fund of Natural Science Foundation of Liaon-
+ing Province of China under Contracts No.
+2019-
+BS-113.
+Scientific research Foundation of Liaon-
+ing Provincial Department of Education under Con-
+tracts No.
+LQN201902.
+Foundation of Innovation
+team 2020, Liaoning Province.
+Opening Founda-
+tion of Songshan Lake Materials Laboratory, Grants
+No.2021SLABFK04.
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+
diff --git a/idE3T4oBgHgl3EQfIwmA/content/tmp_files/load_file.txt b/idE3T4oBgHgl3EQfIwmA/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..94bb19c70e105756d2f043d0add3d6f11d408d4a
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf,len=583
+page_content='Sensitivity of CP Violation of Λ decay in J/ψ → Λ¯Λ at STCF  Yue Xu∗1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Xiaorong Zhou∗2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Xiaodong Shi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Yongxin Guo1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Kuiyong Liu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Li Gong1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  and Xiaoshen Kang†1  1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Liaoning University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Shenyang 110036,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' People’s Republic of China  2Department of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Hefei  230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' People’s Republic of China  3State Key Laboratory of Particle Detection and Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Hefei 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' People’s  Republic of China  Abstract  The process of J/ψ → Λ¯Λ is studied using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0 × 1012  J/ψ Monte Carlo (MC) events at √s=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='097 GeV  with a fast simulation software at future Super Tau  Charm Facility (STCF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The statistical sensitivity  for CP violation is determined to be the order of  O (10−4) by measuring the asymmetric parameters  of the Λ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Furthermore, the decay of J/ψ → Λ¯Λ  also serves as a benchmark process to optimize the de-  tector responses using the interface provided by the  fast simulation software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  1  Introduction  The electromagnetic force, weak nuclear force, and  strong nuclear force are addressed with the Standard  Model (SM), which is established as a well-tested  physics theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Although SM is so successful, there  are still some unresolved issues including the source  of CP violation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' In SM, CP violation can be in-  cluded by introducing a complex phase in the quark  mixing matrix, which is named Cabibbo-Kobayashi-  Maskawa (CKM) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Experimentally, starting in  1964, people subsequently observed CP violation in  the weak decay process of the K, B, and D meson  ∗These authors contributed equally to the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  systems [2, 3, 4, 5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The CKM quark mixing  matrix can give a wonderful explanation of the ob-  served CP violation in the meson systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' However,  the magnitude of CP violation predicted by the SM  cannot explain the matter-antimatter asymmetry in  the universe [8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Moreover, many extensions  of the SM imply that the CKM matrix may not be  the only source of CP violation [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  So more  experimental studies are required to further test the  CP violation mechanism in SM and search for other  sources of CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  In 1956, Lee and Yang first proposed the vio-  lation of parity (P) conservation in the weak de-  cays of baryons [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The degree of violation can  be expressed in terms of the asymmetry parameters,  α = 2Re (s∗p)/ (|s|2 +|p|2), where s and p stand for  the parity-violating s-wave and parity-conserving p-  wave amplitudes in the weak decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' In 1986, theoret-  ical physicist Pakvasa proposed that the observable  quantity of CP violation could be constructed us-  ing asymmetric parameters in the decay of baryons,  and predicted that the CP violation of baryons in  the SM is O (10−5) [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The processes of pionic  decays of hyperons provide a good place to explore  CP violation as they have a large branch ratio close  to 1 [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The CP asymmetry can be described  as ACP = α+¯α  α−¯α, and the asymmetric parameters are  CP-odd for the charge conjugate decay of B/ ¯B (B  is a spin-1/2 baryon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Therefore, if CP is conserved,  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='04337v1  [hep-ph]  11 Jan 2023  α = −¯α, ACP is equal to 0 [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The  Fermilab  has  specially  designed  Hy-  perCP (E871) experiment to study CP violation of  baryons in charged-Ξ and Λ hyperon decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' They  have analyzed 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='7×107 Ξ− → Λπ− → pπ−π−  and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='1×107 Ξ+  →  Λπ+  →  pπ+π+ events to  determine  the  products  αΞαΛ  and  ¯αΞ ¯αΛ  [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The  sum  AΛ  CP +AΞ  CP  was  estimated  to  be (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='1 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='4) × 10−4 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' In 2019, by studying  the quantum entanglement of baryon pairs in the  J/ψ → Λ¯Λ process and using a multi-dimensional  fitting method, the BESIII experiment obtained an  independent measurement of AΛ  CP with matching  precision:  AΛ  CP = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='006 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='012 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='007, under  the statistics of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='4×106 J/ψ → Λ¯Λ → pπ−¯pπ+  events [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  Recently, the asymmetries from the  direct and subsequent J/ψ → Ξ−¯Ξ+ decays were  measured for the first time at BESIII and found  to be AΞ  CP  =  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0029 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0133 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0057 and  ∆φΞ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0075 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0137 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0037 rad [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Despite  these, the CP violation measurement accuracy of the  current experiment still does not meet the prediction  of the SM and is mainly dominated by statistics  uncertainty [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  To test for the existence of new sources of CP  violation other than SM, a hyperon sample with  larger statistics is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The STCF is a futural  high-luminosity collider and also one of major op-  tions for the accelerator-based high-energy project  in China in the post-BEPCII era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The center-of-  mass energy (√s) of the STCF collision will cover  2 ∼ 7 GeV, which has been doubled compared to  BEPCII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The peaking luminosity is expected to be  over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='5 × 1035 cm−2 s−1 or higher at √s = 4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  It is expected to provide more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0 × 1012 J/ψ  events per year and has great potential for improv-  ing luminosity and realizing beam polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' So  STCF will be an ideal place to study CP violation of  Λ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  In this analysis, we performed the sensitivity study  of decay asymmetries of Λ decay and the decay chan-  nel is e+e− → J/ψ → Λ (→ pπ−)¯Λ (→ ¯nπ0) with  the statistics of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0×1012J/ψ MC events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The ampli-  tude of the signal process follows the helicity ampli-  tude method which is described explicitly, as shown  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Furthermore, the final states of Λ → pπ−  decay have one low-momentum π− particle, which  plays a key role in limiting the overall reconstruction  efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Therefore, it is essential to improve the re-  construction efficiency of the low-momentum π− to  get better sensitivity, so the decay of J/ψ → Λ¯Λ is  also used as a benchmark process in this analysis to  perform optimization of detector performance design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  2  Formalism  The production process e+e− → J/ψ → Λ¯Λ is de-  scribed in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' system of J/ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The scattering  angle θ of Λ is defined by  cos θ = ˆp · ˆk,  (1)  where p and k are the three momenta of outgoing  Λ and initial positron, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The scattering  plane with the vector p and k is used to form the xz-  plane, and the corresponding y-axis is perpendicular  to the scattering plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The right-handed coordinate  system is defined as follows:  ex =  1  sin θ (ˆk × ˆp) × ˆp,  ey =  1  sin θ (ˆk × ˆp),  ez = ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  (2)  The spin density matrix for a two spin 1/2 particle  state can be expressed in terms of a set of 4 × 4  matrices obtained from the outer product, ⊗, of σµ  and σ¯ν [21]:  ρ = 1  4  �  µ¯ν  Cµ¯νσΛ  µ ⊗ σ  ¯Λ  ¯ν ,  (3)  where σµ,¯ν with µ, ¯ν= 0, 1, 2, 3, represent spin-1/2  base matrices for baryon Λ/¯Λ in the rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The  2 × 2 matrices are σ0 = 12, σ1 = σx, σ2 = σy, and  σ3 = σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' In particular, the spin matrices σµ and σ¯ν  are given in the helicity frames of the baryons Λ and  ¯Λ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' We define the coordinate system for  Λ¯Λ decay, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The real coefficients  Cµ¯ν for e+e− → J/ψ → Λ¯Λ with non-polarized inject  2  beams are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 4,  Cµ¯ν =  �  �  �  �  1 + α cos2 θ  0  β sin θ cos θ  0  0  sin2 θ  0  γ sin θ cos θ  −β sin θ cos θ  0  α sin2 θ  0  0  −γ sin θ cos θ  0  −α − cos2 θ  �  �  �  �  (4)  ,  where β  =  �  (1 − α2) sin  (∆Φ) and γ  =  �  (1 − α2) cos (∆Φ), are functions of the scatter-  ing angle θ of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' In the real coefficients Cµ¯ν of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 4,  there are two parameters related to the production  process of e+e− → J/ψ → Λ¯Λ, the ratio of two he-  licity amplitudes α, and the relative phase of the two  helicity amplitudes ∆Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  After considering the subsequent two-body weak  decays into pπ−/¯nπ0, the joint angular distribution  of the p/¯n pair is given within the present formalism  as [21]:  Trρp¯n ∝  3  �  µ,¯ν=0  Cµ¯ν (θ)aΛ  µ0a  ¯Λ  ¯ν0,  (5)  where the aΛ  µ0 (θ1, φ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' α1) and a¯Λ  ¯ν0 (θ2, φ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' α2) repre-  sent the correlation of the spin density matrices in  the sequential decays and the full expressions can be  found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' α1/α2 are the decay asymmetries  for Λ → pπ−/¯Λ → ¯nπ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The variables θ1 and φ1  are the proton spherical coordinates in the Λ helicity  frame with the axes x1, y1, z1 defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The  variables θ2 and φ2 are the anti-neutron spherical an-  gles in the ¯Λ helicity frame with the axes x2, y2, z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  "  #"  $%  $&  \'()  +)  ,)  \'(-  +-  ,-  \'(  Figure 1: The reaction system with the defined he-  licity angles in Λ¯Λ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  An event of the reaction e+e− → J/ψ → Λ (→  pπ−)¯Λ (→ ¯nπ0) is specified by the five-dimensional  vector ξ = (θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Ω1 (θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' φ1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Ω2 (θ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' φ2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' and the joint  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='angular distribution W (ξ) can be expressed as:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='W (ξ) = F0 (ξ) + αF5 (ξ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='+ α1α2 (F1 (ξ) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='1 − α2 cos (∆Φ)F2 (ξ) + αF6 (ξ))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='1 − α2 sin (∆Φ) (−α1F3 (ξ) + α2F4 (ξ))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='(6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='with a set of angular functions Fi (ξ) defined as:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='F0 (ξ) = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='F1 (ξ) = sin2 θ sin θ1 sin θ2 cos φ1 cos φ2 − cos2 θ cos θ1 cos θ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='F2 (ξ) = sin θ cos θ (sin θ1 cos θ2 cos φ1 − cos θ1 sin θ2 cos φ2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='F3 (ξ) = sin θ cos θ sin θ1 sin φ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='F4 (ξ) = sin θ cos θ sin θ2 sin φ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='F5 (ξ) = cos2 θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='F6 (ξ) = sin2 θ sin θ1 sin θ2 sin φ1 sin φ2 − cos θ1 cos θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  (7)  There are four terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 6: the first two (F0 +  αF5) describe the production angular distribution,  and the third and fourth terms give the spin correla-  tion and polarization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The polarization  is in the ey direction and is related to the phase ∆Φ  via [22]  Py = −  √  1 − α2 sin θ cos θ  1 + α cos2 θ  sin (∆Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  (8)  The polarization can only occur when ∆Φ is not equal  to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' As a consequence, the decay asymmetries can be  determined with nonzero ∆Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Using this conclusion,  the BESIII experiment used the angular distribution  analysis method to observe the nonzero relative phase  ∆Φ of Λ in the baryon system for the first time, and  then measured the decay asymmetry of Λ decay [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  3  Detector and MC simulations  The design structure of the STCF detector from the  interaction point to the outside mainly includes a  tracking system, a particle identification (PID) sys-  tem, an electromagnetic calorimeter (EMC), a super-  conducting solenoid and a muon detector (MUD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  3  The detailed conceptual design of each sub-detector  can be found in [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The STCF detector and offline software system are  under research and development at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' In order  to study the physical potential of STCF and further  optimize the detector design, a fast simulation soft-  ware package dedicated to STCF detectors has been  developed [23, 24] and it has proven to be a useful  tool for analysis in STCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The fast simulation is sim-  ple to use and can simulate the response of objects  in each sub-detector without Geant4, including vari-  ables such as efficiency, and resolution (space, mo-  mentum, energy, time, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' By default, all the pa-  rameterized parameters for each sub-detector perfor-  mance are based on the BESIII performance [25], but  can be adjusted flexibly by scaling a factor according  to the expected performance of the STCF detector,  or by implementing a special interface to model any  performance described with an external histogram,  an input curve, or a series of discrete data [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' In  this analysis, the default scale factor is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0,  which can be used to optimize the detector design  according to physical requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  4  Analysis of J/ψ → Λ¯Λ with  fast simulation  The J/ψ → Λ¯Λ reaction is identified with the Λ sub-  sequently decaying into pπ− and ¯Λ decay into ¯nπ0 re-  sulting in a final state of pπ−¯nγγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' So, the candidate  events are required to have at least two oppositely  charged tracks and at least three showers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The combination of positive and negative charged  tracks closest to the PDG mass of Λ was chosen as  the Λ candidate [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' In addition, the two daughter  tracks are constrained to originate from a common  decay vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The most energetic shower with en-  ergy deposition greater than 350 MeV is selected as  ¯n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The two showers except the ¯n candidate are con-  sistent with photons and are used to reconstruct the  π0 candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' At least, one good π0 is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' In  order to select the J/ψ → Λ (pπ−)¯Λ (¯nπ0) candidate  events, a two-constrained (2C) kinematic fit was per-  formed, where ¯n is treated as a missed particle with  mass fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='938 GeV [26], and the constraints in-  cluding the four-momentum conservation of J/ψ and  an additional constraint of photon pair to have an in-  variant mass equal to π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Furthermore, θ¯n is required  to be less than 5◦, where θ¯n is defined as the angle  between the ¯n direction obtained from kinematic fit  and the most energetic shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' To further suppress  the background, Λ and ¯Λ candidates are required to  be within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='110 GeV/c2 < Mpπ− < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='120 GeV/c2  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='098 GeV/c2 < M¯nπ0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='127 GeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0 × 106 events of the J/ψ → Λ¯Λ → pπ−¯nπ0  process were generated to optimize the selection cri-  teria and evaluate the selection efficiencies for the  baryon pair production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Based on the above selection  conditions, with the help of fast simulation software,  129575 candidate events of J/ψ → Λ¯Λ → pπ−¯nπ0  were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The step-by-step selection efficiency is  shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  Furthermore, these MC samples also are used to  optimize the detector response and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0 × 1012 events  of signal process were generated to test the sensi-  tivity of CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  To analyze the potential  background process, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0 × 106 events of J/ψ →  anything were generated as the inclusive MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Af-  ter the above event selection criteria were applied  on the inclusive MC and by topology analysis, the  J/ψ → Λ¯Σ0 → pπ−¯nπ0γ process has be shown to be  the dominant background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' So, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0 × 1012 events of  the J/ψ → Λ¯Σ0 → pπ−¯nπ0γ process were generated  to do the background test in the next chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Fur-  thermore, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='7×109 events of the signal process were  generated using the phase space (PHSP) generator to  estimate the normalization coefficient in Maximum  Likelihood (MLL) fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  5  Optimization of detector per-  formance  After the above event selection, the final selection  efficiency is about 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='96%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The performance of the  detector can be optimized from the following aspects:  the selection efficiency of the charged tracks, the mo-  mentum resolution of the charged tracks, and the po-  sition resolution of the photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Utilizing the signal  4  MC sample and with the help of fast simulation soft-  ware tools, the optimized results of the detector re-  sponse are as follows:  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='Tracking efficiency  The charged particles in the final state that can be  identified by the detector include electrons, muons,  pions, kaons, and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  These charged parti-  cles have a wide range of momentum, some can be  as high as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='5 GeV/c, and some can be less than  1 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' This situation requires the detector to have  the ability to cover a large momentum range and  high-reconstruction efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  In the part of track  system design of STCF, different materials or ad-  vanced tracking algorithms can be used to further  improve the ability of low-momentum track recon-  struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The J/ψ → Λ¯Λ → pπ−¯nπ0 decay has low-  momentum final state particle π−, which is a good  choice for optimizing the detector response, improv-  ing the resolution of low-momentum particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  In this analysis, we gradually adjusted the scale  factor of tracking efficiency from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' It can be  seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 2 that the final selection efficiency has  increased significantly in the range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='1  of the scale factor, and the selection efficiency will  increase from 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='96% to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='67%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  Charged track efficiency scale  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='8  2  Efficiency(%)  13  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='2  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='4  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='6  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='8  14  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='2  Figure 2: Charged track efficiency scale versus the  selection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='Momentum resolution of the charged tracks  The momentum resolution of the charged tracks  can also be optimized by the fast simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  σxy  and σz are the spatial resolutions of tracks in the  xy-plane and z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' By default, σxy = 130 µm  and σz=2480 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  Optimizing σxy from 52 µm to  130 µm, and the corresponding σz is optimized from  992 µm to 2480 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' There is no significant change  in efficiency, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  m)  µ  Momentum resolution of charged tracks(  50  60  70  80  90  100  110  120  130  Efficiency(%)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='7  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='8  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='9  13  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='1  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='2  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='3  Figure 3: Momentum resolution of charged tracks  versus the selection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  In addition, the transverse momentum PT and po-  lar angle cos θ are two characteristic quantities of  track reconstruction in MDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' They are related to the  level of track bending and hit positions of tracks in  the MDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The optimization curve of the transverse  momentum of low-momentum π− is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 4,  where the black and red points represent the ratio of  signal efficiency to MC truth before and after all the  above optimization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='Position resolution of photon  The decay of J/ψ → Λ¯Λ → pπ−¯nπ0 has a final  state particle π0, π0 is reconstructed by two photons,  so this process is also very sensitive to the EMC per-  formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' With the increase in the resolution of the  π0, there will be a better signal-to-background ra-  tio and higher detection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Optimizing the  signal-to-background ratio can provide a reference for  the EMC design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' In this analysis, the signal process  5  ) GeV  π  (  T  P  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='3  Trk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' efficiency (%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='9  1  Before optimization  After optimization  Figure 4: The Optimization curve of the transverse  momentum of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  J/ψ → Λ¯Λ → pπ−¯nπ0 and the main background  process J/ψ → Λ¯Σ0 → pπ−¯nπ0γ were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' By  fitting the distribution of the invariant mass of M¯nπ0,  a 3σ mass interval of M¯nπ0 is obtained to further re-  duce the impact of the background process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Figure 5  shows the signal selection efficiency and background  rejection under the change of photon position reso-  lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The scale factor of the position resolution  of photon varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='4 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The red and blue  points represent the case of using the nominal ¯Λ mass  window and the optimized ¯Λ mass window, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  Although this will lose some signal events,  it can reduce more background and make the signal  cleaner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' It is appropriate to set the scale factor to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='7, which corresponds to the position resolution of  4 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The signal selection efficiency will get increase  from 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='96% to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='11%, while the main background  will be reduced from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='27% to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='17%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  After all the optimization of detector responds, the  events of signal MC will increase from 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='96% to  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='97%, while the events of main background (J/ψ →  Λ¯Σ0 → pπ−¯nπ0γ) will reduce from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='27% to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='09%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The selection efficiency is as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  Signal efficiency(%)  13  14  15  16  17  18  Background rejection  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='955  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='965  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='975  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='98  mass window  Λ  Nominal  mass window  Λ  Optimized  Figure 5: The change of signal selection efficiency and  background rejection with position resolution of the  photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  6  Extraction of the parameters  In this analysis, the parameters can be extracted by  applying an unbinned MLL fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The probability den-  sity function of the ith event can be expressed by  P (ξi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' pars) = W (ξi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' pars)ϵ (ξi)/N (pars)  (9)  , where ϵ (ξi) is the efficiency of each event, ξi and  pars are a set of angular vectors and parameters:  ξi = (θ, Ω1, Ω2), pars= (α, α1, α2, ∆Φ), as described  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The joint probability density for observing N  events in the data sample is [27]:  P (ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=', ξN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' pars) =  N  �  i=1  P (ξi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' pars)  =  N  �  i=1  W (ξi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' pars)ϵ (ξi)  N (pars)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  (10)  By taking the natural logarithm of the joint probabil-  ity density, the efficiency function can be separated  lnP (ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=', ξN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' pars) =  N  �  i  lnW (ξi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' pars)  N (pars) +  N  �  i  lnϵ (ξi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  (11)  6  Table 1: Events selection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  No optimized eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' (%)  Optimized eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' (%)  Increased efficiency after  optimization in step (%)  Charged tracks  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='21  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='38  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='17  Λ reconstruction  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='27  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='88  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='61  Good showers  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='73  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='84  π0 1C fit (Nγ ≥2)  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='76  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='93  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='17  Kinematic 2-C fit  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='33  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='31  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='98  Energy deposition of ¯n >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='35 GeV  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='18  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='88  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='70  θ¯n < 5◦  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='54  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='34  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='80  Λ and ¯Λ mass window  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='96  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='97  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='01  Usually, the minimization of -lnL is performed by  using MINUIT [28]  − lnL = −  N  �  i  lnW (ξi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' pars)ϵ (ξi)  N (pars)  (12)  , where N is the normalization factor, given by  N =  �  W (ξ)ϵ (ξi)d cos θdΩ1dΩ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  (13)  For a certain set of pars, N (pars) can be rewritten as  the integration on each Fi term according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  To test the statistical sensitivity, the fitting was ap-  plied on J/ψ samples with different statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The  precision for the decay parameters is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  It is found that the precision of the parameters is pro-  portional to the square root of the J/ψ sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The  correlation matrix among the parameters is shown in  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  Table 2: Correlation Matrix for the parameters, ob-  tained with MINUIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  pars  α  α1  α2  ∆Φ  α  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='089  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='339  α1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='089  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='853  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='120  α2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content='058  ∆Φ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content='000  According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 6, the moment of sin θ1 sin φ1 is  samples(T)  ψ  J/  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content='9  1  Error of MLL fitting  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='003  α  1  α  2  α  Φ  ∆  Figure 6: The statistical sensitivity of J/ψ samples  with different statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  7  given by  ⟨sin θ1 sin φ1⟩ =  1  Nnorm  �  W (ξ) sin θ1 sin φ1dΩ1dΩ2  ≈ −  √  1 − α2α1 sin (∆Φ)  3 + α  sin θ cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  (14)  In the analysis of experimental data, ⟨sin θ1 sin φ1⟩  can be calculated by the average of sin θ1 sin φ1 in  each cos θ bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The moment of sin θ1 sin φ1 can be  connected with the polarization according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 8,  ⟨sin θ1 sin φ1⟩ ≈ (1 + α cos2 θ)α1  3 + α  Py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  (15)  c_signal  θ  cos  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='8  1  Polarization  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content='4  c_signal  Signal MC  PHSP MC  c_signal  Figure 7:  Polarization as a function of cos θ for  J/ψ → Λ¯Λ → pπ−¯nπ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The points with error bars  are the signal MC, and the blue dashed histogram is  the no-polarization scenario of PHSP MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The distribution of polarization versus cos θ as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 7 and the events are not corrected with  detection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  7  Prospect of CP sensitivity at  STCF  The asymmetry parameters used to observe CP vi-  olation are affected by statistics and proportional to  the �NJ/ψ, where NJ/ψ is the number of J/ψ events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  By generating a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0 × 1012 MC sample, after event  selection and detector optimization, the statistical ac-  curacy of CP violation is 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The STCF has great  potential in improving luminosity and realizing beam  polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' It is expected that more than 1 ab−1  experimental data and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='4×1013 J/ψ events will be  obtained per year, with the substantial increase in  statistics, larger data samples will be generated on  STCF, and in the future, it will hopefully reach a  level of accuracy and theoretical prediction compati-  bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  8  Summary and prospect  With the fast simulation software package, the MC  samples of J/ψ → Λ¯Λ → pπ−¯nπ0 process were gen-  erated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  After the optimization of detector perfor-  mance, the events selection efficiency of the signal  process is increased by 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='22% compared to the unop-  timized and the main background process is reduced  by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Furthermore, the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='0 × 1012 J/ψ MC was  used to pre-studied the sensitivity of CP violation  of the J/ψ → Λ¯Λ process at the future STCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The  statistical accuracy of CP violation of Λ hyperon is  10−4, which is close to the prediction of SM of CP  violation in Λ hyperon decay [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  Acknowledgments  The authors would like to thank the USTC Super-  computing Center and the Hefei Comprehensive Na-  tional Science Center for their strong support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' This  work is supported in part by National Natural Science  Foundation of China (NSFC) under Contracts Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  11872030, 11905092, 11972177, 12122509, 11625523,  12105132, 11705078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' The international partnership  program of the Chinese Academy of Sciences un-  der Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 211134KYSB20200057 and by USTC  Research Funds of the Double First-Class Initiative  and the Fundamental Research Funds for the Cen-  tral Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  The Doctoral Scientific Research  Foundation of Liaoning Province No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content='  LQN201902, the Natural Science Foundation  of Liaoning Provincial Department of Education No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  LCJ202003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' China Postdoctoral Science Foundation  under Contracts Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content='  Opening Founda-  tion of Songshan Lake Materials Laboratory, Grants  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='2021SLABFK04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' B  465, 335(1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [5] BaBar Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 87, 091801 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [6] Belle Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' ), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  87, 091802 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [7] LHCb Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' ), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  122, 211803 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Sakharov, Pisma Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 5,  32 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [9] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  Bernreuther,  Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  Notes  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  591,  237 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Canetti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Drewes and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Shaposhnikov,  New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 14, 095012 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Nilles, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 110, 1(1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Haber and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Kane, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 117,  75 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Lee and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Yang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content=' Pakvasa, arXiv:hep-ph/9808472 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Donoghue, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' He and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Pakvasa,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' D 34, 833 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content=' Okubo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content=' Pais, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 3, 242 (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [18] HyperCP  Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  Holmstrom  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' ),  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' 93, 262001 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [19] BESIII Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Ablikim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' ), Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  15, 631 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [20] BESIII  Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  Ablikim  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' ),  arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='11155 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [21] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Perotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' F¨aldt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Kupsc, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Leupold and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content=' Song, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' D 99, 056008 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [22] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' F¨aldt and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Kupsc, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Shi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Qin and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Peng,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Peng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Zheng and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' Zhou, Physics  49, 513 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content='  [25] BESIII Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
+page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQfIwmA/content/2301.04337v1.pdf'}
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+GRAPH NEURAL NETWORKS
+FOR AERODYNAMIC
+FLOW RECONSTRUCTION FROM SPARSE SENSING
+Gregory Duth´e
+ETH Z¨urich
+duthe@ibk.baug.ethz.ch
+Imad Abdallah
+ETH Z¨urich
+abdallah@ibk.baug.ethz.ch
+Sarah Barber
+Eastern Switzerland University of Applied Sciences
+sarah.barber@ost.ch
+Eleni Chatzi
+ETH Z¨urich
+chatzi@ibk.baug.ethz.ch
+ABSTRACT
+Sensing the fluid flow around an arbitrary geometry entails extrapolating from the
+physical quantities perceived at its surface in order to reconstruct the features of
+the surrounding fluid. This is a challenging inverse problem, yet one that if solved
+could have a significant impact on many engineering applications. The exploitation
+of such an inverse logic has gained interest in recent years with the advent of
+widely available cheap but capable MEMS-based sensors. When combined with
+novel data-driven methods, these sensors may allow for flow reconstruction around
+immersed structures, benefiting applications such as unmanned airborne/underwater
+vehicle path planning or control and structural health monitoring of wind turbine
+blades. In this work, we train deep reversible Graph Neural Networks (GNNs) to
+perform flow sensing (flow reconstruction) around two-dimensional aerodynamic
+shapes: airfoils. Motivated by recent work, which has shown that GNNs can be
+powerful alternatives to mesh-based forward physics simulators, we implement a
+Message-Passing Neural Network to simultaneously reconstruct both the pressure
+and velocity fields surrounding simulated airfoils based on their surface pressure
+distributions, whilst additionally gathering useful farfield properties in the form of
+context vectors. We generate a unique dataset of Computational Fluid Dynamics
+simulations by simulating random, yet meaningful combinations of input boundary
+conditions and airfoil shapes. We show that despite the challenges associated
+with reconstructing the flow around arbitrary airfoil geometries in high Reynolds
+turbulent inflow conditions, our framework is able to generalize well to unseen
+cases.
+1
+INTRODUCTION
+Many engineering applications stand to benefit from the ability to sense and reconstruct fluid flow
+features from sparse measurements originating at a structure’s surface. Flow sensing could be crucial
+for improvements in the accuracy and resilience of wind turbine and unmanned aircraft controllers.
+Another possible application is monitoring of wind loaded structures (Barber et al., 2022), where the
+use of cheap micro-electromechanical systems (MEMS) in combination with novel methods for flow
+sensing could lead to robust structural health monitoring solutions. In this work, we focus on common
+aerodynamic structures: we aim to reconstruct the flow around 2-D airfoils. Traditionally, computing
+the flow around an airfoil requires approaches from Computational Fluid Dynamics (CFD), which
+are forward-physics simulators. In CFD, the inflow, outflow and wall boundary conditions are set,
+and over many iterations a solution for the discretized Navier-Stokes PDEs is reached, which then
+yields a pressure distribution at the airfoil surface. We aim to solve the inverse problem: given only
+the pressure distribution at the airfoil surface, a solution for the flow field and farfield boundary
+conditions is to be found. Moreover, our aim is to do so for any airfoil geometry subject to a wide
+variety of turbulent inflows.
+1
+arXiv:2301.03228v1  [cs.CE]  9 Jan 2023
+
+Adopting the notation of Erichson et al. (2020), the problem can be described in the following manner.
+An airfoil equipped with p distributed barometric sensors is placed in a steady flow of air, providing
+surface pressure measurements s ∈ Rp at multiple locations around its perimeter. The sensors sample
+from the surrounding flow field x ∈ Rm through a measurement operator H:
+s = H(x)
+(1)
+The goal is to construct an estimate of the flow field ˆx surrounding the airfoil, by learning from
+training data a function F that approximates the highly nonlinear inverse measurement operator G
+such that:
+F(s) = ˆx ≈ x = G(s)
+(2)
+Meshes are an extremely useful tool, indispensable in many engineering domains and especially
+in CFD. Contrary to Cartesian grid representations, mesh representations offer high flexibility for
+irregular geometries and allow for variable spatial density. This makes them ideal for discretizing
+complex physical problems, where one can balance the trade-off between numerical accuracy and
+computational efficiency in certain regions of interest. Furthermore, meshes can also be described
+in terms of nodes and edges, i.e. as a graph. In this context, the flow reconstruction problem can
+be described as follows. A graph G = (V, E, U) is constructed from an airfoil CFD mesh, with m
+fluid nodes Vf and p airfoil boundary nodes Va. The flow features of all Vf are unknown, whilst the
+features of Va are known. Our aim is then to learn a graph operator F that estimates the information
+at the fluid nodes using the information contained at the airfoil boundary nodes, the input graph-level
+attributes Uin and the edges E:
+ˆ
+Vf = F(Va, E, Uin)
+(3)
+An ancillary goal is to estimate the global context U of the graph, as this contains relevant information
+for applications. Figure 1 provides an description of the flow reconstruction problem in terms of
+graph learning.
+Figure 1: Problem setup. We aim to learn the reconstruction operator F which estimates properties
+of the fluid nodes as well as the graph context. This amounts to reconstructing a solution to the
+Navier-Stokes equations which satisfies the boundary conditions perceived at the airfoil surface.
+From a geometric learning perspective, flow reconstruction is a challenging problem for several
+reasons. The first significant hurdle to overcome is the size of our graphs. We use meshes with high
+densities close to the airfoil in order to achieve good spatial resolution in these critical regions. Thus,
+our dataset contains graphs with a mean of around 55’000 nodes, which is an order of magnitude
+higher than previous mesh-graph learned simulation methods (Pfaff et al., 2020). Moreover, the input
+information is concentrated in a very localized domain of the graph: the airfoil nodes. It is difficult to
+propagate the necessary information to reconstruct nodes far from the airfoil with a shallow Graph
+Neural Network (GNN), meaning that deep GNN architectures with a large number of message-
+passing steps are required to push this ’information barrier’ away from the airfoil nodes. However,
+deep GNNs go hand in hand with other issues such as large memory requirements, over-smoothing,
+and over-squashing.
+In this work, we combine a number of existing graph-learning methods to tackle the aforementioned
+challenges. Our contributions may be summarized as follows:
+• We combine unknown Feature Propagation (Rossi et al., 2021) with very deep Grouped Reversible
+GNNs (Li et al., 2021) to reconstruct flow features at the fluid nodes, whilst additionally gathering
+contextual farfield information.
+2
+
+G = (a,V,&, Uin)
+Uin Estimated
+U Predicted
+input context
+context
+Learned operator
+Known airfoil nodes
+ Unknown fluid nodes
+Known airfoil nodes
+Reconstructed fluid nodes• Generalization of 2D aerodynamic flow field learning with GNNs, including (1) arbitrary airfoil
+geometries, (2) arbitrary turbulent inflow conditions (flow velocity, turbulence intensity, and angle
+of attack), and (3) simultaneous flow field reconstruction and inference of contextual farfield flow
+information based only on sparse pressure data on the surface of the airfoil.
+• Generation of a unique training dataset of OpenFOAM airfoil CFD simulations with many different
+geometries and inflow conditions which are parsed to graph structure and made publicly available.
+• We gather qualitative and quantitative results on unseen airfoil and flow configurations, and perform
+a number of experiments to understand the limitations of this framework. In particular, we test and
+compare three different GNN layer architectures.
+2
+RELATED WORK
+Machine learning methods have recently garnered interest in the fluid mechanics community (Brunton
+et al., 2020). Fluid-related problems are typically nonlinear, complex and generate large amounts
+of data, all of which are conditions under which deep learning approaches thrive. Specifically in
+the context of flow reconstruction from sparse measurements, several neural network approaches
+can be found in the literature. In an article by Erichson et al. (2020), a ”Shallow Neural Network”,
+i.e. a fully-connected network with only two hidden layers, was applied to estimate transient flows
+from sparse measurements. The authors trained the networks on a single specific geometrical flow
+configuration, for example the flow behind a cylinder, and then tested on the same configuration
+at different time steps. We aim to avoid this limitation as our goal is to estimate the flow around
+any airfoil geometry. The authors compare their findings against typically used proper orthogonal
+decomposition (POD) methods and note significant improvements in terms of the reconstruction error.
+This work was then extended to turbulent flow reconstruction around airfoils based on experimental
+data in Carter et al. (2021), where the results were compared to Particle Image Velocimetry (PIV)
+measurements. The viability of neural networks over other approaches was further confirmed in the
+work of Fukami et al. (2020), where multiple methods were pitted against each other to estimate the
+flow behind a cylinder and an airfoil. Again, here the models were trained and tested on a single
+flow configuration, while also being dependent on Cartesian geometrical inputs. In ¨Ozbay & Laizet
+(2022), researchers attempt to avoid this limitation by utilizing Schwarz–Christoffel mappings to
+sample the points at which the flow is reconstructed, thus rendering the method geometry invariant.
+The authors train multiple neural network architectures on a collection of transient flow simulations
+around randomly generated 2-D Bezier shapes at a predefined inflow Reynolds number. As inputs for
+the flow reconstruction, they use multiple pressure sensors on the shapes’ surface as well as velocity
+probes in the wake. Their results indicate that, when compared to a Cartesian sampling strategy, a
+significant performance boost is achieved for all neural network types, especially in the vicinity of
+the immersed shape. While this work demonstrates robustness to various geometric configurations, it
+requires additional velocity sensors and is trained on a singular farfield boundary condition, both of
+which we aim to avoid and improve upon. Another method which avoids geometrical dependency
+is reported in Chen et al. (2021). In this work, to which our approach most closely relates to, the
+authors utilize a Graph Convolutional Network (GCN) (Kipf & Welling, 2016) on graphs constructed
+from CFD meshes of randomly generated Bezier shapes. The GCN is used to predict the flow
+around the shapes at a fixed laminar (Reynolds number of 10) inflow condition without using surface
+measurements. In our approach, we aim to reconstruct a wide variety of turbulent flows given only
+surface readings, a significantly less constrained problem. We also aim to characterize the global
+properties of the flow, similarly to Zhou et al. (2021). To our best knowledge, we are the first to
+attempt to simultaneously reconstruct the flow while estimating turbulent inflow parameters at large
+Reynolds numbers for arbitrary airfoil geometries.
+The dataset that we generate to train our GNN model is similar in terms of the geometries, meshing
+and CFD pipeline to the work of Thuerey et al. (2020), the main differences being the chosen RANS
+model and the post-processing (graph parsing). Other datasets found in the literature focus only on
+the NACA family of airfoils (Schillaci et al., 2021).
+Graph networks are based on the message-passing framework (Gilmer et al., 2017), where a nodes
+features are updated by aggregating messages emanating from its neighbors. Many different types of
+message-passing schemes can be constructed, with some using attention mechanisms (Veliˇckovi´c
+et al., 2017) and others relying on strong theoretical backgrounds (Xu et al., 2018). Graph learning
+3
+
+Figure 2: Illustration of the dataset generation pipeline. Airfoil shapes are selected at random from a
+database, then meshed and simulated in OpenFOAM with random feasible boundary conditions. In
+the last step, the finite-volume scheme is used to parse the simulation mesh into a graph.
+methods are increasingly being applied to a wide variety of physics problems (Sanchez-Gonzalez
+et al., 2018; 2020). In Pfaff et al. (2020), the authors successfully demonstrate how GNNs can learn
+to replicate forward mesh-based physics simulators and are able to predict the evolution of a transient
+solution. Motivated by these results, our approach is constructed upon the the same basic Encode-
+Process-Decode network structure. However, a key difference to note is that, contrary to the next-step
+prediction problem, flow reconstruction has to overcome high amounts of missing information, with
+known features being extremely localized. To address this hurdle, we turn to graph-based feature
+propagation methods (Rossi et al., 2021), which is closely related to matrix completion approaches
+(Monti et al., 2017). Feature propagation is an effective yet computationally inexpensive method for
+initializing graphs with missing features. We use this method as pre-processing step, through which
+graphs are passed before being fed into the rest of the GNN model.
+Training GNNs for very large graphs is challenging, with typical approaches tending toward minimiz-
+ing the number of learnable parameters so that the problem becomes tractable (Chen et al., 2020).
+This often results in relatively shallow GNNs, which could adversely influence the propagation of
+the information contained at the airfoil nodes through sufficient extents of the graph. Making use of
+subgraph sampling strategies (Hamilton et al., 2017) is another possible approach, one which also
+allows for larger/deeper GNNs. However, these methods are not applicable in our case, as we need
+to feed entire graphs in one pass due to the heterogeneity in information localization. Moreover,
+subgraph sampling would yield additional difficulties for our ancillary goal of predicting global graph
+properties for farfield estimation. Recent work by Li et al. (2021) has shown that it is possible to train
+very deep GNNs on large graphs by making use of Grouped Reversible layers, which reduces memory
+requirements at the cost of extra computation. This method forms the core of the processing block of
+the proposed flow reconstruction GNN. Another issue which traditionally characterizes training deep
+GNNs on large graphs is over-squashing (Alon & Yahav, 2020). Over-squashing is a by-product of
+a graph’s structure, where bottlenecks and tree-like structures (Topping et al., 2021) can cause the
+latent representation of certain nodes to be overwhelmed by the amount of information needed to be
+stored. We make use of this information while parsing the simulation meshes into graphs.
+3
+DATASET GENERATION
+In this section we introduce the different elements of our data generation pipeline. In total we generate
+1120 converged simulations, which are separated into train, validation and test datasets in a 80/10/10
+split. Figure 2 depicts an illustration of this pipeline.
+Geometry selection and meshing
+In our dataset generation pipeline, airfoil shapes are drawn at
+random from the UUIC database of airfoils (Selig, 1996). Before passing the shape to the meshing
+algorithm, we carry out some additional interpolation and processing to make sure that the selected
+airfoil has a sufficient amount of points at the leading-edge as well as a properly defined trailing-edge.
+Then, we use Gmsh (Geuzaine & Remacle, 2020) to construct an unstructured O-grid type mesh
+around the selected airfoil. A sizing field is set close to the airfoil in order to make sure that meshes
+with appropriate y+ values for the CFD wall-functions are generated. An overall sizing parameter is
+4
+
+also set for sufficient farfield density, but is adjusted to ensure that an acceptable amount of cells are
+created (< 150′000).
+CFD simulations
+Each mesh is associated with a different inflow configuration. Three parameters
+control the farfield conditions: angle of attack, inflow velocity and turbulence intensity. These
+parameters are drawn from probability distributions reflecting realistic atmospheric flows at Reynolds
+numbers ranging from 2·105 to 6.5·106, with a mean Reynolds number of around 3·106. This mean
+value is well beyond the typical laminar-to-turbulent transition threshold of around 5 · 105 (Incropera
+et al., 1996), which greatly increases the difficulty of the flow reconstruction problem. The farfield
+conditions form the global context vectors of our graphs and are estimated at inference time. We
+simulate the flow around the airfoils using a steady 2-D Reynolds-Averaged Navier–Stokes (RANS)
+CFD solver with the OpenFOAM software package (Jasak et al., 2007). For turbulence modelling,
+we select the K-Omega SST model (Menter & Esch, 2001) along with the standard OpenFOAM
+wall-functions for boundary layer treatment. Only sufficiently converged CFD simulations with
+pressure, velocity and turbulence residuals below 5 · 10−5 are kept.
+Graph parsing
+Contrary to many other mesh-based physics simulators, CFD solvers such as
+OpenFOAM are based on finite volume methods. It is a significant difference that should be reflected
+in the manner in which a mesh is converted into a graph. To do so, we use the cells themselves as
+the nodes, with bidirectional edges being formed between adjacent cells. This allows us to gather
+an additional edge feature which is relevant to the underlying physics. Specifically, the length (or
+surface for 3-D meshes) of the boundary between two cells is used as an edge feature. The benefit
+of this is twofold: a form of sizing is fed to the network and a quantity relevant to flux computation
+is set on the edges. For the nodes, we gather 4 types of features: pressure, x-velocity component,
+y-velocity component and node category (fluid, farfield, wall), the latter of which is inputted as a
+one-hot vector. The global context features are the farfield conditions (turbulence intensity, inflow
+velocity and angle of attack). To avoid unnecessary computational overhead, we do not parse the
+entire CFD domain, which has a radius of 100 airfoil chord lengths, into a graph. Instead we opt to
+only keep cells within a 1 chord circle centered on the airfoil. Furthermore, we set the airfoil nodes to
+be located at the meshed airfoil boundary and add bidirectional edges between adjacent airfoil nodes.
+These additional edges are created with the aim of avoiding tree-like structures in our graphs, as these
+could potentially cause bottlenecks for the learning process (Topping et al., 2021). The graphs of our
+dataset have on average around 55k nodes and around 85k individual edges.
+4
+GRAPH NEURAL NETWORK FRAMEWORK
+4.1
+ARCHITECTURE
+For our GNN architecture, we adopt the Encode-Process-Decode logic that is now popular for
+learning on graph-based physics problems (Sanchez-Gonzalez et al., 2020; Pfaff et al., 2020; Godwin
+et al., 2022), albeit with some notable modifications. Figure 3 shows an overview of the Flow
+Reconstruction GNN.
+Input features
+In the flow reconstruction problem, we assume that the global context of the graph
+is unknown, however some useful physical parameters describing the flow can be estimated. Using
+Bernoulli’s principle, and given that all airfoils are simulated with a zero farfield static pressure, the
+farfield inflow velocity magnitude can be initially approximated as:
+ˆU∞ =
+�2 · p0
+ρ
+(4)
+where ρ is the density of air (constant throughout simulations) and p0 is the total pressure measured
+at the stagnation point, which can be estimated by taking the maximum pressure at the airfoil nodes
+p0 = max(pVa ). While Bernoulli’s principle is not valid for turbulent flows such as the ones we
+try to reconstruct, it serves as a good starting point for farfield velocity estimation. Another useful
+graph property that can be extracted is the normal force coefficient acting on the airfoil. While the
+lift coefficient is usually used to characterize airfoils, it cannot be calculated as the angle of attack is
+unknown (a quantity to be inferred from the learned model). Nevertheless, the normal coefficient is
+5
+
+Figure 3: Overview of the Flow Reconstruction GNN achitecture. Feature Propagation is used to
+initialize the unknown fluid nodes. The graph is then passed through the Encode-Process-Decode
+pipeline, to obtain the reconstructed flow. During the Process step, node features are updated via
+message-passing within a deep Grouped Reversible GNN.
+directly related to the lift coefficient and brings additional physical information which may aid the
+network to reconstruct the flow. It can be estimated via the following equation:
+ˆCn =
+�
+vi∈Va
+pvi · lvi · ny,vi
+p0
+(5)
+where l is the boundary length and ny is the y component of the normal boundary vector, both of
+which are known properties for each mesh cell boundary. We therefore use Uin = ( ˆU∞, ˆCn) as the
+two-dimensional input context vector.
+For the nodes, we only have access to the pressure distribution at the surface of the airfoil, while it is
+set to ’NaN’ values at the fluid nodes. The type of each node is known and is encoded as a one-hot
+vector, bringing the total number of input node features to four. To account for mesh geometry, the
+following four edge features are used as inputs: the x and y components of the relative edge direction
+vector, the edge length, and the cell boundary length value l (see Section 3).
+Pre-processing
+Both the input and target node features are normalized. To avoid biasing the
+normalization, all pressure features are normalized by the mean and standard deviation of the
+known airfoil surface pressure distribution, while both components of the velocity target features
+are normalized by the initial estimated farfield velocity ˆU∞. We use Feature Propagation (Rossi
+et al., 2021) as a preliminary step before feeding a graph to our GNN. This step is an important
+part of our framework as it conditions the input graph into a plausible initial state. Essentially, the
+feature propagator radiates surface pressure information outwards. In most cases, we found 20 feature
+propagation iterations to be sufficient.
+Encoding
+In the encoding layer, ReLU activated MLPs with two hidden layers and LayerNorm are
+used to project the input features of the graph into latent vectors of size N. This encoding layer differs
+to the standard GraphNet encoder (Sanchez-Gonzalez et al., 2020) in that the node encoder MLP
+takes as input the input node features as well as the latent global vector. We make this modification
+so that graph-level attributes are taken into account in the construction of the node latent variables, as
+this is not the case in the processing steps.
+Processing
+For the processing step, we opt to use a deep Grouped Reversible GNN (Li et al.,
+2021) with L message-passing layers. This architecture makes modifications to the typical GNN
+architecture by first splitting the input node feature matrix V across the feature dimension into
+C groups ⟨V1, V2, ..., VC⟩, which are then processed into grouped outputs ⟨V ′
+1, V ′
+2, ..., V ′
+C⟩ with a
+6
+
+OUTPUT
+airfoil
+airfoil
+INPUT
+Latent vectors size: N
+L Message-Passing steps
+Globals encoder 
+Globals decoder
+MLP
+Imi,1
+MLP
+U
+mi3
+Node encoder
+MLP
+Node decoder
+mi,j = Φo (vii
+-1, V'i-1, e'i.
+MLP
+Edge encoder ...:
+MLP
+Encoding
+Processing
+Decoding
+Feature propagationGrouped Reversible GNN layer. These outputs are computed as follows:
+V ′
+0 =
+C
+�
+k=2
+Vk
+V ′
+k = fwk(V ′
+k−1, A, E) + Vk,
+k ∈ {1, . . . , C}
+(6)
+with A the adjacency matrix and E the edge feature matrix.
+The Grouped Reversible framework allows for any type of message-passing architecture to be chosen
+for the GNN layer fwk. We choose to test three popular types of GNN layers: the Graph Attention
+Network (GAT) (Veliˇckovi´c et al., 2017), the modified Graph Isomorphic Network (GIN) (Xu et al.,
+2018) which accounts for edge features (Hu et al., 2019), and the Generalized Aggregation Networks
+(GEN) Li et al. (2020) which modifies the standard GCN with different aggregation schemes while
+also utilizing edge features.
+Decoding
+Only the nodes and global context are decoded back into feature space, as the edges are
+not updated. Both decoding neural networks are MLPs with two hidden layers and ReLU activations
+without any output normalization. At the output of the decoder, we gather for each node the estimated
+pressure and velocity fields. The output context vector is composed of an updated version of the
+farfield velocity, as well as an estimation of the inflow angle (angle of attack) and of the turbulence
+intensity.
+4.2
+TRAINING
+General aspects
+Our models are trained on a dataset composed of 896 graphs. Models are trained
+with the Adam optimizer on a single Nvidia GPU with 10GB of VRAM. Due to the size and nature of
+the graphs, we can only use a batch size of one, albeit with random order shuffling occurring at each
+epoch. The learning rate is initially set at 5 · 10−4 and is exponentially decayed by a factor of 0.97.
+Loss function
+We use a multi-component loss function, in order to minimize both the node feature
+reconstruction error and the context vector prediction error, with L2 losses for both components. An
+additional loss component based on the velocity divergence was also tested but yielded too many
+artefacts and was therefore discarded. The overall loss is:
+L = L2(V, ˆV) + λ · L2(U, ˆU)
+(7)
+where λ is a hyperparameter used to balance the different components. In practice, we usually set λ
+to 1.
+5
+RESULTS
+Our trained models are tested on a dataset comprising of 112 unseen airfoil simulations, each with
+a different combination of turbulent inflow parameters. We show here qualitative and quantitative
+results for our models and perform experiments aiming to investigate limitations and improvements
+of the proposed framework.
+Comparison of GNN layers
+In Table 1, we gather and compare results for the three different types
+of GNN layers used in the Processor: GAT, GIN and GEN. The architecture of the Encoder and
+Decoder networks were kept constant, with a latent size for the node, edge and global features of
+N = 128. In the Grouped Reversible Processor, the number of Layers was set to L = 30, while
+the number of groups was chosen as C = 4. To obtain a consistent number of learnable parameters,
+the hyperparameters for each GNN layer type were carefully selected, more information about the
+different configurations can be found in the appendix. We also study the impact of the depth and
+width on the performance of each model in Appendix D.
+Velocity reconstruction
+One of the more challenging prediction tasks is the estimation of the
+velocity field away from the airfoil. Accurately capturing velocity shear and recirculation regions is
+non-trivial even for CFD simulators and is highly dependant on the airfoil shape and the inflow angle.
+7
+
+Table 1: Root mean squared prediction errors averaged over the test dataset for both the flow
+reconstruction task and the global parameter estimation task. Results are averaged over 3 runs with
+different initializations. All models have latent size of N = 128 and L = 30 layers. While the
+revGAT model performs the best in terms of pressure and y-velocity reconstruction, it is outperformed
+by the revGIN model when it comes to x-velocity reconstruction, and graph-level attribute prediction.
+Node reconstruction RMSE
+Global parameter prediction RMSE
+pressure
+[Pa]
+x-velocity
+[m/s]
+y-velocity
+[m/s]
+farfield velocity
+[m/s]
+angle of attack
+[◦]
+turbulence intensity
+[-]
+revGAT
+77.98 ± 17.99
+7.96 ± 1.19
+2.69 ± 0.53
+0.92 ± 0.17
+4.16 ± 0.10
+0.04 ± 0.003
+revGIN
+158.37 ± 5.16
+6.23 ± 0.32
+4.43 ± 0.22
+0.45 ± 0.06
+4.12 ± 0.09
+0.03 ± 0.003
+revGEN
+137.51 ± 17.31
+10.81 ± 0.25
+5.47 ± 0.05
+0.46 ± 0.05
+4.12 ± 0.01
+0.04 ± 0.002
+To complete this task sucessfully, the GNN needs to be expressive enough to propagate relevant
+information throughout the graph. Table 2 summarizes the prediction errors of the velocity field in
+multiple concentric regions around the airfoil for the three models. We observe that for all three
+models, the error on the x-velocity decreases further away from the airfoil, but this is not the case for
+the y-velocity.
+Table 2: Root mean squared prediction errors of the velocity components for different concentric
+regions of the flow, averaged over the test dataset. Each region is defined as the interior of an ellipse
+with a the length of semi-major axis and b the length of semi-minor (in chord lengths). Results are
+averaged over 3 runs with different initializations.
+x-velocity
+[m/s]
+y-velocity
+[m/s]
+revGAT
+revGIN
+revGEN
+revGAT
+revGIN
+revGEN
+region 1 (a=0.6, b=0.1)
+8.26 ± 1.34
+6.46 ± 0.39
+11.44 ± 0.25
+2.48 ± 0.60
+4.15 ± 0.18
+5.40 ± 0.03
+region 2 (a=0.7, b=0.15)
+7.98 ± 1.25
+6.30 ± 0.35
+11.00 ± 0.26
+2.49 ± 0.43
+4.40 ± 0.23
+5.52 ± 0.05
+region 3 (a=0.8, b=0.2)
+7.95 ± 1.24
+6.27 ± 0.33
+10.92 ± 0.26
+2.61 ± 0.55
+4.44 ± 0.23
+5.53 ± 0.05
+Qualitative results
+Figure 4 displays some qualitative results for our two best performing models
+(revGAT and revGIN), compared to the CFD simulation ground truth. These results highlight the fact
+that the learned model is able to reconstruct flow features well, albeit with some artefacts. As the
+distance to the airfoil increases, these defects become more apparent. Moreover, some parts of the
+flow are not well captured. This is the case for flows exhibiting long wakes. On the other hand, we
+notice that flow features near the leading edge of the airfoil are in general well captured. We provide
+additional examples of reconstructed flows in Appendix E.
+Farfield estimation
+Figure 5 displays the graph-level context prediction results evaluated on the
+test set for the revGIN model. The GNN is able to accurately predict farfield inflow velocity, owing
+to the good initial farfield estimation provided as an input. For the angle of attack estimation, we
+observe good results at small angles but less so for larger positive and negative angles. Prediction of
+the turbulence intensity is however relatively poor, which can be attributed to this variable having a
+lesser impact on the airfoil pressure distribution. Moreover, this variable is not directly set in the CFD
+simulations as it is used to calculate turbulent boundary conditions (kinetic energy k and specific
+rate of dissipation ω of the k − ω turbulence model), which makes it more difficult to retrieve in this
+inverse context.
+6
+DISCUSSION
+Our results indicate that the type of GNN architecture chosen in the Grouped Reversible Processor
+has a clear impact on the flow reconstruction quality. From our comparison, we find that, overall,
+using Graph Attention Network layers usually yields the best reconstructed solutions. However, we
+also observe that the Graph Isomorphic Network layer is better able to capture detached flows (see
+Appendix E). Perhaps a combination of the two could lead to better reconstructed flows, which could
+8
+
+Figure 4: Reconstructed flow around two unseen arbitrary airfoils geometries at different inflow
+configurations for the revGAT and revGIN models. Figure (a) plots the reconstructed pressure field
+while the reconstructed velocity field is shown in Figure (b). For each case the ground truth is shown
+for comparison. Qualitatively, the revGAT model displays fewer artefacts in the solution.
+(a)
+(b)
+(c)
+Figure 5: Comparison of the global properties predicted by our revGIN model against the ground
+truth values, evaluated on the test dataset. The model is able to accurately predict farfield flow velocity
+thanks to a decent initial estimation (a) and to a lesser extent the angle of attack (b), but it falls short
+for farfield turbulence intensity prediction (c).
+be for instance implemented by alternating GIN and GAT layers. Lastly, we find that the performance
+of the GEN layer to be somewhat underwhelming.
+Something to note is that the first few layers of nodes surrounding the airfoil carry a disproportionate
+amount of relevant information which needs to be propagated outwards to an increasing number
+of nodes, thus creating artificial tree-like paths within the graph. While the Grouped Reversible
+framework provides a good way to train deep networks in order to circumvent this, other methods
+may also feasible. One possible solution might be to use hierarchies such as those implemented in
+Martinkus et al. (2021). Another option could be to apply a message-passing GNN in an iterative
+manner, with each step reconstructing increasingly large concentric bands around the airfoil. Physics-
+driven learning methods could also lead to potential improvements. For instance, the message-passing
+framework may well be suited to incorporate elements from the Lattice-Bolzmann method (Chen &
+Doolen, 1998) as it also functions in a similar two step algorithm (collision and streaming). Another
+possible option would be to minimize the gradient of the pressure, as described in Taha & Gonzalez
+(2021).
+9
+
+100
+GNN Prediction
+Bernouli Estimation
+80
+[m/s]
+60
+Pred. U.[
+40
+20
+0
+0
+20
+40
+60
+80
+100
+True U[m/s]20
+10
+Pred. AoA 
+0
+-10
+-20
+-20
+-10
+0
+10
+20
+True AoA [o0.300
+0.275
+Pred. Ti [-]
+0.250
+0.225
+0.200
+0.175
+0.150
+0.15
+0.20
+0.25
+0.30
+True Ti [-]7
+CONCLUSION
+In this work we applied deep graph-based learning techniques to reconstruct pressure and velocity
+fields around arbitrary airfoil geometries subject to high-Reynolds turbulent flows. We show that,
+despite the challenges posed by this problem, such as the large graphs and the very localized
+input information, our Flow Reconstruction GNN framework is able to provide good reconstructed
+solutions, and infer contextual farfield flow information. We compared several message-passing
+architectures within the Grouped Reversible Processor GNN, and found that Graph Attention Network
+layers yielded the best reconstructed solutions. This work provides a flexible framework which may
+easily be applied to other mesh-based inverse physics problems, and which may be of significant
+interest to a number of engineering applications.
+10
+
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+12
+
+APPENDIX
+A
+BENCHMARKING OF THE CFD MODEL
+To ensure that the simulations which constitute our dataset are of sufficient quality, we benchmark
+our CFD pipeline against results reported in the literature for the NACA 0012 airfoil (Krist, 1998;
+Gregory & O’reilly, 1970). In Figure 6 we plot the pressure coefficient for this airfoil at two angles of
+attack. We see that our CFD pipeline matches well with both the previous experimental and numerical
+results.
+(a)
+(b)
+Figure 6: Comparison of our CFD data pipeline against experimental (Gregory & O’reilly, 1970)
+and other CFD results (Krist, 1998). We plot the pressure coefficient distribution on the surface of a
+NACA 0012 airfoil at Re = 6 · 106 at an angle of attack of (a) 10◦ and (b) 15◦.
+B
+ADDITIONAL HYPERPARAMETER DETAILS
+For all models, 30 layers were used in conjunction with a latent space size of 128. The GAT
+model is implemented with 2 attention heads. Moreover a LayerNorm layer is applied to the output
+of each GAT layer, as it was found that this greatly aided training stability. Both the GIN and
+GEN implementation makes use of ReLu activated MLPs with two hidden layers and LayerNorm.
+These choice of parameters ensure that each individual layer has approximately 4.5k to 5k learnable
+parameters. In total for the baseline models with 30 layers and a latent space size of N = 128, we
+obtain models with around 700k trainable parameters.
+13
+
+A0A = 15°
+Ours
+Gregory-O'Reilly
+-10
+CFL3D (NASA)
+-8
+-6
+G
+-4
+-2
+0
+1
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+x/c [-]A0A = 10°
+Ours
+Gregory-O'Reilly
+CFL3D (NASA)
+-3
+G-2
+-1
+0
+1
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+x/c [-]C
+ABLATION OF ESTIMATED GLOBALS
+We plot in Figure 7 the result of removing the estimated farfield conditions ( ˆU∞ and ˆCn) from the
+input of the model. We note a drastic penalty in the quality of the prediction of farfield quantities.
+Figure 7: Relative performance of the model when no estimated farfield parameters are used in the
+input compared to the baseline model.
+D
+DEPTH AND WIDTH STUDY
+We display in Figure 8 the effect of model depth on the reconstruction performance of each reversible
+model. In Figure 9, we plot the relative performance of different latent space sizes (the width) for the
+different models.
+E
+ADDITIONAL RESULTS
+We plot in Figure 10 and Figure 11 some additional results, showcasing the reconstructive abilities of
+our models, as well as some configurations which are not well captured. Notably, detached flows
+are not always properly reconstructed, an issue which may stem from either the dataset not holding
+enough detached flows, or from challenges arising from the GNN architecture.
+14
+
+revGAT (L = 30 N= 128)
+20.0
+With Global Est. Input
+baseline
+17.5
+Without Global Est. Input
+15.0
+ RMSE relative to
+12.5
+10.0
+7.5
+5.0
+Avg
+2.5
+0.0
+pressure x-velocity y-velocity
+Uinf
+AoA
+Ti(a)
+(b)
+(c)
+Figure 8: Comparison of the impact of the depth on reconstruction performance for the different
+layers, relative to the baseline architectures with 30 layers (in green). We also show the relative
+training time for each depth.
+15
+
+revGAT N= 128
+Relative Training Time
+L=10
+ L=20
+L=30
+ L= 40
+1.4
+baseline
+1.3
+1.2
+to
+relative
+1.1
+RMSE
+1.0
+0.8
+x-velocity
+0.0x
+1.0x
+y-velocity
+2.0x
+pressurerevGIN N= 128
+Relative Training Time
+1.20
+L=10
+L=20
+L=30
+ L=40
+1.15
+line
+1.10
+basel
+1.05
+to
+ve
+1.00
+lativ
+@ 0.95
+ISE
+0.90
+0.80
+0.5x
+1.0x
+pressure
+x-velocity
+y-velocity
+0.0x
+1.5xrevGEN N = 128
+Relative Training Time
+1.10
+L=10
+ L=20
+L=30
+L= 40
+ 1.05
+selin
+ba
+1.00
+to
+relative
+0.95
+0.90
+E
+RMSE
+0.80
+x-velocity
+0.5x
+pressure
+y-velocity
+0.0x
+1.0x(a)
+(b)
+(c)
+Figure 9: Comparison of the impact of the size of the latent space on reconstruction performance for
+the different layers, relative to the baseline architectures with a latent space size of N = 128 layers
+(in orange). We also show the relative training time for each width.
+16
+
+revGAT L = 30
+Relative Training Time
+1.3
+I N= 64
+N= 128
+ N=256
+baseline
+1.2
+1.1
+lative
+ 1.0
+e
+RMSE
+0.9
+Avg
+0.8
+x-velocity
+0.5x
+1.0x
+pressure
+y-velocity
+0.0xrevGIN L = 30
+Relative Training Time
+1.15
+N=64
+N=128
+ N=256
+baseline
+1.05
+lative
+0.95
+rel
+E
+0.90
+RM
+0.80
+x-velocity
+y-velocity
+0.0x
+0.5x
+pressure
+1.0xrevGEN L = 30
+Relative Training Time
+1.10
+N=64
+N= 128
+N=256
+ 1.05
+selin
+ba
+1.00
+to
+0.95
+relativ
+0.90
+E
+RMSE
+g 0.85
+Av
+0.80
+x-velocity
+y-velocity
+0.0x
+0.5x
+pressure
+1.0xFigure 10: Reconstructed velocity field magnitude around two unseen arbitrary airfoils geometries at
+different large wake inflow configurations for the revGAT, revGIN and revGEN models. For each
+case the ground truth is shown for comparison. Qualitatively, the revGIN model is able to better
+capture the detached flows in the solution.
+Figure 11: Reconstructed pressure field around two unseen arbitrary airfoils geometries at different
+inflow configurations for the revGAT, revGIN and revGEN models. For each case the ground truth is
+shown for comparison. Qualitatively, the revGAT model displays fewer artefacts.
+17
+
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+page_content='baug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='ch  Sarah Barber  Eastern Switzerland University of Applied Sciences  sarah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='barber@ost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='ch  Eleni Chatzi  ETH Z¨urich  chatzi@ibk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='baug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='ch  ABSTRACT  Sensing the fluid flow around an arbitrary geometry entails extrapolating from the  physical quantities perceived at its surface in order to reconstruct the features of  the surrounding fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' This is a challenging inverse problem, yet one that if solved  could have a significant impact on many engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The exploitation  of such an inverse logic has gained interest in recent years with the advent of  widely available cheap but capable MEMS-based sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' When combined with  novel data-driven methods, these sensors may allow for flow reconstruction around  immersed structures, benefiting applications such as unmanned airborne/underwater  vehicle path planning or control and structural health monitoring of wind turbine  blades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In this work, we train deep reversible Graph Neural Networks (GNNs) to  perform flow sensing (flow reconstruction) around two-dimensional aerodynamic  shapes: airfoils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Motivated by recent work, which has shown that GNNs can be  powerful alternatives to mesh-based forward physics simulators, we implement a  Message-Passing Neural Network to simultaneously reconstruct both the pressure  and velocity fields surrounding simulated airfoils based on their surface pressure  distributions, whilst additionally gathering useful farfield properties in the form of  context vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We generate a unique dataset of Computational Fluid Dynamics  simulations by simulating random, yet meaningful combinations of input boundary  conditions and airfoil shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We show that despite the challenges associated  with reconstructing the flow around arbitrary airfoil geometries in high Reynolds  turbulent inflow conditions, our framework is able to generalize well to unseen  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  1  INTRODUCTION  Many engineering applications stand to benefit from the ability to sense and reconstruct fluid flow  features from sparse measurements originating at a structure’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Flow sensing could be crucial  for improvements in the accuracy and resilience of wind turbine and unmanned aircraft controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Another possible application is monitoring of wind loaded structures (Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2022), where the  use of cheap micro-electromechanical systems (MEMS) in combination with novel methods for flow  sensing could lead to robust structural health monitoring solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In this work, we focus on common  aerodynamic structures: we aim to reconstruct the flow around 2-D airfoils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Traditionally, computing  the flow around an airfoil requires approaches from Computational Fluid Dynamics (CFD), which  are forward-physics simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In CFD, the inflow, outflow and wall boundary conditions are set,  and over many iterations a solution for the discretized Navier-Stokes PDEs is reached, which then  yields a pressure distribution at the airfoil surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We aim to solve the inverse problem: given only  the pressure distribution at the airfoil surface, a solution for the flow field and farfield boundary  conditions is to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Moreover, our aim is to do so for any airfoil geometry subject to a wide  variety of turbulent inflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='03228v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='CE]  9 Jan 2023  Adopting the notation of Erichson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' (2020), the problem can be described in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  An airfoil equipped with p distributed barometric sensors is placed in a steady flow of air, providing  surface pressure measurements s ∈ Rp at multiple locations around its perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The sensors sample  from the surrounding flow field x ∈ Rm through a measurement operator H:  s = H(x)  (1)  The goal is to construct an estimate of the flow field ˆx surrounding the airfoil, by learning from  training data a function F that approximates the highly nonlinear inverse measurement operator G  such that:  F(s) = ˆx ≈ x = G(s)  (2)  Meshes are an extremely useful tool, indispensable in many engineering domains and especially  in CFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Contrary to Cartesian grid representations, mesh representations offer high flexibility for  irregular geometries and allow for variable spatial density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' This makes them ideal for discretizing  complex physical problems, where one can balance the trade-off between numerical accuracy and  computational efficiency in certain regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Furthermore, meshes can also be described  in terms of nodes and edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' as a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In this context, the flow reconstruction problem can  be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' A graph G = (V, E, U) is constructed from an airfoil CFD mesh, with m  fluid nodes Vf and p airfoil boundary nodes Va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The flow features of all Vf are unknown, whilst the  features of Va are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Our aim is then to learn a graph operator F that estimates the information  at the fluid nodes using the information contained at the airfoil boundary nodes, the input graph-level  attributes Uin and the edges E:  ˆ  Vf = F(Va, E, Uin)  (3)  An ancillary goal is to estimate the global context U of the graph, as this contains relevant information  for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Figure 1 provides an description of the flow reconstruction problem in terms of  graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Figure 1: Problem setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We aim to learn the reconstruction operator F which estimates properties  of the fluid nodes as well as the graph context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' This amounts to reconstructing a solution to the  Navier-Stokes equations which satisfies the boundary conditions perceived at the airfoil surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  From a geometric learning perspective, flow reconstruction is a challenging problem for several  reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The first significant hurdle to overcome is the size of our graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We use meshes with high  densities close to the airfoil in order to achieve good spatial resolution in these critical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Thus,  our dataset contains graphs with a mean of around 55’000 nodes, which is an order of magnitude  higher than previous mesh-graph learned simulation methods (Pfaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Moreover, the input  information is concentrated in a very localized domain of the graph: the airfoil nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' It is difficult to  propagate the necessary information to reconstruct nodes far from the airfoil with a shallow Graph  Neural Network (GNN), meaning that deep GNN architectures with a large number of message-  passing steps are required to push this ’information barrier’ away from the airfoil nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' However,  deep GNNs go hand in hand with other issues such as large memory requirements, over-smoothing,  and over-squashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  In this work, we combine a number of existing graph-learning methods to tackle the aforementioned  challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Our contributions may be summarized as follows:  We combine unknown Feature Propagation (Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2021) with very deep Grouped Reversible  GNNs (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2021) to reconstruct flow features at the fluid nodes, whilst additionally gathering  contextual farfield information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  2  G = (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='&,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Uin)  Uin Estimated  U Predicted  input context  context  Learned operator  Known airfoil nodes  Unknown fluid nodes  Known airfoil nodes  Reconstructed fluid nodes• Generalization of 2D aerodynamic flow field learning with GNNs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' including (1) arbitrary airfoil  geometries,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' (2) arbitrary turbulent inflow conditions (flow velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' turbulence intensity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' and angle  of attack),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' and (3) simultaneous flow field reconstruction and inference of contextual farfield flow  information based only on sparse pressure data on the surface of the airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Generation of a unique training dataset of OpenFOAM airfoil CFD simulations with many different  geometries and inflow conditions which are parsed to graph structure and made publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  We gather qualitative and quantitative results on unseen airfoil and flow configurations, and perform  a number of experiments to understand the limitations of this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In particular, we test and  compare three different GNN layer architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  2  RELATED WORK  Machine learning methods have recently garnered interest in the fluid mechanics community (Brunton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Fluid-related problems are typically nonlinear, complex and generate large amounts  of data, all of which are conditions under which deep learning approaches thrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Specifically in  the context of flow reconstruction from sparse measurements, several neural network approaches  can be found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In an article by Erichson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' (2020), a ”Shallow Neural Network”,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' a fully-connected network with only two hidden layers, was applied to estimate transient flows  from sparse measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The authors trained the networks on a single specific geometrical flow  configuration, for example the flow behind a cylinder, and then tested on the same configuration  at different time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We aim to avoid this limitation as our goal is to estimate the flow around  any airfoil geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The authors compare their findings against typically used proper orthogonal  decomposition (POD) methods and note significant improvements in terms of the reconstruction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  This work was then extended to turbulent flow reconstruction around airfoils based on experimental  data in Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' (2021), where the results were compared to Particle Image Velocimetry (PIV)  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The viability of neural networks over other approaches was further confirmed in the  work of Fukami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' (2020), where multiple methods were pitted against each other to estimate the  flow behind a cylinder and an airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Again, here the models were trained and tested on a single  flow configuration, while also being dependent on Cartesian geometrical inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In ¨Ozbay & Laizet  (2022), researchers attempt to avoid this limitation by utilizing Schwarz–Christoffel mappings to  sample the points at which the flow is reconstructed, thus rendering the method geometry invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  The authors train multiple neural network architectures on a collection of transient flow simulations  around randomly generated 2-D Bezier shapes at a predefined inflow Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' As inputs for  the flow reconstruction, they use multiple pressure sensors on the shapes’ surface as well as velocity  probes in the wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Their results indicate that, when compared to a Cartesian sampling strategy, a  significant performance boost is achieved for all neural network types, especially in the vicinity of  the immersed shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' While this work demonstrates robustness to various geometric configurations, it  requires additional velocity sensors and is trained on a singular farfield boundary condition, both of  which we aim to avoid and improve upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Another method which avoids geometrical dependency  is reported in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In this work, to which our approach most closely relates to, the  authors utilize a Graph Convolutional Network (GCN) (Kipf & Welling, 2016) on graphs constructed  from CFD meshes of randomly generated Bezier shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The GCN is used to predict the flow  around the shapes at a fixed laminar (Reynolds number of 10) inflow condition without using surface  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In our approach, we aim to reconstruct a wide variety of turbulent flows given only  surface readings, a significantly less constrained problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We also aim to characterize the global  properties of the flow, similarly to Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' To our best knowledge, we are the first to  attempt to simultaneously reconstruct the flow while estimating turbulent inflow parameters at large  Reynolds numbers for arbitrary airfoil geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  The dataset that we generate to train our GNN model is similar in terms of the geometries, meshing  and CFD pipeline to the work of Thuerey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' (2020), the main differences being the chosen RANS  model and the post-processing (graph parsing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Other datasets found in the literature focus only on  the NACA family of airfoils (Schillaci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Graph networks are based on the message-passing framework (Gilmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2017), where a nodes  features are updated by aggregating messages emanating from its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Many different types of  message-passing schemes can be constructed, with some using attention mechanisms (Veliˇckovi´c  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2017) and others relying on strong theoretical backgrounds (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Graph learning  3  Figure 2: Illustration of the dataset generation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Airfoil shapes are selected at random from a  database, then meshed and simulated in OpenFOAM with random feasible boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In  the last step, the finite-volume scheme is used to parse the simulation mesh into a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  methods are increasingly being applied to a wide variety of physics problems (Sanchez-Gonzalez  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In Pfaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' (2020), the authors successfully demonstrate how GNNs can learn  to replicate forward mesh-based physics simulators and are able to predict the evolution of a transient  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Motivated by these results, our approach is constructed upon the the same basic Encode-  Process-Decode network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' However, a key difference to note is that, contrary to the next-step  prediction problem, flow reconstruction has to overcome high amounts of missing information, with  known features being extremely localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' To address this hurdle, we turn to graph-based feature  propagation methods (Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2021), which is closely related to matrix completion approaches  (Monti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Feature propagation is an effective yet computationally inexpensive method for  initializing graphs with missing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We use this method as pre-processing step, through which  graphs are passed before being fed into the rest of the GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Training GNNs for very large graphs is challenging, with typical approaches tending toward minimiz-  ing the number of learnable parameters so that the problem becomes tractable (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  This often results in relatively shallow GNNs, which could adversely influence the propagation of  the information contained at the airfoil nodes through sufficient extents of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Making use of  subgraph sampling strategies (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2017) is another possible approach, one which also  allows for larger/deeper GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' However, these methods are not applicable in our case, as we need  to feed entire graphs in one pass due to the heterogeneity in information localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Moreover,  subgraph sampling would yield additional difficulties for our ancillary goal of predicting global graph  properties for farfield estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Recent work by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' (2021) has shown that it is possible to train  very deep GNNs on large graphs by making use of Grouped Reversible layers, which reduces memory  requirements at the cost of extra computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' This method forms the core of the processing block of  the proposed flow reconstruction GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Another issue which traditionally characterizes training deep  GNNs on large graphs is over-squashing (Alon & Yahav, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Over-squashing is a by-product of  a graph’s structure, where bottlenecks and tree-like structures (Topping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2021) can cause the  latent representation of certain nodes to be overwhelmed by the amount of information needed to be  stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We make use of this information while parsing the simulation meshes into graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  3  DATASET GENERATION  In this section we introduce the different elements of our data generation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In total we generate  1120 converged simulations, which are separated into train, validation and test datasets in a 80/10/10  split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Figure 2 depicts an illustration of this pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Geometry selection and meshing  In our dataset generation pipeline, airfoil shapes are drawn at  random from the UUIC database of airfoils (Selig, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Before passing the shape to the meshing  algorithm, we carry out some additional interpolation and processing to make sure that the selected  airfoil has a sufficient amount of points at the leading-edge as well as a properly defined trailing-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Then, we use Gmsh (Geuzaine & Remacle, 2020) to construct an unstructured O-grid type mesh  around the selected airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' A sizing field is set close to the airfoil in order to make sure that meshes  with appropriate y+ values for the CFD wall-functions are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' An overall sizing parameter is  4  also set for sufficient farfield density, but is adjusted to ensure that an acceptable amount of cells are  created (< 150′000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  CFD simulations  Each mesh is associated with a different inflow configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Three parameters  control the farfield conditions: angle of attack, inflow velocity and turbulence intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' These  parameters are drawn from probability distributions reflecting realistic atmospheric flows at Reynolds  numbers ranging from 2·105 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='5·106, with a mean Reynolds number of around 3·106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' This mean  value is well beyond the typical laminar-to-turbulent transition threshold of around 5 · 105 (Incropera  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 1996), which greatly increases the difficulty of the flow reconstruction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The farfield  conditions form the global context vectors of our graphs and are estimated at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We  simulate the flow around the airfoils using a steady 2-D Reynolds-Averaged Navier–Stokes (RANS)  CFD solver with the OpenFOAM software package (Jasak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' For turbulence modelling,  we select the K-Omega SST model (Menter & Esch, 2001) along with the standard OpenFOAM  wall-functions for boundary layer treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Only sufficiently converged CFD simulations with  pressure, velocity and turbulence residuals below 5 · 10−5 are kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Graph parsing  Contrary to many other mesh-based physics simulators, CFD solvers such as  OpenFOAM are based on finite volume methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' It is a significant difference that should be reflected  in the manner in which a mesh is converted into a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' To do so, we use the cells themselves as  the nodes, with bidirectional edges being formed between adjacent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' This allows us to gather  an additional edge feature which is relevant to the underlying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Specifically, the length (or  surface for 3-D meshes) of the boundary between two cells is used as an edge feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The benefit  of this is twofold: a form of sizing is fed to the network and a quantity relevant to flux computation  is set on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' For the nodes, we gather 4 types of features: pressure, x-velocity component,  y-velocity component and node category (fluid, farfield, wall), the latter of which is inputted as a  one-hot vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The global context features are the farfield conditions (turbulence intensity, inflow  velocity and angle of attack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' To avoid unnecessary computational overhead, we do not parse the  entire CFD domain, which has a radius of 100 airfoil chord lengths, into a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Instead we opt to  only keep cells within a 1 chord circle centered on the airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Furthermore, we set the airfoil nodes to  be located at the meshed airfoil boundary and add bidirectional edges between adjacent airfoil nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  These additional edges are created with the aim of avoiding tree-like structures in our graphs, as these  could potentially cause bottlenecks for the learning process (Topping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The graphs of our  dataset have on average around 55k nodes and around 85k individual edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  4  GRAPH NEURAL NETWORK FRAMEWORK  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='1  ARCHITECTURE  For our GNN architecture, we adopt the Encode-Process-Decode logic that is now popular for  learning on graph-based physics problems (Sanchez-Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Pfaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Godwin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2022), albeit with some notable modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Figure 3 shows an overview of the Flow  Reconstruction GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Input features  In the flow reconstruction problem, we assume that the global context of the graph  is unknown, however some useful physical parameters describing the flow can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Using  Bernoulli’s principle, and given that all airfoils are simulated with a zero farfield static pressure, the  farfield inflow velocity magnitude can be initially approximated as:  ˆU∞ =  �2 · p0  ρ  (4)  where ρ is the density of air (constant throughout simulations) and p0 is the total pressure measured  at the stagnation point, which can be estimated by taking the maximum pressure at the airfoil nodes  p0 = max(pVa ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' While Bernoulli’s principle is not valid for turbulent flows such as the ones we  try to reconstruct, it serves as a good starting point for farfield velocity estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Another useful  graph property that can be extracted is the normal force coefficient acting on the airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' While the  lift coefficient is usually used to characterize airfoils, it cannot be calculated as the angle of attack is  unknown (a quantity to be inferred from the learned model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Nevertheless, the normal coefficient is  5  Figure 3: Overview of the Flow Reconstruction GNN achitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Feature Propagation is used to  initialize the unknown fluid nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The graph is then passed through the Encode-Process-Decode  pipeline, to obtain the reconstructed flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' During the Process step, node features are updated via  message-passing within a deep Grouped Reversible GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  directly related to the lift coefficient and brings additional physical information which may aid the  network to reconstruct the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' It can be estimated via the following equation:  ˆCn =  �  vi∈Va  pvi · lvi · ny,vi  p0  (5)  where l is the boundary length and ny is the y component of the normal boundary vector, both of  which are known properties for each mesh cell boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We therefore use Uin = ( ˆU∞, ˆCn) as the  two-dimensional input context vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  For the nodes, we only have access to the pressure distribution at the surface of the airfoil, while it is  set to ’NaN’ values at the fluid nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The type of each node is known and is encoded as a one-hot  vector, bringing the total number of input node features to four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' To account for mesh geometry, the  following four edge features are used as inputs: the x and y components of the relative edge direction  vector, the edge length, and the cell boundary length value l (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Pre-processing  Both the input and target node features are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' To avoid biasing the  normalization, all pressure features are normalized by the mean and standard deviation of the  known airfoil surface pressure distribution, while both components of the velocity target features  are normalized by the initial estimated farfield velocity ˆU∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We use Feature Propagation (Rossi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2021) as a preliminary step before feeding a graph to our GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' This step is an important  part of our framework as it conditions the input graph into a plausible initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Essentially, the  feature propagator radiates surface pressure information outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In most cases, we found 20 feature  propagation iterations to be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Encoding  In the encoding layer, ReLU activated MLPs with two hidden layers and LayerNorm are  used to project the input features of the graph into latent vectors of size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' This encoding layer differs  to the standard GraphNet encoder (Sanchez-Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2020) in that the node encoder MLP  takes as input the input node features as well as the latent global vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We make this modification  so that graph-level attributes are taken into account in the construction of the node latent variables, as  this is not the case in the processing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Processing  For the processing step, we opt to use a deep Grouped Reversible GNN (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=',  2021) with L message-passing layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' This architecture makes modifications to the typical GNN  architecture by first splitting the input node feature matrix V across the feature dimension into  C groups ⟨V1, V2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', VC⟩, which are then processed into grouped outputs ⟨V ′  1, V ′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=", V ′  C⟩ with a  6  OUTPUT  airfoil  airfoil  INPUT  Latent vectors size: N  L Message-Passing steps  Globals encoder  Globals decoder  MLP  Imi,1  MLP  U  mi3  Node encoder  MLP  Node decoder  mi,j = Φo (vii  1, V'i-1, e'i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  MLP  Edge encoder .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=':  MLP  Encoding  Processing  Decoding  Feature propagationGrouped Reversible GNN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' These outputs are computed as follows:  V ′  0 =  C  �  k=2  Vk  V ′  k = fwk(V ′  k−1, A, E) + Vk,  k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' , C}  (6)  with A the adjacency matrix and E the edge feature matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  The Grouped Reversible framework allows for any type of message-passing architecture to be chosen  for the GNN layer fwk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We choose to test three popular types of GNN layers: the Graph Attention  Network (GAT) (Veliˇckovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2017), the modified Graph Isomorphic Network (GIN) (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=',  2018) which accounts for edge features (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=', 2019), and the Generalized Aggregation Networks  (GEN) Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' (2020) which modifies the standard GCN with different aggregation schemes while  also utilizing edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Decoding  Only the nodes and global context are decoded back into feature space, as the edges are  not updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Both decoding neural networks are MLPs with two hidden layers and ReLU activations  without any output normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' At the output of the decoder, we gather for each node the estimated  pressure and velocity fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The output context vector is composed of an updated version of the  farfield velocity, as well as an estimation of the inflow angle (angle of attack) and of the turbulence  intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='2  TRAINING  General aspects  Our models are trained on a dataset composed of 896 graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Models are trained  with the Adam optimizer on a single Nvidia GPU with 10GB of VRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Due to the size and nature of  the graphs, we can only use a batch size of one, albeit with random order shuffling occurring at each  epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The learning rate is initially set at 5 · 10−4 and is exponentially decayed by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Loss function  We use a multi-component loss function, in order to minimize both the node feature  reconstruction error and the context vector prediction error, with L2 losses for both components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' An  additional loss component based on the velocity divergence was also tested but yielded too many  artefacts and was therefore discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The overall loss is:  L = L2(V, ˆV) + λ · L2(U, ˆU)  (7)  where λ is a hyperparameter used to balance the different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In practice, we usually set λ  to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  5  RESULTS  Our trained models are tested on a dataset comprising of 112 unseen airfoil simulations, each with  a different combination of turbulent inflow parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We show here qualitative and quantitative  results for our models and perform experiments aiming to investigate limitations and improvements  of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Comparison of GNN layers  In Table 1, we gather and compare results for the three different types  of GNN layers used in the Processor: GAT, GIN and GEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The architecture of the Encoder and  Decoder networks were kept constant, with a latent size for the node, edge and global features of  N = 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In the Grouped Reversible Processor, the number of Layers was set to L = 30, while  the number of groups was chosen as C = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' To obtain a consistent number of learnable parameters,  the hyperparameters for each GNN layer type were carefully selected, more information about the  different configurations can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We also study the impact of the depth and  width on the performance of each model in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Velocity reconstruction  One of the more challenging prediction tasks is the estimation of the  velocity field away from the airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Accurately capturing velocity shear and recirculation regions is  non-trivial even for CFD simulators and is highly dependant on the airfoil shape and the inflow angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  7  Table 1: Root mean squared prediction errors averaged over the test dataset for both the flow  reconstruction task and the global parameter estimation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Results are averaged over 3 runs with  different initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' All models have latent size of N = 128 and L = 30 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' While the  revGAT model performs the best in terms of pressure and y-velocity reconstruction, it is outperformed  by the revGIN model when it comes to x-velocity reconstruction, and graph-level attribute prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Node reconstruction RMSE  Global parameter prediction RMSE  pressure  [Pa]  x-velocity  [m/s]  y-velocity  [m/s]  farfield velocity  [m/s]  angle of attack  [◦]  turbulence intensity  [-]  revGAT  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='98 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='99  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='96 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='19  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='003  revGIN  158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='37 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='16  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='32  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='06  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='003  revGEN  137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='51 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='31  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='05  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='002  To complete this task sucessfully, the GNN needs to be expressive enough to propagate relevant  information throughout the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Table 2 summarizes the prediction errors of the velocity field in  multiple concentric regions around the airfoil for the three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We observe that for all three  models, the error on the x-velocity decreases further away from the airfoil, but this is not the case for  the y-velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Table 2: Root mean squared prediction errors of the velocity components for different concentric  regions of the flow, averaged over the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Each region is defined as the interior of an ellipse  with a the length of semi-major axis and b the length of semi-minor (in chord lengths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Results are  averaged over 3 runs with different initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  x-velocity  [m/s]  y-velocity  [m/s]  revGAT  revGIN  revGEN  revGAT  revGIN  revGEN  region 1 (a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='6, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='1)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='26 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='34  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='39  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='60  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='18  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='03  region 2 (a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='7, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='15)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='98 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='25  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='35  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='26  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='43  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='23  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='05  region 3 (a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='8, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='2)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='95 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='24  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='33  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='26  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='55  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='23  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='05  Qualitative results  Figure 4 displays some qualitative results for our two best performing models  (revGAT and revGIN), compared to the CFD simulation ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' These results highlight the fact  that the learned model is able to reconstruct flow features well, albeit with some artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' As the  distance to the airfoil increases, these defects become more apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Moreover, some parts of the  flow are not well captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' This is the case for flows exhibiting long wakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' On the other hand, we  notice that flow features near the leading edge of the airfoil are in general well captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We provide  additional examples of reconstructed flows in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Farfield estimation  Figure 5 displays the graph-level context prediction results evaluated on the  test set for the revGIN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The GNN is able to accurately predict farfield inflow velocity, owing  to the good initial farfield estimation provided as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' For the angle of attack estimation, we  observe good results at small angles but less so for larger positive and negative angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Prediction of  the turbulence intensity is however relatively poor, which can be attributed to this variable having a  lesser impact on the airfoil pressure distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Moreover, this variable is not directly set in the CFD  simulations as it is used to calculate turbulent boundary conditions (kinetic energy k and specific  rate of dissipation ω of the k − ω turbulence model), which makes it more difficult to retrieve in this  inverse context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  6  DISCUSSION  Our results indicate that the type of GNN architecture chosen in the Grouped Reversible Processor  has a clear impact on the flow reconstruction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' From our comparison, we find that, overall,  using Graph Attention Network layers usually yields the best reconstructed solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' However, we  also observe that the Graph Isomorphic Network layer is better able to capture detached flows (see  Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Perhaps a combination of the two could lead to better reconstructed flows, which could  8  Figure 4: Reconstructed flow around two unseen arbitrary airfoils geometries at different inflow  configurations for the revGAT and revGIN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Figure (a) plots the reconstructed pressure field  while the reconstructed velocity field is shown in Figure (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' For each case the ground truth is shown  for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Qualitatively, the revGAT model displays fewer artefacts in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  (a)  (b)  (c)  Figure 5: Comparison of the global properties predicted by our revGIN model against the ground  truth values, evaluated on the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The model is able to accurately predict farfield flow velocity  thanks to a decent initial estimation (a) and to a lesser extent the angle of attack (b), but it falls short  for farfield turbulence intensity prediction (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  be for instance implemented by alternating GIN and GAT layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Lastly, we find that the performance  of the GEN layer to be somewhat underwhelming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Something to note is that the first few layers of nodes surrounding the airfoil carry a disproportionate  amount of relevant information which needs to be propagated outwards to an increasing number  of nodes, thus creating artificial tree-like paths within the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' While the Grouped Reversible  framework provides a good way to train deep networks in order to circumvent this, other methods  may also feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' One possible solution might be to use hierarchies such as those implemented in  Martinkus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Another option could be to apply a message-passing GNN in an iterative  manner, with each step reconstructing increasingly large concentric bands around the airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Physics-  driven learning methods could also lead to potential improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' For instance, the message-passing  framework may well be suited to incorporate elements from the Lattice-Bolzmann method (Chen &  Doolen, 1998) as it also functions in a similar two step algorithm (collision and streaming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Another  possible option would be to minimize the gradient of the pressure, as described in Taha & Gonzalez  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  9  100  GNN Prediction  Bernouli Estimation  80  [m/s]  60  Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' [  40  20  0  0  20  40  60  80  100  True U[m/s]20  10  Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' AoA  0  10  20  20  10  0  10  20  True AoA [o0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='275  Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Ti [-]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='225  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='175  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='30  True Ti [-]7  CONCLUSION  In this work we applied deep graph-based learning techniques to reconstruct pressure and velocity  fields around arbitrary airfoil geometries subject to high-Reynolds turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We show that,  despite the challenges posed by this problem, such as the large graphs and the very localized  input information, our Flow Reconstruction GNN framework is able to provide good reconstructed  solutions, and infer contextual farfield flow information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We compared several message-passing  architectures within the Grouped Reversible Processor GNN, and found that Graph Attention Network  layers yielded the best reconstructed solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' This work provides a flexible framework which may  easily be applied to other mesh-based inverse physics problems, and which may be of significant  interest to a number of engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  10  REFERENCES  Uri Alon and Eran Yahav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' On the bottleneck of graph neural networks and its practical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
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+page_content='  Douglas W Carter, Francis De Voogt, Renan Soares, and Bharathram Ganapathisubramani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
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+page_content=' Data-Centric  Engineering, 2, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Junfeng Chen, Elie Hachem, and Jonathan Viquerat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Graph neural networks for laminar flow  prediction around random 2d shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
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+page_content=' Deep learning methods for  reynolds-averaged navier–stokes simulations of airfoil flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' AIAA Journal, 58(1):25–36, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Jake Topping, Francesco Di Giovanni, Benjamin Paul Chamberlain, Xiaowen Dong, and Michael M  Bronstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Understanding over-squashing and bottlenecks on graphs via curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' arXiv preprint  arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='14522, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Petar Veliˇckovi´c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua  Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Graph attention networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' arXiv preprint arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='10903, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' How powerful are graph neural  networks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='00826, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Kaiwen Zhou, Luanliang Zhou, Simeng Zhao, Xingyu Qiang, Yingzheng Liu, and Xin Wen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Data-  driven method for flow sensing of aerodynamic parameters using distributed pressure measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  AIAA Journal, 59(9):3504–3516, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  12  APPENDIX  A  BENCHMARKING OF THE CFD MODEL  To ensure that the simulations which constitute our dataset are of sufficient quality, we benchmark  our CFD pipeline against results reported in the literature for the NACA 0012 airfoil (Krist, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Gregory & O’reilly, 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In Figure 6 we plot the pressure coefficient for this airfoil at two angles of  attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We see that our CFD pipeline matches well with both the previous experimental and numerical  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  (a)  (b)  Figure 6: Comparison of our CFD data pipeline against experimental (Gregory & O’reilly, 1970)  and other CFD results (Krist, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We plot the pressure coefficient distribution on the surface of a  NACA 0012 airfoil at Re = 6 · 106 at an angle of attack of (a) 10◦ and (b) 15◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  B  ADDITIONAL HYPERPARAMETER DETAILS  For all models, 30 layers were used in conjunction with a latent space size of 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' The GAT  model is implemented with 2 attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Moreover a LayerNorm layer is applied to the output  of each GAT layer, as it was found that this greatly aided training stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Both the GIN and  GEN implementation makes use of ReLu activated MLPs with two hidden layers and LayerNorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  These choice of parameters ensure that each individual layer has approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='5k to 5k learnable  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In total for the baseline models with 30 layers and a latent space size of N = 128, we  obtain models with around 700k trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content="  13  A0A = 15°  Ours  Gregory-O'Reilly  10  CFL3D (NASA)  8  6  G  4  2  0  1  1  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content="0  x/c [-]A0A = 10°  Ours  Gregory-O'Reilly  CFL3D (NASA)  3  G-2  1  0  1  1  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0  x/c [-]C  ABLATION OF ESTIMATED GLOBALS  We plot in Figure 7 the result of removing the estimated farfield conditions ( ˆU∞ and ˆCn) from the  input of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We note a drastic penalty in the quality of the prediction of farfield quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Figure 7: Relative performance of the model when no estimated farfield parameters are used in the  input compared to the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  D  DEPTH AND WIDTH STUDY  We display in Figure 8 the effect of model depth on the reconstruction performance of each reversible  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' In Figure 9, we plot the relative performance of different latent space sizes (the width) for the  different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  E  ADDITIONAL RESULTS  We plot in Figure 10 and Figure 11 some additional results, showcasing the reconstructive abilities of  our models, as well as some configurations which are not well captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Notably, detached flows  are not always properly reconstructed, an issue which may stem from either the dataset not holding  enough detached flows, or from challenges arising from the GNN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  14  revGAT (L = 30 N= 128)  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0  With Global Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Input  baseline  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='5  Without Global Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Input  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0  RMSE relative to  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0  Avg  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0  pressure x-velocity y-velocity  Uinf  AoA  Ti(a)  (b)  (c)  Figure 8: Comparison of the impact of the depth on reconstruction performance for the different  layers, relative to the baseline architectures with 30 layers (in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We also show the relative  training time for each depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  15  revGAT N= 128  Relative Training Time  L=10  L=20  L=30  L= 40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='4  baseline  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='2  to  relative  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='1  RMSE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='8  x-velocity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0x  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0x  y-velocity  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0x  pressurerevGIN N= 128  Relative Training Time  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='20  L=10  L=20  L=30  L=40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='15  line  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='10  basel  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='05  to  ve  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='00  lativ  @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='95  ISE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='5x  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0x  pressure  x-velocity  y-velocity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0x  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='5xrevGEN N = 128  Relative Training Time  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='10  L=10  L=20  L=30  L= 40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='05  selin  ba  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='00  to  relative  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='90  E  RMSE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='80  x-velocity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='5x  pressure  y-velocity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0x  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0x(a)  (b)  (c)  Figure 9: Comparison of the impact of the size of the latent space on reconstruction performance for  the different layers, relative to the baseline architectures with a latent space size of N = 128 layers  (in orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' We also show the relative training time for each width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  16  revGAT L = 30  Relative Training Time  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='3  I N= 64  N= 128  N=256  baseline  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='1  lative  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0  e  RMSE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='9  Avg  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='8  x-velocity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='5x  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0x  pressure  y-velocity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0xrevGIN L = 30  Relative Training Time  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='15  N=64  N=128  N=256  baseline  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='05  lative  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='95  rel  E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='90  RM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='80  x-velocity  y-velocity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='0x  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
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+page_content='0xrevGEN L = 30  Relative Training Time  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='10  N=64  N= 128  N=256  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='05  selin  ba  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
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+page_content='95  relativ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='90  E  RMSE  g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='85  Av  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='80  x-velocity  y-velocity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
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+page_content='0xFigure 10: Reconstructed velocity field magnitude around two unseen arbitrary airfoils geometries at  different large wake inflow configurations for the revGAT, revGIN and revGEN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' For each  case the ground truth is shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' Qualitatively, the revGIN model is able to better  capture the detached flows in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content='  Figure 11: Reconstructed pressure field around two unseen arbitrary airfoils geometries at different  inflow configurations for the revGAT, revGIN and revGEN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
+page_content=' For each case the ground truth is  shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfgQS0/content/2301.03228v1.pdf'}
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+Analyzing the Representational Geometry of Acoustic Word Embeddings
+Badr M. Abdullah and Dietrich Klakow
+Language Science and Technology (LST), Saarland University, Germany
+Saarland Informatics Campus
+{ babdullah | dietrich }@lsv.uni-saarland.de
+Abstract
+Acoustic word embeddings (AWEs) are vector
+representations such that different acoustic ex-
+emplars of the same word are projected nearby
+in the embedding space. In addition to their
+use in speech technology applications such as
+spoken term discovery and keyword spotting,
+AWE models have been adopted as models of
+spoken-word processing in several cognitively
+motivated studies and have been shown to ex-
+hibit human-like performance in some audi-
+tory processing tasks. Nevertheless, the rep-
+resentational geometry of AWEs remains an
+under-explored topic that has not been studied
+in the literature. In this paper, we take a closer
+analytical look at AWEs learned from English
+speech and study how the choice of the learn-
+ing objective and the architecture shapes their
+representational profile. To this end, we em-
+ploy a set of analytic techniques from machine
+learning and neuroscience in three different
+analyses: embedding space uniformity, word
+discriminability, and representational consis-
+tency. Our main findings highlight the promi-
+nent role of the learning objective on shap-
+ing the representation profile compared to the
+model architecture.
+1
+Introduction
+Due to their ubiquity, word embeddings are nowa-
+days a central component in natural language pro-
+cessing (NLP). Inducing word embeddings from
+text yields representations such that words occur-
+ring in similar contexts are nearby in the vector
+space (Mikolov et al., 2013; Pennington et al.,
+2014). Therefore, the representational geometry of
+text-based word embeddings captures lexical simi-
+larity and semantic relatedness at multiple levels of
+granularity. Word embeddings, and their underly-
+ing distributional semantic models, have also been
+adopted as models of human semantic memory in
+cognitive science research (Pereira et al., 2016; Ne-
+matzadeh et al., 2017; Grand et al., 2022).
+Figure 1: UMAP projection (McInnes et al., 2018) of
+a sample of acoustic word embeddings (AWEs) pro-
+duced by a correspondence autoencoder (CAE) model
+trained on English read speech. AWE models project
+different exemplars of the same word type closer in the
+embedding space while abstracting away from speaker
+and context variability.
+In the speech processing domain, researchers
+have independently developed representations of
+acoustic segments that correspond to linguistic
+units (Levin et al., 2013; Bengio and Heigold, 2014;
+Kamper et al., 2016b; Settle and Livescu, 2016a,
+inter alia). A notable example of such represen-
+tations are acoustic word embeddings (AWEs)—
+vector representations that encode the sound struc-
+ture of words, not their semantic and syntactic
+structure—see Fig. 1. AWEs support voice-based
+speech technology applications such as query-by-
+example spoken term discovery (Zhang and Glass,
+2009; Jansen and Durme, 2012; Metze et al., 2013)
+and keyword spotting (Myers et al., 1980; Rohlicek,
+1995). In addition, AWEs can be leveraged to fa-
+cilitate access to speech recordings of endangered
+spoken languages that might lack standardized writ-
+ing systems (Bird, 2021; San et al., 2021)
+However, there are fundamental differences be-
+arXiv:2301.03012v1  [cs.CL]  8 Jan 2023
+
+iste
+a
+a
+bel
+Datween text-based and speech-based word embed-
+dings that have to do with the degree of variability
+between the two modalities. Contrary to written
+words which have context-invariant orthographic
+realizations,1 spoken words are notoriously vari-
+able.
+The underlying sources of variability in
+speech include speaker-related factors such as vo-
+cal tract shape, gender, age, and dialect. In addition,
+two acoustic instances, or exemplars, of the same
+word will vary in different phonological and se-
+mantic contexts even if they are produced by the
+same speaker (Jurafsky, 2003). Therefore, acous-
+tic word embeddings are not static, but have to be
+computed “on the fly” given a speech segment as
+input. Models of AWEs need to abstract away from
+speaker and context variability to project different
+acoustic exemplars of the same word onto (ideally)
+the same point of the embedding space.
+Nevertheless, AWEs have not yet been exten-
+sively studied in the literature from a neural net-
+work interpretability point of view. We are only
+aware of a few prior efforts in this direction that
+have either analyzed the representational geometry
+of AWEs from a cognitively motivated angle (Ma-
+tusevych et al., 2020a; Abdullah et al., 2021a) or
+from a cross-linguistic perspective (Abdullah et al.,
+2021b). In this paper, we make a contribution in
+this direction and use analytic techniques from ma-
+chine learning and neuroscience in three different
+analytic studies: embedding space uniformity (§4),
+word discriminability (§5), and representational
+consistency (§6).
+2
+Acoustic Word Embedding Models
+Given an acoustic signal that corresponds to a spo-
+ken word represented as a temporal sequence of T
+acoustic feature vectors, i.e., a = (a1, a2, ..., aT ),
+the goal of an AWE model is to transform a into a
+fixed-dimensionality vector representation e. Due
+to the variability in speech production (i.e., speech
+rate, emotional state, etc), the length of the acoustic
+segment T varies between different exemplars, or
+instances, of the same word type. Therefore, this
+task is modeled as a mapping F : A −→ RD, where
+A is the (continuous) space of acoustic sequences
+and D is the dimensionality of the embedding. For-
+mally, transforming a variable-length acoustic in-
+put into a D-dimensional AWE is described as
+e = F(a; θF) ∈ RD
+(1)
+1although some orthographic variation exists in informal,
+user-generated text such as tweets.
+where θF are the parameters of the encoder
+function F.
+In a supervised setting of train-
+ing AWE models, one assumes a dataset D =
+{(a1, w1), (a2, w2), . . . , (aN, wN)} of N spoken
+word instances where wi is the word type, or lexi-
+cal category, of the ith acoustic sample. In this pa-
+per, we experiment with two architectural choices—
+recurrent and convolutional—and employ four dif-
+ferent learning objectives for training AWE models
+from the literature. Next, we formally describe
+each of the objectives.
+2.1
+Correspondence Autoencoder
+In the correspondence autoencoder (CAE) (Kam-
+per, 2019), each training acoustic word sample a
+is paired with another sample that corresponds to
+the same word type a+ = (a1
++, a2
++, ..., aS
++). The
+acoustic encoder F takes a as input and produces
+an embedding e, which is then fed to an acous-
+tic decoder H that aims to sequentially recon-
+struct the corresponding acoustic sequence a+—
+see Fig. 6(a). The objective is to minimize the L2
+distance at each timestep in the decoder, which is
+equivalent to
+J =
+S
+�
+i=1
+∥ai
++ − Hi(e)∥2
+(2)
+where a+
+i is the ground-truth acoustic feature vec-
+tor at timestep i and Hi(e) is the reconstructed
+acoustic vector at timestep i as a function of the
+embedding e. Learning the correspondence be-
+tween different acoustic realizations of the same
+word type seems to encourage the encoder to build
+up speaker-invariant word representations while
+preserving linguistically-relevant phonetic informa-
+tion (Matusevych et al., 2020b). When the target
+acoustic sequence to generate is the same as the
+input signal a, this corresponds to a conventional
+autoencoder (AE) which we consider as one of our
+learning objectives in this paper.
+2.2
+Phonologically Guided Encoder
+The phonologically guided encoder (PGE) is
+trained as component in a sequence-to-sequence
+model to map acoustics into phonology (Abdullah
+et al., 2021a). Given the output of the encoder as
+an embedding e, a phonological decoder G(.; θG)
+is trained to decode the corresponding phonologi-
+cal sequence ϕ = (ϕ1, . . . , ϕτ) of the word-form
+—see Fig. 6(b). The objective is to minimize a cate-
+gorical cross-entropy loss at each decoder timestep,
+
+which is equivalent to minimizing the term
+J = −
+�
+(ai,wi)∈D
+log P
+�
+ϕ|ei; θG
+�
+= −
+�
+(ai,wi)∈D
+τ
+�
+t=1
+log P
+�
+ϕt|t, ei; θG
+�
+(3)
+where P
+�
+ϕt|t, ei; θG
+�
+is the probability of the
+phoneme ϕt at the tth timestep, conditioned on
+the previous phoneme sequence ϕ<t and the AWE
+e, and θG are the parameters of the decoder. The
+intuition of this learning objective is the follow-
+ing: although their acoustic realizations vary due to
+speaker and context variability, different exemplars
+of the same word category would have identical
+phonological sequences. We thus expect the en-
+coder to project exemplars of the same lexical cate-
+gory nearby in the embedding space while embed-
+ding similarity in the vector space should correlate
+with phonological similarity.
+2.3
+Contrastive Siamese Encoder
+The contrastive siamese encoder (CSE) has been ex-
+plored in the context of AWEs with both recurrent
+and convolutional architectures in several studies
+(Settle and Livescu, 2016b; Kamper et al., 2016a;
+Jacobs et al., 2021). Contrary to the previously
+described objectives, the CSE explicitly minimizes
+the distance between exemplar embeddings of the
+same word type—see Fig. 6(c). First, each acoustic
+word instance is paired with another instance of the
+same word type (a, a+). Given their embeddings
+(ea, e+), the objective is then to minimize a triplet
+margin loss
+J = max
+�
+0, m + d(ea, e+) − d(ea, e−)
+�
+(4)
+Here, d(., .) is the cosine distance and e− is an
+AWE that corresponds to a different word type
+sampled from the mini-batch such that the term
+d(ea, e−) is minimized. This objective clusters
+acoustic instances of the same word type closer in
+the embedding space while pushing away instances
+of other word types by a distance defined by the
+margin hyperparameter m.
+3
+Data, Setup, and Intrinsic Evaluation
+3.1
+Experimental Data
+The data in our study is drawn from the the Lib-
+riSpeech dataset which contains read speech record-
+ings of American-English (Panayotov et al., 2015),
+which is a public dataset under the CC BY 4.0 li-
+cense. We sample 384 speakers from for training
+and 128 for evaluation—disjoint sets—and obtain
+word-aligned speech samples using the Montreal
+Forced Aligner (McAuliffe et al., 2017). To make
+our models comparable with prior work, which has
+focused on AWEs for low-resource languages, we
+sample ∼ 39.4k samples for training and ∼ 9.7k
+for evaluation.
+The phonetic transcription for
+each word is produced using the online WebMaus
+G2P tool (Strunk et al., 2014). Then, each acous-
+tic segment is parametrized as a sequence of 39-
+dimensional Mel-frequency spectral coefficients
+of 25ms frames with 15ms overlap—the conven-
+tional feature representation of speech in automatic
+speech recognition (ASR). It is worth pointing out
+that in this paper we consider each morphological
+variant of a lexeme as a separate lexical category.
+For example, different inflections of the lexeme
+MAKE such as {MADE, MAKING, MAKER, etc.}
+represent different lexical categories, each with its
+own exemplars.
+3.2
+Architectures, Hyperparameters, and
+Training Details
+CNN Acoustic Encoder.
+We employ a 3-layer
+temporal convolutional network (1D-CNN) with
+256, 384, and 512 filters and widths of 4, 8, and 16
+for each layer and keep stride step at 1. Following
+each convolutional operation, we apply batch nor-
+malization, ReLU non-linearity, and dropout. We
+apply average pooling to downsample the repre-
+sentation at the end of the convolution block, then
+apply one non-linear layer with Tanh on the CNN
+output, which yields a 512-dimensional AWE.
+RNN Acoustic Encoder.
+We employ a 3-layer
+directional Gated Recurrent Unit (GRU) with a
+hidden state dimension of 512, then apply one non-
+linear layer with Tanh on the GRU output, which
+yields a 512-dimensional AWE. We apply layer-
+wise dropout with a probability of 0.1.
+Phonological Decoder G(.; θG).
+We employ a
+1-layer GRU of 512 units hidden state that takes the
+512-dimensional AWE as the initial hidden state
+and decodes the corresponding phonological se-
+quence without teacher forcing.
+Acoustic Decoder H(.; θH).
+We employ a 1-
+layer GRU of 512 units hidden state that takes the
+512-dimensional AWE as the initial hidden state
+and decodes the corresponding acoustic sequence
+with a teacher forcing ratio of 0.2.
+
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+-0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+20.0
+Density
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+-0.0
+0.2
+0.4
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+Density
+Cosine similarity
+(d) Phonologically-Guided Encoder
+(e) Contrastive Siamese Encoder
+(a) Randomly Initialized Encoder
+(b) Autoencoder
+(c) Correspondence Autoencoder
+Recurrent Models
+Convolutional Models
+Cross category
+Within category
+Figure 2: Distribution of cosine similarity scores of the different models for within category samples (i.e., exemplar
+pairs of the same word type) and cross-category samples (i.e., sample pairs that correspond to different word types).
+Each row in the figure corresponds to one learning objective and each column corresponds to one architecture.
+Contrastive Loss.
+For the CSE, we experiment
+with different values of the margin hyperparameter
+m = {0.2, 0.3, 0.4, 0.5}, out of which 0.4 yields
+the best performance on the validation set.
+Training Details.
+All models in this study are
+randomly initialized with each parameter drawn
+uniformly from [−0.05, 0.05]. Then, each model
+is trained for 100 epochs with a batch size of 256
+using the ADAM optimizer (Kingma and Ba, 2015)
+and an initial learning rate of 0.001. The learning
+rate is reduced by a factor of 0.5 if the mAP on the
+validation set does not improve for 10 epochs.
+Implementation.
+We build our models using
+PyTorch (Paszke et al., 2019) and use FAISS (John-
+son et al., 2017) for efficient similarity search. Our
+code is based on our prior work in building and
+analyzing AWEs (Abdullah et al., 2021a,b).
+3.3
+Quantitative Evaluation
+We conduct an intrinsic evaluation for the AWEs
+to assess the performance of our models using the
+same-different acoustic word discrimination task
+with the mean average precision (mAP) metric
+(Carlin et al., 2011; Kamper et al., 2015; Settle
+et al., 2019; Algayres et al., 2020). This task evalu-
+ates the ability of the model to determine whether
+two given speech segments correspond to the same
+word type—that is, whether or not two acoustic
+segments are exemplars of the same category. The
+results of the evaluation is shown in Fig. 7 in the
+appendix. We observe that each recurrent encoder
+outperforms its convolutional counterpart within
+each objective. Moreover, the performance largely
+depends on the strength of the supervision signal
+where the contrastive encoders outperform other
+objectives that lack explicit loss to group exem-
+plars of the same category closer in the embedding
+space.
+4
+Analysis 1: Embedding Space
+Uniformity
+In our first analysis, we take a closer look at how
+uniform are representational spaces of AWE mod-
+els by analyzing the distribution of cosine similarity
+for each model type and the degree to which the
+embeddings are isotropic.
+4.1
+Distribution of Cosine Similarity
+One way of analyzing the geometry of represen-
+tation spaces in the acoustic domain is by inspect-
+ing the similarity distributions of exemplars of the
+same lexical category (or word type) versus ran-
+domly sampled, cross-category exemplars. We per-
+form this analysis on the training samples and de-
+pict the result in Fig. 2. We observe that the dif-
+ference between the means of the within-category
+and those of cross-category distributions is largely
+dependent on the strength of the supervision sig-
+nal with the randomly initialized encoders (RIE)
+having the smallest mean differences for both archi-
+tectures. The contrastive encoders have the largest
+mean difference–with mean cross-category scores
+centered at the zero—which is intuitive given the
+explicit supervision signal they receive in grouping
+exemplars of the same category closer in the em-
+bedding space. One surprising observation is the
+
+RIE
+AE
+CAE
+PGE
+CSE
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Isotropy score
+RNN
+CNN
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+mAP
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+Isotropy score
+(a)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+mAP
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Isotropy score
+(b)
+Figure 3: (a). The degree of isotropy of AWE for each
+model. (b) Correlation between the word discrimina-
+tion performance measured by mAP and isotropy score
+(Pearson r = 0.89, p < 0.001).
+behavior of the untrained convolutional encoder
+which gives cosine similarity scores very close to
+1 for each input pair. In appendix C, we demon-
+strate that this behavior is mainly caused by the
+unbounded activation function (i.e., ReLU) in the
+convolutional layers.
+4.2
+Degree of Isotropy
+Although inspecting the cosine similarity distribu-
+tions is an insightful analysis, it does not enable us
+to make well-informed judgments about the unifor-
+mity of the representation spaces. Here, we ask two
+questions: (1) do AWE models utilize all dimen-
+sions of the vector space to represent the speech
+samples and separate the categories? and (2) how
+do architecture and learning objective affect the dis-
+tributivity of information in the embedding space?
+To answer these questions, we inspect the degree
+of isotropy in the representation spaces. An embed-
+ding space is said to be maximally isotropic if the
+variance is uniformly distributed across all dimen-
+sions. Prior work in NLP has found that seman-
+tic word embeddings tend to be anisotropic since
+they only utilize a few dimensions of the vector
+space—an effect that has been observed for word
+embeddings that are static (Mimno and Thomp-
+son, 2017; Mu and Viswanath, 2018) as well as
+contextualized (Ethayarajh, 2019; Cai et al., 2020;
+Rudman et al., 2022). The degree of isotropy in
+acoustic embeddings, however, remains so far un-
+known. To inspect the degree of isotropy of the
+AWE vector spaces, we use the IsoScore metric
+recently proposed by Rudman et al. (2022), which
+is—to the best of our knowledge—the only metric
+in the literature that is grounded on the mathemati-
+cal definition of isotropy. The IsoScore metric oper-
+ates on the covariance matrix of the embedding di-
+mensions and returns values between 0 (minimally
+isotropic) and 1 (maximally isotropic). We quan-
+tify the degree of isotropy using IsoScore for each
+model type and show the result in Fig. 3(a). We
+observe that IsoScore returns values that are within
+the range [0.002, 0.095], which indicates that em-
+bedding spaces for all models tend towards being
+minimally isotropic. However, the embeddings
+of untrained, randomly initialized encoders (RIE)
+tend to be extremely anisotropic (i.e., IsoScore val-
+ues close to 0). This observation suggests that the
+anisotropic space does not “emerge” during the
+model training but rather that it is an inherent prop-
+erty of the encoder architecture. We are not aware
+of prior work in NLP that has studied the degree
+of isotropy in untrained NLP models to investigate
+whether anisotropic spaces are an emergent or in-
+herent feature. In our case, training with a learning
+objective that encourages the model to separate
+word categories moves the representation space
+more towards utilizing more dimensions, therefore
+resulting in a higher degree of isotropy. Moreover,
+recurrent encoders tend to be more isotropic than
+their convolutional counterparts within the same
+learning objective.
+Despite the tendency of all models to be
+anisotropic, we find a strong positive correlation be-
+tween the degree of isotropy and the performance
+on word discrimination—see Fig. 3(b). That is, the
+more dimensions the model utilizes in the represen-
+tation space, the better it performs on the intrinsic
+evaluation task.
+5
+Analysis 2: Word Discriminability
+Ideally, AWE models should project exemplars of
+the same word category onto the same point in the
+embedding space. However, there are no strong
+constraints during training to encourage maximal
+separability between different word categories. In
+this analysis, we seek to answer two questions:
+(1) how well-separated are the word categories of
+the training samples? and (2) to what degree do
+lexical properties predict the discriminability of
+word categories?
+5.1
+Category Discriminability Index
+In order to investigate the geometric density of
+each word category in the representation space,
+we need to measure within-category compactness
+and cross-category separability. Inspired by the
+exemplar discriminability index proposed in the
+neuroscience literature (Nili et al., 2020), we define
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+mAP
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Avg. CDI
+RIE
+AE
+CAE
+PGE
+CSE
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Avg. CDI
+RNN
+CNN
+(a)
+(b)
+Figure 4: Averaged Category Discriminability Index
+(CDI) for each AWE model with error bars showing
+standard deviation over word categories.
+(b) Corre-
+lation between the word discrimination performance
+measured by mAP and averaged CDI (Pearson r =
+0.90, p < 0.001).
+category discriminability index (CDI) as a metric
+that operates on within-category and cross-category
+distances. If we consider each lexical category in
+the training set as a set of its exemplar embeddings
+C = {e1, . . . , e|C|}, CDI is defined for a single
+category C as
+CDI(C) = 1
+|C|
+�
+∀ei∈C
+�
+�
+∀ej∼C|j̸=i
+d(ei, ˜ej) − d(ei, ej)
+�
+(5)
+where d(., .) is the cosine distance and ej is a
+within-category sample while ˜ej is an embedding
+sampled from a different category. If we normal-
+ize the embeddings, CDI ∈ [−1, 1] with values
+closer to 1 indicating higher word discriminability.
+We compute CDI for each word category in the
+training set and take the average over categories to
+estimate how well the categories are separated in
+the embedding space of each model type. The re-
+sult of this analysis is shown in Fig. 4(a). For each
+learning objective, we observe that word discrim-
+inability is higher in the recurrent encoders com-
+pared to their convolutional counterparts. Besides
+that, the contrastive objective yields encoders with
+a higher word discriminability index regardless of
+the architecture type—recurrent vs. convolutional.
+Furthermore, we report a strong positive correla-
+tion between average CDI and the performance on
+the evaluation task—see Fig. 4(b), indicating that
+word discrimination performance on future, held-
+out samples can be predicted based on the CDI
+computed on the training samples.
+5.2
+Effect of Frequency and Distinctiveness
+The CDI quantifies the separability and compact-
+ness for each lexical category in the representation
+space. Next, we aim to identify the factors that
+could make a lexical category compact and well-
+separable.
+In this analysis, we study the effect of two lexical
+properties that could be quantified in a data-driven
+approach: word frequency and acoustic distinctive-
+ness. Our initial hypothesis is that a word category
+with many training exemplars becomes more dis-
+criminable in the embedding space as the repeated
+exposure to samples of various degrees of variabil-
+ity should enable the model to learn compact and
+precise representation for categories with high fre-
+quency. Also, words that are acoustically distinct
+have fewer competitors in the perceptual space,
+thus they should be more separable than words with
+many phonological neighbours that sound similar.
+Therefore, we expect word acoustic distinctiveness
+(WAD) to positively correlate with CDI. In this
+analysis, we operationalize WAD using two met-
+rics: word length (i.e., the number of phonemes)
+and phonological distinctiveness. Word length con-
+tributes to WAD since word formation in natural
+languages is a combinatorial process. That is, in-
+creasing the number of phonemes in a word-form
+decreases the likelihood of encountering a simi-
+larly sounding word-form which makes it less con-
+fusable. However, the word formation process is
+governed by language-specific phonotactic rules
+which makes some sound combinations more prob-
+able than others. To capture the probabilistic nature
+of sound sequences, we employ phonological in-
+formation content (PIC), an information-theoretic
+metric that estimates WAD based on its phoneme-
+to-phoneme transition probabilities (Meylan and
+Griffiths, 2017). Given a word-form as a sequence
+of phonemes ϕ = (ϕ1, . . . , ϕτ), PIC is defined as
+PIC(ϕ) = −
+τ
+�
+i=1
+log pθ(ϕi|ϕ<i)
+(6)
+where pθ is a probabilistic phoneme-level language
+model (PLM). We estimate pθ using a trigram PLM
+with the counts of the phonemes in the training
+word categories. Higher values of PIC indicate less
+probable phoneme sequences thus more distinct
+word-forms. Note that PIC is not length normalized
+and therefore shorter words tend to have lower PIC.
+Next, we conduct a correlation analysis between
+word CDI and the three lexical predictors: fre-
+
+Objective
+Arch.
+Frequency
+Length
+PIC
+AE
+CNN
+-0.081†
+0.315†
+0.263†
+RNN
+-0.087†
+0.357†
+0.306†
+CAE
+CNN
+0.021
+0.376†
+0.274†
+RNN
+0.077†
+0.447†
+0.359†
+PGE
+CNN
+0.035
+0.039*
+-0.011
+RNN
+-0.043*
+0.325†
+0.263†
+CSE
+CNN
+0.131†
+0.075†
+0.031
+RNN
+0.109†
+0.100†
+0.030
+Table 1: Pearson correlation (r) between word category
+discriminability index (CDI) and three lexical proper-
+ties: frequency, length, and phonological information
+content (PIC). Statistical significance is marked with *
+and † for p < 0.05 and p < 0.001, respectively.
+quency, length, and PIC. The result of this analysis
+is shown in Table 1. Surprisingly, our correlation
+analysis shows that lexical frequency is a poor pre-
+dictor of CDI. Although in five out of eight mod-
+els the frequency positively correlates with CDI,
+the correlation is rather weak. However, measures
+of acoustic distinctiveness have a stronger corre-
+lation with CDI compared to frequency, and the
+strength of the correlation is more noticeable in all
+decoding-based models—except the convolutional
+PGE—compared to contrastive models. We also
+find it surprising that PIC is not a better predictor of
+CDI than word length. However, it has been shown
+in a related work that autoencoder-based AWEs en-
+code duration as an acoustic feature (Matusevych
+et al., 2021). Taken together with our findings, this
+suggests that the models exploit and rely on acous-
+tic word length as a feature to discriminate between
+the lexical categories. Arguably, word length is a
+more accessible feature to learn from the acoustic
+signal compared to structural phonological regular-
+ities in the training data.
+6
+Analysis 3: Network Representational
+Consistency
+Suppose we train two instances of the same archi-
+tecture and learning objective on the same training
+samples, but each with different random initial-
+izations. Do these two neural network instances
+exhibit differences in their representational geome-
+tries? In this section, we shed light on the repre-
+sentational discrepancies caused by different ini-
+tializations. In other words, we are interested in
+quantifying the degree to which variability in the
+initial conditions affects the way two models sepa-
+rate the same set of speech samples.
+6.1
+Performance Stability
+First, we quantify the effect of the initial weights
+on the evaluation task performance. To this end,
+we train six model instances—in identical setup but
+with different initializations—for each architecture
+and each learning objective, which yields 48 model
+instances in total (6 × 4 RNN runs and 6 × 4 CNN
+runs). We evaluate each model instance on the
+acoustic word discrimination task while observing
+the result variation per model type. The result of
+the performance stability analysis is shown in Ta-
+ble 2 in Appendix D. We observe that all instances
+have converged and the performance is fairly stable
+across different runs.
+6.2
+Representational Discrepancies
+Our previous performance stability analysis has
+demonstrated that different DNN instances exhibit
+only trivial quantitative differences. However, a
+stable performance on the evaluation task does
+not entail an identical representational geometry
+across different instances. That is, two network
+instances could have have an identical performance
+on the evaluation task while each having a distinct
+representational geometry. To closely investigate
+representational discrepancies between network in-
+stances, we employ the representational consis-
+tency (RC) analysis (Mehrer et al., 2020), which
+is a neuroscience-inspired technique based on the
+representational similarity analysis (RSA) frame-
+work (Kriegeskorte et al., 2008). For our analysis,
+we operationalize the RC using linear Centered
+Kernal Alignment (CKA) as a representational sim-
+ilarity measure of two views of the same input sam-
+ples (Kornblith et al., 2019). CKA abstracts away
+from the embeddings themselves and operates on
+pairwise distances between the sample representa-
+tions. Concretely, given K spoken-word samples
+aK
+1 = {a1, . . . , aK}, we embed the samples using
+two encoder instances to obtain two different views
+of the samples X ∈ RK×D and Y ∈ RK×D. Then,
+each view matrix is multiplied by a centering ma-
+trix H = IK−1K/K to make each column’s mean
+equal to zero and obtain centered second moment
+matrices as
+GX = HXX⊤H⊤/D,
+GY = HYY⊤H⊤/D
+(7)
+
+0.6
+0.7
+0.8
+0.9
+1.0
+RIE
+RIE
+CSE
+Recurrent Encoders
+Convolutional Encoders
+AE
+CSE
+AE
+PGE
+CAE
+PGE
+CAE
+Figure 5: Network representational consistency (RC):
+(top) recurrent encoders and (bottom) convolutional en-
+coders. Values closer to 1 indicates higher RC.
+Then, the representational similarity of the two
+views is computed using CKA as
+CKA(X, Y) = ⟨vec(GX), vec(GY)⟩
+||GX||F ||GY||F
+(8)
+where vec(.) is the vector-reshaped matrix, ⟨., .⟩ is
+the inner product, and ||.||F is the Frobenius norm
+to ensure that CKA ∈ [0, 1] where values close to 1
+indicate that the two instances are highly consistent,
+while values close to 0 indicate low consistency.
+Using CKA, we conduct pairwise similarity
+analysis across all six instances which yields 15
+comparisons for each model type. We report the
+mean of the resulting CKA values for each model
+type in Fig. 5. First, we observe that randomly
+initialized encoders (RIE) are highly consistent
+for both architectures (mean CKARIE/RNN
+≈
+mean CKARIE/CNN = 0.98).
+However, after
+training the encoder instances, convolutional net-
+works are more consistent than their recurrent coun-
+terparts. Note that this behaviour cannot be at-
+tributed to a difference in the number of trainable
+parameters between the two architectures since
+they are comparable. Moreover, all decoding-based
+learning objectives return mean CKA values above
+0.87, which indicates that their representational
+profiles are similar despite some noticeable differ-
+ences especially among the recurrent encoders. The
+only exception to this trend are model instances
+trained with contrastive loss since they are sig-
+nificantly less consistent compared to the other
+learning objectives (mean CKACSE/RNN = 0.61
+and mean CKACSE/CNN = 0.74). We empha-
+size that CKA is a second-order isomorphismic
+approach that operates on the similarity of the pair-
+wise sample similarity matrices across different
+views. Therefore, the anisotropic nature of AWEs
+reported in §4 cannot explain their similarity-based
+representational profiles, and by implication, their
+representational consistency.
+7
+Discussion
+Acoustic word embeddings (AWEs) are vector rep-
+resentations that encode the sound structure and
+acoustic-phonetic features of spoken words. AWEs
+are induced from actual acoustic realizations of
+speech, and therefore AWE models have to abstract
+away from non-linguistic dimensions of variabil-
+ity in speech signals (e.g., speaker characteristics,
+speech rate, recording conditions, etc). While an-
+alyzing the representational geometry of seman-
+tic word embeddings is a topic that has received
+a substantial attention in the NLP research com-
+munity, the interpretability of AWEs remains an
+under-explored topic and we are aware of a few
+prior studies in this direction (Matusevych et al.,
+2020b; Abdullah et al., 2021a,b). In this article,
+we made a number contributions in analyzing the
+representational geometry of AWEs and obtained
+research findings which we discuss and summarize
+in this section.
+Learning objective affects the geometry more
+than architecture.
+Our three analyses in this pa-
+per have shown that the learning objective shapes
+the representational geometry of the AWE encoders
+more than their underlying architectures. This find-
+ing suggests that recurrent and convolutional en-
+coders exhibit similar inductive biases while the
+learning process is mainly guided by the loss func-
+tion.
+AWE models tend to be anisotropic.
+Our anal-
+ysis in §4 has shown that AWEs tend towards be-
+ing minimally isotropic. However, this behavior
+is not an emergent property of the training pro-
+cess, but rather an inherent behavior of the neural
+network. Moreover, the degree of isotropy after
+training the model positively correlates with the
+acoustic word discrimination evaluation task. Since
+different models have different degree of isotropy
+and the representation space is not always uniform,
+we conclude that any comparison between differ-
+ent models based on absolute distance metrics such
+cosine distance will definitely lead to inaccurate
+observations.
+Word distinctiveness, but not frequency, pre-
+dicts category discriminability.
+While word
+acoustic distinctiveness has been found in §5 to
+be a good predictor of the degree to which a word
+category is compact and well-separated in the em-
+bedding space, word frequency does not correlate
+with category discriminability. In retrospective,
+
+this finding should not be surprising as frequent
+words tend to have shorter lengths. Shorter words
+have more phonological neighbours that are per-
+ceptually similar in form and thus they are more
+confusable with other words. Future work could
+employ more sophisticated linear mixed effects
+models to analyse the interaction between different
+lexical properties such as frequency, phonological
+neighbourhood density, and word length and their
+effect on word category discriminability.
+AWE models exhibit individual differences.
+Although AWE model instances trained with differ-
+ent random initializations are stable with respect to
+the performance of the evaluation task, they exhibit
+individual differences in their representational pro-
+files as shown in §6. However, the degree of the
+network representational consistency across differ-
+ent initializations depends on both the architecture
+and the learning objective. Contrastive objectives
+are less consistent than decoding-based objectives,
+while recurrent encoder are less consistent than
+their convolutional counterparts.
+Contrastive models have distinct representa-
+tional profiles.
+In the analyses we presented
+in this paper, we observed that the contrastive
+encoders behave differently than other encoders
+trained with non-contrastive losses. For example,
+word distinctiveness has been found to be a weak
+predictor of category discriminability in the em-
+bedding spaces of the contrastive encoders. Recall
+that our contrastive encoders have a stronger con-
+straint in grouping exemplars of the same category
+closer in the embedding space guided by the mar-
+gin hyperparameter, while decoding-based model
+lack this constraint. We hypothesize that this con-
+straint forces the models to emphasize the separa-
+bility of the lexical categories in the embedding
+space. Therefore, a stronger constraint seems to
+make contrastive encoders different compared to
+other learning objectives and different instances of
+the same contrastive encoder are less consistent in
+their representational geometry.
+8
+Conclusion
+In this paper, we have taken a closer, analytical
+look at the representational geometry of acoustic
+word embeddings (AWEs) from three different, but
+complementary perspectives: (1) embedding space
+uniformity, (2) word discriminability, and (3) net-
+work representational consistency. We have shown
+that the representational spaces of AWEs tend to-
+wards being minimally isotropic, or in other words,
+they utilize only a few dimensions of the embed-
+ding space. Another finding was that most AWE
+models rely on word length as a feature to discrim-
+inate between lexical categories since the word dis-
+criminability index positively correlates with the
+number of phonemes in a word. Furthermore, our
+representational consistency analysis have shown
+that AWE models exhibit individual differences in
+their representational profiles, with the contrastive
+encoders being the most inconsistent across differ-
+ent random initializations.
+Even though we focused on acoustic word em-
+beddings in this paper, our analytic methodology
+can also be employed for the interpretability of self-
+supervised speech representation models such as
+contrastive predictive coding (Oord et al., 2018)
+and wav2vec (Schneider et al., 2019). Also, the
+emergent representations of sublexical units such
+phonemes and syllables in speech neural networks
+can be analyzed using the our proposed methodol-
+ogy in this paper.
+9
+Acknowledgements
+The authors would like to thank the anonymous
+reviewers for their encouraging feedback and in-
+sightful comments on the paper. We express our
+heartfelt thanks to Miriam Schulz for proofreading
+the paper. This research is funded by the Deutsche
+Forschungsgemeinschaft (DFG, German Research
+Foundation), Project-ID 232722074 – SFB 1102.
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+and Michael Auli. 2019. wav2vec: Unsupervised
+pre-training for speech recognition. arXiv preprint
+arXiv:1904.05862.
+Shane Settle, Kartik Audhkhasi, Karen Livescu, and
+Michael Picheny. 2019.
+Acoustically grounded
+word embeddings for improved acoustics-to-word
+speech recognition. In Proc. ICASSP.
+Shane Settle and Karen Livescu. 2016a. Discrimina-
+tive acoustic word embeddings: Recurrent neural
+network-based approaches.
+In IEEE Spoken Lan-
+guage Technology Workshop (SLT).
+Shane Settle and Karen Livescu. 2016b. Discrimina-
+tive acoustic word embeddings: Recurrent neural
+network-based approaches. In Proc. IEEE Spoken
+Language Technology Workshop (SLT).
+Jan Strunk, Florian Schiel, and Frank Seifart. 2014.
+Untrained forced alignment of transcriptions and au-
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+Yaodong Zhang and James R Glass. 2009. Unsuper-
+vised spoken keyword spotting via segmental DTW
+on Gaussian posteriorgrams.
+In IEEE Workshop
+on Automatic Speech Recognition & Understanding
+(ASRU).
+
+Acoustic Word
+Embedding
+(a)
+(b)
+(c)
+Figure 6: A visual illustration of the different learning objectives for training AWE encoders: (a) correspondence
+auto-encoder (CAE): a sequence-to-sequence network with an acoustic decoder, (b) phonologically guided encoder
+(PGE): a sequence-to-sequence network with a phonological decoder, and (c) contrastive siamese encoder (CSE):
+a contrastive network trained via triplet margin loss. After training the model, only the encoder component of the
+model F is used to produce AWEs.
+1
+10
+100
+Epoch — log scale
+0
+0.2
+0.4
+0.6
+0.8
+mAP
+(a) Recurrent
+1
+10
+100
+Epoch — log scale
+0
+0.2
+0.4
+0.6
+0.8
+mAP
+(b) Convolutional
+Contrastive Siamese Encoder (CSE)
+Phonologically-Guided Encoder (PGE)
+Correspondence Autoencoder (CAE)
+Autoencoder (AE)
+RIE
+AE
+CAE
+PGE
+CSE
+0.0
+0.2
+0.4
+0.6
+0.8
+mAP
+RNN
+CNN
+(c) Best Epoch
+Figure 7: Evaluation on the same-different acoustic word discrimination task quantified by the word discrimination
+task and the mAP metric: Learning curves of 100 training epochs for (a) the recurrent encoder and (b) convolutional
+encoders. (c) mAP of the best epoch.
+Appendices
+A
+AWE Models
+See Fig. 6 for a visual illustration of the different
+learning objectives in our paper.
+B
+Intrinsic Evaluation
+The results of the intrinsic evaluation—same-
+different word discrimination task quantified by
+the mAP metric—is shown in Fig. 7. Note that the
+CAE model is pre-trained as autoencoder for 10
+epochs, following prior work (Kamper, 2019).
+C
+Randomly Initialized CNN Encoder
+To further investigate the minimally isotropic be-
+havior and the near 1 values of cosine similari-
+ties of the untrained, randomly initialized convo-
+lutional encoder reported in §4, we examine the
+potential contribution of two factors to this obser-
+vation; batch normalization (BN) and the activation
+function of the convolutional layers. The result of
+this analysis is depicted in Fig. 8. We observe that
+removing the BN layer has no effect on the distri-
+butions of cosine similarities as they remain almost
+identical to the encoder that has a BN layer—see
+Fig. 8(b). However, changing the activation func-
+tion from the unbounded ReLU to the bounded
+Tanh in the convolutional layers makes the distri-
+butions of cosine similarities move towards zero
+mean, even though they remain closer to 1 than
+0. Therefore, this behavior seems to be related to
+the inner dynamics of the convolutional operation
+and gets amplified where the activation function
+in the convolutional layers are unbounded. Nev-
+ertheless, identifying the source of this behavior
+requires further investigation with different activa-
+tion functions and a controlled ablation study.
+
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+-0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+20.0
+Density
+Cosine similarity
+Convolutional Encoder (batch norm + ReLU)
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+-0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+10.0
+20.0
+Density
+Cosine similarity
+Convolutional Encoder (only ReLU)
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+-0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+5.0
+10.0
+Density
+Cosine similarity
+Convolutional Encoder (batch norm + Tanh)
+(a)
+(b)
+(c)
+Figure 8: Cosine similarity distributions across three
+different variants of convolutional encoders: (a) con-
+volutional layers with batch normalization and ReLU
+non-linearity, (b) convolutional layers with ReLU non-
+linearity but without batch normalization, and (3) con-
+volutional layers with batch normalization and Tanh
+non-linearity.
+D
+Performance Stability across Different
+Runs
+For the analysis in §6, we have trained six neu-
+ral network instances for each encoder type using
+the same training samples to investigate the per-
+formance stability and representational consistency
+of training runs that differ in their random seeds.
+A summary statistics for the performance on the
+evaluation task measured by the mAP metric is
+shown in Table 2. One can observe only trivial
+differences on the evaluation tasks. Therefore, we
+conclude that the performance of different train-
+ing runs is stable and our findings on the network
+representational consistency reported in §6 cannot
+be explained by quantitative differences, but rather
+by representational discrepancies due to disagree-
+ment in the geometric arrangement of the speech
+samples in the embedding space.
+E
+Qualitative Analysis
+To further inspect the representation space and its
+neighborhood structure, we conduct a qualitative
+analysis by querying the representation space with
+a few word samples. In this analysis, we compute
+word category centroids by averaging the word em-
+beddings of the training samples, then we use a
+word centroid as a query and obtain the top-10
+ranked nearest neighbors. The result of this analy-
+sis is shown in Fig. 3. For the majority of the ex-
+amples in Fig. 3, we observe that there is a strong
+Objective
+Arch.
+mean
+max
+min
+std
+AE
+CNN
+0.137
+0.141
+0.133
+0.0026
+RNN
+0.183
+0.186
+0.179
+0.0024
+CAE
+CNN
+0.505
+0.510
+0.500
+0.0040
+RNN
+0.646
+0.650
+0.643
+0.0029
+PGE
+CNN
+0.595
+0.599
+0.592
+0.0033
+RNN
+0.704
+0.710
+0.687
+0.1000
+CSE
+CNN
+0.676
+0.680
+0.674
+0.0023
+RNN
+0.742
+0.745
+0.739
+0.0027
+Table 2: mAP statistics across six different runs for
+each model type.
+word onset bias where the most similar words are
+those that begin with a similar sounding prefix as
+the query word.
+
+Query (↓)
+Convolutional Encoders (CNN)
+Recurrent Encoders (RNN)
+AE
+CAE
+PGE
+CSE
+AE
+CAE
+PGE
+CSE
+mentioned
+mention
+mention
+mention
+mention
+mention
+mention
+mention
+mention
+wretched
+mansion
+mansion
+mansion
+wretched
+mansion
+mansion
+mansion
+nation
+motion
+legends
+merchant
+nation
+merchant
+merchant
+merchant
+midst
+merchant
+management
+mission
+merchant
+motion
+legends
+mission
+merchants
+making
+merchants
+mental
+motion
+nation
+mountain
+pinching
+motion
+wilson
+magic
+vincent
+merchants
+making
+merchants
+massive
+merchant
+nation
+matrons
+pinching
+midst
+vincent
+mission
+mental
+message
+midst
+mission
+medicine
+milking
+nineteen
+magician
+transient
+regiment
+missing
+merchant
+crouching
+winter
+nature
+motion
+motion
+winter
+nature
+magician
+midst
+vessel
+rachel
+wretched
+hudson
+intellectual
+individual
+introduction
+intellect
+intellect
+individual
+individual
+individual
+introduction
+interesting
+individual
+individual
+adjoining
+interesting
+introduction
+intelligence
+individual
+indifferent
+interrupted
+introduction
+recollection
+neglected
+uncomfortable
+introduction
+immature
+newton
+indifference
+intelligent
+delightful
+petition
+intelligent
+intellect
+objection
+institution
+attraction
+encouragement
+individual
+magician
+intelligence
+uncomfortable
+implacable
+departure
+intellect
+interrupted
+employing
+hokosa
+interesting
+intelligent
+delightful
+imitation
+immature
+intelligence
+impetuous
+compassion
+invisible
+interpretation
+theatrical
+hokosa
+indifferent
+indifferently
+employed
+departure
+interrupted
+industrial
+thoughtful
+encountered
+encouragingly
+unconditional
+natural
+convention
+imperfectly
+incapable
+industrial
+neglected
+implacable
+impetuous
+accumulated
+consulted
+incredible
+insensible
+election
+maker
+labor
+naked
+naked
+baker
+labor
+nature
+baker
+liquor
+liquor
+natured
+liquor
+naked
+nature
+local
+nature
+negro
+labored
+nature
+natured
+negro
+walker
+naked
+liquor
+eaten
+labour
+local
+nature
+liquor
+local
+labour
+labour
+baker
+wicker
+labour
+baker
+local
+naked
+labor
+labor
+nature
+leaping
+major
+major
+native
+labour
+major
+major
+labor
+lifted
+labor
+negro
+nature
+rachel
+natured
+negro
+naked
+walker
+native
+native
+major
+liquor
+baker
+neighbors
+newspaper
+local
+making
+wicker
+matrons
+labored
+liquor
+vapor
+mink
+nature
+navy
+labor
+vigor
+leaping
+negro
+labors
+vigour
+profession
+position
+procession
+procession
+professor
+position
+procession
+procession
+professor
+proceed
+professor
+professor
+sufficient
+professors
+possession
+proportion
+procession
+positions
+position
+position
+procession
+possessions
+position
+perfection
+perfection
+physician
+possession
+petition
+professors
+proceeded
+professor
+possession
+sufficient
+proceeded
+professors
+pushing
+efficiency
+physician
+possessions
+protection
+proposition
+possessed
+pushing
+professors
+efficient
+condition
+permission
+proportions
+proportion
+prison
+perfection
+possession
+petition
+procession
+discussion
+position
+production
+possessions
+positions
+physician
+prevent
+presumption
+positions
+possessions
+petition
+perfect
+discussion
+positions
+position
+protested
+commission
+professor
+compassion
+discussion
+preferred
+precious
+physician
+proceed
+physician
+petition
+pushing
+seized
+ceased
+ceased
+ceased
+thieves
+ceased
+ceased
+ceased
+thieves
+freedom
+season
+seizing
+ceased
+faded
+feasts
+thieves
+ceased
+seated
+thieves
+season
+season
+cities
+scenes
+saves
+fuse
+faded
+saves
+thieves
+feast
+singing
+thieves
+seats
+jesus
+singing
+seems
+saves
+seizing
+scenes
+saves
+seems
+spheres
+scenes
+scenes
+ceasing
+feared
+feeding
+seems
+scenes
+feels
+season
+ceasing
+seems
+ceasing
+season
+feast
+seemed
+cities
+cities
+saints
+feast
+saves
+sweetest
+saints
+feast
+season
+field
+feast
+seats
+species
+seated
+faced
+saved
+seats
+seeming
+sins
+seemed
+speed
+saying
+seemed
+seizing
+scenes
+experiments
+experiment
+experiment
+experiment
+experiment
+experiment
+experiment
+experiment
+experiment
+experience
+experienced
+experience
+experienced
+experienced
+experienced
+experienced
+attendants
+experienced
+experience
+experienced
+garments
+experience
+experience
+experience
+extremities
+experiences
+experiences
+experiences
+extermination
+extinguished
+experiences
+experiences
+islands
+extinguished
+extremities
+expense
+expense
+experiences
+exposed
+expressions
+experienced
+exchange
+established
+embarrassment
+experience
+expected
+extremities
+extermination
+prominence
+extremities
+extraordinary
+expanse
+aramis
+exchange
+expense
+extremities
+edmunds
+expressions
+extinguished
+extraordinary
+disturbance
+expressions
+expanse
+extremity
+instruments
+extremely
+extremity
+extremities
+examined
+extremities
+extinguished
+expression
+attendance
+extremity
+expanse
+expressions
+vanished
+extent
+exclusion
+expensive
+commons
+Table 3: Top-10 nearest word embedding centroids for a word sample.
+
diff --git a/mtE1T4oBgHgl3EQfOQNK/content/tmp_files/load_file.txt b/mtE1T4oBgHgl3EQfOQNK/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf,len=1452
+page_content='Analyzing the Representational Geometry of Acoustic Word Embeddings  Badr M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Abdullah and Dietrich Klakow  Language Science and Technology (LST), Saarland University, Germany  Saarland Informatics Campus  { babdullah | dietrich }@lsv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='uni-saarland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='de  Abstract  Acoustic word embeddings (AWEs) are vector  representations such that different acoustic ex-  emplars of the same word are projected nearby  in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In addition to their  use in speech technology applications such as  spoken term discovery and keyword spotting,  AWE models have been adopted as models of  spoken-word processing in several cognitively  motivated studies and have been shown to ex-  hibit human-like performance in some audi-  tory processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Nevertheless, the rep-  resentational geometry of AWEs remains an  under-explored topic that has not been studied  in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In this paper, we take a closer  analytical look at AWEs learned from English  speech and study how the choice of the learn-  ing objective and the architecture shapes their  representational profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' To this end, we em-  ploy a set of analytic techniques from machine  learning and neuroscience in three different  analyses: embedding space uniformity, word  discriminability, and representational consis-  tency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Our main findings highlight the promi-  nent role of the learning objective on shap-  ing the representation profile compared to the  model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  1  Introduction  Due to their ubiquity, word embeddings are nowa-  days a central component in natural language pro-  cessing (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Inducing word embeddings from  text yields representations such that words occur-  ring in similar contexts are nearby in the vector  space (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Pennington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=',  2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Therefore, the representational geometry of  text-based word embeddings captures lexical simi-  larity and semantic relatedness at multiple levels of  granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Word embeddings, and their underly-  ing distributional semantic models, have also been  adopted as models of human semantic memory in  cognitive science research (Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Ne-  matzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Grand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Figure 1: UMAP projection (McInnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2018) of  a sample of acoustic word embeddings (AWEs) pro-  duced by a correspondence autoencoder (CAE) model  trained on English read speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' AWE models project  different exemplars of the same word type closer in the  embedding space while abstracting away from speaker  and context variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  In the speech processing domain, researchers  have independently developed representations of  acoustic segments that correspond to linguistic  units (Levin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Bengio and Heigold, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Kamper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Settle and Livescu, 2016a,  inter alia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' A notable example of such represen-  tations are acoustic word embeddings (AWEs)—  vector representations that encode the sound struc-  ture of words, not their semantic and syntactic  structure—see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' AWEs support voice-based  speech technology applications such as query-by-  example spoken term discovery (Zhang and Glass,  2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Jansen and Durme, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Metze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2013)  and keyword spotting (Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Rohlicek,  1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In addition, AWEs can be leveraged to fa-  cilitate access to speech recordings of endangered  spoken languages that might lack standardized writ-  ing systems (Bird, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' San et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2021)  However, there are fundamental differences be-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='03012v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='CL]  8 Jan 2023  iste  a  a  bel  Datween text-based and speech-based word embed-  dings that have to do with the degree of variability  between the two modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Contrary to written  words which have context-invariant orthographic  realizations,1 spoken words are notoriously vari-  able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  The underlying sources of variability in  speech include speaker-related factors such as vo-  cal tract shape, gender, age, and dialect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In addition,  two acoustic instances, or exemplars, of the same  word will vary in different phonological and se-  mantic contexts even if they are produced by the  same speaker (Jurafsky, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Therefore, acous-  tic word embeddings are not static, but have to be  computed “on the fly” given a speech segment as  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Models of AWEs need to abstract away from  speaker and context variability to project different  acoustic exemplars of the same word onto (ideally)  the same point of the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Nevertheless, AWEs have not yet been exten-  sively studied in the literature from a neural net-  work interpretability point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We are only  aware of a few prior efforts in this direction that  have either analyzed the representational geometry  of AWEs from a cognitively motivated angle (Ma-  tusevych et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Abdullah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2021a) or  from a cross-linguistic perspective (Abdullah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=',  2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In this paper, we make a contribution in  this direction and use analytic techniques from ma-  chine learning and neuroscience in three different  analytic studies: embedding space uniformity (§4),  word discriminability (§5), and representational  consistency (§6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  2  Acoustic Word Embedding Models  Given an acoustic signal that corresponds to a spo-  ken word represented as a temporal sequence of T  acoustic feature vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', a = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', aT ),  the goal of an AWE model is to transform a into a  fixed-dimensionality vector representation e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Due  to the variability in speech production (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', speech  rate, emotional state, etc), the length of the acoustic  segment T varies between different exemplars, or  instances, of the same word type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Therefore, this  task is modeled as a mapping F : A −→ RD, where  A is the (continuous) space of acoustic sequences  and D is the dimensionality of the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' For-  mally, transforming a variable-length acoustic in-  put into a D-dimensional AWE is described as  e = F(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' θF) ∈ RD  (1)  1although some orthographic variation exists in informal,  user-generated text such as tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  where θF are the parameters of the encoder  function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  In a supervised setting of train-  ing AWE models, one assumes a dataset D =  {(a1, w1), (a2, w2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' , (aN, wN)} of N spoken  word instances where wi is the word type, or lexi-  cal category, of the ith acoustic sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In this pa-  per, we experiment with two architectural choices—  recurrent and convolutional—and employ four dif-  ferent learning objectives for training AWE models  from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Next, we formally describe  each of the objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='1  Correspondence Autoencoder  In the correspondence autoencoder (CAE) (Kam-  per, 2019), each training acoustic word sample a  is paired with another sample that corresponds to  the same word type a+ = (a1  +, a2  +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', aS  +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The  acoustic encoder F takes a as input and produces  an embedding e, which is then fed to an acous-  tic decoder H that aims to sequentially recon-  struct the corresponding acoustic sequence a+—  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The objective is to minimize the L2  distance at each timestep in the decoder, which is  equivalent to  J =  S  �  i=1  ∥ai  + − Hi(e)∥2  (2)  where a+  i is the ground-truth acoustic feature vec-  tor at timestep i and Hi(e) is the reconstructed  acoustic vector at timestep i as a function of the  embedding e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Learning the correspondence be-  tween different acoustic realizations of the same  word type seems to encourage the encoder to build  up speaker-invariant word representations while  preserving linguistically-relevant phonetic informa-  tion (Matusevych et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' When the target  acoustic sequence to generate is the same as the  input signal a, this corresponds to a conventional  autoencoder (AE) which we consider as one of our  learning objectives in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='2  Phonologically Guided Encoder  The phonologically guided encoder (PGE) is  trained as component in a sequence-to-sequence  model to map acoustics into phonology (Abdullah  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Given the output of the encoder as  an embedding e, a phonological decoder G(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' θG)  is trained to decode the corresponding phonologi-  cal sequence ϕ = (ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' , ϕτ) of the word-form  —see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The objective is to minimize a cate-  gorical cross-entropy loss at each decoder timestep,  which is equivalent to minimizing the term  J = −  �  (ai,wi)∈D  log P  �  ϕ|ei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' θG  �  = −  �  (ai,wi)∈D  τ  �  t=1  log P  �  ϕt|t, ei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' θG  �  (3)  where P  �  ϕt|t, ei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' θG  �  is the probability of the  phoneme ϕt at the tth timestep, conditioned on  the previous phoneme sequence ϕ<t and the AWE  e, and θG are the parameters of the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The  intuition of this learning objective is the follow-  ing: although their acoustic realizations vary due to  speaker and context variability, different exemplars  of the same word category would have identical  phonological sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We thus expect the en-  coder to project exemplars of the same lexical cate-  gory nearby in the embedding space while embed-  ding similarity in the vector space should correlate  with phonological similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='3  Contrastive Siamese Encoder  The contrastive siamese encoder (CSE) has been ex-  plored in the context of AWEs with both recurrent  and convolutional architectures in several studies  (Settle and Livescu, 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Kamper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Jacobs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Contrary to the previously  described objectives, the CSE explicitly minimizes  the distance between exemplar embeddings of the  same word type—see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' First, each acoustic  word instance is paired with another instance of the  same word type (a, a+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Given their embeddings  (ea, e+), the objective is then to minimize a triplet  margin loss  J = max  �  0, m + d(ea, e+) − d(ea, e−)  �  (4)  Here, d(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=') is the cosine distance and e− is an  AWE that corresponds to a different word type  sampled from the mini-batch such that the term  d(ea, e−) is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' This objective clusters  acoustic instances of the same word type closer in  the embedding space while pushing away instances  of other word types by a distance defined by the  margin hyperparameter m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  3  Data, Setup, and Intrinsic Evaluation  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='1  Experimental Data  The data in our study is drawn from the the Lib-  riSpeech dataset which contains read speech record-  ings of American-English (Panayotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2015),  which is a public dataset under the CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='0 li-  cense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We sample 384 speakers from for training  and 128 for evaluation—disjoint sets—and obtain  word-aligned speech samples using the Montreal  Forced Aligner (McAuliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' To make  our models comparable with prior work, which has  focused on AWEs for low-resource languages, we  sample ∼ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='4k samples for training and ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='7k  for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  The phonetic transcription for  each word is produced using the online WebMaus  G2P tool (Strunk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Then, each acous-  tic segment is parametrized as a sequence of 39-  dimensional Mel-frequency spectral coefficients  of 25ms frames with 15ms overlap—the conven-  tional feature representation of speech in automatic  speech recognition (ASR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' It is worth pointing out  that in this paper we consider each morphological  variant of a lexeme as a separate lexical category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  For example, different inflections of the lexeme  MAKE such as {MADE, MAKING, MAKER, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='}  represent different lexical categories, each with its  own exemplars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='2  Architectures, Hyperparameters, and  Training Details  CNN Acoustic Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  We employ a 3-layer  temporal convolutional network (1D-CNN) with  256, 384, and 512 filters and widths of 4, 8, and 16  for each layer and keep stride step at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Following  each convolutional operation, we apply batch nor-  malization, ReLU non-linearity, and dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We  apply average pooling to downsample the repre-  sentation at the end of the convolution block, then  apply one non-linear layer with Tanh on the CNN  output, which yields a 512-dimensional AWE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  RNN Acoustic Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  We employ a 3-layer  directional Gated Recurrent Unit (GRU) with a  hidden state dimension of 512, then apply one non-  linear layer with Tanh on the GRU output, which  yields a 512-dimensional AWE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We apply layer-  wise dropout with a probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Phonological Decoder G(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' θG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  We employ a  1-layer GRU of 512 units hidden state that takes the  512-dimensional AWE as the initial hidden state  and decodes the corresponding phonological se-  quence without teacher forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content='0  Density  Cosine similarity  (d) Phonologically-Guided Encoder  (e) Contrastive Siamese Encoder  (a) Randomly Initialized Encoder  (b) Autoencoder  (c) Correspondence Autoencoder  Recurrent Models  Convolutional Models  Cross category  Within category  Figure 2: Distribution of cosine similarity scores of the different models for within category samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', exemplar  pairs of the same word type) and cross-category samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', sample pairs that correspond to different word types).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Each row in the figure corresponds to one learning objective and each column corresponds to one architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Contrastive Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  For the CSE, we experiment  with different values of the margin hyperparameter  m = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='5}, out of which 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='4 yields  the best performance on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Training Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  All models in this study are  randomly initialized with each parameter drawn  uniformly from [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Then, each model  is trained for 100 epochs with a batch size of 256  using the ADAM optimizer (Kingma and Ba, 2015)  and an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The learning  rate is reduced by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='5 if the mAP on the  validation set does not improve for 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  We build our models using  PyTorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2019) and use FAISS (John-  son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2017) for efficient similarity search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Our  code is based on our prior work in building and  analyzing AWEs (Abdullah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2021a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='3  Quantitative Evaluation  We conduct an intrinsic evaluation for the AWEs  to assess the performance of our models using the  same-different acoustic word discrimination task  with the mean average precision (mAP) metric  (Carlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Kamper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Settle  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Algayres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' This task evalu-  ates the ability of the model to determine whether  two given speech segments correspond to the same  word type—that is, whether or not two acoustic  segments are exemplars of the same category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The  results of the evaluation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 7 in the  appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We observe that each recurrent encoder  outperforms its convolutional counterpart within  each objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Moreover, the performance largely  depends on the strength of the supervision signal  where the contrastive encoders outperform other  objectives that lack explicit loss to group exem-  plars of the same category closer in the embedding  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  4  Analysis 1: Embedding Space  Uniformity  In our first analysis, we take a closer look at how  uniform are representational spaces of AWE mod-  els by analyzing the distribution of cosine similarity  for each model type and the degree to which the  embeddings are isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='1  Distribution of Cosine Similarity  One way of analyzing the geometry of represen-  tation spaces in the acoustic domain is by inspect-  ing the similarity distributions of exemplars of the  same lexical category (or word type) versus ran-  domly sampled, cross-category exemplars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We per-  form this analysis on the training samples and de-  pict the result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We observe that the dif-  ference between the means of the within-category  and those of cross-category distributions is largely  dependent on the strength of the supervision sig-  nal with the randomly initialized encoders (RIE)  having the smallest mean differences for both archi-  tectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The contrastive encoders have the largest  mean difference–with mean cross-category scores  centered at the zero—which is intuitive given the  explicit supervision signal they receive in grouping  exemplars of the same category closer in the em-  bedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' One surprising observation is the  RIE  AE  CAE  PGE  CSE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content='10  Isotropy score  RNN  CNN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='10  Isotropy score  (b)  Figure 3: (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The degree of isotropy of AWE for each  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' (b) Correlation between the word discrimina-  tion performance measured by mAP and isotropy score  (Pearson r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='89, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  behavior of the untrained convolutional encoder  which gives cosine similarity scores very close to  1 for each input pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In appendix C, we demon-  strate that this behavior is mainly caused by the  unbounded activation function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', ReLU) in the  convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='2  Degree of Isotropy  Although inspecting the cosine similarity distribu-  tions is an insightful analysis, it does not enable us  to make well-informed judgments about the unifor-  mity of the representation spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Here, we ask two  questions: (1) do AWE models utilize all dimen-  sions of the vector space to represent the speech  samples and separate the categories?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' and (2) how  do architecture and learning objective affect the dis-  tributivity of information in the embedding space?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  To answer these questions, we inspect the degree  of isotropy in the representation spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' An embed-  ding space is said to be maximally isotropic if the  variance is uniformly distributed across all dimen-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Prior work in NLP has found that seman-  tic word embeddings tend to be anisotropic since  they only utilize a few dimensions of the vector  space—an effect that has been observed for word  embeddings that are static (Mimno and Thomp-  son, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Mu and Viswanath, 2018) as well as  contextualized (Ethayarajh, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Rudman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The degree of isotropy in  acoustic embeddings, however, remains so far un-  known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' To inspect the degree of isotropy of the  AWE vector spaces, we use the IsoScore metric  recently proposed by Rudman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' (2022), which  is—to the best of our knowledge—the only metric  in the literature that is grounded on the mathemati-  cal definition of isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The IsoScore metric oper-  ates on the covariance matrix of the embedding di-  mensions and returns values between 0 (minimally  isotropic) and 1 (maximally isotropic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We quan-  tify the degree of isotropy using IsoScore for each  model type and show the result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We  observe that IsoScore returns values that are within  the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='095], which indicates that em-  bedding spaces for all models tend towards being  minimally isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' However, the embeddings  of untrained, randomly initialized encoders (RIE)  tend to be extremely anisotropic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', IsoScore val-  ues close to 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' This observation suggests that the  anisotropic space does not “emerge” during the  model training but rather that it is an inherent prop-  erty of the encoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We are not aware  of prior work in NLP that has studied the degree  of isotropy in untrained NLP models to investigate  whether anisotropic spaces are an emergent or in-  herent feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In our case, training with a learning  objective that encourages the model to separate  word categories moves the representation space  more towards utilizing more dimensions, therefore  resulting in a higher degree of isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Moreover,  recurrent encoders tend to be more isotropic than  their convolutional counterparts within the same  learning objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Despite the tendency of all models to be  anisotropic, we find a strong positive correlation be-  tween the degree of isotropy and the performance  on word discrimination—see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' That is, the  more dimensions the model utilizes in the represen-  tation space, the better it performs on the intrinsic  evaluation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  5  Analysis 2: Word Discriminability  Ideally, AWE models should project exemplars of  the same word category onto the same point in the  embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' However, there are no strong  constraints during training to encourage maximal  separability between different word categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In  this analysis, we seek to answer two questions:  (1) how well-separated are the word categories of  the training samples?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' and (2) to what degree do  lexical properties predict the discriminability of  word categories?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='1  Category Discriminability Index  In order to investigate the geometric density of  each word category in the representation space,  we need to measure within-category compactness  and cross-category separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Inspired by the  exemplar discriminability index proposed in the  neuroscience literature (Nili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2020), we define  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='0  mAP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='0  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' CDI  RIE  AE  CAE  PGE  CSE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='0  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' CDI  RNN  CNN  (a)  (b)  Figure 4: Averaged Category Discriminability Index  (CDI) for each AWE model with error bars showing  standard deviation over word categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  (b) Corre-  lation between the word discrimination performance  measured by mAP and averaged CDI (Pearson r =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='90, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  category discriminability index (CDI) as a metric  that operates on within-category and cross-category  distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' If we consider each lexical category in  the training set as a set of its exemplar embeddings  C = {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' , e|C|}, CDI is defined for a single  category C as  CDI(C) = 1  |C|  �  ∀ei∈C  �  �  ∀ej∼C|j̸=i  d(ei, ˜ej) − d(ei, ej)  �  (5)  where d(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=') is the cosine distance and ej is a  within-category sample while ˜ej is an embedding  sampled from a different category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' If we normal-  ize the embeddings, CDI ∈ [−1, 1] with values  closer to 1 indicating higher word discriminability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  We compute CDI for each word category in the  training set and take the average over categories to  estimate how well the categories are separated in  the embedding space of each model type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The re-  sult of this analysis is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' For each  learning objective, we observe that word discrim-  inability is higher in the recurrent encoders com-  pared to their convolutional counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Besides  that, the contrastive objective yields encoders with  a higher word discriminability index regardless of  the architecture type—recurrent vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' convolutional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Furthermore, we report a strong positive correla-  tion between average CDI and the performance on  the evaluation task—see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 4(b), indicating that  word discrimination performance on future, held-  out samples can be predicted based on the CDI  computed on the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='2  Effect of Frequency and Distinctiveness  The CDI quantifies the separability and compact-  ness for each lexical category in the representation  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Next, we aim to identify the factors that  could make a lexical category compact and well-  separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  In this analysis, we study the effect of two lexical  properties that could be quantified in a data-driven  approach: word frequency and acoustic distinctive-  ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Our initial hypothesis is that a word category  with many training exemplars becomes more dis-  criminable in the embedding space as the repeated  exposure to samples of various degrees of variabil-  ity should enable the model to learn compact and  precise representation for categories with high fre-  quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Also, words that are acoustically distinct  have fewer competitors in the perceptual space,  thus they should be more separable than words with  many phonological neighbours that sound similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Therefore, we expect word acoustic distinctiveness  (WAD) to positively correlate with CDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In this  analysis, we operationalize WAD using two met-  rics: word length (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', the number of phonemes)  and phonological distinctiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Word length con-  tributes to WAD since word formation in natural  languages is a combinatorial process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' That is, in-  creasing the number of phonemes in a word-form  decreases the likelihood of encountering a simi-  larly sounding word-form which makes it less con-  fusable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' However, the word formation process is  governed by language-specific phonotactic rules  which makes some sound combinations more prob-  able than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' To capture the probabilistic nature  of sound sequences, we employ phonological in-  formation content (PIC), an information-theoretic  metric that estimates WAD based on its phoneme-  to-phoneme transition probabilities (Meylan and  Griffiths, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Given a word-form as a sequence  of phonemes ϕ = (ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' , ϕτ), PIC is defined as  PIC(ϕ) = −  τ  �  i=1  log pθ(ϕi|ϕ<i)  (6)  where pθ is a probabilistic phoneme-level language  model (PLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We estimate pθ using a trigram PLM  with the counts of the phonemes in the training  word categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Higher values of PIC indicate less  probable phoneme sequences thus more distinct  word-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Note that PIC is not length normalized  and therefore shorter words tend to have lower PIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Next, we conduct a correlation analysis between  word CDI and the three lexical predictors: fre-  Objective  Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Frequency  Length  PIC  AE  CNN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='081†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='315†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='263†  RNN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='087†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='357†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='306†  CAE  CNN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='021  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='376†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='274†  RNN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='077†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='447†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='359†  PGE  CNN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='039*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='011  RNN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='043*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='325†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='263†  CSE  CNN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='131†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='075†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='031  RNN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='109†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='100†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='030  Table 1: Pearson correlation (r) between word category  discriminability index (CDI) and three lexical proper-  ties: frequency, length, and phonological information  content (PIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Statistical significance is marked with *  and † for p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='05 and p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='001, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  quency, length, and PIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The result of this analysis  is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Surprisingly, our correlation  analysis shows that lexical frequency is a poor pre-  dictor of CDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Although in five out of eight mod-  els the frequency positively correlates with CDI,  the correlation is rather weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' However, measures  of acoustic distinctiveness have a stronger corre-  lation with CDI compared to frequency, and the  strength of the correlation is more noticeable in all  decoding-based models—except the convolutional  PGE—compared to contrastive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We also  find it surprising that PIC is not a better predictor of  CDI than word length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' However, it has been shown  in a related work that autoencoder-based AWEs en-  code duration as an acoustic feature (Matusevych  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Taken together with our findings, this  suggests that the models exploit and rely on acous-  tic word length as a feature to discriminate between  the lexical categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Arguably, word length is a  more accessible feature to learn from the acoustic  signal compared to structural phonological regular-  ities in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  6  Analysis 3: Network Representational  Consistency  Suppose we train two instances of the same archi-  tecture and learning objective on the same training  samples, but each with different random initial-  izations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Do these two neural network instances  exhibit differences in their representational geome-  tries?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In this section, we shed light on the repre-  sentational discrepancies caused by different ini-  tializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In other words, we are interested in  quantifying the degree to which variability in the  initial conditions affects the way two models sepa-  rate the same set of speech samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='1  Performance Stability  First, we quantify the effect of the initial weights  on the evaluation task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' To this end,  we train six model instances—in identical setup but  with different initializations—for each architecture  and each learning objective, which yields 48 model  instances in total (6 × 4 RNN runs and 6 × 4 CNN  runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We evaluate each model instance on the  acoustic word discrimination task while observing  the result variation per model type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The result of  the performance stability analysis is shown in Ta-  ble 2 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We observe that all instances  have converged and the performance is fairly stable  across different runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='2  Representational Discrepancies  Our previous performance stability analysis has  demonstrated that different DNN instances exhibit  only trivial quantitative differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' However, a  stable performance on the evaluation task does  not entail an identical representational geometry  across different instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' That is, two network  instances could have have an identical performance  on the evaluation task while each having a distinct  representational geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' To closely investigate  representational discrepancies between network in-  stances, we employ the representational consis-  tency (RC) analysis (Mehrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2020), which  is a neuroscience-inspired technique based on the  representational similarity analysis (RSA) frame-  work (Kriegeskorte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' For our analysis,  we operationalize the RC using linear Centered  Kernal Alignment (CKA) as a representational sim-  ilarity measure of two views of the same input sam-  ples (Kornblith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' CKA abstracts away  from the embeddings themselves and operates on  pairwise distances between the sample representa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Concretely, given K spoken-word samples  aK  1 = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' , aK}, we embed the samples using  two encoder instances to obtain two different views  of the samples X ∈ RK×D and Y ∈ RK×D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Then,  each view matrix is multiplied by a centering ma-  trix H = IK−1K/K to make each column’s mean  equal to zero and obtain centered second moment  matrices as  GX = HXX⊤H⊤/D,  GY = HYY⊤H⊤/D  (7)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='0  RIE  RIE  CSE  Recurrent Encoders  Convolutional Encoders  AE  CSE  AE  PGE  CAE  PGE  CAE  Figure 5: Network representational consistency (RC):  (top) recurrent encoders and (bottom) convolutional en-  coders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Values closer to 1 indicates higher RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Then, the representational similarity of the two  views is computed using CKA as  CKA(X, Y) = ⟨vec(GX), vec(GY)⟩  ||GX||F ||GY||F  (8)  where vec(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=') is the vector-reshaped matrix, ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='⟩ is  the inner product, and ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='||F is the Frobenius norm  to ensure that CKA ∈ [0, 1] where values close to 1  indicate that the two instances are highly consistent,  while values close to 0 indicate low consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Using CKA, we conduct pairwise similarity  analysis across all six instances which yields 15  comparisons for each model type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We report the  mean of the resulting CKA values for each model  type in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' First, we observe that randomly  initialized encoders (RIE) are highly consistent  for both architectures (mean CKARIE/RNN  ≈  mean CKARIE/CNN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  However, after  training the encoder instances, convolutional net-  works are more consistent than their recurrent coun-  terparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Note that this behaviour cannot be at-  tributed to a difference in the number of trainable  parameters between the two architectures since  they are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Moreover, all decoding-based  learning objectives return mean CKA values above  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='87, which indicates that their representational  profiles are similar despite some noticeable differ-  ences especially among the recurrent encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The  only exception to this trend are model instances  trained with contrastive loss since they are sig-  nificantly less consistent compared to the other  learning objectives (mean CKACSE/RNN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='61  and mean CKACSE/CNN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We empha-  size that CKA is a second-order isomorphismic  approach that operates on the similarity of the pair-  wise sample similarity matrices across different  views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Therefore, the anisotropic nature of AWEs  reported in §4 cannot explain their similarity-based  representational profiles, and by implication, their  representational consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  7  Discussion  Acoustic word embeddings (AWEs) are vector rep-  resentations that encode the sound structure and  acoustic-phonetic features of spoken words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' AWEs  are induced from actual acoustic realizations of  speech, and therefore AWE models have to abstract  away from non-linguistic dimensions of variabil-  ity in speech signals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', speaker characteristics,  speech rate, recording conditions, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' While an-  alyzing the representational geometry of seman-  tic word embeddings is a topic that has received  a substantial attention in the NLP research com-  munity, the interpretability of AWEs remains an  under-explored topic and we are aware of a few  prior studies in this direction (Matusevych et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=',  2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Abdullah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2021a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In this article,  we made a number contributions in analyzing the  representational geometry of AWEs and obtained  research findings which we discuss and summarize  in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Learning objective affects the geometry more  than architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Our three analyses in this pa-  per have shown that the learning objective shapes  the representational geometry of the AWE encoders  more than their underlying architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' This find-  ing suggests that recurrent and convolutional en-  coders exhibit similar inductive biases while the  learning process is mainly guided by the loss func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  AWE models tend to be anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Our anal-  ysis in §4 has shown that AWEs tend towards be-  ing minimally isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' However, this behavior  is not an emergent property of the training pro-  cess, but rather an inherent behavior of the neural  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Moreover, the degree of isotropy after  training the model positively correlates with the  acoustic word discrimination evaluation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Since  different models have different degree of isotropy  and the representation space is not always uniform,  we conclude that any comparison between differ-  ent models based on absolute distance metrics such  cosine distance will definitely lead to inaccurate  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Word distinctiveness, but not frequency, pre-  dicts category discriminability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  While word  acoustic distinctiveness has been found in §5 to  be a good predictor of the degree to which a word  category is compact and well-separated in the em-  bedding space, word frequency does not correlate  with category discriminability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In retrospective,  this finding should not be surprising as frequent  words tend to have shorter lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Shorter words  have more phonological neighbours that are per-  ceptually similar in form and thus they are more  confusable with other words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Future work could  employ more sophisticated linear mixed effects  models to analyse the interaction between different  lexical properties such as frequency, phonological  neighbourhood density, and word length and their  effect on word category discriminability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  AWE models exhibit individual differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Although AWE model instances trained with differ-  ent random initializations are stable with respect to  the performance of the evaluation task, they exhibit  individual differences in their representational pro-  files as shown in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' However, the degree of the  network representational consistency across differ-  ent initializations depends on both the architecture  and the learning objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Contrastive objectives  are less consistent than decoding-based objectives,  while recurrent encoder are less consistent than  their convolutional counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Contrastive models have distinct representa-  tional profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  In the analyses we presented  in this paper, we observed that the contrastive  encoders behave differently than other encoders  trained with non-contrastive losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' For example,  word distinctiveness has been found to be a weak  predictor of category discriminability in the em-  bedding spaces of the contrastive encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Recall  that our contrastive encoders have a stronger con-  straint in grouping exemplars of the same category  closer in the embedding space guided by the mar-  gin hyperparameter, while decoding-based model  lack this constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We hypothesize that this con-  straint forces the models to emphasize the separa-  bility of the lexical categories in the embedding  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Therefore, a stronger constraint seems to  make contrastive encoders different compared to  other learning objectives and different instances of  the same contrastive encoder are less consistent in  their representational geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  8  Conclusion  In this paper, we have taken a closer, analytical  look at the representational geometry of acoustic  word embeddings (AWEs) from three different, but  complementary perspectives: (1) embedding space  uniformity, (2) word discriminability, and (3) net-  work representational consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We have shown  that the representational spaces of AWEs tend to-  wards being minimally isotropic, or in other words,  they utilize only a few dimensions of the embed-  ding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Another finding was that most AWE  models rely on word length as a feature to discrim-  inate between lexical categories since the word dis-  criminability index positively correlates with the  number of phonemes in a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Furthermore, our  representational consistency analysis have shown  that AWE models exhibit individual differences in  their representational profiles, with the contrastive  encoders being the most inconsistent across differ-  ent random initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Even though we focused on acoustic word em-  beddings in this paper, our analytic methodology  can also be employed for the interpretability of self-  supervised speech representation models such as  contrastive predictive coding (Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2018)  and wav2vec (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Also, the  emergent representations of sublexical units such  phonemes and syllables in speech neural networks  can be analyzed using the our proposed methodol-  ogy in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  9  Acknowledgements  The authors would like to thank the anonymous  reviewers for their encouraging feedback and in-  sightful comments on the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We express our  heartfelt thanks to Miriam Schulz for proofreading  the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' This research is funded by the Deutsche  Forschungsgemeinschaft (DFG, German Research  Foundation), Project-ID 232722074 – SFB 1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content='  IEEE  Transactions on Big Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Dan Jurafsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Probabilistic modeling in psy-  cholinguistics: Linguistic comprehension and pro-  duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Probabilistic linguistics, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Kamper, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Wang, and Karen Livescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2016a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Deep convolutional acoustic word embeddings us-  ing word-pair side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Herman Kamper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Truly unsupervised acoustic  word embeddings using weak top-down constraints  in encoder-decoder models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Herman Kamper, Micha Elsner, Aren Jansen, and  Sharon Goldwater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Unsupervised neural net-  work based feature extraction using weak top-down  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Herman Kamper, Weiran Wang, and Karen Livescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  2016b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Deep convolutional acoustic word embed-  dings using word-pair side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Diederik P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Kingma and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Adam: A  method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Simon Kornblith, Mohammad Norouzi, Honglak Lee,  and Geoffrey Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Similarity of neural  network representations revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In International  Conference on Machine Learning, pages 3519–3529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Nikolaus Kriegeskorte, Marieke Mur, and Peter A Ban-  dettini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Representational similarity analysis-  connecting the branches of systems neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Frontiers in systems neuroscience, 2:4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Keith Levin, Katharine Henry, Aren Jansen, and Karen  Livescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Fixed-dimensional acoustic embed-  dings of variable-length segments in low-resource  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In IEEE Workshop on Automatic Speech  Recognition and Understanding (ASRU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Yevgen Matusevych, Herman Kamper, and Sharon  Goldwater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2020a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Analyzing autoencoder-based  acoustic word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  In BAICS Workshop  ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Yevgen Matusevych, Herman Kamper, and Sharon  Goldwater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2020b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Analyzing autoencoder-based  acoustic word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In Bridging AI and Cog-  nitive Science Workshop, ICLR 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Yevgen Matusevych, Herman Kamper, Thomas Schatz,  Naomi Feldman, and Sharon Goldwater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' A  phonetic model of non-native spoken word process-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In Proceedings of the 16th Conference of the  European Chapter of the Association for Computa-  tional Linguistics: Main Volume, pages 1480–1490,  Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Michael McAuliffe, Michaela Socolof, Sarah Mihuc,  Michael Wagner, and Morgan Sonderegger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Montreal Forced Aligner:  Trainable text-speech  alignment using Kaldi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In Interspeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Leland McInnes, John Healy, Nathaniel Saul, and  Lukas Großberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Umap: Uniform mani-  fold approximation and projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Journal of Open  Source Software, 3(29):861.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Johannes Mehrer,  Courtney J Spoerer,  Nikolaus  Kriegeskorte, and Tim C Kietzmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Individ-  ual differences among deep neural network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Nature communications, 11(1):1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Florian Metze, Xavier Anguera, Etienne Barnard,  Marelie Davel, and Guillaume Gravier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The  spoken web search task at MediaEval 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Stephan C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content=' Griffiths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Word forms - not just their lengths- are optimized for  efficient communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' ArXiv, abs/1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content='  Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Cor-  rado, and Jeff Dean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content=' Advances in neural information processing sys-  tems, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The strange  geometry of skip-gram with negative sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In  Proceedings of the 2017 Conference on Empirical  Methods in Natural Language Processing, pages  2873–2878, Copenhagen, Denmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Association for  Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content='  In International Conference on  Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Cory Myers, Lawrence Rabiner, and Andrew Rosen-  berg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content='  In ICASSP’80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content=' Cognitive  Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content='  Shane Settle and Karen Livescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content=' Discrimina-  tive acoustic word embeddings: Recurrent neural  network-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  In IEEE Spoken Lan-  guage Technology Workshop (SLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Shane Settle and Karen Livescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content=' Discrimina-  tive acoustic word embeddings: Recurrent neural  network-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' IEEE Spoken  Language Technology Workshop (SLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Jan Strunk, Florian Schiel, and Frank Seifart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Untrained forced alignment of transcriptions and au-  dio for language documentation corpora using web-  maus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  In Proceedings of the Ninth International  Conference on Language Resources and Evaluation  (LREC’14), pages 3940–3947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Yaodong Zhang and James R Glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Unsuper-  vised spoken keyword spotting via segmental DTW  on Gaussian posteriorgrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  In IEEE Workshop  on Automatic Speech Recognition & Understanding  (ASRU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Acoustic Word  Embedding  (a)  (b)  (c)  Figure 6: A visual illustration of the different learning objectives for training AWE encoders: (a) correspondence  auto-encoder (CAE): a sequence-to-sequence network with an acoustic decoder, (b) phonologically guided encoder  (PGE): a sequence-to-sequence network with a phonological decoder, and (c) contrastive siamese encoder (CSE):  a contrastive network trained via triplet margin loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' After training the model, only the encoder component of the  model F is used to produce AWEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  1  10  100  Epoch — log scale  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='8  mAP  (a) Recurrent  1  10  100  Epoch — log scale  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='8  mAP  (b) Convolutional  Contrastive Siamese Encoder (CSE)  Phonologically-Guided Encoder (PGE)  Correspondence Autoencoder (CAE)  Autoencoder (AE)  RIE  AE  CAE  PGE  CSE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='8  mAP  RNN  CNN  (c) Best Epoch  Figure 7: Evaluation on the same-different acoustic word discrimination task quantified by the word discrimination  task and the mAP metric: Learning curves of 100 training epochs for (a) the recurrent encoder and (b) convolutional  encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' (c) mAP of the best epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  Appendices  A  AWE Models  See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 6 for a visual illustration of the different  learning objectives in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  B  Intrinsic Evaluation  The results of the intrinsic evaluation—same-  different word discrimination task quantified by  the mAP metric—is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Note that the  CAE model is pre-trained as autoencoder for 10  epochs, following prior work (Kamper, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  C  Randomly Initialized CNN Encoder  To further investigate the minimally isotropic be-  havior and the near 1 values of cosine similari-  ties of the untrained, randomly initialized convo-  lutional encoder reported in §4, we examine the  potential contribution of two factors to this obser-  vation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' batch normalization (BN) and the activation  function of the convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' The result of  this analysis is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' We observe that  removing the BN layer has no effect on the distri-  butions of cosine similarities as they remain almost  identical to the encoder that has a BN layer—see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' However, changing the activation func-  tion from the unbounded ReLU to the bounded  Tanh in the convolutional layers makes the distri-  butions of cosine similarities move towards zero  mean, even though they remain closer to 1 than  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Therefore, this behavior seems to be related to  the inner dynamics of the convolutional operation  and gets amplified where the activation function  in the convolutional layers are unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content=' Nev-  ertheless, identifying the source of this behavior  requires further investigation with different activa-  tion functions and a controlled ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content='  E  Qualitative Analysis  To further inspect the representation space and its  neighborhood structure, we conduct a qualitative  analysis by querying the representation space with  a few word samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+page_content='  word onset bias where the most similar words are  those that begin with a similar sounding prefix as  the query word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfOQNK/content/2301.03012v1.pdf'}
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+Operational Research in Engineering Sciences: Theory and Applications 
+Vol. 3, Issue x, 2020, pp. x-x 
+ISSN: 2620-1607 
+eISSN: 2620-1747 
+ DOI: https:// 
+* Corresponding author. 
+hmoghimi@ce.sharif.edu (S.H. Moghimi), jhabibi@sharif.edu (J. Habibi), 
+hamid.jahad@ce.sharif.edu (H. Jahad), fazli@sharif.edu (M.A. Fazli) 
+OPERATIONAL SCHEDULING OF OIL PRODUCTS PIPELINE 
+WITH INTERMEDIATE EVENT OCCURRENCES 
+Seyyed Hamed Moghimi 1*, Jafar Habibi 1, 
+Hamid Jahad 1,Mohammad Amin Fazli1 
+1 Department of Computer Engineering, Sharif University of Technology, Tehran, Iran 
+ 
+Received: 31 December 2022 
+Accepted: n x 2023  
+First online: n x 2023 
+ 
+Research paper 
+Abstract: Oil products are the main source of energy in the world today. 
+Distribution of these products is one of the main issues in the industry. The 
+main tools for this work are pipelines, and along with it, railways, shipping and 
+roads are also used. Optimal planning of pipelines is an example of decision-
+making problem and was the focus of many researchers in the past years. The 
+use of mixed integer linear programming (MILP) is one of the efficient methods 
+to solve this problem. However, models still ignore important operational 
+challenges. Vulnerability to deal with incidents as well as lack of attention to 
+other transportation methods as a complement to the pipeline are among the 
+weak points of the existing models. In this research, we intend to facilitate the 
+decision-making process for experts in the field of distribution of oil products. 
+For this purpose, we must improve the existing MILP methods and modify them 
+for use in the real operational environment in such a way that sufficient 
+flexibility, the possibility of responding to incidents, and the ability to revise 
+the program are added to them. 
+Key words: Pipeline network, Operational Planning, Event Occurrence, Plan Revision. 
+1. Introduction 
+In the last two centuries, one of the main issues in the world has been the provision 
+of energy. Oil and its refined products are considered as the most important fuel 
+sources in the world. One on the important problems about oil products is their 
+distribution. Considering the mechanical and chemical properties of these materials, 
+such as their fluidity, density, as well as their flammability, their transportation is 
+more sensitive, and for this reason, the related costs are also higher. Main tools used 
+to move oil products today are pipelines, vehicles, railways, and vessels. Among those, 
+pipelines have efficiency, higher safety and lower price and that is why many pipelines 
+
+S.H.Moghimi et al. /Oper. Res. Eng. Sci. Theor. Appl. 3 (x) (2020) 1-20 
+ 
+2 
+ 
+have been built around the world. For example, nearly two-thirds of oil products in 
+the United States are distributed through pipelines.  
+Optimal planning in order to make maximum use of the transmission capacity of 
+pipelines is one of the main problems in this industry, which has allocated a significant 
+amount of past and present researches. One of the most important points about this 
+optimization problem is the very high financial turnover that exists in this regard. 
+Considering the global price of oil products, the costs that may be imposed on a system 
+in case of non-optimal planning of pipelines can be about several million dollars per 
+day. Another reason is the strategic importance of fuel distribution. In all the countries 
+of the world, energy carriers are considered as strategic assets, and providing the 
+required amounts for normal use and saving for critical times is considered as one of 
+the strategic goals of the administrations.  
+Answering the optimization problem is not an obvious and simple task, because 
+the distribution network of oil products includes several pipes, different types of tanks 
+and depots, refineries, pump stations and control centers, which together form a 
+complex graph of different nodes and edges with unique characteristics and rules. 
+Furthermore, various oil products are flowing inside the pipes, with different volumes 
+based on the needs of destinations. Several rules, including physical constraints and 
+safety principles, are also effective in making decisions about the order and amount of 
+injection of different products. Therefore, it can be concluded that the nature of such 
+a problem will not be simple and planning for it is a difficult and challenging activity.  
+The main tool used for solving the optimization problem of oil products 
+distribution is mixed-integer linear programming (MILP) models. This method is 
+developed in the last two decades and several researches have focused to solve the 
+problems in this area. In this research, we proposed a novel method to take 
+environmental changes and event occurrences into consideration. Our model can 
+adapt to new environment and continues to execute the distribution plan, event 
+different types of incidents occur and change the surrounding parameters. 
+2. Related Works 
+2.1. Early Researches 
+As of the beginning of the 21st century, researches were carried out in line with the 
+use of the linear programming model for the optimization of multi-product pipelines. 
+The first model presented for this purpose looked at the problem from a discrete point 
+of view (Rejowski & Pinto, 2003). This means that the planning time horizon was 
+divided into small intervals and also, the length of the pipelines were categorized into 
+small volume intervals. This work made it easier to look at the problem in a discrete 
+way, to understand mathematical relationships and equations and inequalities, but on 
+the other hand, both the number of relations and computational complexity of the 
+model increased significantly, and the accuracy and quality of the obtained results 
+were not at a good level. Following that, other activities such as (Neiro & Pinto, 2004) 
+and (Rejowski & Pinto, 2004) presented linear programming models with the same 
+discrete approach.  
+With a little distance from these researches, the first linear programming model 
+with continuous view was presented (Cafaro & Cerda, 2004). Continuous view versus 
+
+Operational scheduling of oil products pipeline with intermediate event occurrences 
+ 
+3 
+ 
+ 
+discrete view means that instead of breaking time and place into small intervals and 
+building model variables from the values of these intervals, the time and place of 
+batches at key frames are considered as model variables. Key frames mean the 
+moments when changes are to be made in the operation environment, for example a 
+new batch is pumped into the pipeline or a batch is received at a station. With this 
+point of view, the number of variables and constraints is significantly reduced and, as 
+a result, the computational performance of the model grows up. The reason for 
+reducing the dimensions of the problem in this model is that in discrete modeling, 
+there are many times when the environment is operating normally in several 
+consecutive time intervals, and practically all these intervals can be assumed as one 
+key frame. After these early researches, a growing trend of researches began with the 
+aim of using linear programming in modeling pipelines for more complex situations 
+and involving more parameters into the problem to be closer to reality. 
+2.2. Topologies 
+In (Rejowski & Pinto, 2003) and (Cafaro & Cerda, 2004), both approaches consider 
+only one refinery as the pipeline input and one or more stations as product depots. 
+After that some researches focus on the variation of pipeline topology. In (MirHassani 
+& Ghorbanalizadeh, 2008), (Mostafaei et al., 2015), and (Taherkhani et al., 2017) tree 
+shaped pipelines are modeled. These tree topologies include the main pipeline and 
+another level of child branches. More complicated topologies have been modeled in 
+(Cafaro & Cerda, 2012), which introduced pipelines with mesh structure. 
+2.3. Input and output stations 
+One of the parameters considered in researches is number of input and output in 
+the pipeline. Early models, consider the refinery as an entry point to the pipeline and 
+some consecutive depots along the pipeline. This approach is also used in many later 
+researches such as (Rejowski & Pinto, 2004), (Rejowski & Pinto, 2008), (Cafaro & 
+Cerda, 2008), (MirHassani & Ghorbanalizadeh, 2008), (MirHassani & Fani Jahromi, 
+2011).  
+In the other hand, some researches simplify this assumption and just model one 
+input and only one output terminal. This change results in less complexity and 
+therefore, the possibility to other parameters being modeled (Relvas et al., 2006), 
+(Moradi & MirHassani, 2014), and (Moradi & MirHassani, 2016). 
+More complex input and output terminals are also modeled. For examples (Cerda 
+& Cafaro, 2009), (Cafaro & Cerda, 2010), (Magatao et al., 2015), (Mostafaei et al., 2015), 
+(Mostafaei et al., 2016), and (Taherkhani, 2020) take into account multiple input 
+nodes and dual-purposed terminals which can both pump new batches and withdraw 
+product from them. 
+2.4. Scheduling time horizon 
+In all the researches around pipeline scheduling, one of the important parameters 
+is the time horizon of planning, indicating the duration of the entire period for which 
+planning is done. The importance of this parameter's value is related to a trade-off 
+between the computational complexity of the model and its application in the real 
+world. The shorter the time horizon results in fewer parameters and formulas and 
+
+S.H.Moghimi et al. /Oper. Res. Eng. Sci. Theor. Appl. 3 (x) (2020) 1-20 
+ 
+4 
+ 
+therefore, a less complex model. On the other hand, a shorter time horizon causes the 
+future state of pipeline not to be considered. So it is possible that the result of the 
+execution of models with shorter time horizon causes the pipeline network not to 
+perform well in the long term, although they have good results in short time.  
+Researches have considered the time horizon as a determining parameter in 
+research classification. There is a focus on three types of time horizon with the aim of 
+different and specific applications: short-term scheduling which focuses to plan for 
+less than a week like (De Souza Filho et al., 2013) and (Zhang et al., 2017), medium-
+term scheduling with the aim to plan for one to two weeks such as (Liao et al., 2018a), 
+(Liao et al., 2018b) and (Chen et al., 2017), and long-term scheduling for horizons up 
+to a month like (Meira et al., 2018), (Cafaro & Cerda, 2017), and (Cafaro et al., 2015). 
+2.5. Revision in scheduling 
+The main target of the proposed method is to achieve a model that can take changes 
+in the operational environment into account and produces a plan that can adapt itself 
+when events and incidents occur at the middle of operation.  
+Only a few number of researches tried to deal with events. Almost the first attempt 
+to revise a scheduling is presented in (Relvas et al., 2007). In this research, alongside 
+with an MILP model for a one-to-one pipeline, a mechanism is introduced to enable 
+revision of the plan. This mechanism works in a simple way. It states that any time 
+during the execution of the plan, if it is determined that there is a need to revise the 
+initial plan, then the execution is stopped, changes are made in the parameters and a 
+new plan will be generated.  
+Some other researches such as (Su et al., 2018) and (Mostafaei et al., 2016) have 
+done focusing on the uncertainty of the model parameters. Another approach that has 
+been investigated in (Taherkhani et al., 2017) in order to respond to the random 
+nature of requirements is modeling with random variables and expectation 
+calculations. 
+3. Proposed Model 
+In this section, we will describe the proposed mathematical model for pipeline 
+planning, with the ability to be aware of and adapt to upcoming events. Our proposed 
+model is based on the model presented in (Cafaro & Cerda, 2010). There are six main 
+sets in this model: The set of pumping runs (𝐾), injected batches (𝐼), oil products (𝑃), 
+input terminals (𝑆), and receiving destinations (𝐽) are also included in the basic 
+models. The last set (𝐸) contains upcoming events in the planning horizon, ordered by 
+time of occurrence. This means that the event 𝑒 must happen before 𝑒′ if 𝑒 <  𝑒′. The 
+Set 𝐸 has at least two elements 𝑒0 and 𝑒𝑚𝑎𝑥, representing the start and the end of the 
+planning horizon respectively. 
+Table 1. Nomenclature 
+(a) Sets 
+𝐾 
+chronologically ordered blocks of pumping runs 
+𝐼 
+product batches (𝐼 = 𝐼old ∪ 𝐼new ) 
+𝐼𝑛𝑒𝑤 
+new batches to be injected during the planning horizon 
+
+Operational scheduling of oil products pipeline with intermediate event occurrences 
+ 
+5 
+ 
+ 
+𝐼𝑜𝑙𝑑 
+old batches in the initial state of pipeline 
+𝑃 
+refined oil products 
+𝑃𝑖 
+refined oil product contained in old batch 𝑖 ∈ 𝐼old  
+𝑆 
+pipeline input terminals 
+𝐽 
+pipeline output terminals 
+𝐽𝑝 
+distribution depots demanding product 𝑝 
+𝐽𝑝,𝑠 
+distribution depots demanding product 𝑝 to be supplied by an 
+upstream source 𝑠 
+𝐸 
+chronologically ordered events. 
+It consists of at least two members, 𝑒0 and 𝑒𝑚𝑎𝑥 representing the start 
+and the end of planning 
+(b) Parameters 
+ℎ𝑚𝑎𝑥 
+planning horizon length 
+𝑃𝑉 
+total pipeline volume from the origin to the farthest depot 
+𝑇𝑒 
+occurrence time of event 𝑒. (𝑇𝑒0 = 0, 𝑇𝑒𝑚𝑎𝑥 = ℎ𝑚𝑎𝑥) 
+𝑐𝑏𝑝,𝑗 
+unit backorder penalty cost for tardy orders of product 𝑝 at output 
+terminal 𝑗 
+𝑐𝑖𝑓𝑝,𝑝′ 
+total reprocessing cost of interface material involving products 𝑝 and 
+𝑝′ 
+𝑐𝑖𝑛𝑝,𝑠 
+pumping cost per unit of product 𝑝 at input terminal 𝑠 
+𝐷𝑚𝑖𝑛, 𝐷𝑚𝑎𝑥 
+minimum/maximum delivery size to output terminals 
+𝐷𝐿𝑝,𝑗 
+minimum request of product 𝑝 at the output terminal 𝑗 
+𝐷𝑈𝑝,𝑗 
+maximum amount of product 𝑝 that can be delivered to output 
+terminal 𝑗 
+𝑄𝑚𝑖𝑛, 𝑄𝑚𝑎𝑥 
+minimum/maximum batch injection size 
+𝑄𝑚𝑎𝑥,𝑝 
+maximum batch injection size for product 𝑝 
+𝑆𝐿𝑝,𝑠 
+lowest amount of product 𝑝 to be shipped from input terminal 𝑠 
+𝑆𝑈𝑝,𝑠 
+maximum amount of product 𝑝 that can be shipped from input 
+terminal 𝑠 
+𝑣𝑏𝑚𝑖𝑛,𝑠, 𝑣𝑏𝑚𝑎𝑥,𝑠 
+minimum/maximum product flow-rates at source 𝑠 
+𝑊𝑖
+0 
+initial volume of old batch 𝑖 
+𝜎𝑗 
+volume coordinate of output terminal 𝑗 from the pipeline origin 
+𝜏𝑠 
+volume coordinate of input terminal 𝑠 from the pipeline origin 
+(c) Continuous variables 
+𝐵𝐶 
+total backorder penalty cost for tardily meet product demands 
+𝐵𝑝,𝑗 
+backorder of product 𝑝 at the output terminal 𝑗 
+𝐶𝑘 
+completion time of run block 𝑘 (measured in time units) 
+𝐿𝑘 
+length of run block 𝑘 
+𝐿𝑘,𝑠 
+length of pumping run 𝑘 taking place at source 𝑠 
+𝐷𝑖,𝑗
+(𝑘) 
+volume of batch 𝑖 diverted to output terminal 𝑗 during run block 𝑘 
+𝐷𝑃𝑖,𝑗,𝑝
+(𝑘)  
+volume of product 𝑝 diverted from batch 𝑖 to terminal 𝑗 during run 
+block 𝑘 
+𝐹𝑖,𝑘 
+upper coordinate of batch 𝑖 from the origin at time 𝐶𝑘 
+
+S.H.Moghimi et al. /Oper. Res. Eng. Sci. Theor. Appl. 3 (x) (2020) 1-20 
+ 
+6 
+ 
+𝑃𝐶𝑘 
+total pumping cost during block 𝑘 
+𝑄𝑖,𝑠
+(𝑘) 
+size of batch 𝑖 shipped from input terminal 𝑠 during run 𝑘 
+𝑄𝑃𝑖,𝑠,𝑝
+(𝑘)  
+size of batch 𝑖 containing product 𝑝 inputted from input terminal 𝑠 
+through run 𝑘 
+𝑅𝐶𝑖 
+total reprocessing cost for the interface between batch 𝑖 and the 
+following (𝑖 +  1) 
+𝑊𝑖,𝑘 
+size of batch 𝑖 at the end of run 𝑘 
+(d) Binary variables 
+𝑣𝑖,𝑠
+(𝑘) 
+variable denoting that a portion of batch 𝑖 is injected from source 𝑠 
+through block 𝑘 
+𝑥𝑖,𝑗
+(𝑘) 
+variable denoting that a portion of batch 𝑖 is diverted to the depot 𝑗 
+during block 𝑘 
+𝑦𝑖,𝑝 
+variable denoting that batch 𝑖 contains product 𝑝 
+𝑏𝑘,𝑒 
+variable denoting that pumping run 𝑘 is executed completely inside 
+of the interval [𝑇𝑒−1, 𝑇𝑒] 
+3.1. Sequencing pumping run 
+A block 𝑘 ∈ 𝐾 must start after the completion of the previous block (𝑘 − 1). Let 𝐶𝑘 
+denote the completion time and 𝐿𝑘 the duration of block 𝑘. 
+𝐶𝑘 − 𝐿𝑘 ≥ 𝐶𝑘−1∀𝑘 ∈ 𝐾(𝑘 > 1) 
+(1) 
+All pumping blocks should also be finished before the planning horizon ℎ𝑚𝑎𝑥. 
+𝐶𝑘 ≤ ℎ𝑚𝑎𝑥∀𝑘 ∈ 𝐾 
+(2) 
+3.2. Aligning pumping runs with events 
+There is an assumption in the model that during the execution of a pumping run 𝑘, 
+the environmental conditions and parameters remain stable. On the other hand, 
+almost all events will change the parameters of the environment when they happen. 
+Therefore, it should be prevented that an event occurs during the execution of a 
+pumping run.  
+For this purpose, we define the binary variable 𝑏𝑘,𝑒 in such a way that its value 
+equals to 1, iff pumping run 𝑘 is executed completely between two consecutive events 
+𝑒 and 𝑒 − 1. It means that pumping run 𝑘 must be started after 𝑇𝑒−1 (the occurrence 
+time of the event 𝑒 − 1) and must be finished before 𝑇𝑒 (the occurrence time of the 
+event 𝑒). Figure 1 shows an example of this situation. In all other conditions, 𝑏𝑘,𝑒 will 
+be 0. 
+ 
+
+Te-1
+7.
+K
+k
+Ck
+CK
+-
+--Operational scheduling of oil products pipeline with intermediate event occurrences 
+ 
+7 
+ 
+ 
+Figure 1. An example of a pumping run fully executed between two consecutive events 
+The following equations are added to the model to ensure the mentioned 
+alignment. First, each pumping run 𝑘 should be executed no more than once. Then: 
+∑
+𝑏𝑘,𝑒
+𝑒∈𝐸
+= 1∀𝑘 ∈ 𝐾, 𝑒 ∈ 𝐸 
+(3) 
+Second, if a pumping run is executed between two events, the start and finish time 
+of the run should be placed between occurrence time of those events.  
+𝑇𝑒−1𝑏𝑘,𝑒 ≤ 𝐶𝑘 − 𝐿𝑘∀𝑘 ∈ 𝐾, 𝑒 ∈ 𝐸 
+(4) 
+𝐶𝑘 ≤ 𝑇𝑒𝑏𝑘,𝑒 + (1 − 𝑏𝑘,𝑒)𝑀𝑇∀𝑘 ∈ 𝐾, 𝑒 ∈ 𝐸 
+(5) 
+The value of constant 𝑀𝑇 should be larger than all time values in the model. So it 
+can be chosen as ℎ𝑚𝑎𝑥 or higher. 
+3.3. Injecting batches in different pumping runs 
+A pumping operation is identified by its corresponding pumping run 𝑘 ∈ 𝐾 and 
+source terminal 𝑠 ∈ 𝑆. We define the binary variable 𝑣𝑖,𝑠
+(𝑘) to state that the batch 𝑖 ∈
+𝐼𝑛𝑒𝑤 in injected to the pipeline from input terminal 𝑠 at the pumping run 𝑘. If 𝑣𝑖,𝑠
+(𝑘) = 1, 
+then the whole or at least some amount of batch 𝑖 is injected from source terminal 𝑠 
+during pumping run 𝑘. To ensure that no more than one batch is injected in the 
+pipeline from each source in each pumping run, we have: 
+∑𝑖∈𝐼
+𝑣𝑖,𝑠
+(𝑘) ≤ 1∀𝑘 ∈ 𝐾, 𝑠 ∈ 𝑆 
+(6) 
+If for a certain pumping run 𝑘, the term (∑
+∑
+𝑣𝑖,𝑠
+(𝑘)
+𝑖∈𝐼
+𝑠∈𝑆
+) is zero, it means that no 
+batches are injected in that run. Since the size of set (𝐾) is normally chosen larger that 
+the number of pumping runs in the optimal solution, this situation is probable for the 
+last members of (𝐾) (named dummy runs).  
+The following Equation ensures that if a pumping run 𝑘 is dummy, all later runs 
+also turn to dummy.  
+∑𝑠∈𝑆
+∑𝑖∈𝐼
+𝑣𝑖,𝑠
+(𝑘) ≤ 𝑀(∑𝑠′∈𝑆
+∑𝑖′∈𝐼
+𝑣𝑖′,𝑠′
+(𝑘−1))∀𝑘 ∈ 𝐾(𝑘 > 1) 
+(7) 
+The constant 𝑀 is a very large number, greater than all values that can be generated 
+in the model. We will use it more in later equations in the model.  
+3.4. Batch injection and size relation 
+Every Batch 𝑖 is injected for the first time in a pumping run 𝑘 at a source terminal 
+𝑠. It is also possible that some portion of 𝑖 is appended to it in later pumping run 𝑘′ at 
+another (or the same) terminal 𝑠′. We say that batch 𝑖 is injected in both runs (𝑘, 𝑠) 
+and (𝑘′, 𝑠′). 
+Let 𝑄𝑖,𝑠
+(𝑘) denote the size of injected product to batch 𝑖 in run (𝑘, 𝑠). 𝑄𝑖,𝑠
+(𝑘) will be 
+positive iff there is actually a real injection in batch 𝑖 at run (𝑘, 𝑠). This means that 
+𝑣𝑖,𝑠
+(𝑘)and 𝑄𝑖,𝑠
+(𝑘) will be either zero or positive simultaneously: 
+
+S.H.Moghimi et al. /Oper. Res. Eng. Sci. Theor. Appl. 3 (x) (2020) 1-20 
+ 
+8 
+ 
+𝑄𝑚𝑖𝑛𝑣𝑖,𝑠
+(𝑘) ≤ 𝑄𝑖,𝑠
+(𝑘) ≤ 𝑄𝑚𝑎𝑥𝑣𝑖,𝑠
+(𝑘)∀𝑖 ∈ 𝐼, 𝑠 ∈ 𝑆, 𝑘 ∈ 𝐾 
+(8) 
+Parameters 𝑄𝑚𝑖𝑛 and 𝑄𝑚𝑎𝑥 present the global lower and upper bound for the 
+volume of product that can be injected during a pumping run.  
+3.5. Choosing lengths of pumping runs 
+Let 𝐿𝑘,𝑠 be the length of the run (𝑘, 𝑠), means the time length during run 𝑘 that 
+terminal 𝑠 is actually pumping products. The whole run 𝑘 can be finished only if all of 
+its sub-runs (𝑘, 𝑠) is finished before. So the length of run 𝑘 will be given by  𝐿𝑘 =
+max
+𝑠∈𝑆 (𝐿𝑘,𝑠) or: 
+𝐿𝑘 ≥ 𝐿𝑘,𝑠∀𝑘 ∈ 𝐾, 𝑠 ∈ 𝑆 
+(9) 
+The injected volume of refined product is also related to the length of pumping run 
+and the rate at which a terminal can perform injection. The total volume of product 
+injected to the pipeline during run (𝑘, 𝑠) is equal to  ∑
+𝑄𝑖,𝑠
+(𝑘)
+𝑖∈𝐼
+. In the base model from 
+(Cafaro & Cerda, 2010) the following equation is presented, where 𝑣𝑏𝑚𝑖𝑛,𝑠 and  𝑣𝑏𝑚𝑎𝑥,𝑠 
+stand for the minimum and maximum pumping rate of terminal 𝑠. 
+𝑣𝑏𝑚𝑖𝑛,𝑠𝐿𝑘,𝑠 ≤ ∑𝑖∈𝐼
+𝑄𝑖,𝑠
+(𝑘) ≤ 𝑣𝑏𝑚𝑎𝑥,𝑠𝐿𝑘,𝑠∀𝑘 ∈ 𝐾, 𝑠 ∈ 𝑆 
+(10) 
+In this model, we want to take the effects of events into consideration. The next 
+section discusses the problem and presents the proposed technique to address this 
+requirement.  
+3.6. Choosing parameters based on event occurrences 
+When an event occurs, it may affect the conditions of pipeline and its surroundings. 
+In our model, this will be done by changing the value of parameters. Therefore, the 
+value of a parameter (same as  𝑣𝑏𝑚𝑖𝑛,𝑠 and  𝑣𝑏𝑚𝑎𝑥,𝑠 from Equation 8) should be chosen 
+according to event occurrences. Since the parameters are assumed to be fixed when 
+solving an MILP model, our technique to address this need is to have a vector of all 
+values that a parameter can take over the planning horizon and using the respective 
+value in the constraint expressions.  
+For example, suppose there is a predetermined maintenance program at terminal 
+𝑠 (represented by event 𝑒) that will cause the maximum pumping rate at the terminal 
+to decrease to half of the previous value, and it remains the same until the maintenance 
+end (represented by event 𝑒′). In such a case, we consider three values for the 
+parameters  𝑣𝑏𝑚𝑎𝑥,𝑠: the value before the occurrence of 𝑒, the value between 𝑒 and 𝑒′, 
+and the value after 𝑒′.  
+Let us denote the value of an arbitrary parameter 𝐴 between events 𝑒 −  1 and 𝑒 
+by the notation 𝐴〈𝑒〉. So the three parameters mentioned before will be denoted by 
+𝑣𝑏𝑚𝑎𝑥,𝑠
+〈𝑒〉
+, 𝑣𝑏𝑚𝑎𝑥,𝑠
+〈𝑒′〉 , and 𝑣𝑏𝑚𝑎𝑥,𝑠
+〈𝑒𝑚𝑎𝑥〉.  
+According to these assumptions and notations, we rewrite the Equation 10: 
+𝑣𝑏𝑚𝑖𝑛,𝑠
+<𝑒> 𝐿𝑘,𝑠 − 𝑀(1 − 𝑏𝑘,𝑒) ≤ ∑𝑖∈𝐼
+𝑄𝑖,𝑠
+(𝑘) ≤ 𝑣𝑏𝑚𝑎𝑥,𝑠
+<𝑒> 𝐿𝑘,𝑠 + 𝑀(1 − 𝑏𝑘,𝑒)∀𝑘 ∈ 𝐾, 𝑠 ∈
+𝑆, 𝑒 ∈ 𝐸(𝑒 > 0)  
+(11) 
+
+Operational scheduling of oil products pipeline with intermediate event occurrences 
+ 
+9 
+ 
+ 
+The term 𝑀(1 − 𝑏𝑘,𝑒) is called event guard and will be noted as 𝐸𝐺𝑘,𝑒 hereafter. It 
+is used to ensure that if the pumping run 𝑘 is executed between events 𝑒 − 1 and 𝑒 
+(means 𝑏𝑘,𝑒), only the corresponding equation (containing parameters 𝐴〈𝑒〉) takes 
+effect and the other ones go trivial. That will be held because an infinity term is added 
+to the larger side of the inequality. 
+3.7. Tracking the batch size over time 
+Let 𝑊𝑖,𝑘 be the size of a batch 𝑖 at the time 𝐶𝑘. If the volume of batch 𝑖 is changed 
+during run 𝑘, it must be either an injection from source 𝑠 or a delivery to depot 𝑗 ∈ 𝐽.  
+𝑊𝑖,𝑘 = 𝑊𝑖,𝑘−1 + ∑𝑠∈𝑆
+𝑄𝑖,𝑠
+(𝑘) − ∑𝑗∈𝐽
+𝐷𝑖,𝑗
+(𝑘)∀𝑖 ∈ 𝐼, 𝑘 ∈ 𝐾 
+(12) 
+Only the injections that are performed at upstream sources can cause older batches 
+to move forward. So the amount of product that can be delivered to depot 𝑗 is at most 
+equal to the amount of injections performed at upstream sources (𝜏𝑠 < 𝜎𝑗).  
+𝐷𝑖,𝑗
+(𝑘) ≤ ∑𝑠∈𝑆:𝜏𝑠<𝜎𝑗
+∑𝑖′∈𝐼
+𝑄𝑖′,𝑠
+(𝑘)∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾 
+(13) 
+3.8. Tracking the batch location over time 
+Let 𝐹𝑖,𝑘 denote the upper volumetric coordinate of batch 𝑖 at the time 𝐶𝑘. So 𝐹𝑖+1,𝑘 
+also represents the upper coordinate of batch (𝑖 + 1), which is equal to the lower 
+coordinate of batch 𝑖 at the same time. Since the difference between lower and upper 
+coordinates of a batch 𝑖 at the end of run 𝑘 is equal to its size 𝑊𝑖,𝑘, we have: 
+𝐹𝑖,𝑘 − 𝑊𝑖,𝑘 = 𝐹𝑖+1,𝑘∀𝑖 ∈ 𝐼, 𝑘 ∈ 𝐾 
+(14) 
+To prevent backward movement of batches during the time: 
+𝐹𝑖,𝑘−1 ≤ 𝐹𝑖,𝑘∀𝑖 ∈ 𝐼, 𝑘 ∈ 𝐾 
+(15) 
+Finally to ensure that all batches stands into pipeline and no batches will ever be 
+located outside the pipeline coordination, Let 𝑃𝑉 stands for the total pipeline volume, 
+then the upper coordinate of any batch 𝑖 must never be greater than 𝑃𝑉 and its lower 
+coordinate must be non-negative.  
+𝐹𝑖,𝑘 ≤ 𝑃𝑉∀𝑖 ∈ 𝐼, 𝑘 ∈ 𝐾 
+(16) 
+𝐹𝑖,𝑘 − 𝑊𝑖,𝑘 ≥ 0∀𝑖 ∈ 𝐼, 𝑘 ∈ 𝐾 
+(17) 
+3.9. Preventing empty spaces and overflows 
+The pipeline must remain full of products at any time over the planning horizon. 
+Therefore, the sum of volumes of all batches in the pipeline must be equal to 𝑃𝑉: 
+∑𝑖∈𝐼
+𝑊𝑖,𝑘 = 𝑃𝑉∀𝑘 ∈ 𝐾 
+(18) 
+Also the amount of injected and delivered products must be equal at the end of any 
+run 𝑘: 
+∑𝑖∈𝐼
+∑𝑠∈𝑆
+𝑄𝑖,𝑠
+(𝑘) = ∑𝑖∈𝐼
+∑𝑗∈𝐽
+𝐷𝑖,𝑗
+(𝑘)∀𝑘 ∈ 𝐾 
+(19) 
+
+S.H.Moghimi et al. /Oper. Res. Eng. Sci. Theor. Appl. 3 (x) (2020) 1-20 
+ 
+10 
+ 
+3.10. Injections and deliveries feasibility 
+Refined product can be injected into batch 𝑖 during pumping run (𝑘, 𝑠) only if the 
+following two conditions are satisfied: 
+(a) Batch 𝑖 has reached the location of the source 𝑠 (𝜏𝑠) before the run 𝑘 begins: 
+𝐹𝑖,𝑘−1 ≥ 𝜏𝑠𝑣𝑖,𝑠
+(𝑘)∀𝑖 ∈ 𝐼, 𝑠 ∈ 𝑆, 𝑘 ∈ 𝐾 
+(20) 
+(b) Also batch 𝑖 has not completely passed the location of source 𝑠 before the run 
+𝑘: 
+𝐹𝑖,𝑘−1 − 𝑊𝑖,𝑘−1 ≤ 𝜏𝑠 + 𝑀(1 − 𝑣𝑖,𝑠
+(𝑘))∀𝑖 ∈ 𝐼, 𝑠 ∈ 𝑆, 𝑘 ∈ 𝐾 
+(21) 
+3.11. Delivery from batches to depots 
+Same as the previous section, delivery of product from batch 𝑖 to depot 𝑗 is also 
+possible, only if: 
+(a) The batch 𝑖 reaches the location of depot 𝑗 (denoted by 𝜎𝑗) before the end of run 
+𝑘, i.e. 𝐹𝑖,𝑘 ≥ σj 
+(b) The batch 𝑖 has not passed the location of depot 𝑗 before the run 𝑘 begins. It 
+means that the lower coordination of the batch must not exceed the location of the 
+depot, i.e. 𝐹𝑖,𝑘−1 − 𝑊𝑖,𝑘−1 ≤ 𝜎𝑗 
+To fulfill these conditions, let 𝑥𝑖,𝑗
+(𝑘) be a binary variable denoting that some product 
+is delivered from batch 𝑖 to depot 𝑗 during run 𝑘. When 𝑥𝑖,𝑗
+(𝑘) = 0, nothing should be 
+transferred from batch 𝑖 to depot 𝑗. It means 𝐷𝑖,𝑗
+(𝑘) = 0. 
+𝐷𝑚𝑖𝑛,𝑗
+<𝑒> 𝑥𝑖,𝑗
+(𝑘) ≤ 𝐷𝑖,𝑗
+(𝑘) ≤ 𝐷𝑚𝑎𝑥,𝑗
+<𝑒> 𝑥𝑖,𝑗
+(𝑘)∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾, 𝑒 ∈ 𝐸(𝑒 > 0) 
+(22) 
+where 𝐷𝑚𝑖𝑛,𝑗
+〈𝑒〉
+ and 𝐷𝑚𝑎𝑥,𝑗
+〈𝑒〉
+ are bounds on the amount of product that can be delivered 
+to depot 𝑗 in timespan [𝑇𝑒−1, 𝑇𝑒]. 
+The following two constrains are used to meet the two conditions mentioned 
+before in this section, respectively.  
+𝐹𝑖,𝑘 ≥ 𝜎𝑗𝑥𝑖,𝑗
+(𝑘)∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾 
+(23) 
+𝐹𝑖,𝑘−1 − 𝑊𝑖,𝑘−1 ≤ 𝜎𝑗 + 𝑀(1 − 𝑥𝑖,𝑗
+(𝑘))∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾 
+(24) 
+As mentioned before, the prerequisite for delivering product from batch 𝑖 to depot 
+𝑗 at run 𝑘 is that the following inequalities are true: 
+𝐹𝑖,𝑘−1 − 𝑊𝑖,𝑘−1 < 𝜎𝑗 ≤ 𝐹𝑖,𝑘 
+(25) 
+In this case, an upper bound on the volume of product that can be delivered is 𝜎𝑗 −
+(𝐹𝑖,𝑘−1 − 𝑊𝑖,𝑘−1), unless more product is injected in batch 𝑖 from upstream sources at 
+the same time that delivery is in action.  
+From the other hand, delivery of product can be done at any depot that is in touch 
+with the batch during the run. So the total amount of deliveries from batch 𝑖 to all the 
+depots from the origin to depot 𝑗, could not be more than the initial available volume 
+
+Operational scheduling of oil products pipeline with intermediate event occurrences 
+ 
+11 
+ 
+ 
+of batch 𝑖 (𝜎𝑗 − (𝐹𝑖,𝑘−1 − 𝑊𝑖,𝑘−1)) added by the total amount of injections from 
+upstream sources to batch 𝑖. Then: 
+∑𝑗
+𝑗′=1
+𝐷𝑖,𝑗′
+(𝑘) ≤ 𝜎𝑗 − (𝐹𝑖,𝑘−1 − 𝑊𝑖,𝑘−1) + ∑ 𝑠∈𝑆
+𝜏𝑠<𝜎𝑗
+𝑄𝑖,𝑠
+(𝑘) + (𝑃𝑉 − 𝜎𝑗)(1 − 𝑥𝑖,𝑗
+(𝑘))
+∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾
+ 
+(26) 
+3.12. Assigning products to batches 
+At most, only one product can be assigned to batch 𝑖. Let 𝑦𝑖,𝑝 be a binary variable 
+denoting that batch 𝑖 contains product 𝑝 whenever 𝑦𝑖,𝑝 = 1. Then: 
+∑𝑝∈𝑃
+𝑦𝑖,𝑝 ≤ 1∀𝑖 ∈ 𝐼 
+(27) 
+If no products is assigned to batch 𝑖, then we say that 𝑖 is fictitious and will not be 
+injected in the pipeline in real life. It means that there is no pumping run (𝑘, 𝑠) 
+injecting any product to batch 𝑖. Then we have 𝑣𝑖,𝑠
+(𝑘) = 0 for all pumping runs.  
+In the other hand, if there is a product assignment to batch 𝑖, it means that this 
+batch is real and at least one pumping run must inject some products to it. This 
+condition can be mathematically written as follows, 
+∑𝑝∈𝑃
+𝑦𝑖,𝑝 ≤ ∑𝑠∈𝑆
+∑𝑘∈𝐾
+𝑣𝑖,𝑠
+(𝑘) ≤ 𝑀 ∑𝑝∈𝑃
+𝑦𝑖,𝑝∀𝑖 ∈ 𝐼𝑛𝑒𝑤 
+(28) 
+Some products should not be injected next to each other because of contamination. 
+Let us assume that (𝑝, 𝑝′) stands for a forbidden sequence of products. Then, 
+𝑦𝑖−1,𝑝 + 𝑦𝑖,𝑝′ ≤ 1∀𝑖 ∈ 𝐼𝑛𝑒𝑤(𝑖 > 1) 
+(29) 
+3.13. Product injection 
+The whole amount of products injected into batch 𝑖 in all pumping runs must be 
+from a single type of product 𝑝, such that 𝑦𝑖,𝑝 = 1. 
+𝑄𝑃𝑖,𝑠,𝑝
+(𝑘) ≤ 𝑄𝑚𝑎𝑥𝑦𝑖,𝑝∀𝑖 ∈ 𝐼, 𝑠 ∈ 𝑆, 𝑘 ∈ 𝐾, 𝑝 ∈ 𝑃 
+(30) 
+∑𝑝∈𝑃
+𝑄𝑃𝑖,𝑠,𝑝
+(𝑘) = 𝑄𝑖,𝑠
+(𝑘)∀𝑖 ∈ 𝐼, 𝑠 ∈ 𝑆, 𝑘 ∈ 𝐾 
+(31) 
+3.14. Product Delivery 
+Same as the previous section, The whole amount of delivered material from batch 
+𝑖 to different depots must be from a single type of product 𝑝 such that 𝑦𝑖,𝑝 = 1.  
+𝐷𝑃𝑖,𝑗,𝑝
+(𝑘) ≤ 𝐷𝑚𝑎𝑥𝑦𝑖,𝑝∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾, 𝑝 ∈ 𝑃 
+(32) 
+∑𝑝∈𝑃
+𝐷𝑃𝑖,𝑗,𝑝
+(𝑘) = 𝐷𝑖,𝑗
+(𝑘)∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾 
+(33) 
+3.15. Fulfilling product demands 
+The model should provide the demanded amount of products for all depots. Let say 
+that the total amount of product 𝑝 needed at the depot 𝑗 is 𝐷𝐿𝑝,𝑗. To let the model 
+continue to work and be feasible, even if it does not fulfill the demand 𝐷𝐿𝑝,𝑗, we 
+
+S.H.Moghimi et al. /Oper. Res. Eng. Sci. Theor. Appl. 3 (x) (2020) 1-20 
+ 
+12 
+ 
+introduce variable 𝐵𝑝,𝑗 as the amount of product shortage. It will be used in the cost 
+function later. 
+At last, to consider the upper bound of storage capacity for product 𝑝 at depot 𝑗, we 
+use parameter 𝐷𝑈𝑝,𝑗. So the total amount of product 𝑝 delivered to depot 𝑗 must be 
+lower that this bound. 
+𝐷𝐿𝑝,𝑗 − 𝐵𝑝,𝑗 ≤ ∑𝑘∈𝐾
+∑𝑖∈𝐼
+𝐷𝑃𝑖,𝑗,𝑝
+(𝑘) ≤ 𝐷𝑈𝑝,𝑗∀𝑝 ∈ 𝑃, 𝑗 ∈ 𝐽 
+(34) 
+3.16. Initial state of problem 
+To initialize the starting position and order of batches in the pipeline, let 𝑊𝑖
+0 be the 
+volume of the old batch 𝑖 ∈ 𝐼𝑜𝑙𝑑. Then, the upper coordinate of batch 𝑖 ∈ 𝐼𝑜𝑙𝑑 is equal 
+to the sum of initial volumes for all succeeding batches 𝑖′ ∈ 𝐼𝑜𝑙𝑑 (𝑖′ > 𝑖) plus its own 
+volume: 
+𝐹𝑖,0 = ∑𝑖′∈𝐼𝑜𝑙𝑑(𝑖′≥𝑖)
+𝑊𝑖′0∀𝑖 ∈ 𝐼𝑜𝑙𝑑 
+(35) 
+Moreover, it is also known the product 𝑃𝑖 contained in every old batch 𝑖 ∈ 𝐼𝑜𝑙𝑑 
+𝑦𝑖,𝑃𝑖 = 1∀𝑖 ∈ 𝐼𝑜𝑙𝑑 
+(36) 
+3.17. Objective function 
+The main goal of the proposed method is to increase the adaptability of the model 
+to changes in the operating environment. Therefore, in this method, in addition to 
+trying to achieve a program that will deliver the demand of depots with at the lowest 
+cost of pumping (𝑃𝐶) and reprocessing of interface material (𝑅𝐶), the cost of shortage 
+in the needs for the products (𝐵𝐶) must be reduced as much as possible. 
+𝑚𝑖𝑛𝑧 = ∑𝑖∈𝐼
+𝑅𝐶𝑖 + ∑𝑘∈𝐾
+𝑃𝐶𝑘 + 𝐵𝐶 
+(37) 
+The following three equations introduce the terms of the objective function: 
+𝑅𝐶𝑖 ≥ cif𝑝,𝑝′(𝑦𝑖,𝑝 + 𝑦𝑖+1,𝑝′ − 1)∀𝑖 ∈ 𝐼; 𝑝, 𝑝′ ∈ 𝑃 
+(38) 
+The variable 𝑅𝐶𝑖 stands for the total cost of reprocessing interface material 
+between batches 𝑖 and 𝑖 + 1 and the parameter 𝑐𝑖𝑓𝑝,𝑝′ is the cost for reprocessing the 
+interface material resulting from blending of products 𝑝 and 𝑝′. 
+𝑃𝐶𝑘 = ∑𝑖∈𝐼
+∑𝑠∈𝑆
+∑𝑝∈𝑃
+cin𝑝,𝑠𝑄𝑃𝑖,𝑠,𝑝
+(𝑘) ∀𝑘 ∈ 𝐾 
+(39) 
+The variable 𝑃𝐶𝑘 is the total pumping cost at run 𝑘 and the coefficient cin𝑝,𝑠 stands 
+for the cost of pumping a unit volume of product 𝑝 in the source 𝑠.  
+𝐵𝐶 = ∑𝑝∈𝑃
+∑𝑗∈𝐽
+cb𝑝,𝑗𝐵𝑝,𝑗 
+(40) 
+The parameter cb𝑝,𝑗 stands for the cost of failing to provide a single unit of product 
+𝑝 at depot 𝑗 on schedule and 𝐵𝑝,𝑗 (as introduced in Equation 34) is the amount of  
+backorder of product 𝑝 at depot 𝑗. 
+
+Operational scheduling of oil products pipeline with intermediate event occurrences 
+ 
+13 
+ 
+ 
+4. Results & Discussion 
+In this section, the results of the implementation of the proposed model and its 
+performance comparison with previous methods are reported. The study case 
+considered for this comparison is based on an example mentioned in (Cafaro & Cerda, 
+2010), which has been modified to meet the requirements of this comparison. 
+4.1. Example configuration 
+In this example, the operational planning of a pipeline network is carried out, 
+which includes two source stations 𝑆1 and 𝑆2 and three depots 𝐷1, 𝐷2 and 𝐷3. This 
+pipeline transfers three refined products 𝐴, 𝐵 and 𝐶 from the sources to depots. 
+Stations 𝑆1 and 𝑆2 are located at volume coordinates 0 and 40 from the beginning of 
+the pipeline, respectively. Stations 𝐷1, 𝐷2, and 𝐷3 are also located at volume 
+coordinates 20, 60, and 80 from the pipeline origin. Between each two consecutive 
+stations, there is a pipeline with a volume of 20 units.  
+The available amount of the products at the sources and the demands of the depots 
+for the products are shown in Table 2. Also, in Table 3, the cost of recycling the 
+interface between each pair of products can be seen. 
+Table 2. Product supplies and demands 
+ 
+Supplies 
+Demands 
+ 
+S1 
+S2 
+D1 
+D2 
+D3 
+A 
+50 
+20 
+60 
+60 
+- 
+B 
+80 
+60 
+- 
+- 
+100 
+C 
+30 
+80 
+- 
+60 
+- 
+Table 3. Interface costs (in 100$) 
+Predecessor 
+Successor 
+A 
+B 
+C 
+A 
+- 
+22 
+23 
+B 
+24 
+- 
+21 
+C 
+30 
+32 
+- 
+Table 4 contains the cost of pumping one volume unit of each product in each of 
+the sources. The feasible range for pumping rate for all products in all sources is set to 
+be 1.2 (units/h). 
+Table 4. Pumping cost in each source (in 100$ per volume unit) 
+Product 
+Source 
+S1 
+S2 
+A 
+29 
+14.5 
+B 
+34 
+17.0 
+C 
+24.5 
+49.0 
+The initial state of the pipeline includes four batches 𝐵1, 𝐵2, 𝐵4 and 𝐵5. The 
+product and the initial volume of these batches is shown in the top row of Figure 2. 
+
+S.H.Moghimi et al. /Oper. Res. Eng. Sci. Theor. Appl. 3 (x) (2020) 1-20 
+ 
+14 
+ 
+ 
+Figure 2. Optimal pipeline schedule for the example when no events occur 
+4.2. Optimal solution in absence of incidents 
+When no events occur during the planning horizon, the optimal schedule for 
+transporting products and meeting the demands of the depots is as shown in Figure 2. 
+In this case, we will have 9 pumping runs 𝑘1 − 𝑘9. A distinct row is drawn in Figure 2 
+for each of pumping runs. The start time, completion time and duration of each run 
+can be seen at the left of each row and the volume of the product injected into the 
+pipeline or received from it during the run is also drawn as arrows. The color pattern 
+of each batch indicates the product related to it, as is defined in the legend of the figure.  
+As can be seen, the time required to execute the optimal schedule is 183.33 hours. 
+The plan is as follow: 
+In the first run (𝑘1), which starts from time 0 and continues for 8.33 hours, 10 units 
+of product 𝐴 are injected from station 𝑆1 to batch 𝐵5. As the batches move forward in 
+the pipeline, depot 𝐷2 takes 10 units of product 𝐴 from batch 𝐵2. In the second run 
+(𝑘2) which takes 16.67 hours long, station 𝑆2 performs the pumping operation and 
+adds 20 units of product 𝐴 to batch 𝐵2 and exactly the same amount of product from 
+the same batch is received again in depot 𝐷2. Run 𝑘3 takes 33.33 hours and during 
+this run, 40 units of product 𝐴 are injected to batch 𝐵5 from station 𝑆1 and the same 
+amount is received in depot 𝐷1. As there is no pumping interaction and batch overlap 
+during this run, station 𝑆2 can also pump 40 units of product 𝐶 in the form of a new 
+batch 𝐵3 simultaneously. This injection causes the depot 𝐷2 to fully receive the 
+remaining 20 units of product 𝐴 in 𝐵2 and, in addition, withdraw 20 units of product 
+𝐶 from batch 𝐵3. Pumping Run 𝑘4, which ends at 91.67 hours, includes two injections 
+of product 𝐵: 20 units from station 𝑆1 to the new batch 𝐵6 and 40 units from station 
+𝑆2 to 𝐵4 that had reached the station at the end of the previous run. As a result of these 
+two injections, 20 units of the product 𝐴 from 𝐵5 enter depot 𝐷1, the remaining 20 
+units of the product 𝐶 in the batch 𝐵3 are received in depot 𝐷2, and 20 units of product 
+𝐵 are added to the inventory of depot 𝐷3.  
+As seen in Figure 2, during runs 𝑘5 to 𝑘8, only one injection and one delivery take 
+place in each run. In the last pumping run (𝑘9), station 𝑆1 injected 20 units of product 
+
+0
+20
+40
+60
+80
+S1
+D1
+S2
+D2
+D3
+B5
+B4
+B2
+B1
+Start (h)
+End (h)
+[10
+LO
+0.00
+8.33
+B5
+B4
+B2
+B1
+20
+20
+8.33
+25.00
+B5
+B4
+B2
+B1
+40
+40
+40
+2020
+25.00
+58.33
+B5
+B4
+B3
+B1
+20
+20
+20
+20
+58.33
+91.67
+B6
+B5
+B4
+30
+[30
+91.67
+116.67
+B7
+B6
+B5
+B4
+10
+10
+116.67
+125.00
+B7
+B6
+B4
+40
+125.00
+B8
+B7
+B6
+40|
+158.33
+10
+[10
+158.33
+166.67
+B8
+B7
+B6
+20
+[10]
+166.67
+183.33
+B8
+B7Operational scheduling of oil products pipeline with intermediate event occurrences 
+ 
+15 
+ 
+ 
+𝐵 into batch 𝐵8, and as a result, depots 𝐷2 and 𝐷3 received respectively, 10 units of 
+product 𝐶 from batch 𝐵7 and 10 units of product 𝐵 from batch 𝐵6. The final state of 
+the pipeline, 183.33 hours after the start of the program is as follow: the batch 𝐵8 
+containing product 𝐵 is location from the beginning of the pipeline to the volumetric 
+coordinate 70, and batch 𝐵7 is placed from the coordinates 70 to 80, containing 10 
+units of the product 𝐶. 
+4.3. Adding incidents to the example 
+It is mentioned earlier that incidents usually lead to unwanted changes in the 
+schedule and as a result, the time and cost of the plan increase according to the severity 
+of the incident. In this example, we consider a case where a maintenance operation 
+will be executed on the pipeline between 𝑆1 and 𝐷1 after 100 hours from the start of 
+the planning horizon. Therefore, during this operation, it will not be possible to inject 
+any product from the source 𝑆1. We also assume that the maintenance will be finished 
+in 30 hours. 
+In none of the previous pipeline planning methods, incidents and changes in the 
+operational environment have been considered. So there is no way to add events to 
+the old models. When using these models in the real world, when an incident occurs, 
+the only way is to stop the plan execution, update the numbers and parameters of the 
+model according to the new state of the environment, and run the model again. If we're 
+lucky, the model will probably lead to a new plan that will definitely cost more than 
+before. But in many cases, it is not possible to achieve any plan due to the constraints 
+of the model, and practically, the model that worked well before the occurrence of the 
+event, after that only reports infeasible results. 
+ 
+Figure 3. State of operational environment at time t=100h 
+However, in this example, we proceed through the mentioned method. From time 
+𝑡 = 0 to 𝑡 = 100ℎ, it will proceed according to the previous plan and the batches will 
+be injected in exactly the same amount and in the same order. Therefore, at 𝑡 = 100ℎ 
+and at the moment of starting the maintenance operation, the situation is as shown in 
+Figure 3. After that, due to the reduction of maximum rate for injection in the pipeline 
+to zero, there is no way to inject any new batches in the pipeline and practically, the 
+state of the pipeline remains constant for next 30 hours (which clearly shows the 
+waste of time and resources). From 𝑡 = 130ℎ, the planned program continues and the 
+rest of the steps and pumping runs are done according to what was assumed in plan. 
+
+0
+20
+40
+60
+80
+S1
+D1
+S2
+D2
+D3
+B5
+B4
+B2
+B1
+Start (h)
+End (h)
+10
+0.00
+8.33
+B5
+B4
+B2
+B1
+[20
+[20
+8.33
+25.00
+B5
+B4
+B2
+B1
+40
+[40
+40
+[20/20
+25.00
+58.33
+B5
+B4
+B3
+B1
+20
+[20
+ot
+[20]
+B6
+B5
+B4
+20
+58.33
+91.67
+10
+[10
+91.67
+100.00
+B7
+B6
+B5
+B4S.H.Moghimi et al. /Oper. Res. Eng. Sci. Theor. Appl. 3 (x) (2020) 1-20 
+ 
+16 
+ 
+Finally, at 𝑡 = 213.33ℎ, which is 30 hours later than the optimal schedule, the 
+distribution of the products finishes. Step by step execution of this plan has been 
+drawn in Figure 4. 
+ 
+Figure 4. The output plan of older methods when the incident happens 
+4.4. Performance of the proposed method in facing the incident 
+The advantage of the proposed method is that during planning, it pays attention to 
+the future changes of the environment and can consider different values for the 
+parameters. It is also possible to consider predictable events (such as the maintenance 
+program mentioned in this example) from the beginning. 
+In the proposed method, we consider two changes compared to previous methods 
+during modeling. First, we define the set of events as follow: 
+𝐸 = {𝑒0, 𝑒1 (Maintenance starts), 𝑒2 (Maintenance ends), 𝑒𝑚𝑎𝑥} 
+(41) 
+As explained earlier, in this set, two events 𝑒0 and 𝑒𝑚𝑎𝑥 are included to represent 
+the beginning and end of the time horizon. Events 𝑒1 and 𝑒2 also represent the start 
+and end of the maintenance program. Therefore, the values of the parameters 𝑇𝑒1 and 
+𝑇𝑒2 are considered to be 100ℎ and 130ℎ, respectively. 
+The second change is to define a vector of values for parameters 𝑣𝑏𝑚𝑖𝑛,𝑠1 and 
+𝑣𝑏𝑚𝑎𝑥,𝑠1. The values in this vector are as follow (values are in units per hour): 
+𝑣𝑏𝑚𝑖𝑛,𝑠1
+〈𝑒1〉
+= 1.0
+𝑣𝑏𝑚𝑖𝑛,𝑠1
+〈𝑒2〉
+= 0.0
+𝑣𝑏𝑚𝑖𝑛,𝑠1
+〈𝑒𝑚𝑎𝑥〉 = 1.0
+𝑣𝑏𝑚𝑎𝑥,𝑠1
+〈𝑒1〉
+= 1.2
+𝑣𝑏𝑚𝑎𝑥,𝑠1
+〈𝑒2〉
+= 0.0
+𝑣𝑏𝑚𝑎𝑥,𝑠1
+〈𝑒𝑚𝑎𝑥〉 = 1.2
+ 
+(42) 
+In this way, the model realizes that in the interval [𝑇𝑒1, 𝑇𝑒2], no pumping should be 
+performed from station 𝑆1. For other parameters, a vector of length |𝐸| −  1 should 
+be used too, but all the values in them will be the same because the only parameters 
+changing through time in this example are 𝑣𝑏𝑚𝑖𝑛,𝑠1 and 𝑣𝑏𝑚𝑎𝑥,𝑠1 
+
+20
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+B7Operational scheduling of oil products pipeline with intermediate event occurrences 
+ 
+17 
+ 
+ 
+The result of our proposed model has a much better performance than the old 
+methods. As can be seen in Figure 5, this method already looks into the future and 
+plans for the time that station 𝑆1 is not pumping, in such a way that the distribution of 
+products from other stations continues. 
+ 
+Figure 5. The output plan of our proposed model 
+The output of our method is same as the output of the previous methods until the 
+moment 𝑡 = 100ℎ. It pumps batches of different products in the pipeline in the same 
+order and with the same amount. But the divergence starts from time 𝑇𝑒1. Considering 
+that from this moment until 𝑇𝑒2 the pumping rate at station 𝑆1 will be zero, the first 
+part of the pipeline (from the beginning to the volume coordinates of 40 units) remains 
+constant. But in the same period of time, station 𝑆2 starts pumping products that 
+satisfy part of the demands of depots 𝐷2 and 𝐷3. In this way, when in 𝑇𝑒2 i.e. 30 hours 
+later, the station 𝑆1 starts pumping again, the remaining amount of products needed 
+in 𝐷2 and 𝐷3 is reduced compared to the start of the maintenance operation. So by 
+pumping next batches, it can be reached the desired condition faster and therefore 
+with lower cost.  
+More precisely, the output of our model consists of 8 pumping run (𝑘1 − 𝑘8), 
+where the runs 𝑘1 to 𝑘4 are performed exactly the same as in the normal case. The 
+only difference is that batch 𝐵5 in normal case is numbered as 𝐵7 here, because of two 
+intermediate batches that will be injected to the middle of the pipeline in future runs.  
+The fifth run (𝑘5), which is forced to be left unfinished in the previous methods, 
+ends consciously at time 𝑇𝑒1 in this model. Run 𝑘6 begins exactly when event 𝑒1 occurs 
+and thus, station 𝑆1 performs no pumping at this run. During the 16.67 hours of run 
+𝑘6, station 𝑆2 pumps 20 units of product 𝐶 as a new batch called 𝐵5 and on the other 
+hand, depot 𝐷3 withdraws 20 units of product 𝐵 from batch 𝐵4. In run 𝑘7, station 𝑆1 
+is still inactive so the station 𝑆2 again injects a new batch named 𝐵6 containing 20 
+units of product 𝐵 into the pipeline and batch 𝐵5 is completely delivered to the depot 
+𝐷2.  
+Run 𝑘8 starts after the maintenance operation is finished, and therefore, station 𝑆1 
+is reactivated in this run. Station 𝑆1 starts pumping a large new batch 𝐵10 with a 
+
+0
+20
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+3
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+102020]
+133.33
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+B10
+B9
+B8S.H.Moghimi et al. /Oper. Res. Eng. Sci. Theor. Appl. 3 (x) (2020) 1-20 
+ 
+18 
+ 
+volume of 60 units of product 𝐵. In this way, the depot 𝐷2 takes the last part of its 
+demand, i.e. 10 units of the product 𝐴 from the batch 𝐵7, and the depot 𝐷3 takes three 
+consecutive withdrawals. The remaining volume of the batches 𝐵4 and 𝐵6 plus a 
+portion of batch 𝐵8 -which are equal to 50 units of product 𝐵 in overall- are delivered 
+to the depot 𝐷3. The final state of the pipeline is very similar to the final state in the 
+normal case. There are 70 units of product 𝐵 and 10 units of product 𝐶 in the pipeline. 
+4.5. Conclusion 
+The execution time of the output plan of our proposed model is surprisingly equal 
+to the required time in normal case. It means that sometimes even severe incidents 
+like pipeline shutdowns, can be managed easily and with the lowest overhead cost.  
+Our proposed model generates a plan which is 30 hours shorter than the time 
+required to execute the program in the previous methods, due to the fact that the 
+pipeline is not stopped totally during the maintenance operation. The total demand of 
+all depots is fulfilled in only 183.33 hours which is 16\% faster than the output plan 
+of older methods, despite the possibility of older methods to go totally infeasible in 
+many other cases. 
+References  
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+discrete MILP formulation, Computers and Chemical Engineering 28 (10) (2004) 
+2053–2068.  
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+pipeline scheduling, Computers and Chemical Engineering 28 (8) (2004) 1511–1528.  
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+pipeline systems with pumping yield rate constraints, Computers and Chemical 
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+ S. Relvas, H. A. Matos, A. P. F. Barbosa-Povoa, J. Fialho, A. S. Pinheiro, Pipeline 
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+
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+ 
+20 
+ 
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+detailed scheduling of products in a pipeline with multiple pump stations, Energy 119 
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+© 2022 by the authors. Submitted for possible open access publication under the 
+terms and conditions of the Creative Commons Attribution (CC BY) 
+license (http://creativecommons.org/licenses/by/4.0/). 
+ 
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf,len=774
+page_content='Operational Research in Engineering Sciences: Theory and Applications  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 3, Issue x, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' x-x  ISSN: 2620-1607  eISSN: 2620-1747  DOI: https://  Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  hmoghimi@ce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='edu (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Moghimi), jhabibi@sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='edu (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Habibi),  hamid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='jahad@ce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='edu (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Jahad), fazli@sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='edu (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Fazli)  OPERATIONAL SCHEDULING OF OIL PRODUCTS PIPELINE  WITH INTERMEDIATE EVENT OCCURRENCES  Seyyed Hamed Moghimi 1*, Jafar Habibi 1,  Hamid Jahad 1,Mohammad Amin Fazli1  1 Department of Computer Engineering, Sharif University of Technology, Tehran, Iran  Received: 31 December 2022  Accepted: n x 2023  First online: n x 2023  Research paper  Abstract: Oil products are the main source of energy in the world today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Distribution of these products is one of the main issues in the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The  main tools for this work are pipelines, and along with it, railways, shipping and  roads are also used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Optimal planning of pipelines is an example of decision-  making problem and was the focus of many researchers in the past years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The  use of mixed integer linear programming (MILP) is one of the efficient methods  to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' However, models still ignore important operational  challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Vulnerability to deal with incidents as well as lack of attention to  other transportation methods as a complement to the pipeline are among the  weak points of the existing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In this research, we intend to facilitate the  decision-making process for experts in the field of distribution of oil products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  For this purpose, we must improve the existing MILP methods and modify them  for use in the real operational environment in such a way that sufficient  flexibility, the possibility of responding to incidents, and the ability to revise  the program are added to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Key words: Pipeline network, Operational Planning, Event Occurrence, Plan Revision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Introduction  In the last two centuries, one of the main issues in the world has been the provision  of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Oil and its refined products are considered as the most important fuel  sources in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' One on the important problems about oil products is their  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Considering the mechanical and chemical properties of these materials,  such as their fluidity, density, as well as their flammability, their transportation is  more sensitive, and for this reason, the related costs are also higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Main tools used  to move oil products today are pipelines, vehicles, railways, and vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Among those,  pipelines have efficiency, higher safety and lower price and that is why many pipelines  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='Moghimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' /Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 3 (x) (2020) 1-20  2  have been built around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' For example, nearly two-thirds of oil products in  the United States are distributed through pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Optimal planning in order to make maximum use of the transmission capacity of  pipelines is one of the main problems in this industry, which has allocated a significant  amount of past and present researches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' One of the most important points about this  optimization problem is the very high financial turnover that exists in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Considering the global price of oil products, the costs that may be imposed on a system  in case of non-optimal planning of pipelines can be about several million dollars per  day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Another reason is the strategic importance of fuel distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In all the countries  of the world, energy carriers are considered as strategic assets, and providing the  required amounts for normal use and saving for critical times is considered as one of  the strategic goals of the administrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Answering the optimization problem is not an obvious and simple task, because  the distribution network of oil products includes several pipes, different types of tanks  and depots, refineries, pump stations and control centers, which together form a  complex graph of different nodes and edges with unique characteristics and rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Furthermore, various oil products are flowing inside the pipes, with different volumes  based on the needs of destinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Several rules, including physical constraints and  safety principles, are also effective in making decisions about the order and amount of  injection of different products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Therefore, it can be concluded that the nature of such  a problem will not be simple and planning for it is a difficult and challenging activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  The main tool used for solving the optimization problem of oil products  distribution is mixed-integer linear programming (MILP) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' This method is  developed in the last two decades and several researches have focused to solve the  problems in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In this research, we proposed a novel method to take  environmental changes and event occurrences into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Our model can  adapt to new environment and continues to execute the distribution plan, event  different types of incidents occur and change the surrounding parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Related Works  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Early Researches  As of the beginning of the 21st century, researches were carried out in line with the  use of the linear programming model for the optimization of multi-product pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  The first model presented for this purpose looked at the problem from a discrete point  of view (Rejowski & Pinto, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' This means that the planning time horizon was  divided into small intervals and also, the length of the pipelines were categorized into  small volume intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' This work made it easier to look at the problem in a discrete  way, to understand mathematical relationships and equations and inequalities, but on  the other hand, both the number of relations and computational complexity of the  model increased significantly, and the accuracy and quality of the obtained results  were not at a good level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Following that, other activities such as (Neiro & Pinto, 2004)  and (Rejowski & Pinto, 2004) presented linear programming models with the same  discrete approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  With a little distance from these researches, the first linear programming model  with continuous view was presented (Cafaro & Cerda, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Continuous view versus  Operational scheduling of oil products pipeline with intermediate event occurrences  3  discrete view means that instead of breaking time and place into small intervals and  building model variables from the values of these intervals, the time and place of  batches at key frames are considered as model variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Key frames mean the  moments when changes are to be made in the operation environment, for example a  new batch is pumped into the pipeline or a batch is received at a station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' With this  point of view, the number of variables and constraints is significantly reduced and, as  a result, the computational performance of the model grows up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The reason for  reducing the dimensions of the problem in this model is that in discrete modeling,  there are many times when the environment is operating normally in several  consecutive time intervals, and practically all these intervals can be assumed as one  key frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' After these early researches, a growing trend of researches began with the  aim of using linear programming in modeling pipelines for more complex situations  and involving more parameters into the problem to be closer to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Topologies  In (Rejowski & Pinto, 2003) and (Cafaro & Cerda, 2004), both approaches consider  only one refinery as the pipeline input and one or more stations as product depots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  After that some researches focus on the variation of pipeline topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In (MirHassani  & Ghorbanalizadeh, 2008), (Mostafaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2015), and (Taherkhani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2017) tree  shaped pipelines are modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' These tree topologies include the main pipeline and  another level of child branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' More complicated topologies have been modeled in  (Cafaro & Cerda, 2012), which introduced pipelines with mesh structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Input and output stations  One of the parameters considered in researches is number of input and output in  the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Early models, consider the refinery as an entry point to the pipeline and  some consecutive depots along the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' This approach is also used in many later  researches such as (Rejowski & Pinto, 2004), (Rejowski & Pinto, 2008), (Cafaro &  Cerda, 2008), (MirHassani & Ghorbanalizadeh, 2008), (MirHassani & Fani Jahromi,  2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  In the other hand, some researches simplify this assumption and just model one  input and only one output terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' This change results in less complexity and  therefore, the possibility to other parameters being modeled (Relvas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2006),  (Moradi & MirHassani, 2014), and (Moradi & MirHassani, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  More complex input and output terminals are also modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' For examples (Cerda  & Cafaro, 2009), (Cafaro & Cerda, 2010), (Magatao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2015), (Mostafaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2015),  (Mostafaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2016), and (Taherkhani, 2020) take into account multiple input  nodes and dual-purposed terminals which can both pump new batches and withdraw  product from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Scheduling time horizon  In all the researches around pipeline scheduling, one of the important parameters  is the time horizon of planning, indicating the duration of the entire period for which  planning is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=" The importance of this parameter's value is related to a trade-off  between the computational complexity of the model and its application in the real  world." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The shorter the time horizon results in fewer parameters and formulas and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='Moghimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' /Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 3 (x) (2020) 1-20  4  therefore, a less complex model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' On the other hand, a shorter time horizon causes the  future state of pipeline not to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' So it is possible that the result of the  execution of models with shorter time horizon causes the pipeline network not to  perform well in the long term, although they have good results in short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Researches have considered the time horizon as a determining parameter in  research classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' There is a focus on three types of time horizon with the aim of  different and specific applications: short-term scheduling which focuses to plan for  less than a week like (De Souza Filho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2013) and (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2017), medium-  term scheduling with the aim to plan for one to two weeks such as (Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2018a),  (Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2018b) and (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2017), and long-term scheduling for horizons up  to a month like (Meira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2018), (Cafaro & Cerda, 2017), and (Cafaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Revision in scheduling  The main target of the proposed method is to achieve a model that can take changes  in the operational environment into account and produces a plan that can adapt itself  when events and incidents occur at the middle of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Only a few number of researches tried to deal with events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Almost the first attempt  to revise a scheduling is presented in (Relvas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In this research, alongside  with an MILP model for a one-to-one pipeline, a mechanism is introduced to enable  revision of the plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' This mechanism works in a simple way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' It states that any time  during the execution of the plan, if it is determined that there is a need to revise the  initial plan, then the execution is stopped, changes are made in the parameters and a  new plan will be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Some other researches such as (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2018) and (Mostafaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2016) have  done focusing on the uncertainty of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Another approach that has  been investigated in (Taherkhani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=', 2017) in order to respond to the random  nature of requirements is modeling with random variables and expectation  calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Proposed Model  In this section, we will describe the proposed mathematical model for pipeline  planning, with the ability to be aware of and adapt to upcoming events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Our proposed  model is based on the model presented in (Cafaro & Cerda, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' There are six main  sets in this model: The set of pumping runs (𝐾), injected batches (𝐼), oil products (𝑃),  input terminals (𝑆), and receiving destinations (𝐽) are also included in the basic  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The last set (𝐸) contains upcoming events in the planning horizon, ordered by  time of occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' This means that the event 𝑒 must happen before 𝑒′ if 𝑒 <  𝑒′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The  Set 𝐸 has at least two elements 𝑒0 and 𝑒𝑚𝑎𝑥, representing the start and the end of the  planning horizon respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Nomenclature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='(a) Sets  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝐾  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='chronologically ordered blocks of pumping runs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝐼  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='product batches (𝐼 = 𝐼old ∪ 𝐼new )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝐼𝑛𝑒𝑤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='new batches to be injected during the planning horizon  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='Operational scheduling of oil products pipeline with intermediate event occurrences  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝐼𝑜𝑙𝑑  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='old batches in the initial state of pipeline  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑃  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='refined oil products  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑃𝑖  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='refined oil product contained in old batch 𝑖 ∈ 𝐼old  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='pipeline input terminals  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝐽  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='pipeline output terminals  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝐽𝑝  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='distribution depots demanding product 𝑝  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝐽𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑠  distribution depots demanding product 𝑝 to be supplied by an  upstream source 𝑠  𝐸  chronologically ordered events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  It consists of at least two members, 𝑒0 and 𝑒𝑚𝑎𝑥 representing the start  and the end of planning  (b) Parameters  ℎ𝑚𝑎𝑥  planning horizon length  𝑃𝑉  total pipeline volume from the origin to the farthest depot  𝑇𝑒  occurrence time of event 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' (𝑇𝑒0 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝑇𝑒𝑚𝑎𝑥 = ℎ𝑚𝑎𝑥)  𝑐𝑏𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑗  unit backorder penalty cost for tardy orders of product 𝑝 at output  terminal 𝑗  𝑐𝑖𝑓𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑝′  total reprocessing cost of interface material involving products 𝑝 and  𝑝′  𝑐𝑖𝑛𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑠  pumping cost per unit of product 𝑝 at input terminal 𝑠  𝐷𝑚𝑖𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝐷𝑚𝑎𝑥  minimum/maximum delivery size to output terminals  𝐷𝐿𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑗  minimum request of product 𝑝 at the output terminal 𝑗  𝐷𝑈𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑗  maximum amount of product 𝑝 that can be delivered to output  terminal 𝑗  𝑄𝑚𝑖𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝑄𝑚𝑎𝑥  minimum/maximum batch injection size  𝑄𝑚𝑎𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑝  maximum batch injection size for product 𝑝  𝑆𝐿𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑠  lowest amount of product 𝑝 to be shipped from input terminal 𝑠  𝑆𝑈𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑠  maximum amount of product 𝑝 that can be shipped from input  terminal 𝑠  𝑣𝑏𝑚𝑖𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝑣𝑏𝑚𝑎𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑠  minimum/maximum product flow-rates at source 𝑠  𝑊𝑖  0  initial volume of old batch 𝑖  𝜎𝑗  volume coordinate of output terminal 𝑗 from the pipeline origin  𝜏𝑠  volume coordinate of input terminal 𝑠 from the pipeline origin  (c) Continuous variables  𝐵𝐶  total backorder penalty cost for tardily meet product demands  𝐵𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑗  backorder of product 𝑝 at the output terminal 𝑗  𝐶𝑘  completion time of run block 𝑘 (measured in time units)  𝐿𝑘  length of run block 𝑘  𝐿𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑠  length of pumping run 𝑘 taking place at source 𝑠  𝐷𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑗  (𝑘)  volume of batch 𝑖 diverted to output terminal 𝑗 during run block 𝑘  𝐷𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑝  (𝑘)  volume of product 𝑝 diverted from batch 𝑖 to terminal 𝑗 during run  block 𝑘  𝐹𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘  upper coordinate of batch 𝑖 from the origin at time 𝐶𝑘  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='Moghimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' /Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 3 (x) (2020) 1-20  6  𝑃𝐶𝑘  total pumping cost during block 𝑘  𝑄𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑠  (𝑘)  size of batch 𝑖 shipped from input terminal 𝑠 during run 𝑘  𝑄𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑝  (𝑘)  size of batch 𝑖 containing product 𝑝 inputted from input terminal 𝑠  through run 𝑘  𝑅𝐶𝑖  total reprocessing cost for the interface between batch 𝑖 and the  following (𝑖 +  1)  𝑊𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘  size of batch 𝑖 at the end of run 𝑘  (d) Binary variables  𝑣𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑠  (𝑘)  variable denoting that a portion of batch 𝑖 is injected from source 𝑠  through block 𝑘  𝑥𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑗  (𝑘)  variable denoting that a portion of batch 𝑖 is diverted to the depot 𝑗  during block 𝑘  𝑦𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑝  variable denoting that batch 𝑖 contains product 𝑝  𝑏𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑒  variable denoting that pumping run 𝑘 is executed completely inside  of the interval [𝑇𝑒−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝑇𝑒]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Sequencing pumping run  A block 𝑘 ∈ 𝐾 must start after the completion of the previous block (𝑘 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Let 𝐶𝑘  denote the completion time and 𝐿𝑘 the duration of block 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝐶𝑘 − 𝐿𝑘 ≥ 𝐶𝑘−1∀𝑘 ∈ 𝐾(𝑘 > 1)  (1)  All pumping blocks should also be finished before the planning horizon ℎ𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝐶𝑘 ≤ ℎ𝑚𝑎𝑥∀𝑘 ∈ 𝐾  (2)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Aligning pumping runs with events  There is an assumption in the model that during the execution of a pumping run 𝑘,  the environmental conditions and parameters remain stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' On the other hand,  almost all events will change the parameters of the environment when they happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Therefore, it should be prevented that an event occurs during the execution of a  pumping run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  For this purpose, we define the binary variable 𝑏𝑘,𝑒 in such a way that its value  equals to 1, iff pumping run 𝑘 is executed completely between two consecutive events  𝑒 and 𝑒 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' It means that pumping run 𝑘 must be started after 𝑇𝑒−1 (the occurrence  time of the event 𝑒 − 1) and must be finished before 𝑇𝑒 (the occurrence time of the  event 𝑒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Figure 1 shows an example of this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In all other conditions, 𝑏𝑘,𝑒 will  be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Te-1  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  K  k  Ck  CK --Operational scheduling of oil products pipeline with intermediate event occurrences  7  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' An example of a pumping run fully executed between two consecutive events  The following equations are added to the model to ensure the mentioned  alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' First, each pumping run 𝑘 should be executed no more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Then:  ∑  𝑏𝑘,𝑒  𝑒∈𝐸  = 1∀𝑘 ∈ 𝐾, 𝑒 ∈ 𝐸  (3)  Second, if a pumping run is executed between two events, the start and finish time  of the run should be placed between occurrence time of those events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝑇𝑒−1𝑏𝑘,𝑒 ≤ 𝐶𝑘 − 𝐿𝑘∀𝑘 ∈ 𝐾, 𝑒 ∈ 𝐸  (4)  𝐶𝑘 ≤ 𝑇𝑒𝑏𝑘,𝑒 + (1 − 𝑏𝑘,𝑒)𝑀𝑇∀𝑘 ∈ 𝐾, 𝑒 ∈ 𝐸  (5)  The value of constant 𝑀𝑇 should be larger than all time values in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' So it  can be chosen as ℎ𝑚𝑎𝑥 or higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Injecting batches in different pumping runs  A pumping operation is identified by its corresponding pumping run 𝑘 ∈ 𝐾 and  source terminal 𝑠 ∈ 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' We define the binary variable 𝑣𝑖,𝑠  (𝑘) to state that the batch 𝑖 ∈  𝐼𝑛𝑒𝑤 in injected to the pipeline from input terminal 𝑠 at the pumping run 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' If 𝑣𝑖,𝑠  (𝑘) = 1,  then the whole or at least some amount of batch 𝑖 is injected from source terminal 𝑠  during pumping run 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' To ensure that no more than one batch is injected in the  pipeline from each source in each pumping run, we have:  ∑𝑖∈𝐼  𝑣𝑖,𝑠  (𝑘) ≤ 1∀𝑘 ∈ 𝐾, 𝑠 ∈ 𝑆  (6)  If for a certain pumping run 𝑘, the term (∑  ∑  𝑣𝑖,𝑠  (𝑘)  𝑖∈𝐼  𝑠∈𝑆  ) is zero, it means that no  batches are injected in that run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Since the size of set (𝐾) is normally chosen larger that  the number of pumping runs in the optimal solution, this situation is probable for the  last members of (𝐾) (named dummy runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  The following Equation ensures that if a pumping run 𝑘 is dummy, all later runs  also turn to dummy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  ∑𝑠∈𝑆  ∑𝑖∈𝐼  𝑣𝑖,𝑠  (𝑘) ≤ 𝑀(∑𝑠′∈𝑆  ∑𝑖′∈𝐼  𝑣𝑖′,𝑠′  (𝑘−1))∀𝑘 ∈ 𝐾(𝑘 > 1)  (7)  The constant 𝑀 is a very large number, greater than all values that can be generated  in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' We will use it more in later equations in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Batch injection and size relation  Every Batch 𝑖 is injected for the first time in a pumping run 𝑘 at a source terminal  𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' It is also possible that some portion of 𝑖 is appended to it in later pumping run 𝑘′ at  another (or the same) terminal 𝑠′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' We say that batch 𝑖 is injected in both runs (𝑘, 𝑠)  and (𝑘′, 𝑠′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Let 𝑄𝑖,𝑠  (𝑘) denote the size of injected product to batch 𝑖 in run (𝑘, 𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝑄𝑖,𝑠  (𝑘) will be  positive iff there is actually a real injection in batch 𝑖 at run (𝑘, 𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' This means that  𝑣𝑖,𝑠  (𝑘)and 𝑄𝑖,𝑠  (𝑘) will be either zero or positive simultaneously:  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='Moghimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' /Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 3 (x) (2020) 1-20  8  𝑄𝑚𝑖𝑛𝑣𝑖,𝑠  (𝑘) ≤ 𝑄𝑖,𝑠  (𝑘) ≤ 𝑄𝑚𝑎𝑥𝑣𝑖,𝑠  (𝑘)∀𝑖 ∈ 𝐼, 𝑠 ∈ 𝑆, 𝑘 ∈ 𝐾  (8)  Parameters 𝑄𝑚𝑖𝑛 and 𝑄𝑚𝑎𝑥 present the global lower and upper bound for the  volume of product that can be injected during a pumping run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Choosing lengths of pumping runs  Let 𝐿𝑘,𝑠 be the length of the run (𝑘, 𝑠), means the time length during run 𝑘 that  terminal 𝑠 is actually pumping products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The whole run 𝑘 can be finished only if all of  its sub-runs (𝑘, 𝑠) is finished before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' So the length of run 𝑘 will be given by  𝐿𝑘 =  max  𝑠∈𝑆 (𝐿𝑘,𝑠) or:  𝐿𝑘 ≥ 𝐿𝑘,𝑠∀𝑘 ∈ 𝐾, 𝑠 ∈ 𝑆  (9)  The injected volume of refined product is also related to the length of pumping run  and the rate at which a terminal can perform injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The total volume of product  injected to the pipeline during run (𝑘, 𝑠) is equal to  ∑  𝑄𝑖,𝑠  (𝑘)  𝑖∈𝐼  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In the base model from  (Cafaro & Cerda, 2010) the following equation is presented, where 𝑣𝑏𝑚𝑖𝑛,𝑠 and  𝑣𝑏𝑚𝑎𝑥,𝑠  stand for the minimum and maximum pumping rate of terminal 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝑣𝑏𝑚𝑖𝑛,𝑠𝐿𝑘,𝑠 ≤ ∑𝑖∈𝐼  𝑄𝑖,𝑠  (𝑘) ≤ 𝑣𝑏𝑚𝑎𝑥,𝑠𝐿𝑘,𝑠∀𝑘 ∈ 𝐾, 𝑠 ∈ 𝑆  (10)  In this model, we want to take the effects of events into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The next  section discusses the problem and presents the proposed technique to address this  requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Choosing parameters based on event occurrences  When an event occurs, it may affect the conditions of pipeline and its surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  In our model, this will be done by changing the value of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Therefore, the  value of a parameter (same as  𝑣𝑏𝑚𝑖𝑛,𝑠 and  𝑣𝑏𝑚𝑎𝑥,𝑠 from Equation 8) should be chosen  according to event occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Since the parameters are assumed to be fixed when  solving an MILP model, our technique to address this need is to have a vector of all  values that a parameter can take over the planning horizon and using the respective  value in the constraint expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  For example, suppose there is a predetermined maintenance program at terminal  𝑠 (represented by event 𝑒) that will cause the maximum pumping rate at the terminal  to decrease to half of the previous value, and it remains the same until the maintenance  end (represented by event 𝑒′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In such a case, we consider three values for the  parameters  𝑣𝑏𝑚𝑎𝑥,𝑠: the value before the occurrence of 𝑒, the value between 𝑒 and 𝑒′,  and the value after 𝑒′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Let us denote the value of an arbitrary parameter 𝐴 between events 𝑒 −  1 and 𝑒  by the notation 𝐴〈𝑒〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' So the three parameters mentioned before will be denoted by  𝑣𝑏𝑚𝑎𝑥,𝑠  〈𝑒〉  , 𝑣𝑏𝑚𝑎𝑥,𝑠  〈𝑒′〉 , and 𝑣𝑏𝑚𝑎𝑥,𝑠  〈𝑒𝑚𝑎𝑥〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  According to these assumptions and notations, we rewrite the Equation 10:  𝑣𝑏𝑚𝑖𝑛,𝑠  <𝑒> 𝐿𝑘,𝑠 − 𝑀(1 − 𝑏𝑘,𝑒) ≤ ∑𝑖∈𝐼  𝑄𝑖,𝑠  (𝑘) ≤ 𝑣𝑏𝑚𝑎𝑥,𝑠  <𝑒> 𝐿𝑘,𝑠 + 𝑀(1 − 𝑏𝑘,𝑒)∀𝑘 ∈ 𝐾, 𝑠 ∈  𝑆, 𝑒 ∈ 𝐸(𝑒 > 0)  (11)  Operational scheduling of oil products pipeline with intermediate event occurrences  9  The term 𝑀(1 − 𝑏𝑘,𝑒) is called event guard and will be noted as 𝐸𝐺𝑘,𝑒 hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' It  is used to ensure that if the pumping run 𝑘 is executed between events 𝑒 − 1 and 𝑒  (means 𝑏𝑘,𝑒), only the corresponding equation (containing parameters 𝐴〈𝑒〉) takes  effect and the other ones go trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' That will be held because an infinity term is added  to the larger side of the inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Tracking the batch size over time  Let 𝑊𝑖,𝑘 be the size of a batch 𝑖 at the time 𝐶𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' If the volume of batch 𝑖 is changed  during run 𝑘, it must be either an injection from source 𝑠 or a delivery to depot 𝑗 ∈ 𝐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝑊𝑖,𝑘 = 𝑊𝑖,𝑘−1 + ∑𝑠∈𝑆  𝑄𝑖,𝑠  (𝑘) − ∑𝑗∈𝐽  𝐷𝑖,𝑗  (𝑘)∀𝑖 ∈ 𝐼, 𝑘 ∈ 𝐾  (12)  Only the injections that are performed at upstream sources can cause older batches  to move forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' So the amount of product that can be delivered to depot 𝑗 is at most  equal to the amount of injections performed at upstream sources (𝜏𝑠 < 𝜎𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝐷𝑖,𝑗  (𝑘) ≤ ∑𝑠∈𝑆:𝜏𝑠<𝜎𝑗  ∑𝑖′∈𝐼  𝑄𝑖′,𝑠  (𝑘)∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾  (13)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Tracking the batch location over time  Let 𝐹𝑖,𝑘 denote the upper volumetric coordinate of batch 𝑖 at the time 𝐶𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' So 𝐹𝑖+1,𝑘  also represents the upper coordinate of batch (𝑖 + 1), which is equal to the lower  coordinate of batch 𝑖 at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Since the difference between lower and upper  coordinates of a batch 𝑖 at the end of run 𝑘 is equal to its size 𝑊𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' we have:  𝐹𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘 − 𝑊𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘 = 𝐹𝑖+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘∀𝑖 ∈ 𝐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝑘 ∈ 𝐾  (14)  To prevent backward movement of batches during the time:  𝐹𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘−1 ≤ 𝐹𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘∀𝑖 ∈ 𝐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝑘 ∈ 𝐾  (15)  Finally to ensure that all batches stands into pipeline and no batches will ever be  located outside the pipeline coordination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Let 𝑃𝑉 stands for the total pipeline volume,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  then the upper coordinate of any batch 𝑖 must never be greater than 𝑃𝑉 and its lower  coordinate must be non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝐹𝑖,𝑘 ≤ 𝑃𝑉∀𝑖 ∈ 𝐼, 𝑘 ∈ 𝐾  (16)  𝐹𝑖,𝑘 − 𝑊𝑖,𝑘 ≥ 0∀𝑖 ∈ 𝐼, 𝑘 ∈ 𝐾  (17)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Preventing empty spaces and overflows  The pipeline must remain full of products at any time over the planning horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Therefore, the sum of volumes of all batches in the pipeline must be equal to 𝑃𝑉:  ∑𝑖∈𝐼  𝑊𝑖,𝑘 = 𝑃𝑉∀𝑘 ∈ 𝐾  (18)  Also the amount of injected and delivered products must be equal at the end of any  run 𝑘:  ∑𝑖∈𝐼  ∑𝑠∈𝑆  𝑄𝑖,𝑠  (𝑘) = ∑𝑖∈𝐼  ∑𝑗∈𝐽  𝐷𝑖,𝑗  (𝑘)∀𝑘 ∈ 𝐾  (19)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='Moghimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' /Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 3 (x) (2020) 1-20  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Injections and deliveries feasibility  Refined product can be injected into batch 𝑖 during pumping run (𝑘, 𝑠) only if the  following two conditions are satisfied:  (a) Batch 𝑖 has reached the location of the source 𝑠 (𝜏𝑠) before the run 𝑘 begins:  𝐹𝑖,𝑘−1 ≥ 𝜏𝑠𝑣𝑖,𝑠  (𝑘)∀𝑖 ∈ 𝐼, 𝑠 ∈ 𝑆, 𝑘 ∈ 𝐾  (20)  (b) Also batch 𝑖 has not completely passed the location of source 𝑠 before the run  𝑘:  𝐹𝑖,𝑘−1 − 𝑊𝑖,𝑘−1 ≤ 𝜏𝑠 + 𝑀(1 − 𝑣𝑖,𝑠  (𝑘))∀𝑖 ∈ 𝐼, 𝑠 ∈ 𝑆, 𝑘 ∈ 𝐾  (21)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Delivery from batches to depots  Same as the previous section, delivery of product from batch 𝑖 to depot 𝑗 is also  possible, only if:  (a) The batch 𝑖 reaches the location of depot 𝑗 (denoted by 𝜎𝑗) before the end of run  𝑘, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝐹𝑖,𝑘 ≥ σj  (b) The batch 𝑖 has not passed the location of depot 𝑗 before the run 𝑘 begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' It  means that the lower coordination of the batch must not exceed the location of the  depot, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝐹𝑖,𝑘−1 − 𝑊𝑖,𝑘−1 ≤ 𝜎𝑗  To fulfill these conditions, let 𝑥𝑖,𝑗  (𝑘) be a binary variable denoting that some product  is delivered from batch 𝑖 to depot 𝑗 during run 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' When 𝑥𝑖,𝑗  (𝑘) = 0, nothing should be  transferred from batch 𝑖 to depot 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' It means 𝐷𝑖,𝑗  (𝑘) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝐷𝑚𝑖𝑛,𝑗  <𝑒> 𝑥𝑖,𝑗  (𝑘) ≤ 𝐷𝑖,𝑗  (𝑘) ≤ 𝐷𝑚𝑎𝑥,𝑗  <𝑒> 𝑥𝑖,𝑗  (𝑘)∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾, 𝑒 ∈ 𝐸(𝑒 > 0)  (22)  where 𝐷𝑚𝑖𝑛,𝑗  〈𝑒〉  and 𝐷𝑚𝑎𝑥,𝑗  〈𝑒〉  are bounds on the amount of product that can be delivered  to depot 𝑗 in timespan [𝑇𝑒−1, 𝑇𝑒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  The following two constrains are used to meet the two conditions mentioned  before in this section, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝐹𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘 ≥ 𝜎𝑗𝑥𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑗  (𝑘)∀𝑖 ∈ 𝐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝑗 ∈ 𝐽,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝑘 ∈ 𝐾  (23)  𝐹𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘−1 − 𝑊𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘−1 ≤ 𝜎𝑗 + 𝑀(1 − 𝑥𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑗  (𝑘))∀𝑖 ∈ 𝐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝑗 ∈ 𝐽,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝑘 ∈ 𝐾  (24)  As mentioned before,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' the prerequisite for delivering product from batch 𝑖 to depot  𝑗 at run 𝑘 is that the following inequalities are true:  𝐹𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘−1 − 𝑊𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘−1 < 𝜎𝑗 ≤ 𝐹𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘  (25)  In this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' an upper bound on the volume of product that can be delivered is 𝜎𝑗 −  (𝐹𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘−1 − 𝑊𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='𝑘−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' unless more product is injected in batch 𝑖 from upstream sources at  the same time that delivery is in action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  From the other hand, delivery of product can be done at any depot that is in touch  with the batch during the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' So the total amount of deliveries from batch 𝑖 to all the  depots from the origin to depot 𝑗, could not be more than the initial available volume  Operational scheduling of oil products pipeline with intermediate event occurrences  11  of batch 𝑖 (𝜎𝑗 − (𝐹𝑖,𝑘−1 − 𝑊𝑖,𝑘−1)) added by the total amount of injections from  upstream sources to batch 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Then:  ∑𝑗  𝑗′=1  𝐷𝑖,𝑗′  (𝑘) ≤ 𝜎𝑗 − (𝐹𝑖,𝑘−1 − 𝑊𝑖,𝑘−1) + ∑ 𝑠∈𝑆  𝜏𝑠<𝜎𝑗  𝑄𝑖,𝑠  (𝑘) + (𝑃𝑉 − 𝜎𝑗)(1 − 𝑥𝑖,𝑗  (𝑘))  ∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾  (26)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Assigning products to batches  At most, only one product can be assigned to batch 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Let 𝑦𝑖,𝑝 be a binary variable  denoting that batch 𝑖 contains product 𝑝 whenever 𝑦𝑖,𝑝 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Then:  ∑𝑝∈𝑃  𝑦𝑖,𝑝 ≤ 1∀𝑖 ∈ 𝐼  (27)  If no products is assigned to batch 𝑖, then we say that 𝑖 is fictitious and will not be  injected in the pipeline in real life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' It means that there is no pumping run (𝑘, 𝑠)  injecting any product to batch 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Then we have 𝑣𝑖,𝑠  (𝑘) = 0 for all pumping runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  In the other hand, if there is a product assignment to batch 𝑖, it means that this  batch is real and at least one pumping run must inject some products to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' This  condition can be mathematically written as follows,  ∑𝑝∈𝑃  𝑦𝑖,𝑝 ≤ ∑𝑠∈𝑆  ∑𝑘∈𝐾  𝑣𝑖,𝑠  (𝑘) ≤ 𝑀 ∑𝑝∈𝑃  𝑦𝑖,𝑝∀𝑖 ∈ 𝐼𝑛𝑒𝑤  (28)  Some products should not be injected next to each other because of contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Let us assume that (𝑝, 𝑝′) stands for a forbidden sequence of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Then,  𝑦𝑖−1,𝑝 + 𝑦𝑖,𝑝′ ≤ 1∀𝑖 ∈ 𝐼𝑛𝑒𝑤(𝑖 > 1)  (29)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Product injection  The whole amount of products injected into batch 𝑖 in all pumping runs must be  from a single type of product 𝑝, such that 𝑦𝑖,𝑝 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝑄𝑃𝑖,𝑠,𝑝  (𝑘) ≤ 𝑄𝑚𝑎𝑥𝑦𝑖,𝑝∀𝑖 ∈ 𝐼, 𝑠 ∈ 𝑆, 𝑘 ∈ 𝐾, 𝑝 ∈ 𝑃  (30)  ∑𝑝∈𝑃  𝑄𝑃𝑖,𝑠,𝑝  (𝑘) = 𝑄𝑖,𝑠  (𝑘)∀𝑖 ∈ 𝐼, 𝑠 ∈ 𝑆, 𝑘 ∈ 𝐾  (31)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Product Delivery  Same as the previous section, The whole amount of delivered material from batch  𝑖 to different depots must be from a single type of product 𝑝 such that 𝑦𝑖,𝑝 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝐷𝑃𝑖,𝑗,𝑝  (𝑘) ≤ 𝐷𝑚𝑎𝑥𝑦𝑖,𝑝∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾, 𝑝 ∈ 𝑃  (32)  ∑𝑝∈𝑃  𝐷𝑃𝑖,𝑗,𝑝  (𝑘) = 𝐷𝑖,𝑗  (𝑘)∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾  (33)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Fulfilling product demands  The model should provide the demanded amount of products for all depots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Let say  that the total amount of product 𝑝 needed at the depot 𝑗 is 𝐷𝐿𝑝,𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' To let the model  continue to work and be feasible, even if it does not fulfill the demand 𝐷𝐿𝑝,𝑗, we  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='Moghimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' /Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 3 (x) (2020) 1-20  12  introduce variable 𝐵𝑝,𝑗 as the amount of product shortage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' It will be used in the cost  function later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  At last, to consider the upper bound of storage capacity for product 𝑝 at depot 𝑗, we  use parameter 𝐷𝑈𝑝,𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' So the total amount of product 𝑝 delivered to depot 𝑗 must be  lower that this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝐷𝐿𝑝,𝑗 − 𝐵𝑝,𝑗 ≤ ∑𝑘∈𝐾  ∑𝑖∈𝐼  𝐷𝑃𝑖,𝑗,𝑝  (𝑘) ≤ 𝐷𝑈𝑝,𝑗∀𝑝 ∈ 𝑃, 𝑗 ∈ 𝐽  (34)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Initial state of problem  To initialize the starting position and order of batches in the pipeline, let 𝑊𝑖  0 be the  volume of the old batch 𝑖 ∈ 𝐼𝑜𝑙𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Then, the upper coordinate of batch 𝑖 ∈ 𝐼𝑜𝑙𝑑 is equal  to the sum of initial volumes for all succeeding batches 𝑖′ ∈ 𝐼𝑜𝑙𝑑 (𝑖′ > 𝑖) plus its own  volume:  𝐹𝑖,0 = ∑𝑖′∈𝐼𝑜𝑙𝑑(𝑖′≥𝑖)  𝑊𝑖′0∀𝑖 ∈ 𝐼𝑜𝑙𝑑  (35)  Moreover, it is also known the product 𝑃𝑖 contained in every old batch 𝑖 ∈ 𝐼𝑜𝑙𝑑  𝑦𝑖,𝑃𝑖 = 1∀𝑖 ∈ 𝐼𝑜𝑙𝑑  (36)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Objective function  The main goal of the proposed method is to increase the adaptability of the model  to changes in the operating environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Therefore, in this method, in addition to  trying to achieve a program that will deliver the demand of depots with at the lowest  cost of pumping (𝑃𝐶) and reprocessing of interface material (𝑅𝐶), the cost of shortage  in the needs for the products (𝐵𝐶) must be reduced as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝑚𝑖𝑛𝑧 = ∑𝑖∈𝐼  𝑅𝐶𝑖 + ∑𝑘∈𝐾  𝑃𝐶𝑘 + 𝐵𝐶  (37)  The following three equations introduce the terms of the objective function:  𝑅𝐶𝑖 ≥ cif𝑝,𝑝′(𝑦𝑖,𝑝 + 𝑦𝑖+1,𝑝′ − 1)∀𝑖 ∈ 𝐼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 𝑝, 𝑝′ ∈ 𝑃  (38)  The variable 𝑅𝐶𝑖 stands for the total cost of reprocessing interface material  between batches 𝑖 and 𝑖 + 1 and the parameter 𝑐𝑖𝑓𝑝,𝑝′ is the cost for reprocessing the  interface material resulting from blending of products 𝑝 and 𝑝′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝑃𝐶𝑘 = ∑𝑖∈𝐼  ∑𝑠∈𝑆  ∑𝑝∈𝑃  cin𝑝,𝑠𝑄𝑃𝑖,𝑠,𝑝  (𝑘) ∀𝑘 ∈ 𝐾  (39)  The variable 𝑃𝐶𝑘 is the total pumping cost at run 𝑘 and the coefficient cin𝑝,𝑠 stands  for the cost of pumping a unit volume of product 𝑝 in the source 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  𝐵𝐶 = ∑𝑝∈𝑃  ∑𝑗∈𝐽  cb𝑝,𝑗𝐵𝑝,𝑗  (40)  The parameter cb𝑝,𝑗 stands for the cost of failing to provide a single unit of product  𝑝 at depot 𝑗 on schedule and 𝐵𝑝,𝑗 (as introduced in Equation 34) is the amount of  backorder of product 𝑝 at depot 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Operational scheduling of oil products pipeline with intermediate event occurrences  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Results & Discussion  In this section, the results of the implementation of the proposed model and its  performance comparison with previous methods are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The study case  considered for this comparison is based on an example mentioned in (Cafaro & Cerda,  2010), which has been modified to meet the requirements of this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Example configuration  In this example, the operational planning of a pipeline network is carried out,  which includes two source stations 𝑆1 and 𝑆2 and three depots 𝐷1, 𝐷2 and 𝐷3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' This  pipeline transfers three refined products 𝐴, 𝐵 and 𝐶 from the sources to depots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Stations 𝑆1 and 𝑆2 are located at volume coordinates 0 and 40 from the beginning of  the pipeline, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Stations 𝐷1, 𝐷2, and 𝐷3 are also located at volume  coordinates 20, 60, and 80 from the pipeline origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Between each two consecutive  stations, there is a pipeline with a volume of 20 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  The available amount of the products at the sources and the demands of the depots  for the products are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Also, in Table 3, the cost of recycling the  interface between each pair of products can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Product supplies and demands  Supplies  Demands  S1  S2  D1  D2  D3  A  50  20  60  60 B  80  60 100  C  30  80 60 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Interface costs (in 100$)  Predecessor  Successor  A  B  C  A 22  23  B  24 21  C  30  32 Table 4 contains the cost of pumping one volume unit of each product in each of  the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The feasible range for pumping rate for all products in all sources is set to  be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='2 (units/h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Pumping cost in each source (in 100$ per volume unit)  Product  Source  S1  S2  A  29  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='5  B  34  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='0  C  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='5  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='0  The initial state of the pipeline includes four batches 𝐵1, 𝐵2, 𝐵4 and 𝐵5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The  product and the initial volume of these batches is shown in the top row of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='Moghimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' /Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 3 (x) (2020) 1-20  14  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Optimal pipeline schedule for the example when no events occur  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Optimal solution in absence of incidents  When no events occur during the planning horizon, the optimal schedule for  transporting products and meeting the demands of the depots is as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  In this case, we will have 9 pumping runs 𝑘1 − 𝑘9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' A distinct row is drawn in Figure 2  for each of pumping runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The start time, completion time and duration of each run  can be seen at the left of each row and the volume of the product injected into the  pipeline or received from it during the run is also drawn as arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The color pattern  of each batch indicates the product related to it, as is defined in the legend of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  As can be seen, the time required to execute the optimal schedule is 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  The plan is as follow:  In the first run (𝑘1), which starts from time 0 and continues for 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33 hours, 10 units  of product 𝐴 are injected from station 𝑆1 to batch 𝐵5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' As the batches move forward in  the pipeline, depot 𝐷2 takes 10 units of product 𝐴 from batch 𝐵2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In the second run  (𝑘2) which takes 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67 hours long, station 𝑆2 performs the pumping operation and  adds 20 units of product 𝐴 to batch 𝐵2 and exactly the same amount of product from  the same batch is received again in depot 𝐷2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Run 𝑘3 takes 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33 hours and during  this run, 40 units of product 𝐴 are injected to batch 𝐵5 from station 𝑆1 and the same  amount is received in depot 𝐷1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' As there is no pumping interaction and batch overlap  during this run, station 𝑆2 can also pump 40 units of product 𝐶 in the form of a new  batch 𝐵3 simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' This injection causes the depot 𝐷2 to fully receive the  remaining 20 units of product 𝐴 in 𝐵2 and, in addition, withdraw 20 units of product  𝐶 from batch 𝐵3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Pumping Run 𝑘4, which ends at 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67 hours, includes two injections  of product 𝐵: 20 units from station 𝑆1 to the new batch 𝐵6 and 40 units from station  𝑆2 to 𝐵4 that had reached the station at the end of the previous run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' As a result of these  two injections, 20 units of the product 𝐴 from 𝐵5 enter depot 𝐷1, the remaining 20  units of the product 𝐶 in the batch 𝐵3 are received in depot 𝐷2, and 20 units of product  𝐵 are added to the inventory of depot 𝐷3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  As seen in Figure 2, during runs 𝑘5 to 𝑘8, only one injection and one delivery take  place in each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In the last pumping run (𝑘9), station 𝑆1 injected 20 units of product  0  20  40  60  80  S1  D1  S2  D2  D3  B5  B4  B2  B1  Start (h)  End (h)  [10  LO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  B5  B4  B2  B1  20  20  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  B5  B4  B2  B1  40  40  40  2020  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  B5  B4  B3  B1  20  20  20  20  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  B6  B5  B4  30  [30  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  B7  B6  B5  B4  10  10  116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  B7  B6  B4  40  125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  B8  B7  B6  40|  158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  10  [10  158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  B8  B7  B6  20  [10]  166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  B8  B7Operational scheduling of oil products pipeline with intermediate event occurrences  15  𝐵 into batch 𝐵8, and as a result, depots 𝐷2 and 𝐷3 received respectively, 10 units of  product 𝐶 from batch 𝐵7 and 10 units of product 𝐵 from batch 𝐵6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The final state of  the pipeline, 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33 hours after the start of the program is as follow: the batch 𝐵8  containing product 𝐵 is location from the beginning of the pipeline to the volumetric  coordinate 70, and batch 𝐵7 is placed from the coordinates 70 to 80, containing 10  units of the product 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Adding incidents to the example  It is mentioned earlier that incidents usually lead to unwanted changes in the  schedule and as a result, the time and cost of the plan increase according to the severity  of the incident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In this example, we consider a case where a maintenance operation  will be executed on the pipeline between 𝑆1 and 𝐷1 after 100 hours from the start of  the planning horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Therefore, during this operation, it will not be possible to inject  any product from the source 𝑆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' We also assume that the maintenance will be finished  in 30 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  In none of the previous pipeline planning methods, incidents and changes in the  operational environment have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' So there is no way to add events to  the old models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' When using these models in the real world, when an incident occurs,  the only way is to stop the plan execution, update the numbers and parameters of the  model according to the new state of the environment, and run the model again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=" If we're  lucky, the model will probably lead to a new plan that will definitely cost more than  before." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' But in many cases, it is not possible to achieve any plan due to the constraints  of the model, and practically, the model that worked well before the occurrence of the  event, after that only reports infeasible results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' State of operational environment at time t=100h  However, in this example, we proceed through the mentioned method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' From time  𝑡 = 0 to 𝑡 = 100ℎ, it will proceed according to the previous plan and the batches will  be injected in exactly the same amount and in the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Therefore, at 𝑡 = 100ℎ  and at the moment of starting the maintenance operation, the situation is as shown in  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' After that, due to the reduction of maximum rate for injection in the pipeline  to zero, there is no way to inject any new batches in the pipeline and practically, the  state of the pipeline remains constant for next 30 hours (which clearly shows the  waste of time and resources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' From 𝑡 = 130ℎ, the planned program continues and the  rest of the steps and pumping runs are done according to what was assumed in plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  0  20  40  60  80  S1  D1  S2  D2  D3  B5  B4  B2  B1  Start (h)  End (h)  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  B5  B4  B2  B1  [20  [20  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  B5  B4  B2  B1  40  [40  40  [20/20  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  B5  B4  B3  B1  20  [20  ot  [20]  B6  B5  B4  20  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  10  [10  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  B7  B6  B5  B4S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='Moghimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' /Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 3 (x) (2020) 1-20  16  Finally, at 𝑡 = 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33ℎ, which is 30 hours later than the optimal schedule, the  distribution of the products finishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Step by step execution of this plan has been  drawn in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The output plan of older methods when the incident happens  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Performance of the proposed method in facing the incident  The advantage of the proposed method is that during planning, it pays attention to  the future changes of the environment and can consider different values for the  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' It is also possible to consider predictable events (such as the maintenance  program mentioned in this example) from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  In the proposed method, we consider two changes compared to previous methods  during modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' First, we define the set of events as follow:  𝐸 = {𝑒0, 𝑒1 (Maintenance starts), 𝑒2 (Maintenance ends), 𝑒𝑚𝑎𝑥}  (41)  As explained earlier, in this set, two events 𝑒0 and 𝑒𝑚𝑎𝑥 are included to represent  the beginning and end of the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Events 𝑒1 and 𝑒2 also represent the start  and end of the maintenance program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Therefore, the values of the parameters 𝑇𝑒1 and  𝑇𝑒2 are considered to be 100ℎ and 130ℎ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  The second change is to define a vector of values for parameters 𝑣𝑏𝑚𝑖𝑛,𝑠1 and  𝑣𝑏𝑚𝑎𝑥,𝑠1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The values in this vector are as follow (values are in units per hour):  𝑣𝑏𝑚𝑖𝑛,𝑠1  〈𝑒1〉  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='0  𝑣𝑏𝑚𝑖𝑛,𝑠1  〈𝑒2〉  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='0  𝑣𝑏𝑚𝑖𝑛,𝑠1  〈𝑒𝑚𝑎𝑥〉 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='0  𝑣𝑏𝑚𝑎𝑥,𝑠1  〈𝑒1〉  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='2  𝑣𝑏𝑚𝑎𝑥,𝑠1  〈𝑒2〉  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='0  𝑣𝑏𝑚𝑎𝑥,𝑠1  〈𝑒𝑚𝑎𝑥〉 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='2  (42)  In this way, the model realizes that in the interval [𝑇𝑒1, 𝑇𝑒2], no pumping should be  performed from station 𝑆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' For other parameters, a vector of length |𝐸| −  1 should  be used too, but all the values in them will be the same because the only parameters  changing through time in this example are 𝑣𝑏𝑚𝑖𝑛,𝑠1 and 𝑣𝑏𝑚𝑎𝑥,𝑠1  20  40  60  80  S1  D1  S2  D2  D3  B5  B4  B2  B1  Start (h)  End (h)  B5  B4  B2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content="33  B1  20  B5  B4  B2  20F  EE'8  25." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  B1  40  140  J40  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  B5  B4  20-20  B1  B6  20  B4  20  120  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  B5  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  B7  B6  B5  B4  10  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  BS  130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  B7  B6  B5  B4  120  B7  146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  B6  B4  40  155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  B8  B7  B6  140  B8  188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  B7  10  B6  20l  196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67  213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  B8  10  B7Operational scheduling of oil products pipeline with intermediate event occurrences  17  The result of our proposed model has a much better performance than the old  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' As can be seen in Figure 5, this method already looks into the future and  plans for the time that station 𝑆1 is not pumping, in such a way that the distribution of  products from other stations continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The output plan of our proposed model  The output of our method is same as the output of the previous methods until the  moment 𝑡 = 100ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' It pumps batches of different products in the pipeline in the same  order and with the same amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' But the divergence starts from time 𝑇𝑒1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Considering  that from this moment until 𝑇𝑒2 the pumping rate at station 𝑆1 will be zero, the first  part of the pipeline (from the beginning to the volume coordinates of 40 units) remains  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' But in the same period of time, station 𝑆2 starts pumping products that  satisfy part of the demands of depots 𝐷2 and 𝐷3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In this way, when in 𝑇𝑒2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 30 hours  later, the station 𝑆1 starts pumping again, the remaining amount of products needed  in 𝐷2 and 𝐷3 is reduced compared to the start of the maintenance operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' So by  pumping next batches, it can be reached the desired condition faster and therefore  with lower cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  More precisely, the output of our model consists of 8 pumping run (𝑘1 − 𝑘8),  where the runs 𝑘1 to 𝑘4 are performed exactly the same as in the normal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The  only difference is that batch 𝐵5 in normal case is numbered as 𝐵7 here, because of two  intermediate batches that will be injected to the middle of the pipeline in future runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  The fifth run (𝑘5), which is forced to be left unfinished in the previous methods,  ends consciously at time 𝑇𝑒1 in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Run 𝑘6 begins exactly when event 𝑒1 occurs  and thus, station 𝑆1 performs no pumping at this run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' During the 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='67 hours of run  𝑘6, station 𝑆2 pumps 20 units of product 𝐶 as a new batch called 𝐵5 and on the other  hand, depot 𝐷3 withdraws 20 units of product 𝐵 from batch 𝐵4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In run 𝑘7, station 𝑆1  is still inactive so the station 𝑆2 again injects a new batch named 𝐵6 containing 20  units of product 𝐵 into the pipeline and batch 𝐵5 is completely delivered to the depot  𝐷2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Run 𝑘8 starts after the maintenance operation is finished, and therefore, station 𝑆1  is reactivated in this run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Station 𝑆1 starts pumping a large new batch 𝐵10 with a  0  20  40  60  80  3  S1  D1  S2  D2  D3  B7  B4  B2  B1  Start (h)  End (h)  10  [L0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  B7  B4  B2  B1  20  20  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='00  B7  B4  B2  B1  40  [40]  40  12020  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
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+page_content='33  188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33  B10  B9  B8S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='Moghimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' /Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 3 (x) (2020) 1-20  18  volume of 60 units of product 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' In this way, the depot 𝐷2 takes the last part of its  demand, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' 10 units of the product 𝐴 from the batch 𝐵7, and the depot 𝐷3 takes three  consecutive withdrawals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The remaining volume of the batches 𝐵4 and 𝐵6 plus a  portion of batch 𝐵8 -which are equal to 50 units of product 𝐵 in overall- are delivered  to the depot 𝐷3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The final state of the pipeline is very similar to the final state in the  normal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' There are 70 units of product 𝐵 and 10 units of product 𝐶 in the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' Conclusion  The execution time of the output plan of our proposed model is surprisingly equal  to the required time in normal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' It means that sometimes even severe incidents  like pipeline shutdowns, can be managed easily and with the lowest overhead cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='  Our proposed model generates a plan which is 30 hours shorter than the time  required to execute the program in the previous methods, due to the fact that the  pipeline is not stopped totally during the maintenance operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content=' The total demand of  all depots is fulfilled in only 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='33 hours which is 16\\% faster than the output plan  of older methods, despite the possibility of older methods to go totally infeasible in  many other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
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+page_content=' Submitted for possible open access publication under the  terms and conditions of the Creative Commons Attribution (CC BY)  license (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
+page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAyT4oBgHgl3EQflfgF/content/2301.00451v1.pdf'}
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+Stick-slip synchronization in stack of elastically coupled frictional interfaces
+Samuel Poincloux,1, 2 Pedro M. Reis,1, ∗ and Tom W.J. de Geus3, †
+1École Polytechnique Fédérale de Lausanne (EPFL),
+Flexible Structures Laboratory, CH-1015 Lausanne, Switzerland
+2Department of Physics, The University of Tokyo, Tokyo, Japan
+3Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
+We perform physical and numerical experiments to study the stick-slip response of a stack of
+slabs in contact through dry frictional interfaces driven in quasistatic shear. The ratio between the
+drive’s stiffness and the slab’s shear stiffness controls the presence or absence of slip synchronization.
+A sufficiently high stiffness ratio leads to full synchronization, comprising periodic slip events in
+which all interfaces slip simultaneously.
+A lower stiffness ratio leads to asynchronous slips and,
+experimentally, to the stick-slip amplitude being broadly distributed as the number of layers in the
+stack increases. We interpret this broadening in light of the combined effect of surface disorder,
+complex loading paths of the asynchronous slips, and creep. The extracted aging rate is found to
+be of the same order of magnitude as existing experimental results on a similar material. Finally,
+we discuss the emergence of slow slips and an increase in creep-rate variations when more slabs are
+added to the stack.
+INTRODUCTION
+Multiple frictional interfaces coupled by elasticity are
+ubiquitous in everyday objects including books [1, 2], tex-
+tiles [3–5], and multilayer composites [6, 7]. At a larger
+scale, in geology, systems comprising multiple frictional
+interfaces are the norm rather than an exception. For
+example, layered rocks such as shales can show multi-
+ple sliding interfaces under shear [8, 9].
+At the even
+larger scales relevant for terrestrial faults, earthquakes
+produced when slipping are usually not isolated but em-
+bedded into complex fault networks [10]. The mechan-
+ical response of such assemblies of frictional interfaces
+involves the coupling between the elastic deformation of
+the layers and the barriers to sliding of the interfaces.
+Predicting the onset of slipping is a long-standing prob-
+lem even for a single frictional interface [11]. Physical
+insight and understanding of this class of problems have
+been driven primarily by high-precision experiments of
+sliding PMMA blocks whose optical transparency en-
+abled the space-temporal tracking of the local contact
+area [12]. These pioneering experiments have elucidated
+that the onset of slip involves a rupture front that ‘un-
+zips’ the interface. A correlation with strain measure-
+ments close to the interface showed that the stress field
+and dynamics of the front are well described by fracture
+mechanics with the fracture energy as the sole fitting
+parameter [13]. However, the mechanism underlying the
+nucleation of the rupture front remains elusive, primarily
+due to experimental limitations, for which novel proto-
+cols are being proposed [14].
+From a theoretical perspective, the most common mod-
+els for the onset of frictional slippage [15–18] capture the
+phenomenology that sliding starts, and then continues
+in a steady state, when the shear force F balances the
+friction forces µN, where N is the normal force and µ
+the ‘friction coefficient’. In these ‘rate-and-state’ models,
+µ depends nonlinearly on the slip rate v and history θ:
+µ = µ(v, θ). At intermediate values of v, the friction coef-
+ficient is usually assumed to display slip weakening (µ is a
+decreasing function of v) such that the interface is unsta-
+ble. During slip nucleation, the elasticity and inertia of
+the bulk have a stabilizing effect [19–22], such that there
+exists an effective flow curve ˜µ = µ + G/(2cs)v (where
+G is the shear modulus of the material and cs the shear-
+wave speed) whose steady-state displays a minimum at
+µc = ˜µ(vc) [19]. Consequently, any perturbation decays
+and vanishes if the applied load is F/N < µc. At higher
+applied loads, a slip patch of linear size L becomes unsta-
+ble if L > Lc ∼ (F/N − µ∗
+c)−ν (µ∗
+c ≥ µc [21]). Once the
+event grows to be larger than Lc, its dynamics are well
+described by a sharp rupture front [19, 21, 23] that can
+be modeled by linear elastic fracture mechanics [13, 24],
+as discussed above.
+If the loading is performed under
+displacement control, the reaction force µN drops macro-
+scopically only once the interface unzips fully.
+A direct consequence of the phenomenology described
+above is that the interface can display stick-slip behavior
+when driven quasistatically (at a rate V ≪ vc). Under
+the framework of the rate-and-state models, the stick-slip
+amplitude thus depends on the exponent ν, µ∗
+c, and the
+mechanism leading to the slip patch at scales L < Lc.
+The relative relevance of all of these parameters is cur-
+rently under debate; e.g. [14, 21, 22, 25–28]. However,
+one of the authors of the present study recently proposed
+an encompassing theory that avalanches nucleate the in-
+stability such that µ∗
+c = µc and ν = 1/(1 − ζ). Here, ζ
+is the roughness exponent resulting from a competition
+between microscale disorder, and elasticity [22] (not to
+be confused with a roughness exponent extracted from
+height-height correlations of the interface profile). Fur-
+thermore, it is a well-known experimental fact that the
+initial onset to sliding is history-dependent and increases
+with the time that the interfaces were at rest [17, 29–31].
+arXiv:2301.13745v1  [cond-mat.soft]  31 Jan 2023
+
+2
+This behavior is associated with creep [32] and described
+by rate-and-state models where the variable θ introduced
+above is regarded as time.
+Creeping of the interfaces
+must affect the stick-slip amplitude, but disentangling
+its contribution is challenging because slip events occur
+at a narrowly distributed interval.
+Beyond single interfaces, when multiple frictional in-
+terfaces are present, the elasticity of the bulk may poten-
+tially lead to a non-trivial coupling. For example, elastic
+interactions between faults may strongly affect their slip
+dynamics [33]. In addition, acoustic waves transmitted
+through the elastic bulk may lead to remote triggering
+of earthquakes [34, 35], though the large temporal sepa-
+ration suggests a complex coupling. Predicting the me-
+chanical response of an assembly of elastic frictional in-
+terfaces is then a formidable but important challenge. In
+particular, identifying the key parameters coupling the
+layers together and elucidating the role of the number of
+interfaces are open questions.
+Here, we report results from a combined experimental
+and numerical investigation on the quasistatic stick-slip
+response of a stack of elastic slabs in contact through
+frictional interfaces. Based on the ratio between the stiff-
+ness of the drive and the shear stiffness of the slab, we
+distinguish two regimes: ‘stiff’ and ‘compliant’ driving.
+In the stiff-driving regime, our numerical results exhibit
+periodic slips involving narrowly distributed force drops;
+this regime is not accessible in our model experiments.
+By contrast, in the compliant-driving regime, we observe
+both numerically and experimentally a decoupling of the
+slip events along the different layers, with interfaces slid-
+ing one by one. In the experiments, we find that this loss
+of periodicity is accompanied by a broadening of the dis-
+tribution of the stick-slip amplitudes with the number of
+layers. The changes in the measured distributions are in-
+terpreted by a coupling between interface disorder and a
+broad distribution of waiting times between slips, expos-
+ing the role of creep. The remaining unsolved broadening
+of the distributions with the number of slabs, including
+an observed increasing fraction of ‘slow slip’, raises the
+question of the interaction between mechanical noise and
+creep. Depending on the drive’s stiffness, the stack of
+frictional interfaces then displays drastically different re-
+sponses of the stack to shear. Overall, we find that, in the
+stiff-driving case, the stack acts as one layer with periodic
+slips, while in the compliant-driving case, a rich coupling
+between the layers makes slips much more unpredictable.
+DEFINITION OF THE PROBLEM
+We assess the shear response of a model system com-
+prising a stack of n ∈ [1, 5] identical slabs of thick-
+ness h resting on a surface whose position is fixed. In
+Fig. 1, we present a schematic diagram of the system
+and a photograph of our experimental setup. Each slab,
+and its lowermost frictional interface, are numbered as
+i = 1, 2, . . . , n from below.
+We impose homogeneous
+shear, with a set rate, between all the slabs by connect-
+ing each slab through identical springs of stiffness K to a
+lever that is driven to rotate around a fixed axis (Fig. 1a).
+The spring connecting to the i-th layer is attached to the
+lever at a distance ih from the rotation axis.
+For the drive, we impose the lever’s top horizontal dis-
+placement U(t) = V t, where t represents time, and V is
+the imposed velocity, taken to be small enough for the in-
+terfaces to display stick-slip (i.e., V < vc [22, 30]). Thus,
+our drive imposes a shear rate ˙γ ≡ V/H, where H is
+the height of the lever, driving each spring i at a veloc-
+ity vi = ih˙γ. During the periods in which the interfaces
+are ‘stuck’, this drive causes a monotonically increasing
+shear stress at each of the interfaces.
+Our study seeks to address the following questions:
+(1) What are the relevant parameters controlling the slip
+synchronization of the interfaces? (2) How does the shear
+response evolve with an increasing number of layers n?
+(a)
+(b)
+springs
+lever
+weight for normal force
+Figure 1.
+(a) Schematic of our model system, shown here
+with n = 4 active (driven) layers.
+The color code of the
+layers is used throughout the figures. (b) Photograph of the
+corresponding experimental apparatus.
+RIGID VS. COMPLIANT DRIVING
+For a system with a single frictional interface, the stick-
+slip instability occurs only if the driving stiffness K is
+sufficiently low and depends on the flow properties of the
+interface and the applied driving rate [15, 30, 36]; a sum-
+mary of the calculation for a rate-and-state model was
+provided in Ref [30]. With multiple frictional interfaces,
+
+O
+?3
+the drive also controls the degree of synchronization as
+we argue next.
+To gain insight into the effect of the drive on the
+shear response of a stack, regard the driving ‘spring’ as
+a parabolic potential energy imposing the mean position
+of each slab, such that the slab is free to build up shear.
+We now discuss what happens in the limit of rigid and
+compliant driving, defined next.
+Rigidity ratio Φ.
+We define the rigidity ratio Φ ≡
+K/Ks, where K is the rigidity of the driving spring, and
+Ks = AG/h is the shear rigidity of the slabs, with G
+the shear modulus, A the surface area of the frictional
+interface, and h the slab’s height.
+This ratio Φ then
+quantifies the relative deformation of the driving springs
+in comparison to the shear deformation of the slabs. Be-
+low, we investigate and discuss two regimes: rigid (high
+Φ) and compliant (low Φ) driving. In the first, significant
+deformations occur within the layers (and the springs are
+rigid), whereas, in the second, the deformations occur in
+the spring (and the layers are rigid).
+Rigid driving (high Φ).
+Suppose that the first inter-
+face (i = 1) starts slipping. As the mean position of slab
+i = 1 is fixed, the slab can only react with a negative
+shear deformation. Consequently, the shear stress on the
+interface above, i = 2, is increased. This can trigger a
+slip at interface i = 2, which can cause a slip on the in-
+terface above for the same reason, which propagates to
+the slipping of other interfaces. The cascade results in
+a multi-slip event that erases all memory of the system.
+In this case, the multi-layered system thus acts as a sys-
+tem with a single interface with effective properties [37],
+showing periodic stick-slip cycles.
+Compliant driving (low Φ).
+The system can respond
+to a slip at interface i = 1 by advancing the mean position
+of slabs i > 1. Therefore, the stress on the interfaces i >
+2 relaxes, making a macroscopic multi-slip event unlikely.
+A sequence of single-slip events is thus to be expected.
+With an increasing number of layers, the slip sequence of
+the multiple interfaces may lose its periodicity.
+NUMERICAL SIMULATIONS
+Numerical model
+We implement the model system shown schematically
+in (Fig. 1a) into numerical simulations. The numerical
+model consists of n + 1 identical elastic layers separated
+by frictional interfaces. Following [26], we idealize the
+frictional contact problem in order to focus on the dis-
+order in the shear response along the frictional interface.
+In particular, we consider a mesoscopic scale on which
+an effective ‘block’ of a finite width resists elastically to
+shear up to a threshold, after which it yields. The lo-
+cal slip then propagates until a new ‘contact’ is formed
+(i.e., it is again elastic but with a new threshold).
+In
+this framework, each block represents a frictional con-
+tact (or a patch of contacts that are so strongly coupled
+by elasticity that they act as an effective contact) that,
+upon yielding, forms a new contact with a new yielding
+threshold.
+The details of the numerical model are as follows:
+each frictional interface consists of nx equal-sized square
+blocks of linear size l0 that are completely elastic under
+volumetric deformation but yield under shear (deviatoric
+deformation) when a set yield stress is reached. Assum-
+ing that the yield threshold is isotropic in principal de-
+viatoric strain space, this model now corresponds to a
+deviatoric potential energy that consists of a sequence of
+parabolic potentials in equivalent deviatoric strain space.
+The disorder arises from independently randomly draw-
+ing the yield strain sequence of each block. We assume
+that the blocks and the bulk have the same elastic mod-
+uli. Finally, differently from [26], we add a parabolic po-
+tential (with curvature K) to the mean position of each
+of the elastic slabs i > 0, thereby prescribing a homo-
+geneous force density.
+The bottom layer is not driven
+through its mean position; instead, the position of the
+bottom edge is fixed. A key feature of the model is that
+shear can be applied according to the quasistatic proto-
+col. In particular, because the elastic response is linear,
+we rotate the lever by a finite amount if no microscopic
+yielding takes place while infinitesimally rotating it to
+trigger a microscopic event, after which we minimize en-
+ergy before loading again. Thus, we run an event-driven
+protocol, allowing us to separate events.
+Geometrically we do not seek to precisely model Fig. 1a
+as its numerical treatment, together with the disorder, re-
+quires an intractably large number of blocks. Therefore,
+we consider periodic boundary conditions in the horizon-
+tal direction. Furthermore, we choose nx = 2×36, which
+is still tractable to simulate, but of the minimal order not
+to be dominated by finite size effects, as we checked for a
+single frictional interface [22, 26]. Furthermore, we take
+h/ℓ0 ≈ nx/4 based on balancing h/(ℓ0nx) small enough
+to have acoustic interactions while avoiding driving the
+blocks in a fixed displacement such that collective effects
+are suppressed if h ≪ nxℓ0 (e.g. [38]).
+The above model predicts stick-slip behavior [26] when
+full inertial dynamics are considered (using overdamped
+dynamics, this model predicts the abundantly studied
+depinning transition [39]). We consider such inertial dy-
+namics by applying the finite element method to dis-
+cretize space. Along the frictional interface(s), elements
+coincide with the mesoscopic blocks. In the elastic slabs
+away from the frictional interface, the elements are coars-
+ened to gain numerical efficiency (such that the height h
+is only approximated as we fix the aspect ratio of ele-
+ments to one, see SI, "Numerical model"). We use the
+velocity Verlet algorithm to integrate discrete time (with
+a time step significantly smaller than the time of a sin-
+gle oscillation in a well in one block). We remark that
+
+4
+assuming periodicity requires us to add a small damp-
+ing term to the inertial dynamics such that waves with
+a wavelength equal to the horizontal size of the system
+(equal to nxl0) are critically damped. Consequently, we
+must take h/ℓ0 < nx to have acoustic coupling between
+the interfaces.
+Numerical results
+Our numerical model allows us to first illustrate the
+simple reflection above on the role of driving. We con-
+sider a driving rigidity such that Φ ≃ 10−3 (rigid driv-
+ing) and Φ ≃ 10−6 (compliant driving).
+In the two-
+dimensional model, A = nxℓ0, such that Ks = 4G for
+our geometry; we use K = 10−3 and K = 10−6 and
+G = 1/2. In Fig. 2a and Fig. 2b, for rigid and compliant
+driving, respectively, we plot a typical macroscopic stress
+Σ (volume-averaged stress) as a function of applied lever
+rotation γ. Note that the stress is shown in units of the
+typical yield stress of one block, and the rotation in units
+of the rotation needed to yield a typical block at i = 1.
+Macroscopic slip events are defined when all blocks
+along one or more layers yield at least once. Below, we
+will refer to sliding interfaces and associated quantities
+by an index a, while i will be kept as the running in-
+dex for the layers. Slip events correspond to macroscopic
+stress drops in Figs. 2a and 2b, and we distinguish be-
+tween ‘single-slip’ events (all blocks on a single layer yield
+at least once) and ‘multi-slip’ events (all blocks on more
+than one layer yield at least once). Stress drops produced
+by single-slip events are labeled following the layer color
+code introduced in Fig. 1, while muti-slip events are kept
+black. These slip events are separated by ‘stick’ intervals
+during which only microscopic events are observed, where
+one or several blocks yield at least once, as indicated with
+markers (black dots).
+The results confirm that rigid driving causes a peri-
+odic stick-slip sequence with slip events corresponding
+to multi-slip events (Fig. 2a) while compliant driving re-
+sults in a seemingly less periodic sequence of single-slip
+events in Fig. 2b.
+Although not reported in Fig. 2a,
+for Φ ≃ 10−3, we also observed periodic sequences on
+single-slip events for a finite fraction of the loading his-
+tory. This finding is supported by plotting the fraction
+of slip events involving s = 1, . . . , n interfaces in Fig. 2c
+for different n (see legend in Fig. 2d). On the one hand,
+rigid driving results in single- and multi-slip events for
+a comparable fraction of loading history (we discuss in
+the SI "Slip sequences - numerics" that sequences of sin-
+gle and multi-slip events alternate). On the other hand,
+compliant driving shows single slip in the large majority
+of slip events.
+A direct measurement of the stress drop along the
+slipping interface a, ∆µa, displays no n dependence in
+Fig. 2d. The quantity µa is defined as the volume aver-
+age stress on the blocks corresponding to weak layer a,
+also shown in units of the typical yield stress of one block.
+Given that, by construction, normal stress plays no role
+in our model, here µa is akin to a friction coefficient.
+As we have shown that, for high Φ (rigid driving), mul-
+tilayer stick-slip is apparently similar to that of a single
+interface, next, we concentrate on the low Φ regime to
+explore a potential influence of the number of plates n.
+1
+2
+3
+4
+s
+0.00
+0.25
+0.50
+0.75
+1.00
+ρ(s)
+(c)
+Φ ≃ 10−3
+Φ ≃ 10−6
+0.0
+0.1
+0.2
+0.3
+0.4
+∆µa
+0
+5
+10
+15
+P(∆µa)
+(d)
+n =
+1
+2
+3
+4
+600
+800
+1000
+1200
+1400
+1600
+1800
+2000
+2200
+γ
+0.0
+0.2
+0.4
+Σ
+∆Σ
+(a) rigid driving: Φ ≃ 10−3
+600
+800
+1000
+1200
+1400
+1600
+1800
+2000
+2200
+γ
+0.0
+0.2
+0.4
+Σ
+∆Σ
+(b) compliant driving: Φ ≃ 10−6
+a = 1
+a = 2
+a = 3
+Figure 2.
+Numerical results: (a, b) Typical steady-state
+global stress Σ response as a function of lever rotation γ for
+n = 3 for (a) rigid driving and (b) compliant driving. We indi-
+cate all microscopic yielding events with a black dot marker.
+Slip events on a single layer are indicated in color (see leg-
+end), while slip events in black involve more than one inter-
+face. (c) The fraction ρ(s) of macroscopic slip events involving
+s = 1, . . . , n layers, for rigid (dashed) and compliant (solid)
+driving; see legend in (d) for color-map and markers. (d) Dis-
+tribution of stress drops at the slipping interface for different
+n (for slip events on a single layer, for which s = 1 in (c)).
+See the main text for definitions and units.
+EXPERIMENTS
+We now propose an experimental realization of the
+sheared-multilayer model system of Fig. 1a, adapted to
+measure the effect of the number of sliding layers n on
+the slip synchronicity and amplitude; see Fig. 1b. Simi-
+larly to the numerical model, the position of each slab is
+driven by connecting it to the driving lever through linear
+springs (see schematic in Fig. 1a). Naturally, connect-
+ing the spring to the edges of the slabs might introduce
+
+5
+boundary effects, but our experimental system is effec-
+tively much larger than our numerical model (given that
+it presumably has much more local contact patches).
+Experimental apparatus
+The experimental setup shown in Fig. 1b comprises
+a stack of frictional plates (color-coded from purple to
+orange), an actuating lever (green), and driving springs
+(pink), as detailed below.
+The stack is made of a set of rectangular PMMA
+slabs (Snow WH10 DC by Röhm), each of dimensions
+h = 10 mm, L = 150 mm and an out-of-plane width of
+80 mm. A normal force N is applied on the topmost slab
+by a dead weight of 5 kg (N = 49 N). To ensure a spa-
+tially homogeneous contacting surface at this relatively
+low normal force (compared to other PMMA-PMMA fric-
+tion experiments [13, 14, 40]), we use acrylic plates whose
+surface was pre-roughened with asperities of size ∼ 25 µm
+that are larger than potential natural height variations of
+PMMA. We assume that the normal force is uniformly
+distributed and that it is the same for each layer (the
+weight of each slab is less than 3% of that of the dead
+weight).
+The stack is sheared by imposing the displacement at
+the top of the lever (H = 100 mm) at a constant speed
+V = 10 µm/s (i.e. ˙γ = 10−4 s−1), using a DC linear actu-
+ator (L-220.70DG, Physiks Instruments) that is attached
+via a steel link assumed rigid. The PMMA lever is suffi-
+ciently wide to not bend while pulling the slabs and ro-
+tates smoothly on ball bearings around its rotation axis.
+The springs connecting the slabs to the lever are curved
+beams laser-cut from PMMA (colored in pink in Fig. 1b),
+with an equivalent stiffness of K = 55 N/mm when pulled
+or compressed along the horizontal axis. The ends of the
+springs are attached to both the slabs and lever via ball
+bearings to ensure a free rotation and, thus, horizontal
+driving forces.
+The experimental set-up, with its springs and PMMA
+slabs, corresponds to the compliant driving limit with
+Φ ≃ 6 × 10−5, of the same order as for the compliant
+regime in the numerics. Indeed, Φ ≡ K/Ks, with Ks =
+AG/h the shear stiffness of the slabs and G ≡ E/(2(1 +
+ν)), with, for PMMA, Young’s modulus E = 2 GPa and
+Poisson’s ratio ν = 0.3.
+The total horizontal force F needed to rotate the
+lever (Fig. 1) is measured using a uniaxial force sensor
+(LRM200 25 lb, Futek) placed between the steel link and
+the actuator.
+The link is also attached via small ball
+bearings to remain horizontal at all times. We directly
+verify that we are imposing the expected kinematics by
+measuring the position of the lever and base slab.
+In addition to the global force measurement F, we also
+measure the local average horizontal position of the slabs
+xi, by tracking a red marker placed on their side (Fig. 1b),
+from photographs taken at a rate of 5 fps using a digi-
+tal camera (Flea3 FL3-U3-20E4C, Flir, linear pixel size:
+70 µm). After color filtering, xi is measured with an ac-
+curacy of 5 µm. The relative displacement between slabs
+is Ri ≡ xi − xi−1 (see Fig. 1a), which serves as a proxy
+for the total slip at the interface i (neglecting the shear
+deformation of the slabs).
+To vary the number of sliding interfaces n, we keep the
+same number of slabs (5) but remove 5−n springs, start-
+ing from the top (see Fig. 1b where n = 4). This proce-
+dure ensures robust image detection and reduces external
+contamination of the interfaces by keeping them in con-
+tact. Each time the slabs are disassembled to vary n, the
+interfaces are cleaned with isopropanol and quickly dried
+using compressed air. For each value of n, we perform 10
+runs during which we drive over a range ∆γ = 0.6 rad,
+starting at γ = −0.30 rad, each time excluding γ be-
+tween −0.3 and −0.27 (300 s) to ensure measuring in a
+steady state. After each run, the lever is reset back to
+γ = −0.30. On average, each connected layer is forced to
+move by a total relative distance of ∆R = h∆γ = 6 mm
+during a run.
+Further details on the apparatus and the experiments
+are given in the SI ("Experimental set-up"), including
+verification that the lever’s displacement is indeed im-
+posed, even during slip events ("Kinematics"). We also
+discuss the influence of the weight of the lever ("Weight
+and position of the lever").
+Experimental measurements
+In Fig. 3a, for the slab system with n = 4, we present
+a typical time series of the force F(t) required to actuate
+the lever (top plot), together with the corresponding rela-
+tive position of the slabs Ri(t) (bottom plot). The experi-
+ments exhibit stick-slip, with stick periods when the slabs
+are immobile (Ri ≈ constant) and F increases monoton-
+ically, punctuated by macroscopic slip events. These slip
+events are identified by a sudden position jump, ∆Ra
+(with a denoting the sliding interface), accompanied by
+an abrupt force drop ∆F > 0, cf. Fig. 3a. On all oc-
+casions, we find that only one layer slips at a time, re-
+covering similar dynamics as in the numerical model in
+the compliant driving regime with a similar value for Φ
+(Fig. 2b). However, we note that during the stick periods,
+we observe what seem to be ‘slow slip’ events where an
+interface moves gradually, leading to a non-linear force
+response.
+These are out of our primary focus but are
+discussed at the end of the section.
+For each value of n, we acquire an ensemble of at least
+100 slip events per layer, such that the slip quantities as-
+sociated can be represented as probability distributions.
+For example, in Fig. 3b, we show the probability distri-
+bution of force drops, P(∆F), occurring on the interface
+a = 1, for all cases of n considered. Starting from the
+
+6
+peaked distribution for n = 1, as n increases, the distri-
+butions broaden and with higher average values.
+In contrast with more classic stick-slip experiments
+with a single interface [29], the global measure ∆F is not
+a direct quantification of the frictional properties of the
+interface but couples with the specific kinematic of the
+lever. Still, the fact that only one interface slips allows
+us to extract, from ∆F, a jump in a friction-like quan-
+tity ∆µa (or stick-slip amplitude). We define the friction
+coefficient µa of a slipping interface a as the horizontal
+force acting on this interface divided by the normal force.
+Considering the horizontal force balance on an interface
+a, the interface has to resist the combined forces of the
+pulling springs of the slabs i ≥ a, such that
+µa =
+n
+�
+i=a
+fi
+N ,
+(1)
+where fi is the force due to the driving spring on slab
+i (see Fig. 1a for a visual representation of µa and fi).
+When the interface a slides by ∆Ra, the relative positions
+of the other interfaces remain unchanged:
+∆Ri =
+�
+∆Ra > 0
+if
+i = a
+0
+if
+i ̸= a .
+(2)
+This sliding induces a drop in the spring forces: ∆fi = 0
+for i < a, and ∆fi = K∆Ra > 0 for i ≥ a. Indeed,
+even if no slip occurs for the interfaces i > a, the abso-
+lute position of the slabs still moves by ∆Ra, reducing
+the extension of the corresponding springs by the same
+amount. Note that for consistency, we define ∆fi to be
+positive. From Eq. (1), we can then express ∆µa as a
+function of the slip distance ∆Ra:
+∆µa = K
+N (n − a + 1)∆Ra.
+(3)
+We proceed by linking ∆Ra to the global force drop
+∆F, using the fact that only one interface slips at a time.
+Through moment balance on the lever, we obtain:
+F =
+n
+�
+i=1
+fi
+ih
+H .
+(4)
+Combining Eq. (2) and Eq. (4), we obtain a relation be-
+tween the global quantity ∆F and the local one ∆Ra:
+∆F =
+n
+�
+i=a
+K∆Ra
+ih
+H = K∆Ra
+Kh
+H
+(n + a)(n − a + 1)
+2
+(5)
+(Note that i is the only varying term in the sum and
+�n
+i=a i = (n(n + 1) − a(a + 1))/2 = (n + a)(n − a +
+1)/2.) This result is verified in Fig. 3c. Indeed, a direct
+measurement of ∆Ra is very close to the inversion of
+Eq. (5), in which ∆Ra follows from the measured ∆F
+without any fitting parameter.
+0
+100
+200
+300
+400
+(a)
+(c)
+0
+100
+200
+300
+400
+0
+2
+4
+6
+0
+0.2
+0.4
+0.6
+5400
+5500
+5600
+5700
+(b)
+Figure 3.
+(a) Top plot: extract of a time series of the macro-
+scopic force F(t) for a system of n = 4 frictional interfaces.
+The color of the force drops ∆F follows the color code in-
+troduced in Fig. 1 and indicates the index a of the slipping
+interface. Bottom plot: corresponding relative displacement
+(total slip) Ri of each interface i. Each slip event is character-
+ized by its interfacial slip ∆Ra. We denote Ta, the time be-
+tween subsequent slip events on the same interface. Note that
+we show i = 5 only for completeness, by definition, R5 = 0
+if n = 4.
+(b) Probability distribution function P(∆F) for
+slip at interface a = 1, and increasing number of layers n.
+(c) Comparison between a direct measurement of ∆Ra, and
+the computed ∆Ra(∆F), obtained through Eq. (5), for each
+detected slip event.
+Finally, we combine Eqs. (3) and (5) to obtain the
+sought relation between ∆µa and ∆F:
+∆µa = H
+h
+2
+(n + a)
+∆F
+N .
+(6)
+We have thereby disentangled the friction properties of
+the interface from the kinematics of the lever.
+Using
+Eq. (6), we can now obtain a measure of the stick-slip
+amplitude of the interfaces ∆µa, extracted directly from
+the global force ∆F. The measurement of ∆Ri is used
+only to identify the slipping interface a.
+The central experimental result.
+Next, we assess the
+effect of having multiple sheared interfaces on their fric-
+tional properties.
+Fig. 4 shows the probability distri-
+butions P(∆µa) associated with the different sliding in-
+terfaces a (different panels) and the increasing number
+of total active interfaces n (different colors). Each inter-
+face is compared to its response when sliding individually
+(n = 1 in black, see SI "Individual sliding" for experimen-
+tal protocol). For all the interfaces, the stack of slabs
+exhibits significantly enriched statistics when compared
+
+7
+(b)
+(c)
+(d)
+(a)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+10
+30
+20
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+10
+30
+20
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+10
+30
+20
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+10
+30
+20
+Figure 4.
+Probability distribution functions of the stick slip
+amplitude P(∆µa) as a function of n for (a) a = 1, (b) a = 2,
+(c) a = 3 and (d) a = 4.
+to a single sliding layer (n = 1, black circle in the plot).
+For stacks with increasing n, the location of the peak
+of P(∆µa) shifts to lower values of ∆µa. Moreover, the
+respective distributions become broader and secondary
+peaks emerge. These experimental results contrast the
+numerical predictions reported above (Fig. 2c), where
+∆µa was independent of n.
+Interpretation
+We seek to interpret the above experimental findings
+evidencing a variation of the frictional properties with n
+(as more layers are added to the stack, Fig. 4), whereas
+they are independent of n in the numerics (Fig. 2d).
+First, we will attribute the finite width of the peaks in the
+P(∆µa) distributions to disorder (i.e., statistical fluctu-
+ations of the contacting interfaces). Then, we will argue
+that the shift of the main peaks and emergence of the
+secondary peaks in P(∆µa) for n > 1 are related to the
+presence of creep in the system. Finally, we speculate
+on the increase of an effective temperature with n and
+rationalize the appearance of slow slip.
+Statistical fluctuations.
+Even when sliding individu-
+ally (n = 1), the frictional properties of the interfaces are
+distributed: P(∆µa) has a finite width, see black curves
+with circles in Fig. 4. These underlying statistical fluctu-
+ations, also present in the numerical model (Fig. 2d), are
+considered to be related to the disorder of the contacting
+interfaces. The rough interface induces a broad distri-
+bution of barriers, such that there are collective events
+whose sizes are non-trivially distributed. These collective
+events nucleate the slip events [22, 26], such that the load
+at which events slip is distributed.
+Let us now verify experimentally that, in the case of
+individually sliding layers (n = 1), the measured fluctua-
+tions of ∆µa correspond to distinct slipping loads. In the
+individual configuration, the spring drives the layer at a
+constant rate of ˙fa = Kh˙γ (the rotation rate of the lever
+is adapted to account for the difference in height, see SI
+"Individual sliding"). The shear applied to the interface
+then grows at a rate ˙µa = ˙fa/N = Kh˙γ/N. As such, we
+expect the stick-slip amplitude to be proportional to the
+time between slips Ta, following ∆µa = TaKh˙γ/N. This
+expectation is consistent with our data in Fig. 5a, thus
+confirming that statistical fluctuations broaden P(∆µa).
+However, these fluctuations do not account for the shifts
+of the peaks and the appearance of secondary peaks in
+the P(∆µa) distributions. In our stack of slabs (n > 1),
+we anticipate that creep plays that role: with increasing
+n, (I) interfaces experience a complex loading path such
+that (II) the creep of the frictional interface becomes sig-
+nificant.
+(I) Complex loading path.
+First, without any events
+on other interfaces, the loading rate at a given inter-
+face increases with n. In particular, using Eq. (1) and
+˙fi = Kih˙γ while no interfaces are sliding, the interface a
+undergoes a loading rate of ˙µa = K ˙γh(n + a)(n − a +
+1)/(2N), which is an increasing function of n.
+With
+this increased loading rate with n, we thus expect the
+time between slips to typically decrease with n following
+T −1
+a
+∼ (n + a)(n − a + 1). Second, for a given interface,
+a sliding event can occur on a layer below it. In that
+case, all the layers above the sliding one will undergo the
+same position shift, leading to a relaxation of their corre-
+sponding springs. Then, even without actual slip on this
+given interface, slips occurring below will induce drops
+in the shear force of this layer. In Fig. 5b, we plot the
+probability distribution function of Ta for a = 1 and in-
+creasing n. Indeed, we find that the peaked distribution
+for n = 1 shifts to lower values of Ta with increasing n as
+the overall loading rate increases. Moreover, secondary
+peaks in P(Ta) start to appear, which we interpret to be
+due to slip events on the other layers. These observations
+are robust for the other interfaces (see SI). The changing
+in loading rate with n together with the complex loading
+path, allows our experimental system to probe a broad
+distribution of Ta on all its interfaces.
+(II) Creep.
+It is a known experimental fact that
+the macroscopic stress required for the onset of slid-
+ing, characterized by µs (the ‘static friction coefficient’
+in Amontons-Coulomb’s terminology [11, 30]), depends
+on the duration T that the interface was static: µs =
+B ln T [17, 29–31], where the aging rate of the interface
+B is a constitutive parameter. Let us now consider ∆µa
+as a proxy for µs, assuming that a slip event unloads the
+interface to a well-defined and constant quantity (µd the
+‘dynamic friction coefficient’ in Amontons-Coulomb’s ter-
+minology [11, 30]), as is supported by [29]. We expect to
+find, for each sliding interface a and over a wide range of
+
+8
+Ta, that the stick-slip amplitude follows the creep trend:
+∆µa = B ln Ta. In Fig. 5c, we assess this expectation
+experimentally by plotting ∆µa versus Ta in a semi-
+logarithmic scale. To capture the general trend, we bin
+logarithmically Ta, corresponding to the black markers
+with corresponding error bars in Fig. 5c. These averaged
+values of ∆µa indeed follow a linear trend in the semi-log
+plot, and we extract the slope B = 0.053±0.005, which is
+of the same order of magnitude as measured in classical
+stop-and-go experiments that are in the 10−2 order [30],
+and of a direct surface observation on PMMA at room
+temperature that reports B = 0.009 ± 0.001 [40]. Creep
+then translates the large peak shifts and the emergence
+of new ones in the Ta distributions (Fig. 5b) into similar
+changes in the ∆µa distributions. The large fluctuations
+of ∆µa in Fig. 5c are then the combination of two ef-
+fects: creep drives the long time trend, while disorder
+dominates in the narrow time range.
+In Fig. 5d, we schematically represent the coupled role
+of (I), (II), and disorder, following the evolution of the
+interfacial stress with time, characterized by µa, starting
+from the last slip event. The layer slips when it reaches
+µa = µs(Ta), whereby µs is distributed in some way for
+fixed Ta because of disorder (illustrated as a red-shaded
+area, whereby, for simplicity, we lump all fluctuations
+in the threshold to sliding) and increases logarithmically
+with time because of creep. For n = 1 (black line), µa
+increases linearly at the same rate for all the events, thus
+exploring only a narrow region of Ta (the shaded red
+region due to disorder). In the case of multiple active
+interfaces (n > 1, green lines), µa increases faster given
+that ˙µa is an increasing function of n; see point (I) above.
+In some cases, µa directly reaches µs, resulting in a lower
+value of Ta and ∆µa. However, if sliding events occur in
+one of the underneath layers, µa will drop before linearly
+increasing again, delaying slip and thus increasing Ta,
+and consequently ∆µa, because of creep.
+Effective temperature.
+During stick intervals, micro-
+scopic events occur on the interfaces, propagating elastic
+waves across the system [26]. As we increase the number
+of interfaces in the system, we can expect that the over-
+all mechanical noise created by the microscopic events
+also increases. If we speculatively interpret this mechan-
+ical noise as an effective temperature, we would expect a
+change of aging rate B with n. Let us define the aging
+rate for a single event as Ba ≡ ∆µa/ ln Ta. Although the
+mean of P(Ba) does not change with n (see SI "Aging
+rate distributions"), we do find that the width of the dis-
+tribution P(Ba) is an increasing function of n mainly for
+the lowermost interfaces (i ≤ 2), as shown in Fig. 6a.
+For the same interfaces (i ≤ 2), we also observe dis-
+tinctly different slip dynamics when n > 1. In particular,
+as n increases, we find that interfaces i = 1 and i = 2 are
+increasingly more subject to slow slip, defined as sliding
+significantly slower than the slip events (see SI "Smooth
+sliding" for precise definition). These events are not ac-
+101
+102
+0
+0.02
+0.04
+(c)
+(a)
+101
+102
+103
+0
+100
+200
+300
+0
+0.25
+0.5
+0
+0.25
+0.5
+time
+(b)
+(d)
+Figure 5.
+(a) For an interface sliding individually (n = 1),
+stick-slip amplitude ∆µa as a function of the waiting time
+since the last slip event Ta. The black line corresponds to
+the prediction that the stress state of the interface increases
+at a constant rate in between slip events. For multiple slid-
+ing interfaces (n ≥ 1): (b) Probability distribution of the
+waiting time Ta between two consecutive slip events at in-
+terface a = 1, for increasing n.
+(c) For each detected slip
+event, correlation between the stress release, characterized by
+∆µa, and its corresponding waiting time since the last slip
+event on that interface Ta (semi-logarithmic scale). The black
+markers correspond to the mean values for a logarithmic bin-
+ning of Ta (error bars indicate the standard deviation for that
+bin), and the dotted line a linear fit of ∆µa = B ln Ta, with
+B = 0.053±0.005. (d) Schematic of the proposed mechanism
+leading to multimodal and wider distributions of Ta (and thus
+∆µa) as n increases.
+companied by a macroscopic stress drop but rather just
+lead to a lower ˙F > 0, see Fig. 3a. In Fig. 6b, we measure
+the proportion of slow slip compared to the total slid-
+ing distance in an experiment, Rslow. It is computed by
+comparing the total sliding distance to the total slip ac-
+cumulated during individual slip events (∆Ra), such that
+Rslow ≡ 1 −
+�
+∆Ra/∆R. Once a slow slip event starts,
+it appears to be stopped only by slip events occurring ei-
+ther on the same interface or on any other interface. An
+increase in the effective temperature of the interface with
+n could also act as a potential destabilization factor of
+the contacts at the interfaces, increasing the occurrence
+of slow slips with n.
+
+9
+(b)
+(a)
+0
+0.2
+0.4
+0.6
+0.8
+1
+1
+2
+3
+4
+5
+1
+2
+3
+4
+1
+2
+3
+4
+5
+Figure 6.
+As a function of n, for each interface a (different
+colors and marker, see legend): (a) Standard deviation of the
+distribution of the aging rate Ba ≡ ∆µa/ ln Ta of individual
+events, normalized by that quantity for n = 1. (b) Fraction
+of slip that corresponds to slow slip.
+DISCUSSION AND CONCLUSION
+Summary
+We have explored the stick-slip response of a system
+with multiple interfaces by proposing a model system
+comprising n vertically stacked slabs, each connected to
+a lever whose rotation is imposed. The interfaces were
+driven in quasistatic (homogeneous) shear. We proposed
+a dimensionless quantity Φ as the ratio between the driv-
+ing stiffness and the stiffness of the elasticity of the slabs.
+We have argued and demonstrated numerically that the
+system displays synchronization if Φ is sufficiently large
+(Φ ≳ 10−3).
+In that case, the system acts close to a
+single frictional interface with effective properties. If Φ
+is small (Φ ∼ 10−6), interfaces slip one by one, as also
+confirmed experimentally.
+We expect non-trivial collective effects with increasing
+n only in the low Φ limit, which we addressed through
+experiments. In the numerics, the stick-slip amplitude of
+the interfaces ∆µa display a distribution with finite width
+because of statistical fluctuations of the interfaces, but no
+measurable changes with n. By contrast, we measured
+experimentally that the probability distribution of stick-
+slip amplitude ∆µa shows a general broadening with n,
+with peaks shifting to lower values and secondary peaks
+appearing. The interfaces are coupled via the lever, ex-
+posing them to a complex loading path, and leading to a
+broad distribution of the waiting times Ta between two
+slip events on an interface. We find that Ta is now span-
+ning two decades, such that creep of the interfaces plays
+a crucial role in the broadening of ∆µa. The complex
+distributions of ∆µa can then be interpreted as the com-
+bined effect of interface disorder and creep. For narrow
+waiting times Ta, multiple slips explore the statistical
+fluctuations of the contacting interfaces, giving a distri-
+bution with finite width. In addition, this distribution
+follows a creep-induced general trend over widely dis-
+tributed Ta.
+Furthermore, we observe that aging rate variations and
+the fraction of slow slip on the bottom layers is an in-
+creasing function of n. We suggest that these additional
+consequences of adding more layers to the stack might
+be an evidence of an increase in an effective interface
+temperature due to the mechanical noise of microscopic
+events.
+In conclusion, the relative rigidity of the drive against
+the layers dictates whether a stack of interfaces responds
+synchronously or not. When layers slide one by one, in-
+creasing their number lead to complex responses, making
+the prediction of the next slip more challenging.
+Limitations and outlook
+It is pertinent to discuss some limitations of our model
+system and provide suggestions for future work.
+Stiffness ratio.
+We have defined the rigid and com-
+pliant driving regimes, as characterized by the relative
+order-of-magnitude estimation of the respective stiffness
+ratio Φ. Identifying an equivalent of Φ in systems with
+more intricate geometries, such as fault networks, might
+contribute to clarifying their dynamics and help slip pre-
+dictions.
+Hence, a more systematic exploration of the
+response of stacks with varying Φ would be of great in-
+terest.
+However, both the current numerics and experiments
+are not suitable to explore large variations of Φ.
+Our
+numerics are currently limited to relatively high Φ due
+to a combination of the assumption of finite rotations and
+finite machine precision. Future work should allow local
+deformation along the frictional interface to be reset to
+a zero average while keeping the identical stress state,
+to be able to continue sliding indefinitely. In contrast,
+the presented experimental setup is limited to exploring
+the low Φ regime. On the one hand, if the stiffness of
+the springs were to be increased, the motor/lever system
+would no longer be considered rigid, thereby invalidating
+Eq. (5), the relation between global measurement (∆F)
+and local stick-slip amplitude (∆µa). On the other hand,
+if the shear stiffness of the slabs were to be reduced, the
+frictional properties would change as soft solids tend to
+be adhesive [41].
+Creep.
+This proposed model system allowed us to
+measure the aging rate B of the interfaces thanks to com-
+plex stick-slip sequences without the need for stop-and-
+go experiments. However, while our measured value of
+B is compatible with previous experiments on PMMA
+[30, 40], it differs by a factor of five. First, within our ex-
+periments, B is measured over slip events spanning the
+n = 5 interfaces, so it includes the natural variability
+of the interfaces. Averaged values of Ba for the differ-
+ent interfaces give an idea of the variations in the aging
+rate between the plates. They are different at most by
+a factor two, thus only partly explaining the mismatch
+with those previous experiments on PMMA. We also note
+
+10
+that the PMMA plates used here have a much higher
+surface roughness than in [30, 40], with unclear conse-
+quences on the aging rate. Future work should explore
+the relationship between B and surface roughness. Sec-
+ond, recent experimental observations find a relationship
+between the aging rate B and the applied shear load to
+the interface [42] with effect even on partial slip. Our
+setup naturally imposes a broad range of shear loads on
+the interface.
+However, to measure the empirical law
+by [42], our setup would need to be augmented to also
+provide stress measurements per interface, by measuring
+individually the internal forces fi of the driving springs.
+The numerical model does not show the phenomenol-
+ogy observed experimentally associated with creep. In
+the context of inhomogeneities introduced by partial slip
+and their role in creep, our model assumes a pressure-
+independent yield stress in shear, neglecting any influ-
+ence of the pressure inhomogeneities that may arise from
+the normal load or partial slip events. A possible solu-
+tion is to define yielding in equivalent stress space, rather
+than equivalent deviatoric stress space.
+Slow slip.
+The origin of the experimentally observed
+slow slip is unknown.
+A tempting hypothesis is that
+smooth sliding is the result of activated yielding events
+due to increasing mechanical noise on other interfaces.
+However, it is not clear why this interpretation would
+lead to slow slip occurring predominantly on the low-
+ermost interfaces. Indeed, this observation is robust to
+changing the order of the slabs (though the fraction of
+slow slip changes); see SI "Smooth sliding". An alter-
+native hypothesis is that, by increasing n, the loading
+rate becomes sufficiently high to drive the interfaces away
+from the stick-slip regime. This second interpretation is
+consistent with slow slip being predominantly observed
+on the lowermost interfaces.
+Our numerical model does not show any slow slip or
+creep, while we could expect that the emitted mechanical
+noise could lead to activation or creep on other layers.
+Future work should clarify if these observations are due to
+our small homogeneous background damping term. The
+latter is currently chosen to avoid nonphysical periodic
+wave propagation. A dedicated experimental campaign
+could potentially clarify at which length scale the emitted
+shear waves thermalize in PMMA and which fraction of
+waves that reach the material boundaries are reflected.
+Fluctuations.
+We have not systematically varied the
+surface roughness of the interfaces. Instead, experimen-
+tally, we have changed the order of the interfaces and
+observed no qualitative changes. Numerically, we arbi-
+trarily draw yield strains from a Weibull distribution (fol-
+lowing Ref. [22, 26]) as the relationship between surface
+roughness and the distribution of yield strains is a major
+open question.
+Fore- and after-shocks.
+In an idealized spring-block
+model (equipped with microscale disorder), fore- and
+after-shocks appear if creep is added to the drive [43, 44].
+A creeping drive is often associated with the high tem-
+peratures in Earth’s core [43].
+Our experimental sys-
+tem displays slow sliding of ‘deep’ layers already at room
+temperature. A key question is if fore- and after-shocks
+appear in the top layers of our system as well. Answer-
+ing this question experimentally would required exposing
+microscopic events, which would likely involve studying
+acoustic emissions (for which PMMA may not be the op-
+timal choice).
+ACKNOWLEDGMENTS
+The authors are grateful to Mathias Lebihain and
+Federica Paglialunga for fruitful discussions.
+We also
+thank Lebo Molefe for providing the microscope images
+of the surface of the slabs.
+S.P. acknowledges finan-
+cial support from the Japanese Society for the Promo-
+tion of Science as a JSPS International Research Fellow.
+T.G. acknowledges support from the Swiss National Sci-
+ence Foundation (SNSF) by the SNSF Ambizione Grant
+PZ00P2_185843.
+∗ pedro.reis@epfl.ch
+† Correspondence email address:tom@geus.me
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+D. dePolo, W. Ellsworth, J. Gomberg, S. Harmsen,
+L. House, S. Jackson, M. Johnston, L. Jones, R. Keller,
+S. Malone, L. Munguia, S. Nava, J. Pechmann, A. San-
+ford, R. Simpson, R. Smith, M. Stark, M. Stickney, A. Vi-
+dal, S. Walter, V. Wong, and J. Zollweg, Seismicity Re-
+motely Triggered by the Magnitude 7.3 Landers, Califor-
+nia, Earthquake, Science 260, 1617 (1993).
+[36] A. Cochard, L. Bureau, and T. Baumberger, Stabiliza-
+tion of Frictional Sliding by Normal Load Modulation, J.
+Appl. Mech. 70, 220 (2003).
+[37] Similar considerations hold for other physical systems.
+For example, the statistics of avalanches in spring-block
+models that are rigidly driven (using overdamped dy-
+namics) differ from those in the case of compliant driv-
+ing [38, 45].
+[38] A. Rosso, P. Le Doussal, and K. Wiese, Avalanche-size
+distribution at the depinning transition: A numerical test
+of the theory, Phys. Rev. B 80, 144204 (2009).
+[39] D. Fisher, Collective transport in random media: From
+superconductors to earthquakes, Phys. Rep. 301, 113
+(1998).
+[40] O. Ben-David, S. Rubinstein, and J. Fineberg, Slip-stick
+and the evolution of frictional strength, Nature 463, 76
+(2010).
+[41] K. Viswanathan and N. Sundaram, Distinct stick-slip
+modes in adhesive polymer interfaces, Wear 376–377,
+1271 (2017).
+[42] S. Dillavou and S. Rubinstein, Shear Controls Frictional
+Aging by Erasing Memory, Phys. Rev. Lett. 124, 085502
+(2020).
+[43] E. Jagla, F. Landes, and A. Rosso, Viscoelastic Effects
+in Avalanche Dynamics: A Key to Earthquake Statistics,
+Phys. Rev. Lett. 112, 174301 (2014).
+[44] D. Houdoux, A. Amon, D. Marsan, J. Weiss, and J. Cras-
+sous, Micro-slips in an experimental granular shear band
+replicate the spatiotemporal characteristics of natural
+earthquakes, Commun Earth Environ 2, 90 (2021).
+[45] A. Rosso, P. Le Doussal, and K. Wiese, Numerical calcu-
+lation of the functional renormalization group fixed-point
+functions at the depinning transition, Phys. Rev. B 75,
+220201 (2007).
+[46] T. de Geus, FrictionQPotFEM (2021).
+
+12
+SUPPLEMENTARY MATERIAL
+Experimental set-up
+Details on the critical components of the experimental set-up are shown in Fig. S1.
+Lever.
+A lever imposes a constant shear rate on the stack of slabs via a set of springs. The lever is made of two
+6 mm thick and 100 mm wide parallel acrylic slabs, ensuring no bending. It is actuated on its top by a rigid arm
+pulled horizontally at a constant speed by a linear actuator (Fig. S1(a.1)). The rigid arm is made of steel to ensure
+that the lever’s position is rigidly imposed. A uniaxial force sensor between the actuator and the arm measures the
+total pulling force F (Fig. S1(a.2)). The sensor is rigidly connected on both sides by being directly threaded into the
+acrylic adaptors.
+Springs.
+The springs between the lever and the slabs are PMMA beams with a rectangular 5× 5 mm cross-section
+and pre-curved with a radius of curvature of 100 mm over a cord of 150 mm in the middle and extended on both sides
+by two 25 mm straight portions.
+Links.
+The rigid arm and the springs are attached to the lever by the links shown in Fig. S1(b). The lever’s
+rotation around its rotation axis is ensured by a traverse steel shaft embedded into ball bearings. The links are
+inserted between the two slabs of the lever and are made of a central steel pin press-fitted at the end of the spring
+(orange). This pin is embedded into small ball bearings on each side (green), allowing a free relative rotation between
+the springs and the lever. The bearings are then press-fitted into acrylic components (pink) with two threaded holes.
+These components are directly screwed on both sides on both slabs of the lever. The rotation axis of the pins is
+exactly aligned along a line passing by the center of rotation of the lever. Figs. S1(b.1) and S1(b.2) show, respectively,
+a side and top-side view of the link; and Fig. S1(b.3) shows the link when not attached to the lever. The other sides
+of the springs are linked to the frictional slabs (Fig. S1(d)).
+The spring-slab links have a similar design as the lever-spring links, but here the acrylic components are press-fit
+into the slabs. The height of the acrylic components is slightly smaller (8 mm vs. 10 mm) to avoid any contact with
+the interfaces. Fig. S1(c.1) and Fig. S1(c.2) show topside views of the links respectively attached and out of a frictional
+slab. These links ensure a rigid connection between all the components, so only the springs deform during actuation.
+The ball bearings also ensure that the force transmission remains horizontal as the lever rotates.
+Slabs.
+Fig. S1(d) shows a close-up of the surface of the frictional slabs (Snow WH10 DC by Röhm), displaying a
+complex roughness profile with the largest asperities of the order of ∼ 25 µm, with Fig. S1(d.1) having a scale bar of
+100 µm, and Fig. S1(d.2) a scale bar of 25 µm. The position of the slabs is measured using a 7 mm wide circular red
+marker glued on the side of each slab, see Fig. S1(d).
+Slip measurements.
+A raw image taken by the camera is shown in Fig. S1(e.1), the pictures have a resolution of
+70 µm per pixel. Using a color threshold, the red markers are extracted in Fig. S1(e.2) and their geometric center
+is located (red crosses). Owing to the large size of the markers (154 mm2, or around 30000 pixels), the location of
+their geometric centers is determined with a sub-pixel resolution of 5 µm. The relative position between the slabs is
+computed by retrieving the horizontal position of a layer to the one below it.
+
+13
+Figure S1.
+Technical details on the main components of the experimental apparatus. Top: overview of the set-up with the
+location of the different parts: (a) force measurement, (b) lever-spring link, (c) spring-slab link, (d) frictional slabs surface and
+(e) position measurement. Relevant details are shown in the bottom panels (numbered (a.1), etc.), see text for details.
+
+50 mm
+a
+?
+(b)
+(a.1)
+50 mm
+b.1
+(a.2)
+20 mm
+10 mm
+(b.3)
+10 mm
+(c.1)(c.2)
+10 mm
+10 mm
+20 mm
+(e.1)
+(e.2)
+(d.2)
+25 μm
+5 mm
+5 mm
+100 μm14
+Kinematics
+We verify that our setup imposes the desired kinematics on the slabs. We record the horizontal displacement of the
+lever at the top, next to its actuated region. We indeed find a smooth linear increase of this horizontal displacement,
+insensitive to the concurrent jump in force measured, see Fig. S2a. The lever thus rigidly imposes a homogeneous shear
+to the slabs. We also verify that the position of the bottom slab x0, onto which the first layer i = 1 is sliding, remains
+still during the experiment (even during force drops) in Fig. S2b. The noise of this measurement is an estimation of
+the position accuracy of our experimental set-up.
+(a)
+(b)
+1400
+1450
+1500
+1550
+1600
+18
+18.5
+19
+19.5
+20
+20.5
+21
+-1
+0
+1
+2
+3
+4
+5
+1600
+1650
+1700
+1750
+-0.01
+-0.005
+0
+0.005
+0.01
+-1
+0
+1
+2
+3
+4
+5
+Figure S2.
+Verification of the kinematics of the experimental set-up. (a) Blue left axis: horizontal position of a point next to
+the top actuated point of the lever as a function of time. Orange right axis: concurrent measurement of F showing stick phases
+separated by rapid slip events. (b) Blue left axis: horizontal position of the bottom slab i = 0 as a function of time. Orange
+right axis: concurrent measurement of F.
+Individual sliding
+The interfaces are tested individually, but still in their stacked configuration. To test the interface a, we only keep
+spring a, while the slabs below it (i < a) are clamped such that they cannot move. That way, only the bottom
+interface of slab a is sliding (the slab is not pulled from the stack). In this case, the slip ∆Ra and the macroscopic
+force drop are related as
+∆F = K∆Ra
+ah
+H ,
+(7)
+which we verify in Fig. S3a. The drop of frictional resistance then reads
+∆µa = K∆Ra
+N
+= ∆F
+N
+H
+ah
+(8)
+In addition, to ensure testing the interface at the same shear rate ˙γ = 10−4s−1 despite the different heights onto which
+they are attached to the lever, the shear rate is adapted for each layer such that ˙γa = ˙γ/a. With ∆Ra = Taha˙γa =
+Tah˙γ, we have the following expression for the friction jump as a function of the time interval between two slips:
+∆µa = KTah˙γ/N
+(9)
+Figs. S3b and S3c show P(∆F) and the extracted P(∆µa), respectively, for a = 1, 2, . . . , 5. The interfaces show
+significant differences in the mean values of ∆µa, ranging from 0.15 to 0.3 (with the differences in mean values
+bigger than the scatter of the different P(∆µa)) even though the material, preparation protocol, and environmental
+conditions are the same for each of these interfaces/experiments.
+
+15
+(a)
+(c)
+(b)
+0
+100
+200
+300
+400
+0
+100
+200
+300
+400
+Figure S3.
+(a) Verification of ∆Ra(∆F) (Eq. (7)) for individual interfaces. (b) The probability distributions of the drop
+of force ∆F for the different interfaces a are shown using a different color and marker. (c) Probability distributions of the
+corresponding drop in frictional resistance ∆µa as estimated from ∆F using Eq. (8).
+Weight and position of the lever
+We comment on the limitations due to the intrinsic weight of the lever and actuating arm. The required rigidity of
+the lever and arm connecting it to the linear actuator results in a non-negligible weight. This weight manifests itself
+in the measure of F because of the vertical motion induced by the rotation of the lever and adds a slow decreasing
+linear component in the measured signal, as shown in Fig. S4a. This additional component makes a precise measure
+of the absolute values of the force before and after a drop impossible. However, this slow component does not affect
+the measure of the force difference ∆F. Indeed, in Fig. S4b, we show that the probability distribution of ∆µa (we
+arbitrarily show ∆µ1 for n = 5) is not influenced by the angular position of the lever, and hence by the set-up weight
+component in F.
+Figure S4.
+Effect of the weight of the actuating lever and arm. (a) Full time series of F for n = 1 (purple), n = 3 (green),
+and n = 5 (yellow). Particularly noticeable for n = 1, the force signal shows a slow linearly decreasing component due to the
+intrinsic weight of the lever arm and its vertical motion upon rotation of the lever. (b) During an experiment, the lever is
+rotated between γ = [−0.3, +0.3] rad. We represent the probability distribution of ∆µ1 for n = 5, for the entire run (squares),
+and biased for γ < 0 (left triangles) and for γ > 0 (right triangles).
+
+ä n=5
+ n=5<0
+ n=5 >016
+Robustness of experimental distributions
+For each n, 10 runs are performed one after the other.
+One run lasts 6000 s, while the lever rotates between
+γ = [−0.3, +0.3] rad. Fig. S5 shows the distributions of ∆µa (we arbitrarily show ∆µ1 for n = 5) we obtained for
+each individual run (black lines), which are compared to the distribution obtained by combining the data from all the
+runs. All runs display a similar overall distribution, which attests the reproducibility of the experiments.
+single run
+merged runs
+0
+0.1
+0.2
+0.3
+0
+5
+10
+15
+20
+Figure S5.
+Robustness of the distribution of ∆µ. Comparison between the distributions of ∆µ1 of each individual run (black
+lines) and the distribution obtained by compiling all the events of the 10 runs (orange squares). The system has n = 5.
+Results from position measurements
+We confirm that our central result in Fig. 4 remains unchanged if we ∆µa from Ra (instead of ∆F) in Fig. S6.
+(a)
+(b)
+(c)
+(d)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+5
+10
+15
+20
+25
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+5
+10
+15
+20
+25
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+5
+10
+15
+20
+25
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+5
+10
+15
+20
+25
+Figure S6.
+Same as Fig. 4 but obtained using ∆Ra instead of ∆F.
+Waiting time distribution for a > 1
+In addition to Fig. 5c we show the waiting time between consecutive sliding events for interfaces a = 2, 3, 4, 5 in
+Fig. S7. For all the layers, we recover the same qualitative behavior with increasing n: the distribution becomes
+multimodal with the main peak shifting to lower values.
+
+17
+101
+102
+0
+0.005
+0.01
+0.015
+0.02
+101
+102
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+101
+102
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+101
+102
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+(c)
+(d)
+(a)
+(b)
+Figure S7.
+Same as Fig. 5c for (a) a = 2, (b) a = 3, (c) a = 4, (d) a = 5.
+Aging rate distributions
+For each slip event, we quantify its effective aging rate Ba ≡ ∆µa/ ln(Ta). The corresponding probability distribu-
+tions P(Ba) are plotted for each layer a for different n in Fig. S8.
+0.02
+0.04
+0.06
+0.08
+0
+100
+200
+300
+0.02
+0.04
+0.06
+0.08
+0
+100
+200
+300
+0.02
+0.04
+0.06
+0.08
+0
+100
+200
+300
+0.02
+0.04
+0.06
+0.08
+0
+100
+200
+300
+0.02
+0.04
+0.06
+0.08
+0
+100
+200
+300
+(c)
+(d)
+(e)
+(a)
+(b)
+Figure S8.
+Probability distribution of the effective aging rate Ba ≡ ∆µa/ ln(Ta) for each event for a given interface a and
+increasing number of layers n. (a) a = 1, (b) a = 2, (c) a = 3, (d) a = 4, (e) a = 5. The distribution for the isolated interface
+(n = 1) is shown in black.
+Slow slip
+The interfaces seldom show slow sliding with velocities much lower than for the typical slip events (see for example
+Fig. S9a). To discriminate slip events from slow ones, we use a velocity criteria. The velocity of each slab is calculated
+as (Ri(t + ∆t) − Ri(t))/∆t, with ∆t the time between frames (∆t = 0.2 s), and smoothed by a running average over
+five frames (Fig. S9b). Sliding events are then defined by all clusters of at least three frames having a velocity greater
+than 0.02 mm/s (identified by black dots in Fig. S9a and Fig. S9b). The smoothing notably allows capturing the
+frames just before and after the events, allowing for a precise measure of the sliding distances ∆Ra. In Fig. S9c we
+verify that smooth sliding is consistently observed on the lowermost interfaces even if the slabs are shuffled.
+
+18
+(b)
+(a)
+(c)
+0
+20
+40
+60
+80
+100
+1
+2
+3
+4
+5
+0
+0.2
+0.4
+0.6
+0.8
+5000
+5100
+5200
+5300
+0
+0.05
+0.1
+0.15
+0.2
+5000
+5100
+5200
+5300
+Figure S9.
+(a) Time series of the relative position Ri(t) of the slabs for n = 5. The frames detected as belonging to a
+sliding event are marked by a black dot. (b) Corresponding time series of the velocity of the slabs, after applying a running
+average over five frames. The horizontal dashed line marks the limit above which we consider that sliding is fast enough to
+be categorized as a slip event. (c) Same as Fig. 6b with an additional measurement for n = 5 where the slabs are shuffled
+(1, 2, 3, 4, 5 to 4, 3, 2, 1, 5). The resulting data points are shown at n ≈ 5.3 for visibility. After shuffling, it is still the lowest
+interfaces that show most of the slow sliding.
+Numerical model
+• All implementation details are identical to [26], and available in its supplementary material; furthermore the
+entire implementation is open-source, see [46] and its dependencies. Compared to [26] the only differences are:
+– The discretisation, that now comprises n frictional interfaces, see Fig. S10.
+– The boundary conditions are such that the position of the bottom nodes is fixed, while the top nodes are
+completely free. The displacement of the right nodes is set equal to that of the left nodes corresponding to
+periodic boundary conditions in the horizontal direction. Finally, all elements that constitute to an elastic
+layer i = 1, 2, . . . , n are connected to a parabolic potential of curvature K that has its minimum at the
+lever’s position fixed to ˙γiht as in the experiment. This potential is implemented as a force density, and
+added to each node using the node’s volume as weight factor. We use K = 10−3 for rigid driving, and
+K = 10−6 for compliant driving.
+– We use a bulk modulus that if 4.5 times larger than the shear modulus, in accordance with PMMA (see [26]
+for the definitions of the bulk and shear modulus).
+• Macroscopic stress Σ, defined as the equivalent deviatoric part of the macroscopic stress tensor Σij ≡
+�
+σij(⃗r)d⃗r
+with σij(⃗r) the stress at a material point (i.e. at an integration point in our spatial discretisation), and ⃗r the
+position. Thereby Σ2 ≡ 2ΣijΣij. All reported stresses have been normalised by the typical yield stress.
+• The rotation of the lever γ is defined in Fig. 1. The numerical results are normalised by γ0, which is defined as
+the rotation corresponding to a yield event in a system in which all blocks of the first interface (i = 1) have a
+yield strain equal to the typical yield strain.
+
+19
+Figure S10. Finite element mesh of the numerical model for n = 4.
+Slip sequences – numerics
+In the main text, we discussed how rigid driving causes sequences multi-slip events, while compliant driving results
+in sequences of single-slip events. We support this statement using the increment in lever rotation ∆γ between single-
+and multi-slip events in Fig. S11. For stiff driving the multi-slip events are mostly periodic, i.e. distributed around
+a well-defined mean that is independent of n, see Fig. S11b. Single-slip events are slightly less frequent while the
+increment between them is broadly distributed around a mean that is n dependent, see Fig. S11a. For compliant
+driving, multi-slip events are almost absent, see Fig. S11d. Also in this case, the increment between single-slip events
+is broadly distributed and n dependent, see Fig. S11c.
+We comment on the fraction of events that are part of a sequence of single- or multi-slip events using Fig. S12. For
+rigid driving, in Fig. S12a, sequences of single- and multi-slip events alternate. As n increases such sequences alternate
+more often reflected in decreasing fraction of events that are part of any sequence (blue line). Of these sequences,
+there is an equal partitioning in the number of single- and multi-slip sequences (the two highlighted regions have an
+approximately equal height for every n). For compliant driving situation is simple: there are almost exclusively single
+slip events, see Fig. S12b.
+0
+300
+∆γ/γ0
+0
+0.02
+P(∆γ/γ0)
+(a) stiff driving, single-slip
+0
+300
+∆γ/γ0
+(b) stiff driving, multi-slip
+0
+300
+∆γ/γ0
+0
+0.02
+P(∆γ/γ0)
+(c) compliant driving, single-slip
+0
+300
+∆γ/γ0
+(d) compliant driving, multi-slip
+n =
+1
+2
+3
+4
+Figure S11.
+Loading interval expressed as increment in lever angle ∆γ, between (a, c) single- and (b, d) multi-slip events for
+(a, b) rigid and (c, d) compliant driving. The right panels (b, d) show single-slip events for a single slab (n = 1) as reference
+using a dashed line. Note that the distributions are normalised such that the integral of single- plus multi-slip events (i.e. the
+distribution of the left plus that of the right panel) is equal to one.
+
+20
+1
+2
+3
+4
+n
+0.00
+0.25
+0.50
+0.75
+1.00
+φ
+(a) stiff driving
+single-slip
+multi-slip
+1
+2
+3
+4
+n
+(b) compliant driving
+Figure S12.
+Fraction of slip events that are part of a sequence of single- or multi-slip events for (a) stiff and (b) compliant
+driving. The plot is constructed such that the blue line (square markers) corresponds to the total fraction of events that is
+part of a sequence of consecutive single- or multi-slip events. The red line (circular markers) is the fraction of events that are
+part of sequences of single-slip events. The area between the red and blue curves, highlighted in blue, thus corresponds to the
+fraction of events that are part of a sequence of multi-slip events.
+Slip vs. stress drop
+Naturally, the slips and the stress drops are proportional, as was validated experimentally in Fig. 3c. In Fig. S13,
+we verify that this is the case also for the numerical results. From Figs. S13b and S13d, we observe that interfaces
+are characterised by a slightly different compliance, whereby the i = 1 differs from the other interfaces because the
+position of slab i = 0 is fixed. Note that the trend between Figs. S13a and S13c suggests that the driving for our
+numerical results is more rigid than in the experiments, cf. Fig. 3c. This is indeed a known limitation of our numerical
+model as we discussed in the main text.
+0.0
+0.1
+0.2
+0.3
+0.4
+∆µ
+0
+50
+100
+Si/nx
+(a) stiff driving
+n =
+1
+2
+3
+4
+0.0
+0.1
+0.2
+0.3
+0.4
+∆µ
+0
+50
+100
+Si/nx
+(b) stiff driving, n = 4
+i =
+1
+2
+3
+4
+0.0
+0.1
+0.2
+0.3
+0.4
+∆µ
+0
+50
+100
+Si/nx
+(c) compliant driving
+0.0
+0.1
+0.2
+0.3
+0.4
+∆µ
+0
+50
+100
+Si/nx
+(d) compliant driving, n = 4
+Figure S13.
+Slip Si per block as a function of the stress drop on the slipping interface denoted as ∆µ for different (left) n, and
+(right) i for n = 4, for (top) rigid and (bottom) compliant driving. Cf. Fig. 3c where Si ∼ ∆Ri, note that we do not dispose
+of means to seek a quantitative comparison of ∆µ).
+
diff --git a/otFST4oBgHgl3EQfMzga/content/tmp_files/load_file.txt b/otFST4oBgHgl3EQfMzga/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f57a755e7f506bf8c706b11a474db05280dc5fad
--- /dev/null
+++ b/otFST4oBgHgl3EQfMzga/content/tmp_files/load_file.txt
@@ -0,0 +1,1173 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf,len=1172
+page_content='Stick-slip synchronization in stack of elastically coupled frictional interfaces  Samuel Poincloux,1, 2 Pedro M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Reis,1, ∗ and Tom W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' de Geus3, †  1École Polytechnique Fédérale de Lausanne (EPFL),  Flexible Structures Laboratory, CH-1015 Lausanne, Switzerland  2Department of Physics, The University of Tokyo, Tokyo, Japan  3Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland  We perform physical and numerical experiments to study the stick-slip response of a stack of  slabs in contact through dry frictional interfaces driven in quasistatic shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The ratio between the  drive’s stiffness and the slab’s shear stiffness controls the presence or absence of slip synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  A sufficiently high stiffness ratio leads to full synchronization, comprising periodic slip events in  which all interfaces slip simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  A lower stiffness ratio leads to asynchronous slips and,  experimentally, to the stick-slip amplitude being broadly distributed as the number of layers in the  stack increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We interpret this broadening in light of the combined effect of surface disorder,  complex loading paths of the asynchronous slips, and creep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The extracted aging rate is found to  be of the same order of magnitude as existing experimental results on a similar material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Finally,  we discuss the emergence of slow slips and an increase in creep-rate variations when more slabs are  added to the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  INTRODUCTION  Multiple frictional interfaces coupled by elasticity are  ubiquitous in everyday objects including books [1, 2], tex-  tiles [3–5], and multilayer composites [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' At a larger  scale, in geology, systems comprising multiple frictional  interfaces are the norm rather than an exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' For  example, layered rocks such as shales can show multi-  ple sliding interfaces under shear [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  At the even  larger scales relevant for terrestrial faults, earthquakes  produced when slipping are usually not isolated but em-  bedded into complex fault networks [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The mechan-  ical response of such assemblies of frictional interfaces  involves the coupling between the elastic deformation of  the layers and the barriers to sliding of the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Predicting the onset of slipping is a long-standing prob-  lem even for a single frictional interface [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Physical  insight and understanding of this class of problems have  been driven primarily by high-precision experiments of  sliding PMMA blocks whose optical transparency en-  abled the space-temporal tracking of the local contact  area [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' These pioneering experiments have elucidated  that the onset of slip involves a rupture front that ‘un-  zips’ the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' A correlation with strain measure-  ments close to the interface showed that the stress field  and dynamics of the front are well described by fracture  mechanics with the fracture energy as the sole fitting  parameter [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' However, the mechanism underlying the  nucleation of the rupture front remains elusive, primarily  due to experimental limitations, for which novel proto-  cols are being proposed [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  From a theoretical perspective, the most common mod-  els for the onset of frictional slippage [15–18] capture the  phenomenology that sliding starts, and then continues  in a steady state, when the shear force F balances the  friction forces µN, where N is the normal force and µ  the ‘friction coefficient’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In these ‘rate-and-state’ models,  µ depends nonlinearly on the slip rate v and history θ:  µ = µ(v, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' At intermediate values of v, the friction coef-  ficient is usually assumed to display slip weakening (µ is a  decreasing function of v) such that the interface is unsta-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' During slip nucleation, the elasticity and inertia of  the bulk have a stabilizing effect [19–22], such that there  exists an effective flow curve ˜µ = µ + G/(2cs)v (where  G is the shear modulus of the material and cs the shear-  wave speed) whose steady-state displays a minimum at  µc = ˜µ(vc) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Consequently, any perturbation decays  and vanishes if the applied load is F/N < µc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' At higher  applied loads, a slip patch of linear size L becomes unsta-  ble if L > Lc ∼ (F/N − µ∗  c)−ν (µ∗  c ≥ µc [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Once the  event grows to be larger than Lc, its dynamics are well  described by a sharp rupture front [19, 21, 23] that can  be modeled by linear elastic fracture mechanics [13, 24],  as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  If the loading is performed under  displacement control, the reaction force µN drops macro-  scopically only once the interface unzips fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  A direct consequence of the phenomenology described  above is that the interface can display stick-slip behavior  when driven quasistatically (at a rate V ≪ vc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Under  the framework of the rate-and-state models, the stick-slip  amplitude thus depends on the exponent ν, µ∗  c, and the  mechanism leading to the slip patch at scales L < Lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The relative relevance of all of these parameters is cur-  rently under debate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' [14, 21, 22, 25–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' However,  one of the authors of the present study recently proposed  an encompassing theory that avalanches nucleate the in-  stability such that µ∗  c = µc and ν = 1/(1 − ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Here, ζ  is the roughness exponent resulting from a competition  between microscale disorder, and elasticity [22] (not to  be confused with a roughness exponent extracted from  height-height correlations of the interface profile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Fur-  thermore, it is a well-known experimental fact that the  initial onset to sliding is history-dependent and increases  with the time that the interfaces were at rest [17, 29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='13745v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='soft]  31 Jan 2023  2  This behavior is associated with creep [32] and described  by rate-and-state models where the variable θ introduced  above is regarded as time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Creeping of the interfaces  must affect the stick-slip amplitude, but disentangling  its contribution is challenging because slip events occur  at a narrowly distributed interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Beyond single interfaces, when multiple frictional in-  terfaces are present, the elasticity of the bulk may poten-  tially lead to a non-trivial coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' For example, elastic  interactions between faults may strongly affect their slip  dynamics [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In addition, acoustic waves transmitted  through the elastic bulk may lead to remote triggering  of earthquakes [34, 35], though the large temporal sepa-  ration suggests a complex coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Predicting the me-  chanical response of an assembly of elastic frictional in-  terfaces is then a formidable but important challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In  particular, identifying the key parameters coupling the  layers together and elucidating the role of the number of  interfaces are open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Here, we report results from a combined experimental  and numerical investigation on the quasistatic stick-slip  response of a stack of elastic slabs in contact through  frictional interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Based on the ratio between the stiff-  ness of the drive and the shear stiffness of the slab, we  distinguish two regimes: ‘stiff’ and ‘compliant’ driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  In the stiff-driving regime, our numerical results exhibit  periodic slips involving narrowly distributed force drops;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  this regime is not accessible in our model experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  By contrast, in the compliant-driving regime, we observe  both numerically and experimentally a decoupling of the  slip events along the different layers, with interfaces slid-  ing one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In the experiments, we find that this loss  of periodicity is accompanied by a broadening of the dis-  tribution of the stick-slip amplitudes with the number of  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The changes in the measured distributions are in-  terpreted by a coupling between interface disorder and a  broad distribution of waiting times between slips, expos-  ing the role of creep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The remaining unsolved broadening  of the distributions with the number of slabs, including  an observed increasing fraction of ‘slow slip’, raises the  question of the interaction between mechanical noise and  creep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Depending on the drive’s stiffness, the stack of  frictional interfaces then displays drastically different re-  sponses of the stack to shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Overall, we find that, in the  stiff-driving case, the stack acts as one layer with periodic  slips, while in the compliant-driving case, a rich coupling  between the layers makes slips much more unpredictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  DEFINITION OF THE PROBLEM  We assess the shear response of a model system com-  prising a stack of n ∈ [1, 5] identical slabs of thick-  ness h resting on a surface whose position is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1, we present a schematic diagram of the system  and a photograph of our experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Each slab,  and its lowermost frictional interface, are numbered as  i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' , n from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  We impose homogeneous  shear, with a set rate, between all the slabs by connect-  ing each slab through identical springs of stiffness K to a  lever that is driven to rotate around a fixed axis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The spring connecting to the i-th layer is attached to the  lever at a distance ih from the rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  For the drive, we impose the lever’s top horizontal dis-  placement U(t) = V t, where t represents time, and V is  the imposed velocity, taken to be small enough for the in-  terfaces to display stick-slip (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=', V < vc [22, 30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Thus,  our drive imposes a shear rate ˙γ ≡ V/H, where H is  the height of the lever, driving each spring i at a veloc-  ity vi = ih˙γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' During the periods in which the interfaces  are ‘stuck’, this drive causes a monotonically increasing  shear stress at each of the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Our study seeks to address the following questions:  (1) What are the relevant parameters controlling the slip  synchronization of the interfaces?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (2) How does the shear  response evolve with an increasing number of layers n?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (a)  (b)  springs  lever  weight for normal force  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (a) Schematic of our model system, shown here  with n = 4 active (driven) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The color code of the  layers is used throughout the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (b) Photograph of the  corresponding experimental apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  RIGID VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' COMPLIANT DRIVING  For a system with a single frictional interface, the stick-  slip instability occurs only if the driving stiffness K is  sufficiently low and depends on the flow properties of the  interface and the applied driving rate [15, 30, 36];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' a sum-  mary of the calculation for a rate-and-state model was  provided in Ref [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' With multiple frictional interfaces,  O  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3  the drive also controls the degree of synchronization as  we argue next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  To gain insight into the effect of the drive on the  shear response of a stack, regard the driving ‘spring’ as  a parabolic potential energy imposing the mean position  of each slab, such that the slab is free to build up shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  We now discuss what happens in the limit of rigid and  compliant driving, defined next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Rigidity ratio Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  We define the rigidity ratio Φ ≡  K/Ks, where K is the rigidity of the driving spring, and  Ks = AG/h is the shear rigidity of the slabs, with G  the shear modulus, A the surface area of the frictional  interface, and h the slab’s height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  This ratio Φ then  quantifies the relative deformation of the driving springs  in comparison to the shear deformation of the slabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Be-  low, we investigate and discuss two regimes: rigid (high  Φ) and compliant (low Φ) driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In the first, significant  deformations occur within the layers (and the springs are  rigid), whereas, in the second, the deformations occur in  the spring (and the layers are rigid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Rigid driving (high Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Suppose that the first inter-  face (i = 1) starts slipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' As the mean position of slab  i = 1 is fixed, the slab can only react with a negative  shear deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Consequently, the shear stress on the  interface above, i = 2, is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' This can trigger a  slip at interface i = 2, which can cause a slip on the in-  terface above for the same reason, which propagates to  the slipping of other interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The cascade results in  a multi-slip event that erases all memory of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  In this case, the multi-layered system thus acts as a sys-  tem with a single interface with effective properties [37],  showing periodic stick-slip cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Compliant driving (low Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The system can respond  to a slip at interface i = 1 by advancing the mean position  of slabs i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Therefore, the stress on the interfaces i >  2 relaxes, making a macroscopic multi-slip event unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  A sequence of single-slip events is thus to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  With an increasing number of layers, the slip sequence of  the multiple interfaces may lose its periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  NUMERICAL SIMULATIONS  Numerical model  We implement the model system shown schematically  in (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1a) into numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The numerical  model consists of n + 1 identical elastic layers separated  by frictional interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Following [26], we idealize the  frictional contact problem in order to focus on the dis-  order in the shear response along the frictional interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  In particular, we consider a mesoscopic scale on which  an effective ‘block’ of a finite width resists elastically to  shear up to a threshold, after which it yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The lo-  cal slip then propagates until a new ‘contact’ is formed  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=', it is again elastic but with a new threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  In  this framework, each block represents a frictional con-  tact (or a patch of contacts that are so strongly coupled  by elasticity that they act as an effective contact) that,  upon yielding, forms a new contact with a new yielding  threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The details of the numerical model are as follows:  each frictional interface consists of nx equal-sized square  blocks of linear size l0 that are completely elastic under  volumetric deformation but yield under shear (deviatoric  deformation) when a set yield stress is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Assum-  ing that the yield threshold is isotropic in principal de-  viatoric strain space, this model now corresponds to a  deviatoric potential energy that consists of a sequence of  parabolic potentials in equivalent deviatoric strain space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The disorder arises from independently randomly draw-  ing the yield strain sequence of each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We assume  that the blocks and the bulk have the same elastic mod-  uli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Finally, differently from [26], we add a parabolic po-  tential (with curvature K) to the mean position of each  of the elastic slabs i > 0, thereby prescribing a homo-  geneous force density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The bottom layer is not driven  through its mean position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' instead, the position of the  bottom edge is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' A key feature of the model is that  shear can be applied according to the quasistatic proto-  col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In particular, because the elastic response is linear,  we rotate the lever by a finite amount if no microscopic  yielding takes place while infinitesimally rotating it to  trigger a microscopic event, after which we minimize en-  ergy before loading again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Thus, we run an event-driven  protocol, allowing us to separate events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Geometrically we do not seek to precisely model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1a  as its numerical treatment, together with the disorder, re-  quires an intractably large number of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Therefore,  we consider periodic boundary conditions in the horizon-  tal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Furthermore, we choose nx = 2×36, which  is still tractable to simulate, but of the minimal order not  to be dominated by finite size effects, as we checked for a  single frictional interface [22, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Furthermore, we take  h/ℓ0 ≈ nx/4 based on balancing h/(ℓ0nx) small enough  to have acoustic interactions while avoiding driving the  blocks in a fixed displacement such that collective effects  are suppressed if h ≪ nxℓ0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The above model predicts stick-slip behavior [26] when  full inertial dynamics are considered (using overdamped  dynamics, this model predicts the abundantly studied  depinning transition [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We consider such inertial dy-  namics by applying the finite element method to dis-  cretize space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Along the frictional interface(s), elements  coincide with the mesoscopic blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In the elastic slabs  away from the frictional interface, the elements are coars-  ened to gain numerical efficiency (such that the height h  is only approximated as we fix the aspect ratio of ele-  ments to one, see SI, "Numerical model").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We use the  velocity Verlet algorithm to integrate discrete time (with  a time step significantly smaller than the time of a sin-  gle oscillation in a well in one block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We remark that  4  assuming periodicity requires us to add a small damp-  ing term to the inertial dynamics such that waves with  a wavelength equal to the horizontal size of the system  (equal to nxl0) are critically damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Consequently, we  must take h/ℓ0 < nx to have acoustic coupling between  the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Numerical results  Our numerical model allows us to first illustrate the  simple reflection above on the role of driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We con-  sider a driving rigidity such that Φ ≃ 10−3 (rigid driv-  ing) and Φ ≃ 10−6 (compliant driving).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  In the two-  dimensional model, A = nxℓ0, such that Ks = 4G for  our geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' we use K = 10−3 and K = 10−6 and  G = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2b, for rigid and compliant  driving, respectively, we plot a typical macroscopic stress  Σ (volume-averaged stress) as a function of applied lever  rotation γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Note that the stress is shown in units of the  typical yield stress of one block, and the rotation in units  of the rotation needed to yield a typical block at i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Macroscopic slip events are defined when all blocks  along one or more layers yield at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Below, we  will refer to sliding interfaces and associated quantities  by an index a, while i will be kept as the running in-  dex for the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Slip events correspond to macroscopic  stress drops in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2a and 2b, and we distinguish be-  tween ‘single-slip’ events (all blocks on a single layer yield  at least once) and ‘multi-slip’ events (all blocks on more  than one layer yield at least once).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Stress drops produced  by single-slip events are labeled following the layer color  code introduced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1, while muti-slip events are kept  black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' These slip events are separated by ‘stick’ intervals  during which only microscopic events are observed, where  one or several blocks yield at least once, as indicated with  markers (black dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The results confirm that rigid driving causes a peri-  odic stick-slip sequence with slip events corresponding  to multi-slip events (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2a) while compliant driving re-  sults in a seemingly less periodic sequence of single-slip  events in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Although not reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2a,  for Φ ≃ 10−3, we also observed periodic sequences on  single-slip events for a finite fraction of the loading his-  tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' This finding is supported by plotting the fraction  of slip events involving s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' , n interfaces in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2c  for different n (see legend in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' On the one hand,  rigid driving results in single- and multi-slip events for  a comparable fraction of loading history (we discuss in  the SI "Slip sequences - numerics" that sequences of sin-  gle and multi-slip events alternate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' On the other hand,  compliant driving shows single slip in the large majority  of slip events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  A direct measurement of the stress drop along the  slipping interface a, ∆µa, displays no n dependence in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The quantity µa is defined as the volume aver-  age stress on the blocks corresponding to weak layer a,  also shown in units of the typical yield stress of one block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Given that, by construction, normal stress plays no role  in our model, here µa is akin to a friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  As we have shown that, for high Φ (rigid driving), mul-  tilayer stick-slip is apparently similar to that of a single  interface, next, we concentrate on the low Φ regime to  explore a potential influence of the number of plates n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  1  2  3  4  s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='00  ρ(s)  (c)  Φ ≃ 10−3  Φ ≃ 10−6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  ∆µa  0  5  10  15  P(∆µa)  (d)  n =  1  2  3  4  600  800  1000  1200  1400  1600  1800  2000  2200  γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  Σ  ∆Σ  (a) rigid driving: Φ ≃ 10−3  600  800  1000  1200  1400  1600  1800  2000  2200  γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  Σ  ∆Σ  (b) compliant driving: Φ ≃ 10−6  a = 1  a = 2  a = 3  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Numerical results: (a, b) Typical steady-state  global stress Σ response as a function of lever rotation γ for  n = 3 for (a) rigid driving and (b) compliant driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We indi-  cate all microscopic yielding events with a black dot marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Slip events on a single layer are indicated in color (see leg-  end), while slip events in black involve more than one inter-  face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (c) The fraction ρ(s) of macroscopic slip events involving  s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' , n layers, for rigid (dashed) and compliant (solid)  driving;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' see legend in (d) for color-map and markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (d) Dis-  tribution of stress drops at the slipping interface for different  n (for slip events on a single layer, for which s = 1 in (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  See the main text for definitions and units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  EXPERIMENTS  We now propose an experimental realization of the  sheared-multilayer model system of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1a, adapted to  measure the effect of the number of sliding layers n on  the slip synchronicity and amplitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Simi-  larly to the numerical model, the position of each slab is  driven by connecting it to the driving lever through linear  springs (see schematic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Naturally, connect-  ing the spring to the edges of the slabs might introduce  5  boundary effects, but our experimental system is effec-  tively much larger than our numerical model (given that  it presumably has much more local contact patches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Experimental apparatus  The experimental setup shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1b comprises  a stack of frictional plates (color-coded from purple to  orange), an actuating lever (green), and driving springs  (pink), as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The stack is made of a set of rectangular PMMA  slabs (Snow WH10 DC by Röhm), each of dimensions  h = 10 mm, L = 150 mm and an out-of-plane width of  80 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' A normal force N is applied on the topmost slab  by a dead weight of 5 kg (N = 49 N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' To ensure a spa-  tially homogeneous contacting surface at this relatively  low normal force (compared to other PMMA-PMMA fric-  tion experiments [13, 14, 40]), we use acrylic plates whose  surface was pre-roughened with asperities of size ∼ 25 µm  that are larger than potential natural height variations of  PMMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We assume that the normal force is uniformly  distributed and that it is the same for each layer (the  weight of each slab is less than 3% of that of the dead  weight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The stack is sheared by imposing the displacement at  the top of the lever (H = 100 mm) at a constant speed  V = 10 µm/s (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' ˙γ = 10−4 s−1), using a DC linear actu-  ator (L-220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='70DG, Physiks Instruments) that is attached  via a steel link assumed rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The PMMA lever is suffi-  ciently wide to not bend while pulling the slabs and ro-  tates smoothly on ball bearings around its rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The springs connecting the slabs to the lever are curved  beams laser-cut from PMMA (colored in pink in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1b),  with an equivalent stiffness of K = 55 N/mm when pulled  or compressed along the horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The ends of the  springs are attached to both the slabs and lever via ball  bearings to ensure a free rotation and, thus, horizontal  driving forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The experimental set-up, with its springs and PMMA  slabs, corresponds to the compliant driving limit with  Φ ≃ 6 × 10−5, of the same order as for the compliant  regime in the numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Indeed, Φ ≡ K/Ks, with Ks =  AG/h the shear stiffness of the slabs and G ≡ E/(2(1 +  ν)), with, for PMMA, Young’s modulus E = 2 GPa and  Poisson’s ratio ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The total horizontal force F needed to rotate the  lever (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1) is measured using a uniaxial force sensor  (LRM200 25 lb, Futek) placed between the steel link and  the actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The link is also attached via small ball  bearings to remain horizontal at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We directly  verify that we are imposing the expected kinematics by  measuring the position of the lever and base slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  In addition to the global force measurement F, we also  measure the local average horizontal position of the slabs  xi, by tracking a red marker placed on their side (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1b),  from photographs taken at a rate of 5 fps using a digi-  tal camera (Flea3 FL3-U3-20E4C, Flir, linear pixel size:  70 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' After color filtering, xi is measured with an ac-  curacy of 5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The relative displacement between slabs  is Ri ≡ xi − xi−1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1a), which serves as a proxy  for the total slip at the interface i (neglecting the shear  deformation of the slabs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  To vary the number of sliding interfaces n, we keep the  same number of slabs (5) but remove 5−n springs, start-  ing from the top (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1b where n = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' This proce-  dure ensures robust image detection and reduces external  contamination of the interfaces by keeping them in con-  tact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Each time the slabs are disassembled to vary n, the  interfaces are cleaned with isopropanol and quickly dried  using compressed air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' For each value of n, we perform 10  runs during which we drive over a range ∆γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='6 rad,  starting at γ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='30 rad, each time excluding γ be-  tween −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='27 (300 s) to ensure measuring in a  steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' After each run, the lever is reset back to  γ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' On average, each connected layer is forced to  move by a total relative distance of ∆R = h∆γ = 6 mm  during a run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Further details on the apparatus and the experiments  are given in the SI ("Experimental set-up"), including  verification that the lever’s displacement is indeed im-  posed, even during slip events ("Kinematics").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We also  discuss the influence of the weight of the lever ("Weight  and position of the lever").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Experimental measurements  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 3a, for the slab system with n = 4, we present  a typical time series of the force F(t) required to actuate  the lever (top plot), together with the corresponding rela-  tive position of the slabs Ri(t) (bottom plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The experi-  ments exhibit stick-slip, with stick periods when the slabs  are immobile (Ri ≈ constant) and F increases monoton-  ically, punctuated by macroscopic slip events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' These slip  events are identified by a sudden position jump, ∆Ra  (with a denoting the sliding interface), accompanied by  an abrupt force drop ∆F > 0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' On all oc-  casions, we find that only one layer slips at a time, re-  covering similar dynamics as in the numerical model in  the compliant driving regime with a similar value for Φ  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' However, we note that during the stick periods,  we observe what seem to be ‘slow slip’ events where an  interface moves gradually, leading to a non-linear force  response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  These are out of our primary focus but are  discussed at the end of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  For each value of n, we acquire an ensemble of at least  100 slip events per layer, such that the slip quantities as-  sociated can be represented as probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 3b, we show the probability distri-  bution of force drops, P(∆F), occurring on the interface  a = 1, for all cases of n considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Starting from the  6  peaked distribution for n = 1, as n increases, the distri-  butions broaden and with higher average values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  In contrast with more classic stick-slip experiments  with a single interface [29], the global measure ∆F is not  a direct quantification of the frictional properties of the  interface but couples with the specific kinematic of the  lever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Still, the fact that only one interface slips allows  us to extract, from ∆F, a jump in a friction-like quan-  tity ∆µa (or stick-slip amplitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We define the friction  coefficient µa of a slipping interface a as the horizontal  force acting on this interface divided by the normal force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Considering the horizontal force balance on an interface  a, the interface has to resist the combined forces of the  pulling springs of the slabs i ≥ a, such that  µa =  n  �  i=a  fi  N ,  (1)  where fi is the force due to the driving spring on slab  i (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1a for a visual representation of µa and fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  When the interface a slides by ∆Ra, the relative positions  of the other interfaces remain unchanged:  ∆Ri =  �  ∆Ra > 0  if  i = a  0  if  i ̸= a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (2)  This sliding induces a drop in the spring forces: ∆fi = 0  for i < a, and ∆fi = K∆Ra > 0 for i ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Indeed,  even if no slip occurs for the interfaces i > a, the abso-  lute position of the slabs still moves by ∆Ra, reducing  the extension of the corresponding springs by the same  amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Note that for consistency, we define ∆fi to be  positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (1), we can then express ∆µa as a  function of the slip distance ∆Ra:  ∆µa = K  N (n − a + 1)∆Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (3)  We proceed by linking ∆Ra to the global force drop  ∆F, using the fact that only one interface slips at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Through moment balance on the lever, we obtain:  F =  n  �  i=1  fi  ih  H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (4)  Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (2) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (4), we obtain a relation be-  tween the global quantity ∆F and the local one ∆Ra:  ∆F =  n  �  i=a  K∆Ra  ih  H = K∆Ra  Kh  H  (n + a)(n − a + 1)  2  (5)  (Note that i is the only varying term in the sum and  �n  i=a i = (n(n + 1) − a(a + 1))/2 = (n + a)(n − a +  1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=') This result is verified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Indeed, a direct  measurement of ∆Ra is very close to the inversion of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (5), in which ∆Ra follows from the measured ∆F  without any fitting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  0  100  200  300  400  (a)  (c)  0  100  200  300  400  0  2  4  6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='6  5400  5500  5600  5700  (b)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (a) Top plot: extract of a time series of the macro-  scopic force F(t) for a system of n = 4 frictional interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The color of the force drops ∆F follows the color code in-  troduced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1 and indicates the index a of the slipping  interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Bottom plot: corresponding relative displacement  (total slip) Ri of each interface i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Each slip event is character-  ized by its interfacial slip ∆Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We denote Ta, the time be-  tween subsequent slip events on the same interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Note that  we show i = 5 only for completeness, by definition, R5 = 0  if n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (b) Probability distribution function P(∆F) for  slip at interface a = 1, and increasing number of layers n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (c) Comparison between a direct measurement of ∆Ra, and  the computed ∆Ra(∆F), obtained through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (5), for each  detected slip event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Finally, we combine Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (3) and (5) to obtain the  sought relation between ∆µa and ∆F:  ∆µa = H  h  2  (n + a)  ∆F  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (6)  We have thereby disentangled the friction properties of  the interface from the kinematics of the lever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Using  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (6), we can now obtain a measure of the stick-slip  amplitude of the interfaces ∆µa, extracted directly from  the global force ∆F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The measurement of ∆Ri is used  only to identify the slipping interface a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The central experimental result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Next, we assess the  effect of having multiple sheared interfaces on their fric-  tional properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 4 shows the probability distri-  butions P(∆µa) associated with the different sliding in-  terfaces a (different panels) and the increasing number  of total active interfaces n (different colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Each inter-  face is compared to its response when sliding individually  (n = 1 in black, see SI "Individual sliding" for experimen-  tal protocol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' For all the interfaces, the stack of slabs  exhibits significantly enriched statistics when compared  7  (b)  (c)  (d)  (a)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  0  10  30  20  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  0  10  30  20  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  0  10  30  20  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  0  10  30  20  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Probability distribution functions of the stick slip  amplitude P(∆µa) as a function of n for (a) a = 1, (b) a = 2,  (c) a = 3 and (d) a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  to a single sliding layer (n = 1, black circle in the plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  For stacks with increasing n, the location of the peak  of P(∆µa) shifts to lower values of ∆µa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Moreover, the  respective distributions become broader and secondary  peaks emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' These experimental results contrast the  numerical predictions reported above (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2c), where  ∆µa was independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Interpretation  We seek to interpret the above experimental findings  evidencing a variation of the frictional properties with n  (as more layers are added to the stack, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 4), whereas  they are independent of n in the numerics (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  First, we will attribute the finite width of the peaks in the  P(∆µa) distributions to disorder (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=', statistical fluctu-  ations of the contacting interfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Then, we will argue  that the shift of the main peaks and emergence of the  secondary peaks in P(∆µa) for n > 1 are related to the  presence of creep in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Finally, we speculate  on the increase of an effective temperature with n and  rationalize the appearance of slow slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Statistical fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Even when sliding individu-  ally (n = 1), the frictional properties of the interfaces are  distributed: P(∆µa) has a finite width, see black curves  with circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' These underlying statistical fluctu-  ations, also present in the numerical model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 2d), are  considered to be related to the disorder of the contacting  interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The rough interface induces a broad distri-  bution of barriers, such that there are collective events  whose sizes are non-trivially distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' These collective  events nucleate the slip events [22, 26], such that the load  at which events slip is distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Let us now verify experimentally that, in the case of  individually sliding layers (n = 1), the measured fluctua-  tions of ∆µa correspond to distinct slipping loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In the  individual configuration, the spring drives the layer at a  constant rate of ˙fa = Kh˙γ (the rotation rate of the lever  is adapted to account for the difference in height, see SI  "Individual sliding").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The shear applied to the interface  then grows at a rate ˙µa = ˙fa/N = Kh˙γ/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' As such, we  expect the stick-slip amplitude to be proportional to the  time between slips Ta, following ∆µa = TaKh˙γ/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' This  expectation is consistent with our data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 5a, thus  confirming that statistical fluctuations broaden P(∆µa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  However, these fluctuations do not account for the shifts  of the peaks and the appearance of secondary peaks in  the P(∆µa) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In our stack of slabs (n > 1),  we anticipate that creep plays that role: with increasing  n, (I) interfaces experience a complex loading path such  that (II) the creep of the frictional interface becomes sig-  nificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (I) Complex loading path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  First, without any events  on other interfaces, the loading rate at a given inter-  face increases with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In particular, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (1) and  ˙fi = Kih˙γ while no interfaces are sliding, the interface a  undergoes a loading rate of ˙µa = K ˙γh(n + a)(n − a +  1)/(2N), which is an increasing function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  With  this increased loading rate with n, we thus expect the  time between slips to typically decrease with n following  T −1  a  ∼ (n + a)(n − a + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Second, for a given interface,  a sliding event can occur on a layer below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In that  case, all the layers above the sliding one will undergo the  same position shift, leading to a relaxation of their corre-  sponding springs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Then, even without actual slip on this  given interface, slips occurring below will induce drops  in the shear force of this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 5b, we plot the  probability distribution function of Ta for a = 1 and in-  creasing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Indeed, we find that the peaked distribution  for n = 1 shifts to lower values of Ta with increasing n as  the overall loading rate increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Moreover, secondary  peaks in P(Ta) start to appear, which we interpret to be  due to slip events on the other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' These observations  are robust for the other interfaces (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The changing  in loading rate with n together with the complex loading  path, allows our experimental system to probe a broad  distribution of Ta on all its interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (II) Creep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  It is a known experimental fact that  the macroscopic stress required for the onset of slid-  ing, characterized by µs (the ‘static friction coefficient’  in Amontons-Coulomb’s terminology [11, 30]), depends  on the duration T that the interface was static: µs =  B ln T [17, 29–31], where the aging rate of the interface  B is a constitutive parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Let us now consider ∆µa  as a proxy for µs, assuming that a slip event unloads the  interface to a well-defined and constant quantity (µd the  ‘dynamic friction coefficient’ in Amontons-Coulomb’s ter-  minology [11, 30]), as is supported by [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We expect to  find, for each sliding interface a and over a wide range of  8  Ta, that the stick-slip amplitude follows the creep trend:  ∆µa = B ln Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 5c, we assess this expectation  experimentally by plotting ∆µa versus Ta in a semi-  logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' To capture the general trend, we bin  logarithmically Ta, corresponding to the black markers  with corresponding error bars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' These averaged  values of ∆µa indeed follow a linear trend in the semi-log  plot, and we extract the slope B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='053±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='005, which is  of the same order of magnitude as measured in classical  stop-and-go experiments that are in the 10−2 order [30],  and of a direct surface observation on PMMA at room  temperature that reports B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='009 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='001 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Creep  then translates the large peak shifts and the emergence  of new ones in the Ta distributions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 5b) into similar  changes in the ∆µa distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The large fluctuations  of ∆µa in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 5c are then the combination of two ef-  fects: creep drives the long time trend, while disorder  dominates in the narrow time range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 5d, we schematically represent the coupled role  of (I), (II), and disorder, following the evolution of the  interfacial stress with time, characterized by µa, starting  from the last slip event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The layer slips when it reaches  µa = µs(Ta), whereby µs is distributed in some way for  fixed Ta because of disorder (illustrated as a red-shaded  area, whereby, for simplicity, we lump all fluctuations  in the threshold to sliding) and increases logarithmically  with time because of creep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' For n = 1 (black line), µa  increases linearly at the same rate for all the events, thus  exploring only a narrow region of Ta (the shaded red  region due to disorder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In the case of multiple active  interfaces (n > 1, green lines), µa increases faster given  that ˙µa is an increasing function of n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' see point (I) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  In some cases, µa directly reaches µs, resulting in a lower  value of Ta and ∆µa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' However, if sliding events occur in  one of the underneath layers, µa will drop before linearly  increasing again, delaying slip and thus increasing Ta,  and consequently ∆µa, because of creep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  During stick intervals, micro-  scopic events occur on the interfaces, propagating elastic  waves across the system [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' As we increase the number  of interfaces in the system, we can expect that the over-  all mechanical noise created by the microscopic events  also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' If we speculatively interpret this mechan-  ical noise as an effective temperature, we would expect a  change of aging rate B with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Let us define the aging  rate for a single event as Ba ≡ ∆µa/ ln Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Although the  mean of P(Ba) does not change with n (see SI "Aging  rate distributions"), we do find that the width of the dis-  tribution P(Ba) is an increasing function of n mainly for  the lowermost interfaces (i ≤ 2), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  For the same interfaces (i ≤ 2), we also observe dis-  tinctly different slip dynamics when n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In particular,  as n increases, we find that interfaces i = 1 and i = 2 are  increasingly more subject to slow slip, defined as sliding  significantly slower than the slip events (see SI "Smooth  sliding" for precise definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' These events are not ac-  101  102  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='04  (c)  (a)  101  102  103  0  100  200  300  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  time  (b)  (d)  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (a) For an interface sliding individually (n = 1),  stick-slip amplitude ∆µa as a function of the waiting time  since the last slip event Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The black line corresponds to  the prediction that the stress state of the interface increases  at a constant rate in between slip events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' For multiple slid-  ing interfaces (n ≥ 1): (b) Probability distribution of the  waiting time Ta between two consecutive slip events at in-  terface a = 1, for increasing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (c) For each detected slip  event, correlation between the stress release, characterized by  ∆µa, and its corresponding waiting time since the last slip  event on that interface Ta (semi-logarithmic scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The black  markers correspond to the mean values for a logarithmic bin-  ning of Ta (error bars indicate the standard deviation for that  bin), and the dotted line a linear fit of ∆µa = B ln Ta, with  B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='053±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (d) Schematic of the proposed mechanism  leading to multimodal and wider distributions of Ta (and thus  ∆µa) as n increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  companied by a macroscopic stress drop but rather just  lead to a lower ˙F > 0, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 6b, we measure  the proportion of slow slip compared to the total slid-  ing distance in an experiment, Rslow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' It is computed by  comparing the total sliding distance to the total slip ac-  cumulated during individual slip events (∆Ra), such that  Rslow ≡ 1 −  �  ∆Ra/∆R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Once a slow slip event starts,  it appears to be stopped only by slip events occurring ei-  ther on the same interface or on any other interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' An  increase in the effective temperature of the interface with  n could also act as a potential destabilization factor of  the contacts at the interfaces, increasing the occurrence  of slow slips with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  9  (b)  (a)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='8  1  1  2  3  4  5  1  2  3  4  1  2  3  4  5  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  As a function of n, for each interface a (different  colors and marker, see legend): (a) Standard deviation of the  distribution of the aging rate Ba ≡ ∆µa/ ln Ta of individual  events, normalized by that quantity for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (b) Fraction  of slip that corresponds to slow slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  DISCUSSION AND CONCLUSION  Summary  We have explored the stick-slip response of a system  with multiple interfaces by proposing a model system  comprising n vertically stacked slabs, each connected to  a lever whose rotation is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The interfaces were  driven in quasistatic (homogeneous) shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We proposed  a dimensionless quantity Φ as the ratio between the driv-  ing stiffness and the stiffness of the elasticity of the slabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  We have argued and demonstrated numerically that the  system displays synchronization if Φ is sufficiently large  (Φ ≳ 10−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  In that case, the system acts close to a  single frictional interface with effective properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' If Φ  is small (Φ ∼ 10−6), interfaces slip one by one, as also  confirmed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  We expect non-trivial collective effects with increasing  n only in the low Φ limit, which we addressed through  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In the numerics, the stick-slip amplitude of  the interfaces ∆µa display a distribution with finite width  because of statistical fluctuations of the interfaces, but no  measurable changes with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' By contrast, we measured  experimentally that the probability distribution of stick-  slip amplitude ∆µa shows a general broadening with n,  with peaks shifting to lower values and secondary peaks  appearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The interfaces are coupled via the lever, ex-  posing them to a complex loading path, and leading to a  broad distribution of the waiting times Ta between two  slip events on an interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We find that Ta is now span-  ning two decades, such that creep of the interfaces plays  a crucial role in the broadening of ∆µa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The complex  distributions of ∆µa can then be interpreted as the com-  bined effect of interface disorder and creep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' For narrow  waiting times Ta, multiple slips explore the statistical  fluctuations of the contacting interfaces, giving a distri-  bution with finite width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In addition, this distribution  follows a creep-induced general trend over widely dis-  tributed Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Furthermore, we observe that aging rate variations and  the fraction of slow slip on the bottom layers is an in-  creasing function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We suggest that these additional  consequences of adding more layers to the stack might  be an evidence of an increase in an effective interface  temperature due to the mechanical noise of microscopic  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  In conclusion, the relative rigidity of the drive against  the layers dictates whether a stack of interfaces responds  synchronously or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' When layers slide one by one, in-  creasing their number lead to complex responses, making  the prediction of the next slip more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Limitations and outlook  It is pertinent to discuss some limitations of our model  system and provide suggestions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Stiffness ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  We have defined the rigid and com-  pliant driving regimes, as characterized by the relative  order-of-magnitude estimation of the respective stiffness  ratio Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Identifying an equivalent of Φ in systems with  more intricate geometries, such as fault networks, might  contribute to clarifying their dynamics and help slip pre-  dictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Hence, a more systematic exploration of the  response of stacks with varying Φ would be of great in-  terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  However, both the current numerics and experiments  are not suitable to explore large variations of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Our  numerics are currently limited to relatively high Φ due  to a combination of the assumption of finite rotations and  finite machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Future work should allow local  deformation along the frictional interface to be reset to  a zero average while keeping the identical stress state,  to be able to continue sliding indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In contrast,  the presented experimental setup is limited to exploring  the low Φ regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' On the one hand, if the stiffness of  the springs were to be increased, the motor/lever system  would no longer be considered rigid, thereby invalidating  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (5), the relation between global measurement (∆F)  and local stick-slip amplitude (∆µa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' On the other hand,  if the shear stiffness of the slabs were to be reduced, the  frictional properties would change as soft solids tend to  be adhesive [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Creep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  This proposed model system allowed us to  measure the aging rate B of the interfaces thanks to com-  plex stick-slip sequences without the need for stop-and-  go experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' However, while our measured value of  B is compatible with previous experiments on PMMA  [30, 40], it differs by a factor of five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' First, within our ex-  periments, B is measured over slip events spanning the  n = 5 interfaces, so it includes the natural variability  of the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Averaged values of Ba for the differ-  ent interfaces give an idea of the variations in the aging  rate between the plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' They are different at most by  a factor two, thus only partly explaining the mismatch  with those previous experiments on PMMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We also note  10  that the PMMA plates used here have a much higher  surface roughness than in [30, 40], with unclear conse-  quences on the aging rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Future work should explore  the relationship between B and surface roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Sec-  ond, recent experimental observations find a relationship  between the aging rate B and the applied shear load to  the interface [42] with effect even on partial slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Our  setup naturally imposes a broad range of shear loads on  the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  However, to measure the empirical law  by [42], our setup would need to be augmented to also  provide stress measurements per interface, by measuring  individually the internal forces fi of the driving springs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The numerical model does not show the phenomenol-  ogy observed experimentally associated with creep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In  the context of inhomogeneities introduced by partial slip  and their role in creep, our model assumes a pressure-  independent yield stress in shear, neglecting any influ-  ence of the pressure inhomogeneities that may arise from  the normal load or partial slip events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' A possible solu-  tion is to define yielding in equivalent stress space, rather  than equivalent deviatoric stress space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Slow slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The origin of the experimentally observed  slow slip is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  A tempting hypothesis is that  smooth sliding is the result of activated yielding events  due to increasing mechanical noise on other interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  However, it is not clear why this interpretation would  lead to slow slip occurring predominantly on the low-  ermost interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Indeed, this observation is robust to  changing the order of the slabs (though the fraction of  slow slip changes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' see SI "Smooth sliding".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' An alter-  native hypothesis is that, by increasing n, the loading  rate becomes sufficiently high to drive the interfaces away  from the stick-slip regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' This second interpretation is  consistent with slow slip being predominantly observed  on the lowermost interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Our numerical model does not show any slow slip or  creep, while we could expect that the emitted mechanical  noise could lead to activation or creep on other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Future work should clarify if these observations are due to  our small homogeneous background damping term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The  latter is currently chosen to avoid nonphysical periodic  wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' A dedicated experimental campaign  could potentially clarify at which length scale the emitted  shear waves thermalize in PMMA and which fraction of  waves that reach the material boundaries are reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  We have not systematically varied the  surface roughness of the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Instead, experimen-  tally, we have changed the order of the interfaces and  observed no qualitative changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Numerically, we arbi-  trarily draw yield strains from a Weibull distribution (fol-  lowing Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' [22, 26]) as the relationship between surface  roughness and the distribution of yield strains is a major  open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Fore- and after-shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  In an idealized spring-block  model (equipped with microscale disorder), fore- and  after-shocks appear if creep is added to the drive [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  A creeping drive is often associated with the high tem-  peratures in Earth’s core [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Our experimental sys-  tem displays slow sliding of ‘deep’ layers already at room  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' A key question is if fore- and after-shocks  appear in the top layers of our system as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Answer-  ing this question experimentally would required exposing  microscopic events, which would likely involve studying  acoustic emissions (for which PMMA may not be the op-  timal choice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors are grateful to Mathias Lebihain and  Federica Paglialunga for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  We also  thank Lebo Molefe for providing the microscope images  of the surface of the slabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' acknowledges finan-  cial support from the Japanese Society for the Promo-  tion of Science as a JSPS International Research Fellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' acknowledges support from the Swiss National Sci-  ence Foundation (SNSF) by the SNSF Ambizione Grant  PZ00P2_185843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  ∗ pedro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='reis@epfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='ch  † Correspondence email address:tom@geus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
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+page_content=' de Geus, FrictionQPotFEM (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  12  SUPPLEMENTARY MATERIAL  Experimental set-up  Details on the critical components of the experimental set-up are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Lever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  A lever imposes a constant shear rate on the stack of slabs via a set of springs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The lever is made of two  6 mm thick and 100 mm wide parallel acrylic slabs, ensuring no bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' It is actuated on its top by a rigid arm  pulled horizontally at a constant speed by a linear actuator (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The rigid arm is made of steel to ensure  that the lever’s position is rigidly imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' A uniaxial force sensor between the actuator and the arm measures the  total pulling force F (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The sensor is rigidly connected on both sides by being directly threaded into the  acrylic adaptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Springs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The springs between the lever and the slabs are PMMA beams with a rectangular 5× 5 mm cross-section  and pre-curved with a radius of curvature of 100 mm over a cord of 150 mm in the middle and extended on both sides  by two 25 mm straight portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The rigid arm and the springs are attached to the lever by the links shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The lever’s  rotation around its rotation axis is ensured by a traverse steel shaft embedded into ball bearings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The links are  inserted between the two slabs of the lever and are made of a central steel pin press-fitted at the end of the spring  (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' This pin is embedded into small ball bearings on each side (green), allowing a free relative rotation between  the springs and the lever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The bearings are then press-fitted into acrylic components (pink) with two threaded holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  These components are directly screwed on both sides on both slabs of the lever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The rotation axis of the pins is  exactly aligned along a line passing by the center of rotation of the lever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1) and S1(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2) show, respectively,  a side and top-side view of the link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3) shows the link when not attached to the lever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The other sides  of the springs are linked to the frictional slabs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The spring-slab links have a similar design as the lever-spring links, but here the acrylic components are press-fit  into the slabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The height of the acrylic components is slightly smaller (8 mm vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 10 mm) to avoid any contact with  the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2) show topside views of the links respectively attached and out of a frictional  slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' These links ensure a rigid connection between all the components, so only the springs deform during actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The ball bearings also ensure that the force transmission remains horizontal as the lever rotates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Slabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(d) shows a close-up of the surface of the frictional slabs (Snow WH10 DC by Röhm), displaying a  complex roughness profile with the largest asperities of the order of ∼ 25 µm, with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1) having a scale bar of  100 µm, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2) a scale bar of 25 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The position of the slabs is measured using a 7 mm wide circular red  marker glued on the side of each slab, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Slip measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  A raw image taken by the camera is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1), the pictures have a resolution of  70 µm per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Using a color threshold, the red markers are extracted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S1(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2) and their geometric center  is located (red crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Owing to the large size of the markers (154 mm2, or around 30000 pixels), the location of  their geometric centers is determined with a sub-pixel resolution of 5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The relative position between the slabs is  computed by retrieving the horizontal position of a layer to the one below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  13  Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Technical details on the main components of the experimental apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Top: overview of the set-up with the  location of the different parts: (a) force measurement, (b) lever-spring link, (c) spring-slab link, (d) frictional slabs surface and  (e) position measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Relevant details are shown in the bottom panels (numbered (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' ), see text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  50 mm  a  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (b)  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1)  50 mm  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2)  20 mm  10 mm  (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3)  10 mm  (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1)(c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2)  10 mm  10 mm  20 mm  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1)  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2)  (d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2)  25 μm  5 mm  5 mm  100 μm14  Kinematics  We verify that our setup imposes the desired kinematics on the slabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We record the horizontal displacement of the  lever at the top, next to its actuated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We indeed find a smooth linear increase of this horizontal displacement,  insensitive to the concurrent jump in force measured, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The lever thus rigidly imposes a homogeneous shear  to the slabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We also verify that the position of the bottom slab x0, onto which the first layer i = 1 is sliding, remains  still during the experiment (even during force drops) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The noise of this measurement is an estimation of  the position accuracy of our experimental set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (a)  (b)  1400  1450  1500  1550  1600  18  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  19  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  20  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  21  1  0  1  2  3  4  5  1600  1650  1700  1750  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='005  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='01  1  0  1  2  3  4  5  Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Verification of the kinematics of the experimental set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (a) Blue left axis: horizontal position of a point next to  the top actuated point of the lever as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Orange right axis: concurrent measurement of F showing stick phases  separated by rapid slip events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (b) Blue left axis: horizontal position of the bottom slab i = 0 as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Orange  right axis: concurrent measurement of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Individual sliding  The interfaces are tested individually, but still in their stacked configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' To test the interface a, we only keep  spring a, while the slabs below it (i < a) are clamped such that they cannot move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' That way, only the bottom  interface of slab a is sliding (the slab is not pulled from the stack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In this case, the slip ∆Ra and the macroscopic  force drop are related as  ∆F = K∆Ra  ah  H ,  (7)  which we verify in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The drop of frictional resistance then reads  ∆µa = K∆Ra  N  = ∆F  N  H  ah  (8)  In addition, to ensure testing the interface at the same shear rate ˙γ = 10−4s−1 despite the different heights onto which  they are attached to the lever, the shear rate is adapted for each layer such that ˙γa = ˙γ/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' With ∆Ra = Taha˙γa =  Tah˙γ, we have the following expression for the friction jump as a function of the time interval between two slips:  ∆µa = KTah˙γ/N  (9)  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S3b and S3c show P(∆F) and the extracted P(∆µa), respectively, for a = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The interfaces show  significant differences in the mean values of ∆µa, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='15 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3 (with the differences in mean values  bigger than the scatter of the different P(∆µa)) even though the material, preparation protocol, and environmental  conditions are the same for each of these interfaces/experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  15  (a)  (c)  (b)  0  100  200  300  400  0  100  200  300  400  Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (a) Verification of ∆Ra(∆F) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (7)) for individual interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (b) The probability distributions of the drop  of force ∆F for the different interfaces a are shown using a different color and marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (c) Probability distributions of the  corresponding drop in frictional resistance ∆µa as estimated from ∆F using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Weight and position of the lever  We comment on the limitations due to the intrinsic weight of the lever and actuating arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The required rigidity of  the lever and arm connecting it to the linear actuator results in a non-negligible weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' This weight manifests itself  in the measure of F because of the vertical motion induced by the rotation of the lever and adds a slow decreasing  linear component in the measured signal, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' This additional component makes a precise measure  of the absolute values of the force before and after a drop impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' However, this slow component does not affect  the measure of the force difference ∆F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Indeed, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S4b, we show that the probability distribution of ∆µa (we  arbitrarily show ∆µ1 for n = 5) is not influenced by the angular position of the lever, and hence by the set-up weight  component in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Effect of the weight of the actuating lever and arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (a) Full time series of F for n = 1 (purple), n = 3 (green),  and n = 5 (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Particularly noticeable for n = 1, the force signal shows a slow linearly decreasing component due to the  intrinsic weight of the lever arm and its vertical motion upon rotation of the lever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (b) During an experiment, the lever is  rotated between γ = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3] rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We represent the probability distribution of ∆µ1 for n = 5, for the entire run (squares),  and biased for γ < 0 (left triangles) and for γ > 0 (right triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  ä n=5  n=5<0  n=5 >016  Robustness of experimental distributions  For each n, 10 runs are performed one after the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  One run lasts 6000 s, while the lever rotates between  γ = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3] rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S5 shows the distributions of ∆µa (we arbitrarily show ∆µ1 for n = 5) we obtained for  each individual run (black lines), which are compared to the distribution obtained by combining the data from all the  runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' All runs display a similar overall distribution, which attests the reproducibility of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  single run  merged runs  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3  0  5  10  15  20  Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Robustness of the distribution of ∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Comparison between the distributions of ∆µ1 of each individual run (black  lines) and the distribution obtained by compiling all the events of the 10 runs (orange squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The system has n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Results from position measurements  We confirm that our central result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 4 remains unchanged if we ∆µa from Ra (instead of ∆F) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (a)  (b)  (c)  (d)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  0  5  10  15  20  25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  0  5  10  15  20  25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  0  5  10  15  20  25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5  0  5  10  15  20  25  Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 4 but obtained using ∆Ra instead of ∆F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Waiting time distribution for a > 1  In addition to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 5c we show the waiting time between consecutive sliding events for interfaces a = 2, 3, 4, 5 in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' For all the layers, we recover the same qualitative behavior with increasing n: the distribution becomes  multimodal with the main peak shifting to lower values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  17  101  102  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02  101  102  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='05  101  102  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  101  102  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  (c)  (d)  (a)  (b)  Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 5c for (a) a = 2, (b) a = 3, (c) a = 4, (d) a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Aging rate distributions  For each slip event, we quantify its effective aging rate Ba ≡ ∆µa/ ln(Ta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The corresponding probability distribu-  tions P(Ba) are plotted for each layer a for different n in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='08  0  100  200  300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='08  0  100  200  300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='08  0  100  200  300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='08  0  100  200  300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='08  0  100  200  300  (c)  (d)  (e)  (a)  (b)  Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Probability distribution of the effective aging rate Ba ≡ ∆µa/ ln(Ta) for each event for a given interface a and  increasing number of layers n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (a) a = 1, (b) a = 2, (c) a = 3, (d) a = 4, (e) a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The distribution for the isolated interface  (n = 1) is shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Slow slip  The interfaces seldom show slow sliding with velocities much lower than for the typical slip events (see for example  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S9a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' To discriminate slip events from slow ones, we use a velocity criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The velocity of each slab is calculated  as (Ri(t + ∆t) − Ri(t))/∆t, with ∆t the time between frames (∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2 s), and smoothed by a running average over  five frames (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Sliding events are then defined by all clusters of at least three frames having a velocity greater  than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02 mm/s (identified by black dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S9a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The smoothing notably allows capturing the  frames just before and after the events, allowing for a precise measure of the sliding distances ∆Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S9c we  verify that smooth sliding is consistently observed on the lowermost interfaces even if the slabs are shuffled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  18  (b)  (a)  (c)  0  20  40  60  80  100  1  2  3  4  5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='8  5000  5100  5200  5300  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='2  5000  5100  5200  5300  Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  (a) Time series of the relative position Ri(t) of the slabs for n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The frames detected as belonging to a  sliding event are marked by a black dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (b) Corresponding time series of the velocity of the slabs, after applying a running  average over five frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The horizontal dashed line marks the limit above which we consider that sliding is fast enough to  be categorized as a slip event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' (c) Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 6b with an additional measurement for n = 5 where the slabs are shuffled  (1, 2, 3, 4, 5 to 4, 3, 2, 1, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The resulting data points are shown at n ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='3 for visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' After shuffling, it is still the lowest  interfaces that show most of the slow sliding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Numerical model  All implementation details are identical to [26], and available in its supplementary material;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' furthermore the  entire implementation is open-source, see [46] and its dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Compared to [26] the only differences are:  – The discretisation, that now comprises n frictional interfaces, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  – The boundary conditions are such that the position of the bottom nodes is fixed, while the top nodes are  completely free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The displacement of the right nodes is set equal to that of the left nodes corresponding to  periodic boundary conditions in the horizontal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Finally, all elements that constitute to an elastic  layer i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' , n are connected to a parabolic potential of curvature K that has its minimum at the  lever’s position fixed to ˙γiht as in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' This potential is implemented as a force density, and  added to each node using the node’s volume as weight factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We use K = 10−3 for rigid driving, and  K = 10−6 for compliant driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  – We use a bulk modulus that if 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='5 times larger than the shear modulus, in accordance with PMMA (see [26]  for the definitions of the bulk and shear modulus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Macroscopic stress Σ, defined as the equivalent deviatoric part of the macroscopic stress tensor Σij ≡  �  σij(⃗r)d⃗r  with σij(⃗r) the stress at a material point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' at an integration point in our spatial discretisation), and ⃗r the  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Thereby Σ2 ≡ 2ΣijΣij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' All reported stresses have been normalised by the typical yield stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  The rotation of the lever γ is defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The numerical results are normalised by γ0, which is defined as  the rotation corresponding to a yield event in a system in which all blocks of the first interface (i = 1) have a  yield strain equal to the typical yield strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  19  Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Finite element mesh of the numerical model for n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Slip sequences – numerics  In the main text, we discussed how rigid driving causes sequences multi-slip events, while compliant driving results  in sequences of single-slip events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' We support this statement using the increment in lever rotation ∆γ between single-  and multi-slip events in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' For stiff driving the multi-slip events are mostly periodic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' distributed around  a well-defined mean that is independent of n, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S11b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Single-slip events are slightly less frequent while the  increment between them is broadly distributed around a mean that is n dependent, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S11a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' For compliant  driving, multi-slip events are almost absent, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S11d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Also in this case, the increment between single-slip events  is broadly distributed and n dependent, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S11c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  We comment on the fraction of events that are part of a sequence of single- or multi-slip events using Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' For  rigid driving, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S12a, sequences of single- and multi-slip events alternate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' As n increases such sequences alternate  more often reflected in decreasing fraction of events that are part of any sequence (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Of these sequences,  there is an equal partitioning in the number of single- and multi-slip sequences (the two highlighted regions have an  approximately equal height for every n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' For compliant driving situation is simple: there are almost exclusively single  slip events, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S12b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  0  300  ∆γ/γ0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02  P(∆γ/γ0)  (a) stiff driving, single-slip  0  300  ∆γ/γ0  (b) stiff driving, multi-slip  0  300  ∆γ/γ0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='02  P(∆γ/γ0)  (c) compliant driving, single-slip  0  300  ∆γ/γ0  (d) compliant driving, multi-slip  n =  1  2  3  4  Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Loading interval expressed as increment in lever angle ∆γ, between (a, c) single- and (b, d) multi-slip events for  (a, b) rigid and (c, d) compliant driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The right panels (b, d) show single-slip events for a single slab (n = 1) as reference  using a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Note that the distributions are normalised such that the integral of single- plus multi-slip events (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' the  distribution of the left plus that of the right panel) is equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  20  1  2  3  4  n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='00  φ  (a) stiff driving  single-slip  multi-slip  1  2  3  4  n  (b) compliant driving  Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Fraction of slip events that are part of a sequence of single- or multi-slip events for (a) stiff and (b) compliant  driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The plot is constructed such that the blue line (square markers) corresponds to the total fraction of events that is  part of a sequence of consecutive single- or multi-slip events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The red line (circular markers) is the fraction of events that are  part of sequences of single-slip events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' The area between the red and blue curves, highlighted in blue, thus corresponds to the  fraction of events that are part of a sequence of multi-slip events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Slip vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' stress drop  Naturally, the slips and the stress drops are proportional, as was validated experimentally in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S13,  we verify that this is the case also for the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' From Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S13b and S13d, we observe that interfaces  are characterised by a slightly different compliance, whereby the i = 1 differs from the other interfaces because the  position of slab i = 0 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Note that the trend between Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' S13a and S13c suggests that the driving for our  numerical results is more rigid than in the experiments, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' This is indeed a known limitation of our numerical  model as we discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
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+page_content='4  ∆µ  0  50  100  Si/nx  (b) stiff driving, n = 4  i =  1  2  3  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
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+page_content='4  ∆µ  0  50  100  Si/nx  (c) compliant driving  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
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+page_content='4  ∆µ  0  50  100  Si/nx  (d) compliant driving, n = 4  Figure S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content='  Slip Si per block as a function of the stress drop on the slipping interface denoted as ∆µ for different (left) n, and  (right) i for n = 4, for (top) rigid and (bottom) compliant driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
+page_content=' 3c where Si ∼ ∆Ri, note that we do not dispose  of means to seek a quantitative comparison of ∆µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFST4oBgHgl3EQfMzga/content/2301.13745v1.pdf'}
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+Enabling Ice Core Science on Mars and Ocean Worlds
+Alexander G. Chipps
+School of Mechanical Engineering
+Georgia Institute of Technology
+Atlanta, GA 30332
+achipps3@gatech.edu
+Cassius B. Tunis
+School of Aerospace Engineering
+Georgia Institute of Technology
+Atlanta, GA 30332
+ctunis3@gatech.edu
+Nathan Chellman
+Desert Research Institute
+2215 Raggio Parkway
+Reno, NV 89512
+nathan.chellman@dri.edu
+Joseph R. McConnell
+Desert Research Institute
+2215 Raggio Parkway
+Reno, NV 89512
+joe.mcconnell@dri.edu
+Bruce Hammer
+University of Minnesota
+Center for Magnetic Resonance Research
+2021 6th Street SE
+Minneapolis, MN 55455
+hammer@umn.edu
+Christopher E. Carr
+School of Aerospace Engineering &
+School of Earth and Atm. Sciences
+620 Cherry St NW, Room G10
+Atlanta, GA 30312
+cecarr@gatech.edu
+Abstract—Ice deposits on Earth provide an extended record
+of volcanism, planetary climate, and life.
+On Mars, such a
+record may extend as far back as tens to hundreds of millions
+of years (My), compared to only a few My on Earth.
+Here,
+we propose and demonstrate a compact instrument, the Melter-
+Sublimator for Ice Science (MSIS), and describe its potential
+use cases. Similar to current use in the analysis of ice cores,
+linking MSIS to downstream elemental, chemical, and biological
+analyses could address whether Mars is, or was in the recent
+past, volcanically active, enable the creation of a detailed climate
+history of the late Amazonian, and seek evidence of subsurface
+life preserved in ice sheets.
+The sublimation feature can not
+only serve as a preconcentrator for in-situ analyses, but also
+enable the collection of rare material such as cosmogenic nu-
+clides, which could be returned to Earth and used to confirm
+and expand the record of nearby supernovas and long-term
+trends in space weather.
+Missions to Ocean Worlds such as
+Europa or Enceladus will involve ice processing, and there MSIS
+would deliver liquid samples for downstream wet chemistry
+analyses. Our combined melter-sublimator system can thus help
+to address diverse questions in heliophysics, habitability, and
+astrobiology.
+TABLE OF CONTENTS
+1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+1
+2. MELTER-SUBLIMATOR FOR ICE SCIENCE . . . . . . .
+2
+3. DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+4. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+BIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+1. INTRODUCTION
+Ice deposits on Earth provide an extended record of volcan-
+ism, cosmic events, and subsurface life. On Mars, records
+entrained in polar and mid-latitude ice may extend as far back
+as tens to hundreds of My [1], [2], compared to only a few
+My (and less than one million years contiguous) on Earth
+[3]. Martian ice investigations would provide a significant
+extension of our knowledge from Earth ice analysis, and
+could serve as a precursor to the exploration of icy bodies
+such as Enceladus and Europa. Such ice, found in deposits
+more than 100 meters thick in mid-latitudes [4], may also
+represent an important resource for human exploration.
+978-1-6654-9032-0/23/$31.00 ©2023 IEEE
+The layers present within ice cores record historical precip-
+itation (e.g., snow), aerosols, and trap atmospheric gases as
+the overburden ice accumulates and the pore space is closed
+off (around 50-80 m deep on Earth).
+One common strategy for reducing ice core contamination is
+to physically remove the outer part of a core, as was done
+for a recent study assessing the contamination of Arctic ice
+cores as a measure of the potential of forward contamination
+during future life detection missions [5]. This process is labor
+intensive and challenging to automate. Melters, which heat an
+ice core or section of an ice core (ice stick) above its melting
+temperature at the base, offer an alternative that facilitates
+automatic processing, and can reduce contamination by per-
+mitting automatic disposal of the meltwater from the outer
+part of the core as described in the next section.
+Melters permit sampling of these layers at high resolution, in-
+cluding by coupling meltwater to real-time downstream anal-
+yses such as inductively-coupled plasma mass spectrometry
+(ICP-MS) for elemental abundance [6], laser spectrometers
+for gas (e.g., methane) detection [7], black carbon or other
+compounds [8], or by collection of fractions for additional
+analyses [9].
+Thus, we propose and demonstrate initial work toward a com-
+pact Melter-Sublimator for Ice Science (MSIS) instrument
+that would enable convenient ice core processing on Mars.
+MSIS is built upon an extensively validated melter design [6],
+[8], [7], [9], [10], and it has a compact and thermally-efficient
+design to meet the requirements of planetary exploration
+(size, mass, power, etc.).
+Through a combination of in-
+situ and sample return processes, pairing an MSIS-processed
+sample with downstream chemical analysis instruments may
+assist in several potential use cases:
+1) Addressing when and where Mars was last volcanically
+active.
+2) Confirming and expanding the record of space weather
+events over a timescale not possible with Earth ice.
+3) Characterizing the inventory of organic molecules on
+Mars in support of the search for life beyond Earth.
+In this paper, we describe the MSIS instrument concept,
+design, and current performance. We also discuss the lim-
+itations and future development goals for MSIS, as well as
+the applications of MSIS to the potential use cases proposed
+above.
+1
+arXiv:2301.01869v1  [physics.ins-det]  5 Jan 2023
+
+2. MELTER-SUBLIMATOR FOR ICE SCIENCE
+Concept of Operations
+The MSIS instrument can be best described as having two
+main components: Melter and Sublimator (Figure 1).
+On Earth, drilling, extraction, and handling processes result
+in contamination of an ice core’s outer surface [11]. Access-
+ing pure, uncontaminated regions of the core and rejecting
+contaminated areas is a requirement for some downstream
+(elemental, chemical, biological) analyses [12]. Thus, like
+the melter design from McConnell et al. [6], [8], [7], [9], [10],
+the MSIS Melter component divides an ice stick into three
+distinct cross-sectional areas as it melts, providing individual
+access to the inner, middle, and outer core sections.
+MSIS is currently designed to accommodate 3.3 cm x 3.3 cm
+ice sticks, following [6], but in principle could be adapted to
+any core cross section. Using the three-ring design adopted
+from [6], water from the outer 70% of the core is discarded
+under ideal operating conditions. The inner, most pristine
+10% is collected separately from the middle 20% to facilitate
+downstream analysis applications requiring different levels of
+purity and volume flow rate. The geometry of these rings
+can be redesigned to suit any specific set of downstream flow
+requirements.
+The Sublimator component, sitting atop the Melter, enables
+preconcentration of material for in-situ analyses or Earth
+return of residues, avoiding the complexities involved in
+storing and returning large samples to Earth. MSIS will be
+able to preconcentrate organic samples through near-full sub-
+limation then controllably melt a reduced sample, allowing
+for sensitive in-situ characterization of non-volatile materials
+such as organics.
+Melter Design & Implementation
+The Melt Head is a stainless steel, 3D-printed structure that
+sits compressed against a heated copper plate with a layer
+of thermal paste at the interface. The Melt Head has two
+3.5-mm-tall partitioning walls that separate the ice stick as
+it melts into the inner, middle, and outer Melt Head cavities
+(Figure 2). Copper was not used for these partitioning walls
+because it would be chemically reactive. Also, a stainless
+steel Melt Head does not permit trace metal-clean operation
+like a silicon carbide one would; however, stainless steel is
+an inexpensive compromise for our initial testing. Melt Head
+cavity dimensions were adapted from the dimensions of the
+McConnell et al. design [6], [8], [7], [9], [10], with the outer
+cavity having a large enough area to melt through a 3.3 cm x
+3.3 cm ice stick. Within each cavity are drainage holes that
+lead into the fluidic channels and to the outlet ports on the
+sides of the Melt Head.
+The Frame and Bottom Cover are Formlabs Clear resin-
+printed structures that compress the Melt Head to the Heating
+Plate via bolts and thumb nuts. The Frame also serves as
+a mounting point for the Fluidic Inserts. The Melt Head is
+press-fitted by its four corners into the Frame and held only at
+those locations. Gaps were designed in the Frame to separate
+it from the Melt Head and the Heating Plate, minimizing heat
+transfer from these components to the plastic; This is for safe
+handling of the MSIS system and to conduct as much heat
+directly from the Heating Plate to the Melt Head as possible.
+For similar reasons, the Bottom Cover minimizes its contact
+with the Heating Plate while maintaining structural integrity.
+The Fluidic Inserts interface with the outlet ports of the
+Melt Head. They were designed to mimic nuts commonly
+used with fluidic tubing and ferrules. There are four Fluidic
+Inserts: two host 1 fluid line, one hosts 2 fluid lines, and
+another hosts 3 fluid lines. Fluidic tubing in this design is
+1/8-in-outer diameter (OD), 1/16-in-inner diameter (ID), flu-
+orinated ethylene propylene (FEP) tubing from IDEX Health
+& Science. A custom design for hosting the fluidic tubing
+was preferred over a commercial product because it can
+easily be adapted for various design configurations and tubing
+sizes.
+Threaded inserts and bolts are used to mount the
+Fluidic Inserts to the Melter Frame. When tightened down,
+the Fluidic Inserts compress the tubing with custom ferrules
+against the Melt Head so that meltwater can be transported
+from the Melter component to downstream devices via the
+tubing. O-rings between each custom ferrule and the Melt
+Head add another measure in preventing meltwater leakage
+in the system.
+Four pumps were added along the fluid lines to provide addi-
+tional control over the flow rate and to prevent the meltwater
+from overflowing beyond the Melt Head.
+Maintaining a
+higher flow rate at the outer cavity and lower flow rates in
+the inner two cavities is important for reducing contamination
+of the inner cavities [12]. Two small, 3 V diaphragm pumps
+yield higher flow rates and are appropriate for draining the
+outer cavities, which nominally send contaminated meltwater
+to waste.
+Peristaltic pumps are preferred for the middle
+and inner cavities because the meltwater does not contact
+the pump internals, preventing sample contamination. One
+peristaltic pump is used for the middle cavity drain lines,
+and another for the inner cavity. Each Melter pump can be
+powered by varying voltages for different flow rates.
+The Heating Plate is a 1/2-in-thick piece of copper manu-
+factured to hold up to five, 1/4-in-diameter cartridge heating
+elements. The interfaces between the cartridge heaters and
+the Heating Plate are covered with thermal paste to lower
+any contact resistance.
+Copper was selected as the Heat-
+ing Plate material because it has high thermal conductivity,
+k ≈ 385
+W
+m·K , while having a relatively reasonable price
+compared to more thermally conductive materials like silver
+or diamond. The MSIS Heating Plate is much thinner than
+that of other melter designs, meaning the thermal load is
+lower for the cartridge heaters; This allows for increased
+thermal efficiency in our instrument. Also, the Heating Plate
+tends to oxidize with use, so semi-frequent sanding of the
+surface is done to reduce the thermal load.
+We now describe the electrical system that controls the Melter
+(Figure 3).
+Inside the Heating Plate sits the 1/4 in DC
+cartridge heaters, powered with an external power supply up
+to 24 V, which is expected on a typical spacecraft bus. In 12
+V operation, each of the five cartridge heaters produces 14
+W, therefore the theoretical maximum power supplied to the
+melter system is 70 W; 12 V was selected for operating the
+cartridge heaters in the lab, as 70 W of power was adequate
+for MSIS benchtop testing, and working with lower voltages
+is good safety practice.
+Per the slim Heating Plate, the
+cartridge heaters lie horizontally, in contrast to their vertical
+orientation in other melter designs [12]. A NOYITO 5 V relay
+board is used to switch the cartridge heaters via an Arduino
+Uno microcontroller. Each relay can switch loads up to 10
+A and 30 VDC. The relay board in this design was selected
+because it offers low and high-level triggering, accepts serial
+commands, and has enough channels to switch five cartridge
+heaters.
+Temperature is monitored and controlled by a closed-loop
+2
+
+Primary 
+Moisture Trap
+Secondary 
+Moisture Trap
+Vacuum Pump
+Meltwater Collection
+Vapor
+Concentrated Meltwater
+Solenoid Valve
+Sublimator 
+Melter
+Melt Head
+Ice 
+Core
+Bottom Cover
+Heating Plate
+B
+A
+Figure 1. Melter-Sublimator for Ice Science (MSIS) Overview. (A) CAD model. (B) Cross-sectional view illustrating
+both melting and sublimation modes of operation.
+A
+B
+Frame
+Melt Head
+Copper Heating Plate 
+(Beneath Melt Head)
+Fluidic Insert
+Bolt + Thumbscrew
+Inner 
+Cavity
+Middle 
+Cavity
+Outer 
+Cavity
+Middle 
+Cavity
+Outer 
+Cavity
+Outer 
+Cavity
+Outer 
+Cavity
+Cartridge Heater Leads
+Mock Ice Core
+Tubing
+Figure 2. (A) Top view of Melter component. Cartridge heaters not shown. (B) Melter component melting lab-made
+ice core on benchtop.
+3
+
+Proportional-Derivative (PD) system that relies on tempera-
+ture measurements taken by two thermistors mounted to the
+Melt Head (Figure 5). Each thermistor is first in series with a
+100 kΩ resistor and powered by 2.5 V. As the temperature at
+the thermistor rises, its internal resistance is lowered, and the
+Arduino microcontroller can measure a larger analog voltage.
+The 10-bit analog-to-digital converter in the microcontroller
+processes this analog signal using a lookup table, comparing
+a given voltage value to an experimentally determined range
+of voltages versus temperatures.
+A temperature setpoint is specified in the microcontroller.
+The control system evaluates its current temperature versus
+the setpoint and begins heating.
+Then, the PD controller
+detects when the current rise in temperature is sufficient
+to turn off the heaters, at which point the control system
+maintains the temperature setpoint with varying bursts of
+heat. This PD controller is adaptable to different setpoints
+and starting temperatures.
+Sublimator Design & Implementation
+The Sublimator component of MSIS (Figure 4) consists of
+a resin-printed cylindrical sublimation chamber that adapts
+to the top of the Melter component.
+The Sublimator is
+sealed against the Melter by a face-seal O-ring. This allows
+containment of the ice sample, whether it be fragments or
+a full ice stick. The current design can house an ice stick
+measuring 3.3 cm × 3.3 cm at the base and a height of up
+to 15 cm. The sublimation chamber is sealed at the top by
+a cap that screws into the chamber, and three horizontally
+mounted Lee Company (Westbrook, Connecticut) solenoid
+valves are attached. The solenoid valves expose the ice core
+to vacuum when opened. Valve opening is controlled by three
+solid-state relays connected to the Arduino microcontroller.
+Through serial command, the operator specifies the number
+of valves they wish to open, depending on the desired outflow.
+A CoolCube 50R “hit-and-hold” circuit is placed in series
+between each relay and each valve. The valves require 12 V
+to actuate but do not require this voltage to remain actuated.
+The “hit-and-hold” circuitry reduces the operating voltage to
+6 V to prevent overheating while keeping the valve open for
+extended periods (e.g., during sublimation).
+The outlet to the solenoid valves extends into barb connectors
+to allow the channeling of water vapor into a common mois-
+ture trap, which consists of a sealed basin with a cylindrical
+water and particulates filter. This helps protect the down-
+stream vacuum pump to which the vapor is being vented.
+Such an adapter may not be necessary in a low-pressure
+atmosphere, as on Mars, where additional vacuum may be
+superfluous depending upon the local pressure.
+In its operation, the Sublimator would reduce the pressure in
+the sublimation chamber to below the triple point of water
+and raise the temperature of the Melt Head to 0◦C (Figure
+5).
+To simulate Martian conditions more accurately, the
+Sublimator would reduce the pressure to 1 Torr.
+Lower
+pressures can be attained with strong vacuum sources, but
+pressures significantly lower than 0.75 Torr can negatively
+impact the sublimation rate by hindering convective heat
+transfer into the ice [13]. Higher sample temperatures also
+favor faster sublimation rates; however, we must maintain
+a lower temperature at the Melt Head throughout the initial
+exposure to vacuum to prevent premature melting [13]. Since
+sublimation is endothermic, the temperature of the ice stick
+will decrease throughout the process. Our control over the
+Melt Head temperature allows us to set temperatures appro-
+priate for any state of the ice stick during the sublimation
+process.
+Depending on the application, the Sublimator does not re-
+quire all the functionalities of the Melter. The Sublimator
+component is designed to both interface with the entire Melter
+component or only the Heating Plate. This allows for the use
+of a thermal plate inside a vacuum chamber to heat a sample
+through the Melter Heating Plate, thereby eliminating the
+need for a cartridge heater-relay setup where space is limited
+(Figure 5).
+For other applications, only a reduced solid
+sample may be desired, so melting may not be necessary.
+Proof of Principle Testing
+Thermal Efficiency of Melting— Thermal efficiency of the
+MSIS Melter component is desired to optimize power usage
+onboard a spacecraft or in an extraterrestrial environment.
+Cold environments, such as the Antarctic ice sheet, Mars, or
+Europa will work against the ease of melting, both demanding
+more power from electronics and potentially lowering the
+melt rate. To improve thermal efficiency, we maximize the
+amount of conductive surface contact between the cartridge
+heaters and the Melt Head and minimize contact with all other
+parts. The heaters can individually attain 100◦C in 60 seconds
+when powered by 12 V. This heating time is sufficiently
+fast to justify using 12 V as opposed to 24 V to power the
+cartridge heaters, which would otherwise increase the power
+consumption of the system.
+To characterize the thermal efficiency of the Melter, three
+tests were run with varying changes.
+The PD feedback
+control system was used in each test, with a 50 ◦C setpoint.
+We determined that the cartridge heaters consume a combined
+69.96 W of power. We then calculated the lossless amount of
+heat required to raise the Melt Head from room temperature
+to 50 ◦C. From this number and power consumption, we
+found the theoretical number of seconds that it would take
+to reach this temperature without losses.
+For each test,
+we measured the amount of time it took to reach 50 ◦C,
+calculated the heat used, then compared this calculation to
+the lossless heat required to raise the Melt Head to 50 ◦C. The
+first test was conducted without any thermal paste, the second
+with thermal paste between the Melt Head and Heating Plate,
+and the final test with thermal paste between the Melt Head
+and Heating Plate, as well as on each cartridge heater. As
+expected, the final test yielded the highest thermal efficiency
+with a value of 59.7%.
+3. DISCUSSION
+Limitations and Future Development
+The MSIS instrument has thus far demonstrated melting of
+a lab-made ice core sample in a benchtop setting. Upon the
+success of 3D printing the Melt Head, a future MSIS design
+could involve inner fluidic channels with more complex ge-
+ometry, with outlet ports on a single side of the instrument;
+this modification would make for a more organized system
+in the lab and for future space missions.
+The next Melt
+Head design could include additional measures for preventing
+contamination of the inner core sections. For instance, the
+Melt Head partitioning walls could be made taller to ensure
+separation of the central core from the outer portion. Also,
+ultraviolet or infrared LEDs could be placed in a ring pattern
+at the outer base of the Melt Head to melt the contaminated
+outer region of the ice core before the inner regions, reducing
+the risk of meltwater mixing across Melt Head cavities.
+The next critical step in instrument development would be
+4
+
+Thermistor 1
+Thermistor 2
+100kΩ
+100kΩ
++2.5V
++2.5V
++12V
+USB
+Micro-
+controller
+NO
+COM
+NO
+COM
+NO
+COM
+NO
+COM
+NO
+COM
+Relay 
+Board
+DC+
+DC-
+In1
+In2
+In3
+In4
+In5
+D1
+D2
+D3
+D4
+D5
+GND
++5V
+Heater 5
+Heater 4
+Heater 3
+Heater 2
+Heater 1
+AREF
+Peristaltic Pump 2
+Peristaltic Pump 1
++12V
+Diaphragm Pump 2
+Diaphragm Pump 1
+A2
+A1
++3V
+Figure 3. Melter Electronic Wiring Diagram
+Sublimator cap
+Sublimator 
+head/chamber
+Solenoid 
+valves
+Vacuum 
+lines
+Moisture trap 
+barb adapter
+Figure 4. Sublimator hardware, excluding relay board
+and electronics. Sublimator pictured without Melter.
+processing a Greenland or Antarctic ice core sample in the lab
+with MSIS, integrated with downstream analytical devices for
+processing validation. Upon such validation and necessary
+modifications, the MSIS instrument could begin utilization in
+the lab or in the field for Earth-based ice core studies.
+At its current state, MSIS requires human operation and
+sample loading. A future vision is to make the MSIS system
+autonomous and compatible with a rover platform exploring
+ice deposits on other bodies. Automating a sublimator would
+require robust monitoring and data acquisition capabilities.
+MSIS currently relies on a user judging when enough subli-
+mation has taken place based on the size of the sublimated
+ice sample.
+To process samples back-to-back, a cleaning
+process would be needed within its fluid lines. A one-size-
+fits-all approach to cleaning debris of different sizes would be
+challenging. However, the Sublimator component of MSIS
+does not limit itself to processing fully intact ice cores.
+Considering the deposition of dust and debris on Mars,
+subsurface Martian ice has the potential to be stratified like
+permafrost ice deposits on Earth [11]. The drilling process
+would inevitably accrue regolith in the ice cores which may
+pose a problem to the MSIS plumbing network. In addition,
+near-surface Mars ice could be more porous due to lower
+gravity and there may be a commensurate increase in the
+depth of pore close-off; this could lead to more fragile ice.
+Rougher drilling techniques tend to produce ice “chips,”
+5
+
+A
+B
+C
+D
+Figure 5. (A) Sublimator being tested inside thermal vacuum chamber, full melter not present. (B) Sublimator inside
+thermal vacuum chamber, melter present. (C) Pressure-Temperature diagram showing conditions induced by the
+Sublimator to begin the concentration sublimation process. (D) Tuned vs. untuned Proportional-Derivative (PD)
+temperature feedback results given a 50 ◦C setpoint.
+which are fragments defined to have a 6 mm maximum
+dimension [11]. These chips lower the resolution and are
+difficult to decontaminate but are still considered useful for
+science.
+Sublimation is an experimentally inefficient process [14]
+which does not pose an issue for benchtop experimentation
+on Earth but complicates its context in a space instrument.
+The convenient presence of low atmospheric pressure on
+Mars and other celestial bodies [13] may allow for more
+efficient sublimation if it reduced or eliminated the need for
+a vacuum pump, as is needed on Earth. Without a vacuum
+pump, the sublimator instrument would save upwards of 1
+kW of power and consume an estimated 76 W at its maximum
+power setting.
+Up to 300 W of power consumption in
+a space instrument is realistic for the capabilities of some
+rovers (e.g., Mars Oxygen ISRU Experiment, or MOXIE,
+instrument on Perseverance [15]) and serves as our metric
+when characterizing the MSIS power consumption.
+Integration with Future Human Lunar/Mars Missions
+The long-term objective is for future versions of MSIS to
+operate beyond Earth. With the Artemis III crew proposed
+to target regions near the Moon’s South Pole2, lunar water
+ice analyses are likely to take place. MSIS could contribute
+to Artemis III demonstrations of in-situ resource utilization
+for pre-processing lunar water ice for oxygen and hydrogen
+extraction to be used in fuel and life support systems. This
+would serve as a precursor to Mars missions with similar
+aims in processing ice to serve as an ingredient for propellant
+production while potentially searching for traces of life [16].
+Also, in the context of preparing for the first human mission to
+Mars, potentially to mid-latitude regions in the 2030s, NASA
+has identified sampling subsurface ice as a critical science
+requirement for searching for life and investigating the evo-
+lution of Mars’ climate [11]. If the Artemis III and Mars
+crews have the means to retrieve and manually handle ice
+core-like samples, MSIS—at its current state—could prove
+beneficial in preparing lunar water ice samples for in-situ
+2https://www.nasa.gov/press-release/nasa-identifies-candidate-regions-
+for-landing-next-americans-on-moon
+6
+
+Vacuum
+Lines
+Sublimator
+Melter
+Fluidic LinesCopperHeating PlateSolid
+760.0
+Liquid
+On-Earth
+storage
+conditions
+Pressure (Torr)
+Triple Point
+On-Mars
+naturalstorage
+conditions
+6.0
+Vapor
+1.0
+Sublimator-
+induced conditions
+200210243
+273
+Temperature (Kelvin)60
+55
+50
+45
+40
+35
+UntunedPD
+Controller
+Tuned PD
+30
+Controller
+25
+20
+15
+200
+400
+600
+800
+1000
+1200
+1400
+Time (seconds)refueling demonstrations, as well as helping meet NASA’s
+Mars human mission goals.
+Volcanism, Climate, and Solar Activity from Mars Ice
+An important caveat of our proposed analyses is that the
+formation and preservation mechanisms for different ice de-
+posits on Mars is under study and without ground truth [17].
+At present, this means the availability of contiguous ice core
+records on Mars is unknown.
+This could complicate our
+ability to establish a high-resolution contiguous history from
+Mars ice.
+Mars’ obliquity, the angle between its orbital plane and its
+spin axis, is currently around 25◦, and it varies on a 120-
+thousand-year (Ky) timescale, with the amplitude of this
+cycling varying chaotically on 1.2 My timescales [18], [19].
+During periods of high obliquity, polar ice is mobilized and
+falls as snow in mid-latitude and equatorial regions [20]. This
+process has resulted in repeated mid-latitude “ice ages” for
+an estimated total of 680 My out of the last 3.6 billion years
+(Gy). Thus, it is reasonable to assume that current remnants
+of these ice ages are mainly locally-contiguous deposits, even
+if a global contiguous record is not expected.
+Detection of Recent Volcanism on Mars— Mars has been
+geologically active until the recent past, and while no active
+volcanism has been observed, it is unknown when it ceased.
+Ice cores on Earth are typically synchronized using volcanic
+events. Emissions of SO2 are oxidized to sulfuric acid and
+measured in ice cores as acidity and sulfate (SO2–
+4 ) pulses
+above the climate background [21]. The present-day atmo-
+sphere of Mars is highly oxidizing and has been so for billions
+of years, except during brief warmer and wetter reducing
+periods [22]. Thus, volcanism on Mars could leave signatures
+of volcanic events within Mars ice similar to those we analyze
+on Earth (e.g., [23], [24]).
+Characterization of Mars Climate in the Late Amazonian—
+Mars has been largely dry, cold, and oxidizing for the last
+3 Gy, a period denoted as the Amazonian. Ice core records
+on Mars, potentially going back tens to hundreds of My
+[1], [2], can help establish recent variability of the climate
+during this period. While the exquisite layering of the Mars
+polar ice caps contains important climate information, these
+deposits may only date to the last 4 My [25]. While young,
+polar deposits may offer a more contiguous, although dusty,
+record during this recent period compared to mid-latitude ice
+deposits. However, these deposits are much more difficult
+to access due to orbital mechanics and harsher conditions,
+making mid-latitude ice a compelling target.
+Seeking Evidence of Subsurface Life— Life on Mars, if it
+exists, could be related to Earth. A Mars origin for Earth life
+could resolve significant inconsistencies between the inferred
+history of life and Earth’s geologic history [26]. If life got
+started on Mars, there is little reason to expect it is now
+extinct, given the significant overlap between the temperature
+and pressure regimes occupied by life on Earth and those
+available on Mars [27]. In short, the most likely place for
+life on Mars today is in the deep subsurface.
+Given this,
+and that current (e.g., Curiosity, Perseverance) and future
+(e.g., Rosalind Franklin) missions are limited to cm to m drill
+depths, how could we sample the deep subsurface in the near
+term, without the mass required to drill to km-scale depths?
+On Earth,
+phreatomagmatic eruptions,
+those involving
+magma interacting with water, yield explosive events, which
+can result in the dispersal of even large microorganisms such
+as diatoms up to nearly 1000 km [28].
+Evidence of such
+eruptions has been identified on Mars, representing interac-
+tions between magma and ground ice. For example, explosive
+volcanism in Elysium Planitia [29] may have occurred as
+recently as 46 to 222 Ky ago [30], [31]. This region, home
+to the InSight Lander, could thus possibly harbor active
+volcanism, which could offer an abode for life.
+Therefore, explosive eruptions, as well as large impact events,
+could eject life or signs of life from deeper within the Martian
+crust, and their remnants could be deposited on accumulat-
+ing ice sheets during periods of high obliquity.
+Because
+such explosive events are occurring up to the present day,
+geologically speaking, such events have likely been ongoing
+over the entire history of Mars mid-latitude ice emplacement.
+Burial within accumulating ice deposits would enhance the
+preservation of organics relative to long-term exposure at the
+surface, due to shielding from space radiation and a lack of
+thermal cycling.
+Such remnants of life, if present, would
+likely be present at extremely low abundance.
+However,
+they could be concentrated, for example as described by [16]
+during large-scale processing of ice for propellant production.
+Returned Samples for Cosmogenic Nuclide Analyses—Cos-
+mogenic nuclides are isotopes produced by interactions be-
+tween atoms and high energy galactic cosmic rays (GCR).
+On Earth, Beryllium-10 (10Be) is used to infer historical
+solar activity from its abundance in ice cores. Cosmogenic
+radionuclides can also reveal extreme solar events [32] and
+nearby supernovas [33].
+On Earth, once 10Be forms and attaches to aerosols, it takes
+1 to 2 years for deposition to occur [34]. Similar modeling
+of transport and scavenging processes, as well as important
+baseline measurements, would be required for Mars to inter-
+pret a record of cosmogenic isotopes recovered from Mars ice
+[35]. Recently, the potential for 10Be analysis from ice chips,
+permitting much faster drilling, has been demonstrated [36].
+While the 1.39 My half-life of 10Be permits its use through
+the full range of ice ages on Earth, alternative cosmogenic
+isotopes would be needed for older ice on Mars. Candidates
+include 53Mn, with a half-life of 3.7 My, and 129I, with a
+half-life of 15.7 My.
+53Mn can be produced within supernova events, and has been
+detected co-located with 60Fe, also produced during super-
+novas, within marine sediments. This has been used to date
+supernova events [33].
+53Mn is also produced terrestrially,
+through cosmic ray bombardment and in-situ production [37].
+The only important target element for 53Mn production in
+situ is iron [37], which is abundant in the dust that undergoes
+local to global aeolian transport on Mars. Dust aloft might
+experience higher GCR rates due to a lack of atmospheric
+shielding, although this effect could be marginal due to
+the relatively thin atmosphere of Mars (1% of Earth in the
+geologically recent past). Dust deposited on ice sheets would
+continue experiencing cosmogenic production of 53Mn dur-
+ing initial burial.
+At the Greenland summit (e.g., GISP2
+core), historical accumulation rates are from 15 to 40 cm/y
+ice equivalent. On Mars, accumulation rates are likely much
+lower, perhaps 10 mm/y [38], but are not well constrained.
+This means isotopic signals on Mars might be spread out over
+broader (> annual) periods than on Earth.
+7
+
+For even older ice, 129I analysis may be useful.
+129I is
+mainly produced by the fission of uranium, with production
+by cosmic rays also contributing. On Earth, production is
+estimated to be 10−14 grams of 129I per gram of 127I [39].
+It has been proposed that Mars may have had a naturally-
+occurring fission reactor in the Mare Acidalium region [40].
+If true, this could have resulted in an explosion on the order
+of 1 Gy ago, which would have scattered radioactive iodine
+across the planet’s surface.
+This could make it harder to
+attribute radioactive iodine to cosmic ray sources.
+The most sensitive approach known to measure 53Mn and
+129I is accelerator mass spectrometry (AMS) [41], [42]. Un-
+fortunately, AMS facilities are large and complex and would
+be unlikely candidates for in-situ analysis. Instead, ice could
+be processed in situ, sublimated, with retention of isotopes
+bound to particulates, and those residues could be returned
+for AMS analysis on Earth. Searching for rare isotopes might
+require processing large volumes of ice; for example, 500 kg
+of Antarctic ice was analyzed to identify interstellar 60Fe by
+AMS [43].
+Enhancing Limit of Detection for Ocean Worlds
+Based on analysis of nutrient-limited Earth environments,
+modeling of energy limitations, and other factors, estimates
+for cell abundances at Ocean Worlds such as Enceladus and
+Europa may be as low or lower than 1000 to 100 cells/ml
+[44]. Most instrumentation flown or under development may
+not have adequate analytical limits of detection without con-
+centration. Concentration approaches include filtration, flow
+focusing, and sublimation, among others. A prototype subli-
+mator was previously developed and was able to demonstrate
+the freeze-drying of a 5 ml sample in 5 hours at under 5 W
+power [13]. This was achieved by accounting for the available
+cold temperatures and low pressure expected to be available
+under Europa or Enceladus surface conditions.
+Even so,
+sublimating large volumes of ice is likely infeasible due to
+the high energy cost of sublimation; we can expect to have
+higher energy budgets on Mars where higher power operation
+could process larger volumes via sublimation. Cold and low-
+pressure conditions are also available on Mars, where surface
+pressure is 600 to 700 Pa; a vacuum pump will be required
+to maintain pressure below the triple point of water, 610 Pa.
+4. CONCLUSIONS
+From processing true ice cores to automating some capabil-
+ities, there is much more subsystem development to occur
+before MSIS is used with human or robotic missions to
+Mars and icy bodies. Fortunately, MSIS has equally diverse
+applications on Earth and its functionalities can be verified
+with Earth-based ice samples. In the long-term, MSIS would
+greatly simplify the processes needed for ice science in other
+planetary environments.
+Our initial work toward a space-
+mission-compatible ice core processing instrument could
+someday enable the examination of extraterrestrial ice cores
+to help address fundamental and diverse questions in plane-
+tary processes, habitability, astrobiology, and heliophysics.
+ACKNOWLEDGMENTS
+Funded by faculty startup funds from the Georgia Institute of
+Technology to C.E.C.
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+mission concepts for the search for evidence of life at
+enceladus,” Astrobiology, vol. 22, no. 6, pp. 685–712,
+2022, pMID: 35290745.
+BIOGRAPHY[
+Alexander G. Chipps is an under-
+graduate researcher at Georgia Tech,
+set to graduate with a B.S. degree in
+Mechanical Engineering in 2023.
+He
+aims to partake in graduate Aerospace
+Engineering research to further develop
+space instrumentation for life detec-
+tion.
+Cassius B. Tunis is an undergraduate
+researcher at Georgia Tech pursuing a
+B.S. degree in Aerospace Engineering
+with a minor in Earth and Atmospheric
+Science. He is interested in building in-
+struments to better understand planetary
+processes.
+Nathan Chellman holds a B.Sc.
+in
+Geology/Biology from Brown Univer-
+sity, and M.S. and Ph.D. degrees in Hy-
+drology from the University of Nevada,
+Reno. He is an Assistant Research Pro-
+fessor and snow and ice hydrologist in
+the Division of Hydrologic Sciences at
+the Desert Research Institute (DRI). He
+specializes in the collection, processing
+and analysis of ice cores.
+Joseph R. McConnell received a B.A.
+in Geology and Geophysics from Yale,
+an M.S. in Exploration Geophysics from
+Stanford University, and a Ph.D. in Hy-
+drogeology and Water Resources from
+the University of Arizona. He is a Re-
+search Professor in the Division of Hy-
+drologic Sciences at the Desert Research
+Institute (DRI). Current NSF, NASA, and
+internally funded research projects in-
+clude ice core chemistry-based studies in Greenland, Antarc-
+tica, and the Americas, using his group’s unique ultra-trace
+ice core analytical laboratory.
+Bruce Hammer received the B.S. in
+BiologyPhysics from the State Univer-
+sity of New York in 1974, the M.S.
+in Biomedical Engineering from North-
+western University in 1976, and the
+Ph.D. in Biomedical Engineering from
+Northwestern University in 1981.
+He
+is a Professor of Radiology & Adjunct
+Professor of Entrepreneurial Studies at
+the University of Minnesota. His work
+focuses on hardware development and its implementation in
+support of biomedical and astrobiology applications, includ-
+ing a NASA-funded sublimator device for Ocean Worlds.
+Christopher E. Carr received his B.S.
+degree in Aero/Astro and Electrical En-
+gineering in 1999, his Masters degree
+in Aero/Astro in 2001, and his Sc.D.
+degree in Medical Physics in 2005, all
+from MIT. He is an Assistant Profes-
+sor in the Daniel Guggenheim School of
+Aerospace Engineering and the School
+of Earth and Atmospheric Sciences at
+the Georgia Institute of Technology. He
+is broadly interested in searching for and expanding the
+presence of life beyond Earth.
+10
+
diff --git a/rNAzT4oBgHgl3EQf5_6l/content/tmp_files/load_file.txt b/rNAzT4oBgHgl3EQf5_6l/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6919e1f30409317188b5d8e3b76fb60a5bfd5841
--- /dev/null
+++ b/rNAzT4oBgHgl3EQf5_6l/content/tmp_files/load_file.txt
@@ -0,0 +1,1194 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf,len=1193
+page_content='Enabling Ice Core Science on Mars and Ocean Worlds  Alexander G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Chipps  School of Mechanical Engineering  Georgia Institute of Technology  Atlanta, GA 30332  achipps3@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='edu  Cassius B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Tunis  School of Aerospace Engineering  Georgia Institute of Technology  Atlanta, GA 30332  ctunis3@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='edu  Nathan Chellman  Desert Research Institute  2215 Raggio Parkway  Reno, NV 89512  nathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='chellman@dri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='edu  Joseph R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' McConnell  Desert Research Institute  2215 Raggio Parkway  Reno, NV 89512  joe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='mcconnell@dri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='edu  Bruce Hammer  University of Minnesota  Center for Magnetic Resonance Research  2021 6th Street SE  Minneapolis, MN 55455  hammer@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='edu  Christopher E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Carr  School of Aerospace Engineering &  School of Earth and Atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Sciences  620 Cherry St NW, Room G10  Atlanta, GA 30312  cecarr@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='edu  Abstract—Ice deposits on Earth provide an extended record  of volcanism, planetary climate, and life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  On Mars, such a  record may extend as far back as tens to hundreds of millions  of years (My), compared to only a few My on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Here,  we propose and demonstrate a compact instrument, the Melter-  Sublimator for Ice Science (MSIS), and describe its potential  use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Similar to current use in the analysis of ice cores,  linking MSIS to downstream elemental, chemical, and biological  analyses could address whether Mars is, or was in the recent  past, volcanically active, enable the creation of a detailed climate  history of the late Amazonian, and seek evidence of subsurface  life preserved in ice sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The sublimation feature can not  only serve as a preconcentrator for in-situ analyses, but also  enable the collection of rare material such as cosmogenic nu-  clides, which could be returned to Earth and used to confirm  and expand the record of nearby supernovas and long-term  trends in space weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Missions to Ocean Worlds such as  Europa or Enceladus will involve ice processing, and there MSIS  would deliver liquid samples for downstream wet chemistry  analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Our combined melter-sublimator system can thus help  to address diverse questions in heliophysics, habitability, and  astrobiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  TABLE OF CONTENTS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' INTRODUCTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' MELTER-SUBLIMATOR FOR ICE SCIENCE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' DISCUSSION .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
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+page_content='  8  BIOGRAPHY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' INTRODUCTION  Ice deposits on Earth provide an extended record of volcan-  ism, cosmic events, and subsurface life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' On Mars, records  entrained in polar and mid-latitude ice may extend as far back  as tens to hundreds of My [1], [2], compared to only a few  My (and less than one million years contiguous) on Earth  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Martian ice investigations would provide a significant  extension of our knowledge from Earth ice analysis, and  could serve as a precursor to the exploration of icy bodies  such as Enceladus and Europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Such ice, found in deposits  more than 100 meters thick in mid-latitudes [4], may also  represent an important resource for human exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  978-1-6654-9032-0/23/$31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='00 ©2023 IEEE  The layers present within ice cores record historical precip-  itation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=', snow), aerosols, and trap atmospheric gases as  the overburden ice accumulates and the pore space is closed  off (around 50-80 m deep on Earth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  One common strategy for reducing ice core contamination is  to physically remove the outer part of a core, as was done  for a recent study assessing the contamination of Arctic ice  cores as a measure of the potential of forward contamination  during future life detection missions [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This process is labor  intensive and challenging to automate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Melters, which heat an  ice core or section of an ice core (ice stick) above its melting  temperature at the base, offer an alternative that facilitates  automatic processing, and can reduce contamination by per-  mitting automatic disposal of the meltwater from the outer  part of the core as described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Melters permit sampling of these layers at high resolution, in-  cluding by coupling meltwater to real-time downstream anal-  yses such as inductively-coupled plasma mass spectrometry  (ICP-MS) for elemental abundance [6], laser spectrometers  for gas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=', methane) detection [7], black carbon or other  compounds [8], or by collection of fractions for additional  analyses [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Thus, we propose and demonstrate initial work toward a com-  pact Melter-Sublimator for Ice Science (MSIS) instrument  that would enable convenient ice core processing on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  MSIS is built upon an extensively validated melter design [6],  [8], [7], [9], [10], and it has a compact and thermally-efficient  design to meet the requirements of planetary exploration  (size, mass, power, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Through a combination of in-  situ and sample return processes, pairing an MSIS-processed  sample with downstream chemical analysis instruments may  assist in several potential use cases:  1) Addressing when and where Mars was last volcanically  active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  2) Confirming and expanding the record of space weather  events over a timescale not possible with Earth ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  3) Characterizing the inventory of organic molecules on  Mars in support of the search for life beyond Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  In this paper, we describe the MSIS instrument concept,  design, and current performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' We also discuss the lim-  itations and future development goals for MSIS, as well as  the applications of MSIS to the potential use cases proposed  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='01869v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='ins-det]  5 Jan 2023  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' MELTER-SUBLIMATOR FOR ICE SCIENCE  Concept of Operations  The MSIS instrument can be best described as having two  main components: Melter and Sublimator (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  On Earth, drilling, extraction, and handling processes result  in contamination of an ice core’s outer surface [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Access-  ing pure, uncontaminated regions of the core and rejecting  contaminated areas is a requirement for some downstream  (elemental, chemical, biological) analyses [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Thus, like  the melter design from McConnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' [6], [8], [7], [9], [10],  the MSIS Melter component divides an ice stick into three  distinct cross-sectional areas as it melts, providing individual  access to the inner, middle, and outer core sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  MSIS is currently designed to accommodate 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='3 cm x 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='3 cm  ice sticks, following [6], but in principle could be adapted to  any core cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Using the three-ring design adopted  from [6], water from the outer 70% of the core is discarded  under ideal operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The inner, most pristine  10% is collected separately from the middle 20% to facilitate  downstream analysis applications requiring different levels of  purity and volume flow rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The geometry of these rings  can be redesigned to suit any specific set of downstream flow  requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The Sublimator component, sitting atop the Melter, enables  preconcentration of material for in-situ analyses or Earth  return of residues, avoiding the complexities involved in  storing and returning large samples to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' MSIS will be  able to preconcentrate organic samples through near-full sub-  limation then controllably melt a reduced sample, allowing  for sensitive in-situ characterization of non-volatile materials  such as organics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Melter Design & Implementation  The Melt Head is a stainless steel, 3D-printed structure that  sits compressed against a heated copper plate with a layer  of thermal paste at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The Melt Head has two  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='5-mm-tall partitioning walls that separate the ice stick as  it melts into the inner, middle, and outer Melt Head cavities  (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Copper was not used for these partitioning walls  because it would be chemically reactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Also, a stainless  steel Melt Head does not permit trace metal-clean operation  like a silicon carbide one would;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' however, stainless steel is  an inexpensive compromise for our initial testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Melt Head  cavity dimensions were adapted from the dimensions of the  McConnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' design [6], [8], [7], [9], [10], with the outer  cavity having a large enough area to melt through a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='3 cm x  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='3 cm ice stick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Within each cavity are drainage holes that  lead into the fluidic channels and to the outlet ports on the  sides of the Melt Head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The Frame and Bottom Cover are Formlabs Clear resin-  printed structures that compress the Melt Head to the Heating  Plate via bolts and thumb nuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The Frame also serves as  a mounting point for the Fluidic Inserts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The Melt Head is  press-fitted by its four corners into the Frame and held only at  those locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Gaps were designed in the Frame to separate  it from the Melt Head and the Heating Plate, minimizing heat  transfer from these components to the plastic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This is for safe  handling of the MSIS system and to conduct as much heat  directly from the Heating Plate to the Melt Head as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  For similar reasons, the Bottom Cover minimizes its contact  with the Heating Plate while maintaining structural integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The Fluidic Inserts interface with the outlet ports of the  Melt Head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' They were designed to mimic nuts commonly  used with fluidic tubing and ferrules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' There are four Fluidic  Inserts: two host 1 fluid line, one hosts 2 fluid lines, and  another hosts 3 fluid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Fluidic tubing in this design is  1/8-in-outer diameter (OD), 1/16-in-inner diameter (ID), flu-  orinated ethylene propylene (FEP) tubing from IDEX Health  & Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' A custom design for hosting the fluidic tubing  was preferred over a commercial product because it can  easily be adapted for various design configurations and tubing  sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Threaded inserts and bolts are used to mount the  Fluidic Inserts to the Melter Frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' When tightened down,  the Fluidic Inserts compress the tubing with custom ferrules  against the Melt Head so that meltwater can be transported  from the Melter component to downstream devices via the  tubing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' O-rings between each custom ferrule and the Melt  Head add another measure in preventing meltwater leakage  in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Four pumps were added along the fluid lines to provide addi-  tional control over the flow rate and to prevent the meltwater  from overflowing beyond the Melt Head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Maintaining a  higher flow rate at the outer cavity and lower flow rates in  the inner two cavities is important for reducing contamination  of the inner cavities [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Two small, 3 V diaphragm pumps  yield higher flow rates and are appropriate for draining the  outer cavities, which nominally send contaminated meltwater  to waste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Peristaltic pumps are preferred for the middle  and inner cavities because the meltwater does not contact  the pump internals, preventing sample contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' One  peristaltic pump is used for the middle cavity drain lines,  and another for the inner cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Each Melter pump can be  powered by varying voltages for different flow rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The Heating Plate is a 1/2-in-thick piece of copper manu-  factured to hold up to five, 1/4-in-diameter cartridge heating  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The interfaces between the cartridge heaters and  the Heating Plate are covered with thermal paste to lower  any contact resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Copper was selected as the Heat-  ing Plate material because it has high thermal conductivity,  k ≈ 385  W  m·K , while having a relatively reasonable price  compared to more thermally conductive materials like silver  or diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The MSIS Heating Plate is much thinner than  that of other melter designs, meaning the thermal load is  lower for the cartridge heaters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This allows for increased  thermal efficiency in our instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Also, the Heating Plate  tends to oxidize with use, so semi-frequent sanding of the  surface is done to reduce the thermal load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  We now describe the electrical system that controls the Melter  (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Inside the Heating Plate sits the 1/4 in DC  cartridge heaters, powered with an external power supply up  to 24 V, which is expected on a typical spacecraft bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' In 12  V operation, each of the five cartridge heaters produces 14  W, therefore the theoretical maximum power supplied to the  melter system is 70 W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' 12 V was selected for operating the  cartridge heaters in the lab, as 70 W of power was adequate  for MSIS benchtop testing, and working with lower voltages  is good safety practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Per the slim Heating Plate, the  cartridge heaters lie horizontally, in contrast to their vertical  orientation in other melter designs [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' A NOYITO 5 V relay  board is used to switch the cartridge heaters via an Arduino  Uno microcontroller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Each relay can switch loads up to 10  A and 30 VDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The relay board in this design was selected  because it offers low and high-level triggering, accepts serial  commands, and has enough channels to switch five cartridge  heaters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Temperature is monitored and controlled by a closed-loop  2  Primary  Moisture Trap  Secondary  Moisture Trap  Vacuum Pump  Meltwater Collection  Vapor  Concentrated Meltwater  Solenoid Valve  Sublimator  Melter  Melt Head  Ice  Core  Bottom Cover  Heating Plate  B  A  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Melter-Sublimator for Ice Science (MSIS) Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' (A) CAD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' (B) Cross-sectional view illustrating  both melting and sublimation modes of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  A  B  Frame  Melt Head  Copper Heating Plate  (Beneath Melt Head)  Fluidic Insert  Bolt + Thumbscrew  Inner  Cavity  Middle  Cavity  Outer  Cavity  Middle  Cavity  Outer  Cavity  Outer  Cavity  Outer  Cavity  Cartridge Heater Leads  Mock Ice Core  Tubing  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' (A) Top view of Melter component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Cartridge heaters not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' (B) Melter component melting lab-made  ice core on benchtop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  3  Proportional-Derivative (PD) system that relies on tempera-  ture measurements taken by two thermistors mounted to the  Melt Head (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Each thermistor is first in series with a  100 kΩ resistor and powered by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' As the temperature at  the thermistor rises, its internal resistance is lowered, and the  Arduino microcontroller can measure a larger analog voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The 10-bit analog-to-digital converter in the microcontroller  processes this analog signal using a lookup table, comparing  a given voltage value to an experimentally determined range  of voltages versus temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  A temperature setpoint is specified in the microcontroller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The control system evaluates its current temperature versus  the setpoint and begins heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Then, the PD controller  detects when the current rise in temperature is sufficient  to turn off the heaters, at which point the control system  maintains the temperature setpoint with varying bursts of  heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This PD controller is adaptable to different setpoints  and starting temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Sublimator Design & Implementation  The Sublimator component of MSIS (Figure 4) consists of  a resin-printed cylindrical sublimation chamber that adapts  to the top of the Melter component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The Sublimator is  sealed against the Melter by a face-seal O-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This allows  containment of the ice sample, whether it be fragments or  a full ice stick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The current design can house an ice stick  measuring 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='3 cm × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='3 cm at the base and a height of up  to 15 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The sublimation chamber is sealed at the top by  a cap that screws into the chamber, and three horizontally  mounted Lee Company (Westbrook, Connecticut) solenoid  valves are attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The solenoid valves expose the ice core  to vacuum when opened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Valve opening is controlled by three  solid-state relays connected to the Arduino microcontroller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Through serial command, the operator specifies the number  of valves they wish to open, depending on the desired outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  A CoolCube 50R “hit-and-hold” circuit is placed in series  between each relay and each valve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The valves require 12 V  to actuate but do not require this voltage to remain actuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The “hit-and-hold” circuitry reduces the operating voltage to  6 V to prevent overheating while keeping the valve open for  extended periods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=', during sublimation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The outlet to the solenoid valves extends into barb connectors  to allow the channeling of water vapor into a common mois-  ture trap, which consists of a sealed basin with a cylindrical  water and particulates filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This helps protect the down-  stream vacuum pump to which the vapor is being vented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Such an adapter may not be necessary in a low-pressure  atmosphere, as on Mars, where additional vacuum may be  superfluous depending upon the local pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  In its operation, the Sublimator would reduce the pressure in  the sublimation chamber to below the triple point of water  and raise the temperature of the Melt Head to 0◦C (Figure  5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  To simulate Martian conditions more accurately, the  Sublimator would reduce the pressure to 1 Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Lower  pressures can be attained with strong vacuum sources, but  pressures significantly lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='75 Torr can negatively  impact the sublimation rate by hindering convective heat  transfer into the ice [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Higher sample temperatures also  favor faster sublimation rates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' however, we must maintain  a lower temperature at the Melt Head throughout the initial  exposure to vacuum to prevent premature melting [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Since  sublimation is endothermic, the temperature of the ice stick  will decrease throughout the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Our control over the  Melt Head temperature allows us to set temperatures appro-  priate for any state of the ice stick during the sublimation  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Depending on the application, the Sublimator does not re-  quire all the functionalities of the Melter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The Sublimator  component is designed to both interface with the entire Melter  component or only the Heating Plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This allows for the use  of a thermal plate inside a vacuum chamber to heat a sample  through the Melter Heating Plate, thereby eliminating the  need for a cartridge heater-relay setup where space is limited  (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  For other applications, only a reduced solid  sample may be desired, so melting may not be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Proof of Principle Testing  Thermal Efficiency of Melting— Thermal efficiency of the  MSIS Melter component is desired to optimize power usage  onboard a spacecraft or in an extraterrestrial environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Cold environments, such as the Antarctic ice sheet, Mars, or  Europa will work against the ease of melting, both demanding  more power from electronics and potentially lowering the  melt rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' To improve thermal efficiency, we maximize the  amount of conductive surface contact between the cartridge  heaters and the Melt Head and minimize contact with all other  parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The heaters can individually attain 100◦C in 60 seconds  when powered by 12 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This heating time is sufficiently  fast to justify using 12 V as opposed to 24 V to power the  cartridge heaters, which would otherwise increase the power  consumption of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  To characterize the thermal efficiency of the Melter, three  tests were run with varying changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The PD feedback  control system was used in each test, with a 50 ◦C setpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  We determined that the cartridge heaters consume a combined  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='96 W of power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' We then calculated the lossless amount of  heat required to raise the Melt Head from room temperature  to 50 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' From this number and power consumption, we  found the theoretical number of seconds that it would take  to reach this temperature without losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  For each test,  we measured the amount of time it took to reach 50 ◦C,  calculated the heat used, then compared this calculation to  the lossless heat required to raise the Melt Head to 50 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The  first test was conducted without any thermal paste, the second  with thermal paste between the Melt Head and Heating Plate,  and the final test with thermal paste between the Melt Head  and Heating Plate, as well as on each cartridge heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' As  expected, the final test yielded the highest thermal efficiency  with a value of 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' DISCUSSION  Limitations and Future Development  The MSIS instrument has thus far demonstrated melting of  a lab-made ice core sample in a benchtop setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Upon the  success of 3D printing the Melt Head, a future MSIS design  could involve inner fluidic channels with more complex ge-  ometry, with outlet ports on a single side of the instrument;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  this modification would make for a more organized system  in the lab and for future space missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The next Melt  Head design could include additional measures for preventing  contamination of the inner core sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' For instance, the  Melt Head partitioning walls could be made taller to ensure  separation of the central core from the outer portion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Also,  ultraviolet or infrared LEDs could be placed in a ring pattern  at the outer base of the Melt Head to melt the contaminated  outer region of the ice core before the inner regions, reducing  the risk of meltwater mixing across Melt Head cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The next critical step in instrument development would be  4  Thermistor 1  Thermistor 2  100kΩ  100kΩ  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='5V  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='5V  +12V  USB  Micro-  controller  NO  COM  NO  COM  NO  COM  NO  COM  NO  COM  Relay  Board  DC+  DC-  In1  In2  In3  In4  In5  D1  D2  D3  D4  D5  GND  +5V  Heater 5  Heater 4  Heater 3  Heater 2  Heater 1  AREF  Peristaltic Pump 2  Peristaltic Pump 1  +12V  Diaphragm Pump 2  Diaphragm Pump 1  A2  A1  +3V  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Melter Electronic Wiring Diagram  Sublimator cap  Sublimator  head/chamber  Solenoid  valves  Vacuum  lines  Moisture trap  barb adapter  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Sublimator hardware, excluding relay board  and electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Sublimator pictured without Melter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  processing a Greenland or Antarctic ice core sample in the lab  with MSIS, integrated with downstream analytical devices for  processing validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Upon such validation and necessary  modifications, the MSIS instrument could begin utilization in  the lab or in the field for Earth-based ice core studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  At its current state, MSIS requires human operation and  sample loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' A future vision is to make the MSIS system  autonomous and compatible with a rover platform exploring  ice deposits on other bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Automating a sublimator would  require robust monitoring and data acquisition capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  MSIS currently relies on a user judging when enough subli-  mation has taken place based on the size of the sublimated  ice sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  To process samples back-to-back, a cleaning  process would be needed within its fluid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' A one-size-  fits-all approach to cleaning debris of different sizes would be  challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' However, the Sublimator component of MSIS  does not limit itself to processing fully intact ice cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Considering the deposition of dust and debris on Mars,  subsurface Martian ice has the potential to be stratified like  permafrost ice deposits on Earth [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The drilling process  would inevitably accrue regolith in the ice cores which may  pose a problem to the MSIS plumbing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' In addition,  near-surface Mars ice could be more porous due to lower  gravity and there may be a commensurate increase in the  depth of pore close-off;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' this could lead to more fragile ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Rougher drilling techniques tend to produce ice “chips,”  5  A  B  C  D  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' (A) Sublimator being tested inside thermal vacuum chamber, full melter not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' (B) Sublimator inside  thermal vacuum chamber, melter present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' (C) Pressure-Temperature diagram showing conditions induced by the  Sublimator to begin the concentration sublimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' (D) Tuned vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' untuned Proportional-Derivative (PD)  temperature feedback results given a 50 ◦C setpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  which are fragments defined to have a 6 mm maximum  dimension [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' These chips lower the resolution and are  difficult to decontaminate but are still considered useful for  science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Sublimation is an experimentally inefficient process [14]  which does not pose an issue for benchtop experimentation  on Earth but complicates its context in a space instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The convenient presence of low atmospheric pressure on  Mars and other celestial bodies [13] may allow for more  efficient sublimation if it reduced or eliminated the need for  a vacuum pump, as is needed on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Without a vacuum  pump, the sublimator instrument would save upwards of 1  kW of power and consume an estimated 76 W at its maximum  power setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Up to 300 W of power consumption in  a space instrument is realistic for the capabilities of some  rovers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=', Mars Oxygen ISRU Experiment, or MOXIE,  instrument on Perseverance [15]) and serves as our metric  when characterizing the MSIS power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Integration with Future Human Lunar/Mars Missions  The long-term objective is for future versions of MSIS to  operate beyond Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' With the Artemis III crew proposed  to target regions near the Moon’s South Pole2, lunar water  ice analyses are likely to take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' MSIS could contribute  to Artemis III demonstrations of in-situ resource utilization  for pre-processing lunar water ice for oxygen and hydrogen  extraction to be used in fuel and life support systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This  would serve as a precursor to Mars missions with similar  aims in processing ice to serve as an ingredient for propellant  production while potentially searching for traces of life [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Also, in the context of preparing for the first human mission to  Mars, potentially to mid-latitude regions in the 2030s, NASA  has identified sampling subsurface ice as a critical science  requirement for searching for life and investigating the evo-  lution of Mars’ climate [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' If the Artemis III and Mars  crews have the means to retrieve and manually handle ice  core-like samples, MSIS—at its current state—could prove  beneficial in preparing lunar water ice samples for in-situ  2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='gov/press-release/nasa-identifies-candidate-regions-  for-landing-next-americans-on-moon  6  Vacuum  Lines  Sublimator  Melter  Fluidic LinesCopperHeating PlateSolid  760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='0  Liquid  On-Earth  storage  conditions  Pressure (Torr)  Triple Point  On-Mars  naturalstorage  conditions  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='0  Vapor  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='0  Sublimator-  induced conditions  200210243  273  Temperature (Kelvin)60  55  50  45  40  35  UntunedPD  Controller  Tuned PD  30  Controller  25  20  15  200  400  600  800  1000  1200  1400  Time (seconds)refueling demonstrations, as well as helping meet NASA’s  Mars human mission goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Volcanism, Climate, and Solar Activity from Mars Ice  An important caveat of our proposed analyses is that the  formation and preservation mechanisms for different ice de-  posits on Mars is under study and without ground truth [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  At present, this means the availability of contiguous ice core  records on Mars is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  This could complicate our  ability to establish a high-resolution contiguous history from  Mars ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Mars’ obliquity, the angle between its orbital plane and its  spin axis, is currently around 25◦, and it varies on a 120-  thousand-year (Ky) timescale, with the amplitude of this  cycling varying chaotically on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='2 My timescales [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  During periods of high obliquity, polar ice is mobilized and  falls as snow in mid-latitude and equatorial regions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This  process has resulted in repeated mid-latitude “ice ages” for  an estimated total of 680 My out of the last 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='6 billion years  (Gy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Thus, it is reasonable to assume that current remnants  of these ice ages are mainly locally-contiguous deposits, even  if a global contiguous record is not expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Detection of Recent Volcanism on Mars— Mars has been  geologically active until the recent past, and while no active  volcanism has been observed, it is unknown when it ceased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Ice cores on Earth are typically synchronized using volcanic  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Emissions of SO2 are oxidized to sulfuric acid and  measured in ice cores as acidity and sulfate (SO2–  4 ) pulses  above the climate background [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' The present-day atmo-  sphere of Mars is highly oxidizing and has been so for billions  of years, except during brief warmer and wetter reducing  periods [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Thus, volcanism on Mars could leave signatures  of volcanic events within Mars ice similar to those we analyze  on Earth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=', [23], [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Characterization of Mars Climate in the Late Amazonian—  Mars has been largely dry, cold, and oxidizing for the last  3 Gy, a period denoted as the Amazonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Ice core records  on Mars, potentially going back tens to hundreds of My  [1], [2], can help establish recent variability of the climate  during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' While the exquisite layering of the Mars  polar ice caps contains important climate information, these  deposits may only date to the last 4 My [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' While young,  polar deposits may offer a more contiguous, although dusty,  record during this recent period compared to mid-latitude ice  deposits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' However, these deposits are much more difficult  to access due to orbital mechanics and harsher conditions,  making mid-latitude ice a compelling target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Seeking Evidence of Subsurface Life— Life on Mars, if it  exists, could be related to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' A Mars origin for Earth life  could resolve significant inconsistencies between the inferred  history of life and Earth’s geologic history [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' If life got  started on Mars, there is little reason to expect it is now  extinct, given the significant overlap between the temperature  and pressure regimes occupied by life on Earth and those  available on Mars [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' In short, the most likely place for  life on Mars today is in the deep subsurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Given this,  and that current (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=', Curiosity, Perseverance) and future  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=', Rosalind Franklin) missions are limited to cm to m drill  depths, how could we sample the deep subsurface in the near  term, without the mass required to drill to km-scale depths?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  On Earth,  phreatomagmatic eruptions,  those involving  magma interacting with water, yield explosive events, which  can result in the dispersal of even large microorganisms such  as diatoms up to nearly 1000 km [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Evidence of such  eruptions has been identified on Mars, representing interac-  tions between magma and ground ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' For example, explosive  volcanism in Elysium Planitia [29] may have occurred as  recently as 46 to 222 Ky ago [30], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This region, home  to the InSight Lander, could thus possibly harbor active  volcanism, which could offer an abode for life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Therefore, explosive eruptions, as well as large impact events,  could eject life or signs of life from deeper within the Martian  crust, and their remnants could be deposited on accumulat-  ing ice sheets during periods of high obliquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Because  such explosive events are occurring up to the present day,  geologically speaking, such events have likely been ongoing  over the entire history of Mars mid-latitude ice emplacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Burial within accumulating ice deposits would enhance the  preservation of organics relative to long-term exposure at the  surface, due to shielding from space radiation and a lack of  thermal cycling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Such remnants of life, if present, would  likely be present at extremely low abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  However,  they could be concentrated, for example as described by [16]  during large-scale processing of ice for propellant production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Returned Samples for Cosmogenic Nuclide Analyses—Cos-  mogenic nuclides are isotopes produced by interactions be-  tween atoms and high energy galactic cosmic rays (GCR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  On Earth, Beryllium-10 (10Be) is used to infer historical  solar activity from its abundance in ice cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Cosmogenic  radionuclides can also reveal extreme solar events [32] and  nearby supernovas [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  On Earth, once 10Be forms and attaches to aerosols, it takes  1 to 2 years for deposition to occur [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Similar modeling  of transport and scavenging processes, as well as important  baseline measurements, would be required for Mars to inter-  pret a record of cosmogenic isotopes recovered from Mars ice  [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Recently, the potential for 10Be analysis from ice chips,  permitting much faster drilling, has been demonstrated [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  While the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='39 My half-life of 10Be permits its use through  the full range of ice ages on Earth, alternative cosmogenic  isotopes would be needed for older ice on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Candidates  include 53Mn, with a half-life of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='7 My, and 129I, with a  half-life of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='7 My.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  53Mn can be produced within supernova events, and has been  detected co-located with 60Fe, also produced during super-  novas, within marine sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This has been used to date  supernova events [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  53Mn is also produced terrestrially,  through cosmic ray bombardment and in-situ production [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The only important target element for 53Mn production in  situ is iron [37], which is abundant in the dust that undergoes  local to global aeolian transport on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Dust aloft might  experience higher GCR rates due to a lack of atmospheric  shielding, although this effect could be marginal due to  the relatively thin atmosphere of Mars (1% of Earth in the  geologically recent past).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Dust deposited on ice sheets would  continue experiencing cosmogenic production of 53Mn dur-  ing initial burial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  At the Greenland summit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=', GISP2  core), historical accumulation rates are from 15 to 40 cm/y  ice equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' On Mars, accumulation rates are likely much  lower, perhaps 10 mm/y [38], but are not well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  This means isotopic signals on Mars might be spread out over  broader (> annual) periods than on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  7  For even older ice, 129I analysis may be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  129I is  mainly produced by the fission of uranium, with production  by cosmic rays also contributing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' On Earth, production is  estimated to be 10−14 grams of 129I per gram of 127I [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  It has been proposed that Mars may have had a naturally-  occurring fission reactor in the Mare Acidalium region [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  If true, this could have resulted in an explosion on the order  of 1 Gy ago, which would have scattered radioactive iodine  across the planet’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  This could make it harder to  attribute radioactive iodine to cosmic ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  The most sensitive approach known to measure 53Mn and  129I is accelerator mass spectrometry (AMS) [41], [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Un-  fortunately, AMS facilities are large and complex and would  be unlikely candidates for in-situ analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Instead, ice could  be processed in situ, sublimated, with retention of isotopes  bound to particulates, and those residues could be returned  for AMS analysis on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Searching for rare isotopes might  require processing large volumes of ice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' for example, 500 kg  of Antarctic ice was analyzed to identify interstellar 60Fe by  AMS [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Enhancing Limit of Detection for Ocean Worlds  Based on analysis of nutrient-limited Earth environments,  modeling of energy limitations, and other factors, estimates  for cell abundances at Ocean Worlds such as Enceladus and  Europa may be as low or lower than 1000 to 100 cells/ml  [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Most instrumentation flown or under development may  not have adequate analytical limits of detection without con-  centration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Concentration approaches include filtration, flow  focusing, and sublimation, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' A prototype subli-  mator was previously developed and was able to demonstrate  the freeze-drying of a 5 ml sample in 5 hours at under 5 W  power [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' This was achieved by accounting for the available  cold temperatures and low pressure expected to be available  under Europa or Enceladus surface conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Even so,  sublimating large volumes of ice is likely infeasible due to  the high energy cost of sublimation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' we can expect to have  higher energy budgets on Mars where higher power operation  could process larger volumes via sublimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Cold and low-  pressure conditions are also available on Mars, where surface  pressure is 600 to 700 Pa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' a vacuum pump will be required  to maintain pressure below the triple point of water, 610 Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' CONCLUSIONS  From processing true ice cores to automating some capabil-  ities, there is much more subsystem development to occur  before MSIS is used with human or robotic missions to  Mars and icy bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Fortunately, MSIS has equally diverse  applications on Earth and its functionalities can be verified  with Earth-based ice samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' In the long-term, MSIS would  greatly simplify the processes needed for ice science in other  planetary environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Our initial work toward a space-  mission-compatible ice core processing instrument could  someday enable the examination of extraterrestrial ice cores  to help address fundamental and diverse questions in plane-  tary processes, habitability, astrobiology, and heliophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  Funded by faculty startup funds from the Georgia Institute of  Technology to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
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+page_content=' Jiang, “Ams measurement  of 53mn and its initial application at ciae,” Nuclear  Instruments and Methods in Physics Research Section  B: Beam Interactions with Materials and Atoms, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
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+page_content=' AMANO, “High sensitivity measure-  ment of iodine-129/iodine-127 ratio by accelerator mass  spectrometry,” Journal of Nuclear Science and Technol-  ogy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
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+page_content=' Kipfstuhl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
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+page_content='  [44] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
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+page_content=' He is a Re-  search Professor in the Division of Hy-  drologic Sciences at the Desert Research  Institute (DRI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Current NSF, NASA, and  internally funded research projects in-  clude ice core chemistry-based studies in Greenland, Antarc-  tica, and the Americas, using his group’s unique ultra-trace  ice core analytical laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Bruce Hammer received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' in  BiologyPhysics from the State Univer-  sity of New York in 1974, the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  in Biomedical Engineering from North-  western University in 1976, and the  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' in Biomedical Engineering from  Northwestern University in 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  He  is a Professor of Radiology & Adjunct  Professor of Entrepreneurial Studies at  the University of Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' His work  focuses on hardware development and its implementation in  support of biomedical and astrobiology applications, includ-  ing a NASA-funded sublimator device for Ocean Worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  Christopher E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' Carr received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  degree in Aero/Astro and Electrical En-  gineering in 1999, his Masters degree  in Aero/Astro in 2001, and his Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  degree in Medical Physics in 2005, all  from MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' He is an Assistant Profes-  sor in the Daniel Guggenheim School of  Aerospace Engineering and the School  of Earth and Atmospheric Sciences at  the Georgia Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content=' He  is broadly interested in searching for and expanding the  presence of life beyond Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
+page_content='  10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQf5_6l/content/2301.01869v1.pdf'}
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+arXiv:2301.01654v1  [math.NT]  4 Jan 2023
+A SELBERG TRACE FORMULA FOR GL3(Fp)\ GL3(Fq)/K
+DAKSH AGGARWAL, ASGHAR GHORBANPOUR, MASOUD KHALKHALI, JIYUAN LU, BALÁZS NÉMETH,
+AND C SHIJIA YU
+Abstract. In this paper, we prove a discrete analog of the Selberg Trace Formula for the group
+GL3(Fq). By considering a cubic extension of the finite field Fq, we define an analogue of the upper
+half space and an action of GL3(Fq) on it. To compute the orbital sums we explicitly identify the
+double coset spaces and fundamental domains in our upper half space. To understand the spectral
+side of the trace formula we decompose the induced representation ρ = IndG
+Γ 1 for G = GL3(Fq)
+and Γ = GL3(Fp).
+Contents
+1.
+Introduction
+1
+Acknowledgments
+2
+2.
+A finite upper half-space
+3
+3.
+Trace formula
+4
+3.1.
+The pre-trace formula
+4
+3.2.
+Double cosets and fundamental domains
+6
+4.
+Central and Hyperbolic Terms
+7
+4.1.
+Central terms
+7
+4.2.
+Hyperbolic terms of the first kind
+7
+4.3.
+Hyperbolic terms of the second kind
+8
+5.
+Parabolic Terms
+9
+5.1.
+Parabolic terms of the first kind
+9
+5.2.
+Parabolic terms of the second kind
+9
+5.3.
+Parabolic terms of the third kind
+10
+6.
+Elliptic Terms
+10
+6.1.
+Elliptic terms of the first kind
+10
+6.2.
+Elliptic terms of the second kind
+12
+7.
+Examples
+13
+8.
+Character of the representation ρ = IndG
+Γ 1
+14
+9.
+Decomposition of ρ when 2, 3 ∤ n
+15
+References
+22
+10.
+Appendix A: Fundamental Domains and Orbital Sums
+23
+1. Introduction
+The Selberg trace formula [7] is one of the most celebrated mathematical results of the past
+century. In its original form, it relates the spectrum of the Laplacian on a hyperbolic surface to
+lengths of closed geodesics on the surface. It can be viewed as a non-abelian generalization of the
+well-known Poisson summation formula in Fourier analysis and has found numerous applications in
+many fields, for example in number theory and the theory of automorphic forms (see Sarnak [6]).
+1
+
+The Selberg trace formula deals with surfaces which can be represented as double cosets of the
+form Γ\ SL2(R)/ SO2(R), where Γ is a discrete subgroup of SL2(R). It is a natural question to
+work out the trace formula for higher-dimensional matrix groups. To quote Dennis Hejhal: “Of
+course, we won’t really understand the trace formula until it is written down for SL4(Z).” Dorothy
+Wallace [10] has worked out the explicit details of the trace formula for SL3(Z)\ SL3(R)/ SO3(R)
+but the four-dimensional case still remains mysterious. A further generalization is the so-called
+Arthur-Selberg trace formula [3], which plays an important role in the Langlands program.
+Selberg type trace formulae have been derived for discrete spaces like graphs as well. By realizing
+a k-regular graph as a quotient of the infinite k-regular tree, Ahumada [1] found such a formula for
+k-regular graphs. Audrey Terras [9] introduced the concept of finite upper half-planes as a finite
+analogue of the Poincaré upper half-plane model where the complex number field is replaced by a
+quadratic extension of a finite field Fq, where q = pr is a prime power. She constructed a family of
+Ramanujan graphs [5] using these upper-half planes and developed a representation-theoretic trace
+formula on GL2(Fp)\ GL2(Fq)/K, where K is the finite analogue of SO2(R).
+The aim of this paper is to present an analogous trace formula for GL3(Fq). Our main result in
+this paper is as follows. Consider the pre-trace formula
+�
+π∈ ˆG
+m(π, ρ) Tr( ˆf(π)) =
+�
+{γ}∈CΓ
+|Gγ|
+|Γγ| IG(f, γ),
+(1.1)
+where G is a finite group, Γ is a subgroup of G, ρ = IndG
+Γ 1 is the induced representation of the
+trivial representation of Γ, and IG(f, γ) is the orbital sum of f at γ. These terms are defined in
+Section 3 and a proof of the pre-trace formula is recalled there as well. Note that (1.1) can be
+viewed as a representation-theoretic analogue of Selberg’s trace formula [7, 9]: the left hand side
+can be thought of as the spectral side, summing over the irreducible representations of G and their
+multiplicities in ρ, while the right hand side can be interpreted as the geometric side, summing over
+conjugacy classes in G. In this paper, to obtain a trace formula for the finite upper half-space we
+take G = GL3(Fq) and Γ = GL3(Fp). Also, f : G → C is chosen to be a K−bi-invariant function
+(K is the stabilizer of p0 as above), i.e. ∀k, h ∈ K and ∀x ∈ G we have f(kxh) = f(x). One can
+also think of f as a function on G/K ≃ Hq which is invariant under left multiplication by elements
+in K.
+This paper is organized as follows.
+In Section 2, we define an analogue, denoted Hq, of the
+upper half plane for the group GL3(Fq) by considering a cubic extension of the finite field Fq.
+We define an action of GL3(Fq) on Hq and this sets the stage for our discrete trace formula. In
+Section 3, we recall a standard pre-trace formula for finite groups. In Sections 4-6, we compute each
+term on the right hand side of the pre-trace formula (3.4) (the so called geometric side) using the
+explicit description of conjugacy classes of GL3(Fq) given in [2]. The conjugacy classes are separated
+into central, hyperbolic, parabolic, and elliptic terms. Each of these types of terms are explicitly
+identified. It would be helpful to have a more conceptual understanding of these computations. In
+Section 7, we give some simple examples of the application of our formula in which the left side of
+the pre-trace formula is known a priori. In Sections 8-9, we calculate the left side of the pre-trace
+formula by computing the character of the induced representation ρ = IndG
+Γ 1 and then using that to
+compute its decomposition into irreducible representations in the case that 2, 3 ∤ n. The calculation
+of the decomposition in the other cases is similar but involve more cases and hence they are not
+incorporated in the arXiv version of this paper. They are available on demand.
+Acknowledgments. We would like to thank the Fields Institue for organizing the Fields Under-
+graduate Summer Research Program in 2021. Daksh Aggarwal, Jiyuan Lu, Balazs Németh, and C
+Yu were participants in this program while working on this paper. The project was proposed and
+supervised by Masoud Khalkhali and Asghar Ghorbanpour.
+2
+
+2. A finite upper half-space
+To generalize Terras’s results to the GL3(Fq) case, we must first discuss the generalization of the
+finite upper half-space. Let p be a prime number and q = pn for a positive integer n. The definition
+of the upper half space Hq comes from considering the cubic extension of Fq. The construction was
+first given by Martinez in [4]. Here we describe it explicitly for our special case of GL3(Fq).
+Lemma 2.1. Fq has a cubic nonresidue if and only if q ≡ 1 (mod 3).
+Proof. Let q = pn. Consider the cube map x �→ x3 : F×
+q → F×
+q . This map has as kernel the subgroup
+µ3 consisting of the cube roots of unity in F×
+q . Let α be a generator of F×
+q . Then if q ≡ 1 (mod 3),
+we have µ3 = {1, α(q−1)/3, α2(q−1)/3}, in which case there are (q − 1)/3 cubic residues. Otherwise,
+µ3 = {1}, and so each element of F×
+q is a cubic residue.
+□
+From now on, we assume that q ≡ 1 (mod 3). Take a cubic nonresidue δ ∈ Fq and consider the
+cubic extension Fq
+�
+3√
+δ
+�
+≃ Fq3. A basis for Fq
+�
+3√
+δ
+�
+/Fq is {1, δ1/3, δ2/3}. For an element α, we
+will write (α1, α2, α3) to denote its components, i.e.,
+α = α1 + α2δ1/3 + α3δ2/3.
+We define
+Hq =
+�
+(α, β) ∈ Fq(
+3√
+δ) × Fq(
+3√
+δ) :
+����
+α2
+α3
+β2
+β3
+���� ̸= 0
+�
+.
+Lemma 2.2. The following map defines an action of GL3(Fq) on Hq :
+
+
+a
+b
+c
+d
+e
+f
+r
+s
+t
+
+ (α, β) =
+�aα + bβ + c
+rα + sβ + t, dα + eβ + f
+rα + sβ + t
+�
+.
+Proof. The compatibility of the action with multiplication in GL3(Fq) follows from the fact that
+GL3(Fq) has an action on the projective space P2(Fq3). So, we need to check that the action is
+closed in Hq. This is a routine verification, which can be found in Appendix A.
+□
+Our distinguished point will be p0 = (δ2/3, δ1/3) ∈ Hq. The action of Aff3(Fq) ≤ GL3(Fq) is
+transitive on Hq, where
+Aff3(Fq) :=
+
+
+
+
+
+a
+b
+c
+d
+e
+f
+0
+0
+1
+
+
+
+
+ .
+With this action, we calculate that
+K := stab(p0) =
+
+
+
+
+
+a
+cδ
+bδ
+b
+a
+cδ
+c
+b
+a
+
+
+
+
+ .
+Indeed, we find that
+K
+∼
+−→ Fq(δ1/3)× :
+
+
+a
+cδ
+bδ
+b
+a
+cδ
+c
+b
+a
+
+ �→ a + bδ1/3 + cδ2/3.
+Also, we have the homogeneous space Hq ≃ GL3(Fq)/K.
+As a quick check note that we have
+| GL3(Fq)| = (q3 − 1)(q3 − q)(q3 − q2) and |K| = q3 − 1 (from the canonical isomorphism), so we
+should have |Hq| equal to
+(q3 − 1)(q3 − q)(q3 − q2)
+q3 − 1
+= (q2 − 1)(q − 1)q3.
+3
+
+Indeed, from the definition we have
+|Hq| = | GL2(Fq)| · q2 = (q2 − 1)(q2 − q)q2 = (q2 − 1)(q − 1)q3.
+Remark 2.3. Observe that the above construction is a natural generalization of the finite upper
+half-plane defined by Terras [9] (here δ is a nonsquare in the finite field Fq):
+Hq =
+�
+x + y
+√
+δ : x, y ∈ Fq, y ̸= 0
+�
+.
+The condition y ̸= 0 is equivalent to 1 and x + y
+√
+δ being linearly independent elements of Fq
+�√
+δ
+�
+over Fq.
+3. Trace formula
+3.1. The pre-trace formula. Our trace formula relies on interpretation of a known result, the
+pre-trace formula. We now derive the pre-trace formula and contextualize it in our setting of the
+group G = GL3(Fq). The idea behind the pre-trace formula is to compute the trace of a specific
+operator in two different bases. We may construct the relevant information for this operator as
+follows:
+Suppose that G is a finite group and Γ ≤ G is a subgroup. Let f : G → C be a complex-valued
+function on G. For a representation κ : G → GL(V ) of G (where V is a finite-dimensional complex
+vector space endowed with a linear G-action), define the Fourier transform of f at κ by:
+ˆf(κ) =
+�
+g∈G
+f(g)κ
+�
+g−1�
+.
+Observe that the Fourier transform ˆf(κ) is End V -valued. The pre-trace formula is now obtained
+by computing the trace of ˆf(ρ) in two different bases with ρ = IndG
+Γ 1. First, we can decompose ρ
+as a direct sum of irreducible representations of G to get:
+Tr
+�
+ˆf(ρ)
+�
+=
+�
+π∈ ˆG
+m(π, ρ) Tr
+�
+ˆf(π)
+�
+.
+(3.1)
+Here ˆG is the set of irreducible representations of G and m(π, ρ) is the multiplicity of the irre-
+ducible representation π ∈ ˆG in ρ. Let
+V = {f : G → C | f(γg) = f(g)
+∀γ ∈ Γ, g ∈ G} = L2(Γ\G).
+As G acts on linearly on V by right multiplication, the corresponding representation is precisely
+ρ = IndG
+Γ 1. More concretely, for φ ∈ V , g, x ∈ G we have
+[ρ(g)φ] (x) = φ(xg).
+Thus:
+�
+ˆf(ρ)φ
+�
+(x) =
+�
+y∈G
+f(y)φ
+�
+xy−1�
+=
+�
+u∈G
+f
+�
+u−1x
+�
+φ(u).
+Further, as φ is Γ-invariant,
+�
+ˆf(ρ)φ
+�
+(x) =
+�
+y∈Γ\G
+�
+γ∈Γ
+f
+�
+y−1γx
+�
+φ(y).
+4
+
+Now we are ready to compute the trace of ˆf(ρ) in another way: consider the indicator basis
+{δy}y∈Γ\G of V 1; with respect to this basis, ˆf(ρ) is represented by a matrix where the entry corre-
+sponding to x, y ∈ Γ\G is �
+γ∈Γ f
+�
+y−1γx
+�
+. Therefore,
+Tr
+�
+ˆf(ρ)
+�
+=
+�
+x∈Γ\G
+�
+γ∈Γ
+f
+�
+x−1γx
+�
+.
+(3.2)
+We can rewrite the right-hand side as a sum over conjugacy classes in G. Let:
+Γγ =
+�
+x ∈ Γ | x−1γx = γ
+�
+= centralizer of γ in Γ
+Gγ =
+�
+x ∈ G | x−1γx = γ
+�
+= centralizer of γ in G
+{γ} =
+�
+x−1γx | x ∈ G
+�
+= conjugacy class of γ in G
+CΓ = set of conjugacy classes in Γ.
+Note that there is a 1-to-1 correspondence between right cosets Γγ\Γ and elements of {γ} given
+by Γγx �→ x−1γx. Since conjugacy classes partition Γ we can rewrite the inner sum as:
+�
+x∈Γ\G
+�
+γ∈Γ
+f(x−1γx) =
+�
+x∈Γ\G
+�
+{γ}∈CΓ
+�
+u∈Γγ\Γ
+f
+�
+x−1u−1γux
+�
+.
+Set y = ux, then we can rewrite this as a sum over right cosets y ∈ Γγ\G:
+�
+x∈Γ\G
+�
+γ∈Γ
+f(x−1γx) =
+�
+y∈Γγ\G
+�
+{γ}∈CΓ
+f(y−1γy).
+Observe that Γγ ≤ Gγ hence we can write y = ts for t ∈ Γγ\Gγ and s ∈ Gγ\G:
+�
+x∈Γ\G
+�
+γ∈Γ
+f(x−1γx) =
+�
+t∈Γγ\Gγ
+�
+s∈Gγ\G
+�
+{γ}∈CΓ
+f
+�
+s−1t−1γts
+�
+=
+�
+{γ}∈CΓ
+|Gγ|
+|Γγ|
+�
+s∈Gγ\G
+f(s−1γs).
+(3.3)
+Define the orbital sum IG(f, γ) of f at γ to be:
+IG(f, γ) :=
+�
+s∈Gγ\G
+f(s−1γs).
+The pre-trace formula now follows from combining (3.1)−(3.3):
+�
+π∈ ˆG
+m(π, ρ) Tr( ˆf(π)) =
+�
+{γ}∈CΓ
+|Gγ|
+|Γγ| IG(f, γ),
+(3.4)
+where ρ = IndG
+Γ 1 and IG(f, γ) is the orbital sum of f at γ as defined above. Note that (3.4) can
+be viewed as a representation-theoretic analogue of Selberg’s trace formula [7, 9]: the left hand side
+can be thought of as the spectral side, summing over the irreducible representations of G and their
+multiplicities in ρ, while the right hand side can be interpreted as the geometric side, summing over
+conjugacy classes in G. For further reference see [9, Ch. 22].
+To obtain the trace formula for the finite upper half-space we take G = GL3(Fq) and Γ = GL3(Fp).
+Also, f : G → C is chosen to be a K−bi-invariant function (K is the stabilizer of p0 as above), i.e.
+∀k, h ∈ K and ∀x ∈ G we have f(kxh) = f(x). One can also think of f as a function on G/K ≃ Hq
+which is invariant under left multiplication by elements in K.
+1For y ∈ Γ\G, δy is the function on Γ\G which outputs 1 if the input is y and 0 otherwise.
+5
+
+Remark 3.5. The left-hand side of (3.4) can be expressed solely in terms of the characters χπ of the
+group G:
+Tr
+�
+ˆf(π)
+�
+= Tr
+
+�
+g∈G
+f(g)π
+�
+g−1�
+
+ =
+�
+g∈G
+f(g) Tr
+�
+π
+�
+g−1��
+=
+=
+�
+g∈G
+f(g)χπ
+�
+g−1�
+=
+�
+g∈G
+f(g)χπ(g) = ⟨f, χπ⟩.
+Here ⟨x, y⟩ denotes the usual Hermitian inner product on the complex vector space L2(G):
+⟨x, y⟩ =
+�
+g∈G
+x(g)y(g)
+We will, in Sections 4-6, compute each term in the sum on the right hand side of the pre-
+trace formula (3.4) using the explicit description of conjugacy classes of GL3(Fp) given in [2]. The
+conjugacy classes are separated into the cases of central, hyperbolic, parabolic, and elliptic terms.
+It is helpful to introduce one more tool that will help us simplify these computations.
+3.2. Double cosets and fundamental domains. To calculate the orbital sums it will be con-
+venient to identify Gγ\G with Gγ\G/K × K/Z (when possible), where Z is the center of G, the
+subgroup of diagonal matrices {aI : a ∈ F×
+q }. Let {ti}l
+i=1 be representatives of double cosets in
+Gγ\G/K and consider the map:
+Gγ\G/K × K/Z → Gγ\G
+(GγtiK, kZ) �→ Gγtik.
+We first show that this map is well-defined and onto. If z ∈ Z then Gγtikz = Gγztik = Gγtik
+because z ∈ Gγ as z commutes with every element of G. Now let Gγx ∈ G\Gγ, then since double
+cosets GγtiK partition G and they are unions of right cosets Gγy we can find a representative ti ∈ G
+and k ∈ K such that Gγx = Gγtik.
+As for injectivity, let ti, tj be representatives of two double cosets in Gγ\G/K and k, k′ be
+representatives of two left cosets in K/Z. Suppose that Gγtik = Gγtjk′. Since double cosets are
+either disjoint or identical this is only possible if ti = tj, let t = ti. So, we have,
+Gγ = Gγtk′k−1t−1,
+(3.6)
+which happens if and only if tk′k−1t−1 ∈ Gγ, equivalently, if k′k−1 ∈ tGγt−1 ∩ K. This can also be
+stated as kH = k′H where H = tGγt−1 ∩ K. Note that Z ≤ K and Z ≤ tGγt−1 = Gtγt−1 as Z
+is the intersection of all centralizers in G, hence we always have Z ≤ K ∩ tGγt−1 = H. If Z = H,
+then (3.6) is equivalent to kZ = k′Z, which implies that the above map is injective. In our specific
+example, the centralizers Gγ contain no non-central elements conjugate to some matrix in K unless
+γ itself is conjugate to a member of K. (This can be seen e.g. by taking simple representatives γ
+of conjugacy classes in G and looking at the characteristic polynomials and eigenvalues of elements
+in Gγ, which are preserved by conjugation.)
+Thus we have two cases:
+Case 1: γ is not conjugate to an element of K. Then we have:
+IG(f, γ) =
+�
+s∈Gγ\G
+f(s−1γs) =
+�
+t∈Gγ\G/K
+�
+k∈K/Z
+f((tk)−1γtk) = |K/Z|
+�
+t∈Gγ\G/K
+f(t−1γt).
+6
+
+(The last equality follows from the fact that f is K−bi-invariant.) We can identify Gγ\G/K
+with Gγ\Hq. In order to calculate orbital sums we will find fundamental domains for the action of
+Gγ on Hq, i.e. subsets of Hq which contain exactly one element from each Gγ-orbit.
+Case 2: γ is conjugate to an element of K. In this case, the above map is not necessarily injective.
+However, Gγ = G or K depending on whether γ is conjugate to a central element of K or not. Then
+either Gγ\G is trivial or it can be identified with Hq.
+4. Central and Hyperbolic Terms
+4.1. Central terms. Let γ =
+
+
+a
+0
+0
+0
+a
+0
+0
+0
+a
+
+ for a ∈ F×
+p . In this case, Gγ = G and Γγ = Γ. So,
+IG(f, γ) = f(p0) and the total contribution from such γ is
+|G|
+|Γ| f(p0) · (p − 1) = (q3 − 1)(q2 − 1)(q − 1)q3
+(p3 − 1)(p2 − 1)p3
+f(p0).
+4.2. Hyperbolic terms of the first kind. Let γ =
+
+
+a
+0
+0
+0
+a
+0
+0
+0
+b
+
+ , a ̸= b ∈ F×
+p . Here
+Gγ =
+
+
+
+
+
+x
+y
+0
+w
+z
+0
+0
+0
+t
+
+
+
+
+ ≃ GL2(Fq) × F×
+q
+and similarly for Γγ.
+A fundamental domain for the action of Gγ is given by
+Gγ\Hq = {(u + δ1/3, v + δ2/3) : u, v ∈ Fq}.
+(4.1)
+We verify this in the next proposition for the sake of illustration. In the next sections, the verifici-
+ation of the fundamental domains is relegated to Appendix A.
+Proposition 4.2. A fundamental domain for Gγ is given by (4.1).
+Proof. First to show the uniqueness of each representative, suppose there exists m ∈ Gγ such that
+m · (u1 + δ1/3, v1 + δ2/3) = (u2 + δ1/3, v2 + δ2/3).
+Then we deduce that y = 0 = w, and x = t = z. So, u1 = u2 and v1 = v2.
+Next, to show the completeness of this fundamental domain, take an arbitrary element (α, β) ∈
+Hq. We wish to find m ∈ Gγ, u, v ∈ Fq such that m · (u + δ1/3, v + δ2/3) = (α, β). Set x = α2, y =
+α3, w = β2, z = β3, and t = 1. Then we see u, v is a solution to
+�
+α2
+α3
+β2
+β3
+� �
+u
+v
+�
+=
+�
+α1
+β1
+�
+,
+which has a solution since the left matrix is invertible.
+□
+7
+
+Thus,
+IG(f, γ) =
+�
+t∈Gγ\G/K
+�
+u∈K/{aI}
+f((tu)−1γ(tu))
+=
+�
+t∈Gγ\Hq
+|K|
+|{aI}|f(t−1γt)
+= q3 − 1
+q − 1
+�
+x,y∈Fq
+f
+
+
+
+
+0
+1
+−y
+1
+0
+−x
+0
+0
+1
+
+ γ
+
+
+0
+1
+x
+1
+0
+y
+0
+0
+1
+
+ p0
+
+
+= (q2 + q + 1)
+�
+x,y∈Fq
+f
+
+
+
+
+a
+0
+(a − b)y
+0
+a
+(a − b)x
+0
+0
+b
+
+ p0
+
+
+= (q2 + q + 1)
+�
+x,y∈Fq
+f
+�
+(x + a
+b δ2/3, y + a
+b δ1/3)
+�
+.
+Let us define the horocycle transform Hf : GL2(Fq) → C of f ∈ L2(K\G/K) by
+Hf(κ) :=
+�
+x,y∈Fq
+�
+ξ∈{κ}
+f
+
+
+
+
+ξ11
+ξ12
+x
+ξ21
+ξ22
+y
+0
+0
+1
+
+ p0
+
+ ,
+where the inner sum runs over ξ ∈ GL2(Fq) in the conjugacy class of κ. So, this hyperbolic term
+is equal to
+(q2 + q + 1)
+�
+a,b∈F×
+p
+a̸=b
+(q2 − 1)(q2 − q)(q − 1)
+(p2 − 1)(p2 − p)(p − 1) · Hf
+�a
+b I
+�
+= (q3 − 1)(q2 − 1)(q2 − q)
+(p2 − 1)(p2 − p)(p − 1)
+�
+a∈F×
+p
+a̸=1
+(p − 1) · Hf(aI)
+= (q3 − 1)(q2 − 1)(q2 − q)
+(p2 − 1)(p2 − p)
+�
+a∈F×
+p
+a̸=1
+Hf(aI).
+We have shown the full computation for the orbital sums for this case both to demonstrate the
+computational technique and to motivate and introduce the horocycle transform.
+For the next
+cases we omit detailed computations for orbital sums, but included some in Appendix A for the
+curious reader.
+4.3. Hyperbolic terms of the second kind. Let γ =
+
+
+a
+0
+0
+0
+b
+0
+0
+0
+c
+
+ with a, b, c ∈ F×
+p distinct. Here
+Gγ is the set of diagonal matrices and:
+Gγ\Hq = {(x + δ1/3 + yδ2/3, r + sδ1/3 + δ2/3) : ys ̸= 1} ⊔ {(x + δ2/3, r + δ1/3 + sδ2/3) : s ̸= 0}
+⊔ {(x + yδ1/3 + δ2/3, r + δ1/3) : y ̸= 0} ⊔ {(x + δ2/3, y + δ1/3)}
+8
+
+We have
+IG(f, γ) = (q2 + q + 1) · Hf
+��
+a/c
+0
+0
+b/c
+��
+,
+where a, b, c ∈ F×
+p comes from a change of variables outlined in Appendix A.
+The total contribution from the second hyperbolic term is:
+1
+6(q2 + q + 1)(q − 1)3
+(p − 1)3
+�
+a,b,c∈F×
+p
+a̸=b̸=c
+Hf
+��
+a/c
+0
+0
+b/c
+��
+= 1
+6(q2 + q + 1)(q − 1)3
+(p − 1)2
+�
+a,b∈F×
+p \{1}
+a̸=b
+Hf
+��
+a
+0
+0
+b
+��
+=
+= 1
+3(q3 − 1)(q − 1)2
+(p − 1)2
+�
+{a,b}⊆F×
+p \{1}
+Hf
+��
+a
+0
+0
+b
+��
+.
+5. Parabolic Terms
+5.1. Parabolic terms of the first kind. Let γ =
+
+
+a
+0
+a
+0
+a
+0
+0
+0
+a
+
+ , a ∈ F×
+p . In this case,
+Gγ =
+
+
+
+
+
+d
+y
+x
+0
+c
+b
+0
+0
+d
+
+ : c, d ∈ F×
+q , b, x, y ∈ Fq
+
+
+ .
+Proposition 5.1. A fundamental domain for Gγ is
+Gγ\Hq = {(uδ1/3, vδ1/3 + δ2/3) : u, v ∈ Fq, u ̸= 0} ⊔ {(δ1/3 + uδ2/3, δ1/3) : u ∈ F×
+q }.
+Proof. See Appendix A.
+□
+Following the computations outlined in Appendix A, the orbital sum in this case is
+IG(f, γ) = (q2 + q + 1) (Hf(I) − f(p0)) .
+And in total, the contribution from the parabolic terms of the first kind is
+q3(q − 1)2
+p3(p − 1)2 (q2 + q + 1) (Hf(I) − f(p0)) · (p − 1) = q3(q3 − 1)(q − 1)
+p3(p − 1)
+(Hf(I) − f(p0)) .
+5.2. Parabolic terms of the second kind. Let γ =
+
+
+a
+a
+0
+0
+a
+a
+0
+0
+a
+
+ , a ∈ F×
+p . We find that
+Gγ = B =
+
+
+
+
+
+z
+y
+x
+0
+c
+b
+0
+0
+d
+
+ : d, c, z ∈ F×
+q , b, x, y ∈ Fq
+
+
+ .
+Here B is the Borel subgroup of upper-triangular matrices.
+Proposition 5.2. A fundamental domain for B is
+B\Hq = {(δ1/3, vδ1/3 + δ2/3) : v ∈ Fq} ⊔ {(δ2/3, δ1/3)}.
+Proof. See Appendix A.
+□
+9
+
+We find the orbital sum (see Appendix A) is
+IG(f, γ) = (q2+q+1)
+
+f
+�
+δ2/3 + δ1/3, δ1/3 + 1
+�
++
+�
+v∈Fq
+f
+�
+(1 − v)δ2/3 − v2δ1/3 + 1, δ2/3 + (v + 1)δ1/3�
+
+ .
+Note that the orbital sum is independent of a, and so we denote each IG(f, γ) by ˜IG(f). Thus, in
+total, the contribution of these parabolic terms is
+q3(q − 1)2
+p3(p − 1)2 ˜IG(f) · (p − 1) = q3(q − 1)2
+p3(p − 1)
+˜IG(f).
+5.3. Parabolic terms of the third kind. Let γ =
+
+
+a
+0
+0
+a
+a
+0
+0
+0
+b
+
+, with a ̸= b ∈ F×
+p . Here
+Gγ =
+
+
+
+
+
+x
+0
+0
+y
+x
+0
+0
+0
+z
+
+ : x, z ∈ F×
+q , y ∈ Fq
+
+
+ .
+Again, via a change of variables, we find the orbital sum (see Appendix A)
+IG(f, γ) = (q2 + q + 1) · Hf
+��
+a/b
+1
+0
+a/b
+��
+.
+The overall contribution of these terms is
+�
+a,b∈F×
+p
+a̸=b
+q(q − 1)2
+p(p − 1)2 (q2 + q + 1) · Hf
+��
+a/b
+1
+0
+a/b
+��
+= q(q − 1)2
+p(p − 1)2 (q2 + q + 1)
+�
+c∈F×
+p
+c̸=1
+(p − 1) · Hf
+��
+c
+1
+0
+c
+��
+= q(q − 1)(q3 − 1)
+p(p − 1)
+�
+c∈F×
+p
+c̸=1
+Hf
+��
+c
+1
+0
+c
+��
+.
+6. Elliptic Terms
+6.1. Elliptic terms of the first kind. Irreducible γ. The characteristic polynomial of such a γ is
+irreducible in Fp[t]. Recall that q = pn. Depending on whether n ≡ 0 (mod 3), there are two cases
+to consider.
+• n ≡ 0 (mod 3). In this case, we have the tower of field extensions Fq/Fp3/Fp. So, γ is
+similar to a diagonal matrix in GL3(Fq). But that is not the case in GL3(Fp), which is
+why this case is different from the second hyperbolic term. Therefore Gγ is the subgroup of
+diagonal matrices over Fq while Γγ is K with entries in Fp, i.e. |Γγ| = p3 −1. Since Gγ is the
+same, we can reuse the fundamental domain for Gγ from earlier, in the second hyperbolic
+case.
+Suppose γ is similar to
+� ξ1 0
+0
+0 ξ2 0
+0
+0 ξ3
+�
+in GL3(Fq), where ξ1, ξ2, ξ3 ∈ Fp3\Fp. Similarly as for
+the second hyperbolic term the orbital sum will be:
+IG(f, γ) = |K/Z| · Hf
+��
+ξ1/ξ3
+0
+0
+ξ2/ξ3
+��
+.
+Since ξ1, ξ2, ξ3 are Galois conjugates without loss of generality ξ1 = ξp
+3 and ξ2 = ξp2
+3 . The
+total contribution of the first elliptic terms is therefore:
+10
+
+1
+3 · (q − 1)3
+p3 − 1 · q3 − 1
+q − 1
+�
+ξ∈F×
+p3\F×
+p
+Hf
+��ξp−1
+0
+0
+ξp2−1
+��
+.
+(The factor of 1
+3 comes from the fact that each γ is counted three times in the sum.)
+• n ̸≡ 0 (mod 3). In this case, there is no cubic extension intermediate to Fq/Fp, and so
+Fq(δ1/3) is the minimal field (of course, in the sense of containment, not size) over which the
+characteristic polynomial of γ has a root. Suppose an eigenvalue of γ is α1 +α2δ1/3 +α3δ2/3
+with eigenvector x1 + x2δ1/3 + x3δ2/3, where αi ∈ Fq and xi ∈ F3
+q. Then,
+γx1 = α1x1 + α3δx2 + α2δx3,
+γx2 = α2x1 + α1x2 + α3δx3,
+γx3 = α3x1 + α2x2 + α1x3.
+Thus, viewed in GL3(Fq), γ is similar to a matrix in K. Conversely, non-diagonal matrices
+in K give us irreducible elements in GL3(Fp). Further, in this case Gγ is precisely K, and
+so Gγ\G = Hq. After identifying elements y ∈ Hq with elements y =
+
+
+a
+b
+c
+d
+e
+f
+0
+0
+1
+
+ ∈ Aff3(Fq)
+and writing γ =
+
+
+α
+εδ
+βδ
+β
+α
+εδ
+ε
+β
+α
+
+, we can compute:
+IG(f, γ) =
+�
+y∈Hq
+f(y−1γy)
+=
+�
+a,b,c,d,e,f∈Fq
+f
+
+
+1
+bd − ae
+
+
+−e
+b
+ce − bf
+d
+−a
+af − cd
+0
+0
+bd − ae
+
+
+
+
+α
+εδ
+βδ
+β
+α
+εδ
+ε
+β
+α
+
+
+
+
+a
+b
+c
+d
+e
+f
+0
+0
+1
+
+ p0
+
+
+=
+�
+a,b,c,d,e,f∈Fq
+f
+
+
+1
+bd − ae
+
+
+x11
+x12
+x13
+x21
+x22
+x23
+x31
+x32
+x33
+
+ p0
+
+ ,
+where the entries of the matrix, by column, are as follows:
+⋄ x11 = α(bd − ae) + β(ab + cde − bdf) + ε(ace − abf − deδ)
+⋄ x21 = β(adf − a2 − cd2) + ε(a2f − acd + d2δ)
+⋄ x31 = βd + εa
+⋄ x12 = β(ce2 − bef + b2) + ε(bce − b2f − e2δ)
+⋄ x22 = α(bd − ae) + β(aef − cde − ab) + ε(abf − bcd + deδ)
+⋄ x32 = βe + εb
+⋄ x13 = β(cef + bc − bf 2 − eδ)
+⋄ x23 = β(af 2 − cdf − ac + dδ) + ε(acf − ac2 + dfδ − aδ)
+⋄ x33 = α + βf + εc
+Unfortunately, there does not seem to be an obvious way to simplify this expression further.
+When computing this case for a specific field Fq, it seems that applying this formula may
+not be more helpful than directly computing.
+11
+
+6.2. Elliptic terms of the second kind. γ =
+
+
+a
+b
+0
+c
+d
+0
+0
+0
+e
+
+, where g =
+�
+a
+b
+c
+d
+�
+is irreducible, and
+e ∈ F×
+p . Also,
+Γγ =
+
+
+
+
+
+a
+bξ
+0
+b
+a
+0
+0
+0
+c
+
+ : c(a2 − b2ξ) ̸= 0
+
+
+ ,
+where ξ is a nonsquare in Fp.
+Once again, depending on whether n ≡ 0 (mod 2), i.e. whether n is even or odd, there are two
+cases to consider.
+• n is even. In this case, Fp ⊂ Fp2 ⊆ Fq and γ is diagonalizable in GL3(Fq). Gγ is a subgroup
+of diagonal matrices over Fq, so the fundamental domain Gγ\Hq is the same as for the
+second hyperbolic case.
+Suppose γ is similar to
+� η1 0 0
+0 η2 0
+0
+0 e
+�
+in G, where η1, η2 ∈ Fp2\Fp and e ∈ F×
+p . The orbital
+sum will be (analogously to the second hyperbolic case):
+IG(f, γ) = |K/Z| · Hf
+��
+η1/e
+0
+0
+η2/e
+��
+.
+As η1, η2 are Galois conjugates without loss of generality we may assume that η1 = ηp
+2.
+Therefore the total contribution of the second elliptic terms will be:
+1
+2 ·
+(q − 1)3
+(p − 1)(p2 − 1) · q3 − 1
+q − 1
+�
+e∈F×
+p
+�
+η∈F×
+p2\F×
+p
+Hf
+��
+η/e
+0
+0
+ηp/e
+��
+=
+= 1
+2 · (q − 1)3
+(p2 − 1) · q3 − 1
+q − 1
+�
+η∈F×
+p2\F×
+p
+Hf
+��
+η
+0
+0
+ηp
+��
+.
+• n is odd. In this case γ would not be diagonalizable in GL3(Fq). In general, it will be
+conjugate to an element of the form:
+
+
+k
+lξ
+0
+l
+k
+0
+0
+0
+m
+
+ ,
+where k ∈ Fp, m, l, ξ ∈ F×
+p such that ξ is a nonsquare in Fp.
+(This ensures that the
+determinant m(k2 − l2ξ) is nonzero.) In this case we have:
+Gγ =
+
+
+
+
+
+a
+bξ
+0
+b
+a
+0
+0
+0
+c
+
+ : a, b, c ∈ Fq, c(a2 − b2ξ) ̸= 0
+
+
+ .
+We seek a fundamental domain for this specific Gγ.
+Proposition 6.1. A fundamental domain for Gγ is given by:
+Gγ\Hq =
+��
+x + uδ1/3 + vδ2/3, y + δ1/3�
+: v ∈ F×
+q , x, y, u ∈ Fq
+�
+.
+The corresponding orbital sum is
+IG(f, γ) = (q3 − 1)(q2 − 1)
+(p − 1)(p2 − 1) Hf
+��
+k/m
+lξ/m
+l/m
+k/m
+��
+.
+12
+
+The total contribution from the second elliptic terms in this case is
+(q3 − 1)(q2 − 1)
+(p2 − 1)
+�
+s∈Fp
+t∈F×
+p
+Hf
+��
+s
+tξ
+t
+s
+��
+.
+A detailed proof of both the proposition and the orbital sum calculation can be found in
+Appendix A.
+7. Examples
+Throughout this section, subscripts under multiplicities indicate which group is the domain of
+the corresponding representations. Set ρ = IndG
+Γ 1 and κ = IndG
+K 1.
+Example 7.1. Let f : G → C be the constant one function. The right-hand side of the trace
+formula will be |G| while the left-hand side becomes (by orthogonality of characters):
+�
+π∈ ˆG
+m(π, ρ)G⟨f, χπ⟩ =
+�
+π∈ ˆG
+m(π, ρ)G⟨χ1, χπ⟩ = |G| · m(1, ρ)G.
+So the trace formula says |G| · m(1, ρ) = |G| which is a manifestation of Frobenius reciprocity:
+m(1, ρ)G =
+1
+|G|⟨1, IndG
+Γ 1⟩G = 1
+|Γ|⟨ResG
+Γ 1, 1⟩Γ = m(1, 1)Γ = 1.
+Example 7.2. Let f : Hq → C be the indicator function δp0 which is 1 on p0 and zero otherwise.
+It is K−bi-invariant because stab(p0) = K. On the left-hand side of the trace formula we have:
+�
+π∈ ˆG
+m(π, ρ)G⟨f, χπ⟩ =
+�
+π∈ ˆG
+m(π, ρ)G
+�
+g∈G
+f(g · p0)χπ
+�
+g−1�
+=
+�
+π∈ ˆG
+m(π, ρ)G
+�
+k∈K
+χπ
+�
+k−1�
+=
+=
+�
+π∈ ˆG
+m(π, ρ)G⟨1, χπ⟩K = |K| ·
+�
+π∈ ˆG
+m(π, ρ)G · m(1, ResG
+K π)K =
+= |K| ·
+�
+π∈ ˆG
+m(π, ρ)G · m(π, κ)G.
+On the right-hand side, the only nonzero terms are the central terms and the first elliptic terms
+when n is not divisible by 3. If that is the case, we claim that for a non-central γ in K we have
+IG(f, γ) = 3.
+Suppose γ corresponds to multiplication by ξ ∈ Fq
+�
+δ1/3�× under the isomorphism K ≃ Fq
+�
+δ1/3�×.
+When viewed as an element of GL3(Fq), the eigenvalues of γ are the Galois conjugates of ξ, which
+are ξ, ξq and ξq2 in our setting. This means that the conjugates of γ lying in K are precisely γq and
+γq2, hence there will be exactly three non-zero terms in each orbital sum IG(f, γ).
+The total contribution of the first elliptic terms will be:
+1
+3 · q3 − 1
+p3 − 1
+�
+γ∈K
+γ̸∈Z
+IG(f, γ) = (q3 − 1)(q3 − q)
+p3 − 1
+.
+Thus we get:
+Case 1: n is divisible by 3. Then we have
+�
+π∈ ˆG
+m(π, ρ)G · m(π, κ)G = q3(q2 − 1)(q − 1)
+p3(p2 − 1)(p3 − 1).
+13
+
+Case 2: n is not divisible by 3. Then we have
+�
+π∈ ˆG
+m(π, ρ)G · m(π, κ)G = q3(q2 − 1)(q − 1)
+p3(p2 − 1)(p3 − 1) + q3 − q
+p3 − 1.
+Remark 7.3. Note that the sum on the left-hand side can also be written as:
+�
+π∈ ˆG
+m(π, ρ)G · m(π, κ)G =
+1
+|G|⟨χρ, χκ⟩.
+8. Character of the representation ρ = IndG
+Γ 1
+Having effectively computed the right-hand side of the pre-trace formula (3.4), we now turn to
+the left-hand side. For this, we require the character χρ of ρ, which we now calculate. We use the
+usual formula for the character of an induced representation (see [9, Ch. 16]):
+χρ(γ) = 1
+|Γ|
+�
+x∈G
+1Γ(xγx−1).
+We treat each conjugacy class. By H1,q, H2,q, P1,q, . . . , E2,q we denote the size of various types of
+conjugacy classes in GL3(Fq). Given a set S, we use
+1S to denote the indicator function for S.
+• Central class: γ =
+
+
+a
+0
+0
+0
+a
+0
+0
+0
+a
+
+ , a ∈ F×
+q . Since γ ∈ Z(G), we have xγx−1 = γ. So,
+χρ(γ) = |G|
+|Γ|
+1Fp(a).
+• First hyperbolic class: γ =
+
+
+a
+0
+0
+0
+a
+0
+0
+0
+b
+
+ with a ̸= b ∈ F×
+q . Now, γ is similar to an element of
+Γ only when a, b are roots of a polynomial over Fp. This occurs only when a, b ∈ F×
+p ; for
+instance, this can be seen by considering the determinant Det(γ) = a2b. Being the constant
+coefficient of charγ ∈ Fp[x], we have Det(γ) ∈ Fp, which implies Det(γ) is invariant under
+the Frobenius automorphism (as Gal(Fq/Fp) is generated by the Frobenius). Since Fq/Fp
+is Galois, if a, b ̸∈ Fp, they must be Galois conjugates, i.e., ap = b and bp = a. But this
+forces a2b = (a2b)p = b2a, i.e., a = b, a contradiction. So, by a simple application of the
+orbit-stabilizer theorem, it follows that
+χρ(γ) = |G|
+|Γ|
+H1,p
+H1,q
+1Γ(γ).
+• Second hyperbolic class: γ =
+
+
+a
+0
+0
+0
+b
+0
+0
+0
+c
+
+ with a, b, c ∈ F×
+q distinct. There are three cases
+which allow γ to be similar to an element of Γ: either all three are Galois conjugates in
+Fp3/Fp (only possible if 3 | n), or a, b are Galois conjugates in Fp2/Fp and c ∈ Fp (only
+possible if 2 | n), or all three belong to Fp. These cases give rise to the three terms in the
+sum:
+χρ(γ) =
+|G|
+|Γ| · H2,q
+�
+E2,p ·
+1Fp3\Fp(a)δb(ap)δc(ap2) + E1,p ·
+1Fp2\Fp(a)δb(ap)1Fp(c) + H2,p ·
+1Γ(γ)
+�
+.
+14
+
+• First parabolic class: γ =
+
+
+a
+1
+0
+0
+a
+0
+0
+0
+a
+
+ , a ∈ F×
+q . If a ̸∈ Fp was the root of a polynomial over
+Fp, its Galois conjugate ap ̸= a would also be a root, which is clearly not the case. So,
+χρ(γ) = |G|
+|Γ|
+P1,p
+P1,q
+1Fp(a).
+• Second parabolic class: γ =
+
+
+a
+1
+0
+0
+a
+1
+0
+0
+a
+
+ , a ∈ F×
+q . Similar to the previous class,
+χρ(γ) = |G|
+|Γ|
+P2,p
+P2,q
+1Fp(a).
+• Third parabolic class: γ =
+
+
+a
+1
+0
+0
+a
+0
+0
+0
+b
+
+ with a ̸= b ∈ F×
+q . By a similar argument as for the
+first hyperbolic class, we have
+χρ(g) = |G|
+|Γ|
+P3,p
+P3,q
+1Γ(g).
+• First elliptic class: γ =
+
+
+w
+0
+0
+0
+wq
+0
+0
+0
+r
+
+ with w ∈ Fq2 \ Fq and r ∈ F×
+q . Here γ is similar to
+an element of Γ if and only if w, wq are roots of a degree-2 irreducible polynomial over Fp.
+However, if 2 | n, then Fp2 is an intermediate extension in Fq/Fp and so roots of any degree-2
+polynomial over Fp are included in Fq. Thus, χρ(γ) = 0 if n is divisible by 2, otherwise
+χρ(γ) = |G|
+|Γ|
+E1,p
+E1,q
+1Fp2\Fp(w)1Fp(r).
+• Second elliptic class: γ =
+
+
+w
+0
+0
+0
+wq
+0
+0
+0
+wq2
+
+ , w ∈ Fq(δ1/3) \ Fq. As for the previous class,
+χρ(γ) = 0 if n is divisible by 3, otherwise
+χρ(γ) = |G|
+|Γ|
+E2,p
+E2,q
+1Fp3\Fp(w).
+9. Decomposition of ρ when 2, 3 ∤ n
+In this section, we compute the decomposition of ρ when neither 2 nor 3 divide n; thus elliptic
+elements in GL3(Fp) remain elliptic in GL3(Fq).
+We use the character table for GL3(Fq) in Terras [9, pp. 383-4] and Steinberg [8]. As usual, to
+calculate multiplicity m(π, ρ) of an irreducible representation π present in ρ, we compute the inner
+product of their respective characters. By H1, H2, P1 . . . , E2 we denote the size of different types of
+conjugacy classes in GL3(Fp). By NmE/F, we denote the norm of the field extension E/F.
+15
+
+(1) α, a character of F×
+q . Then,
+m(α, ρ) = 1
+|Γ|
+� �
+a∈F×
+p
+α(a3) + H1
+�
+a,b∈F×
+p
+a̸=b
+α(a2b) + H2
+6
+�
+a,b,c∈F×
+p
+a,b,c distinct
+α(abc) + P1
+�
+a∈F×
+p
+α(a3) + P2
+�
+a∈F×
+p
+α(a3)
++ P3
+�
+a,b∈F×
+p
+a̸=b
+α(a2b) + E1
+2
+�
+w∈Fp2\Fp
+r∈F×
+p
+α(r NmFq2/Fq(w)) + E2
+3
+�
+w∈Fp3\Fp
+α(NmFq3/Fq(w))
+�
+.
+We can simplify these sums on the basis of three cases:
+• α3 |F×
+p ̸= 1. Each of the sums is zero and so m(α, ρ) = 0.
+• α3 |F×
+p = 1 but α |F×
+p ̸= 1. A short calculation yields
+m(α, ρ) = 1
+|Γ|
+�
+1 − H1 + H2
+3 + P1 + P2 − P3 − E2
+3
+�
+(p − 1) = 0.
+• α |F×
+p = 1. Then simply
+m(α, ρ) = 1
+|Γ|
+�
+{g}∈CΓ
+|{g}| = 1.
+(2) πα, where α is a character of F×
+q .
+m(πα, ρ) = 1
+|Γ|
+�
+(q2 + q)
+�
+a∈F×
+p
+α(a3) + (q + 1)H1
+�
+a,b∈F×
+p
+a̸=b
+α(a2b) + H2
+3
+�
+a,b,c∈F×
+p
+a,b,c distinct
+α(abc) + qP1
+�
+a∈F×
+p
+α(a3)
++ 0 + P3
+�
+a,b∈F×
+p
+a̸=b
+α(a2b) + 0 − E2
+3
+�
+w∈Fp3\Fp
+α(NmFq3/Fq(w))
+�
+.
+• α3 |F×
+p ̸= 1. Here again all the sums are zero, giving m(πα, ρ) = 0.
+• α3 |F×
+p = 1 but α |F×
+p ̸= 1.
+m(πα, ρ) = 1
+|Γ|
+�
+(q2 + q) − (q + 1)H1 + 2
+3H2 + qP1 − P3 + E2
+3
+�
+(p − 1)
+=
+(q − p)(q − p2)
+p3(p − 1)2(p + 1)(p2 + p + 1).
+• α |F×
+p = 1.
+m(πα, ρ) = 1
+|Γ|
+�
+(q2 + q)(p − 1) + (q + 1)H1(p − 1)(p − 2) + (p − 1)(p − 2)(p − 3)H2
+3
++ qP1(p − 1) + (p − 1)(p − 2)P3 − E2
+3 (p3 − p)
+�
+=
+(q − p)(q + p5 − 2p2)
+p3(p − 1)2(p + 1)(p2 + p + 1).
+As a sanity check, showing this is actually an integer is not very hard.
+16
+
+(3) π′
+α, where α is a character of F×
+q .
+m(π′
+α, ρ) = 1
+|Γ|
+�
+q3 �
+a∈F×
+p
+α(a3) + qH1
+�
+a,b∈F×
+p
+a̸=b
+α(a2b) + H2
+6
+�
+a,b,c∈F×
+p
+a,b,c distinct
+α(abc) + 0 + 0 + 0
+− E1
+2
+�
+w∈Fp2\Fp
+r∈F×
+p
+α(r NmFq2/Fq(w)) + E2
+3
+�
+w∈Fp3\Fp
+α(NmFq3/Fq(w))
+�
+.
+• α3 |F×
+p ̸= 1. m(πα, ρ) = 0.
+• α3 |F×
+p = 1 but α |F×
+p ̸= 1.
+m(π′
+α, ρ) = 1
+|Γ|
+�
+q3 − qH1 + 1
+3H2 − E2
+3
+�
+(p − 1) =
+(q − p)(q − p2)(q + p2 + p)
+p3(p − 1)2(p + 1)(p2 + p + 1).
+• α |F×
+p = 1.
+m(π′
+α, ρ) = 1
+|Γ|
+�
+q3(p − 1) + q(p − 1)(p − 2)H1 + (p − 1)(p − 2)(p − 3)H2
+6
+− (p2 − p)(p − 1)E1
+2 + (p3 − p)E2
+3
+�
+= (q − p)(q2 + pq + p5 − p4 − p3 − p2)
+p3(p − 1)2(p + 1)(p2 + p + 1)
+.
+(4) πα,β for distinct characters α and β of F×
+q .
+m(πα,β, ρ) = 1
+|Γ|
+�
+(q2 + q + 1)
+�
+a∈F×
+p
+α(a2)β(a) + H1
+�
+a,b∈F×
+p
+a̸=b
+((q + 1)α(ab)β(a) + α(a2)β(b))
++ H2
+6
+�
+a,b,c∈F×
+p
+a,b,c distinct
+(β(a)α(bc) + β(b)α(ac) + β(c)α(ab)) + (q + 1)P1
+�
+a∈F×
+p
+α(a2)β(a)
++ P2
+�
+a∈F×
+p
+α(a2)β(a) + P3
+�
+a,b∈F×
+p
+a̸=b
+(α(ab)β(a) + α(a2)β(b))
++ E1
+2
+�
+w∈Fp2\Fp
+r∈F×
+p
+α(NmFq2/Fq(w))β(r) + 0
+�
+.
+• α2β |F×
+p ̸= 1. m(πα,β, ρ) = 0.
+17
+
+• α2β |F×
+p = 1 but α2 |F×
+p ̸= 1 and β |F×
+p ̸= 1.
+m(πα,β, ρ) = 1
+|Γ|
+�
+(q2 + q + 1)(p − 1) − (q + 2)(p − 1)H1 + (p − 1)H2
++ (q + 1)(p − 1)P1 + (p − 1)P2 − 2(p − 1)P3
+�
+=
+(q − p)(q − p2)
+p3(p − 1)2(p + 1)(p2 + p + 1).
+• α2 |F×
+p = 1 and β |F×
+p = 1 but α |F×
+p ̸= 1.
+m(πα,β, ρ) = 1
+|Γ|
+�
+(q2 + q + 1)(p − 1) − (p − 1)(q − p + 3)H1 − (p − 1)(p − 3)H2
+2
++ (q + 1)(p − 1)P1 + (p − 1)P2 + (p − 1)(p − 3)P3 − (p − 1)2 E1
+2
+�
+=
+(q − p)(q − p2)
+p3(p − 1)2(p + 1)(p2 + p + 1).
+• α |F×
+p = 1 and β |F×
+p = 1.
+m(πα,β, ρ) = 1
+|Γ|
+�
+(q2 + q + 1)(p − 1) + (q + 2)(p − 1)(p − 2)H1 + (p − 1)(p − 2)(p − 3)H2
+2
++ (q + 1)(p − 1)P1 + (p − 1)P2 + 2(p − 1)(p − 2)P3 + (p − 1)(p2 − p)E1
+2
+�
+= q(q + p5 − 2p2 − p) + p3(p5 − 2p3 − p2 + 3)
+p3(p − 1)2(p + 1)(p2 + p + 1)
+.
+(5) π′
+α,β for distinct characters α and β of F×
+q .
+m(π′
+α,β, ρ) = 1
+|Γ|
+�
+q(q2 + q + 1)
+�
+a∈F×
+p
+α(a2)β(a) + H1
+�
+a,b∈F×
+p
+a̸=b
+((q + 1)α(ab)β(a) + qα(a2)β(b))
++ H2
+6
+�
+a,b,c∈F×
+p
+a,b,c distinct
+(β(a)α(bc) + β(b)α(ac) + β(c)α(ab)) + qP1
+�
+a∈F×
+p
+α(a2)β(a)
++ 0 + P3
+�
+a,b∈F×
+p
+a̸=b
+α(ab)β(a) − E1
+2
+�
+w∈Fp2\Fp
+r∈F×
+p
+α(NmFq2/Fq(w))β(r) + 0
+�
+.
+• α2β |F×
+p ̸= 1. m(π′
+α,β, ρ) = 0.
+18
+
+• α2β |F×
+p = 1 but α2 |F×
+p ̸= 1 and β |F×
+p ̸= 1.
+m(π′
+α,β, ρ) = 1
+|Γ|
+�
+q(q2 + q + 1)(p − 1) − (2q + 1)(p − 1)H1 + (p − 1)H2
++ q(p − 1)P1 − (p − 1)P3
+�
+= (q − p)(q − p2)(q + p2 + p + 1)
+p3(p − 1)2(p + 1)(p2 + p + 1) .
+• α2 |F×
+p = 1 and β |F×
+p = 1 but α |F×
+p ̸= 1.
+m(πα,β, ρ) = 1
+|Γ|
+�
+q(q2 + q + 1)(p − 1) + (p − 1)(qp − 3q − 1)H1 − (p − 1)(p − 3)H2
+2
++ q(p − 1)P1 − (p − 1)P3 + (p − 1)2 E1
+2
+�
+= (q − p)(q(q + p + 1) + p2(p3 − p2 − p − 2))
+p3(p − 1)2(p + 1)(p2 + p + 1)
+.
+• α |F×
+p = 1 and β |F×
+p = 1.
+m(πα,β, ρ) = 1
+|Γ|
+�
+q(q2 + q + 1)(p − 1) + (2q + 1)(p − 1)(p − 2)H1 + (p − 1)(p − 2)(p − 3)H2
+2
++ q(p − 1)P1 + (p − 1)(p − 2)P3 − (p − 1)(p2 − p)E1
+2
+�
+= (q − p)(q(q + p + 1) + p2(2p − 3)(p2 + p + 1))
+p3(p − 1)2(p + 1)(p2 + p + 1)
+.
+(6) πα,β,γ for distinct characters α, β, γ of F×
+q .
+m(πα,β,γ, ρ) = 1
+|Γ|
+�
+(q + 1)(q2 + q + 1)
+�
+a∈F×
+p
+α(a)β(a)γ(a) + H2
+�
+a,b,c∈F×
+p
+a,b,c distinct
+α(a)β(b)γ(c)
++ (q + 1)H1
+�
+a,b∈F×
+p
+a̸=b
+(α(b)β(a)γ(a) + β(b)γ(a)α(a) + γ(b)α(a)β(a))
++ (2q + 1)P1
+�
+a∈F×
+p
+α(a)β(a)γ(a) + P2
+�
+a∈F×
+p
+α(a)β(a)γ(a)
++ P3
+�
+a,b∈F×
+p
+a̸=b
+(α(b)β(a)γ(a) + β(b)γ(a)α(a) + γ(b)α(a)β(a)) + 0 + 0
+�
+.
+• αβγ |F×
+p ̸= 1. m(πα,β,γ, ρ) = 0.
+19
+
+• αβγ |F×
+p = 1 but none of α, β, or γ restrict to the trivial character on F×
+p .
+m(πα,β,γ, ρ) = 1
+|Γ|
+�
+(q + 1)(q2 + q + 1)(p − 1) + 2(p − 1)H2 − 3(q + 1)(p − 1)H1
++ (2q + 1)(p − 1)P1 + (p − 1)P2 − 3(p − 1)P3
+�
+= (q − p)(q − p2)(q + p2 + p + 2)
+p3(p − 1)2(p + 1)(p2 + p + 1) .
+• αβ |F×
+p = 1 and γ |F×
+p = 1 but α and β are nontrivial on F×
+p . Note that since m(πα,β,γ, ρ)
+is symmetric in α, β, γ, the calculation here also holds for the other two permutations
+of this case.
+m(πα,β,γ, ρ) = 1
+|Γ|
+�
+(q + 1)(q2 + q + 1)(p − 1) − (p − 1)(p − 3)H2 + (q + 1)(p − 1)(p − 4)H1
++ (2q + 1)(p − 1)P1 + (p − 1)P2 + (p − 1)(p − 4)P3
+�
+= (q − p)(q(q + p + 2) + p2(p3 − p2 − p − 3))
+p3(p − 1)2(p + 1)(p2 + p + 1)
+.
+• All three α, β, γ restrict to the trivial character on F×
+p .
+m(πα,β,γ, ρ) = 1
+|Γ|
+�
+(q + 1)(q2 + q + 1)(p − 1) + (p − 1)(p − 2)(p − 3)H2 + 3(q + 1)(p − 1)(p − 2)H1
++ (2q + 1)(p − 1)P1 + (p − 1)P2 + 3(p − 1)(p − 2)P3
+�
+= (q(q2 + 2q + 3p5 − p4 − p3 − 6p2 − 2p) + p3(p5 − 4p3 + p + 6))
+p3(p − 1)2(p + 1)(p2 + p + 1)
+.
+(7) ρα,ν for α a character of F×
+q and ν a character of F×
+q2 such that νq ̸= ν (i.e., it is nondecom-
+posable).
+m(ρα,ν, ρ) = 1
+|Γ|
+�
+(q − 1)(q2 + q + 1)
+�
+a∈F×
+p
+α(a)ν(a) + (q − 1)H1
+�
+a,b∈F×
+p
+a̸=b
+α(b)ν(a) + 0
+− P1
+�
+a∈F×
+p
+α(a)ν(a) − P2
+�
+a∈F×
+p
+α(a)ν(a) − P3
+�
+a,b∈F×
+p
+a̸=b
+α(b)ν(a)
+− E1
+2
+�
+w∈Fp2\Fp
+r∈F×
+p
+α(r)(ν(w) + ν(wp)) + 0
+�
+.
+Note that the last nonzero sum can be written as −E1
+�
+w∈Fp2\Fp
+r∈F×
+p
+α(r)ν(w).
+• αν |F×
+p ̸= 1. m(ρα,ν, ρ) = 0.
+20
+
+• αν |F×
+p = 1 but α |F×
+p ̸= 1 and ν |F×
+p ̸= 1.
+m(ρα,ν, ρ) = 1
+|Γ|
+�
+(q − 1)(q2 + q + 1)(p − 1) − (q − 1)(p − 1)H1
+− (p − 1)P1 − (p − 1)P2 + (p − 1)P3
+�
+=
+(q − p)(q − p2)(q + p2 + p)
+p3(p − 1)2(p + 1)(p2 + p + 1).
+• α |F×
+p = 1, ν |F×
+p3̸= 1 but ν |F×
+p = 1.
+m(ρα,ν, ρ) = 1
+|Γ|
+�
+(q − 1)(q2 + q + 1)(p − 1) + (q − 1)(p − 1)(p − 2)H1
+− (p − 1)P1 − (p − 1)P2 − (p − 1)(p − 2)P3 + (p − 1)2E1
+�
+= (q − p)(q(q + p) + p2(p3 − p2 − p − 1))
+(p − 1)2p3(p + 1)(p2 + p + 1)
+.
+• α |F×
+p = 1 and ν |F×
+p3= 1.
+m(ρα,ν, ρ) = 1
+|Γ|
+�
+(q − 1)(q2 + q + 1)(p − 1) + (q − 1)(p − 1)(p − 2)H1
+− (p − 1)P1 − (p − 1)P2 − (p − 1)(p − 2)P3 − (p − 1)(p2 − p)E1
+�
+= q(q2 + p5 − p4 − p3 − 2p2) + p4(−p4 + 2p + 1)
+p3(p − 1)2(p + 1)(p2 + p + 1)
+.
+(8) σµ for µ a character of F×
+q3 such that µq ̸= µ.
+m(σµ, ρ) = 1
+|Γ|
+�
+(q − 1)2(q + 1)
+�
+a∈F×
+p
+µ(a) + 0 + 0 − (q − 1)P1
+�
+a∈F×
+p
+µ(a) + P2
+�
+a∈F×
+p
+µ(a) + 0
++ 0 + E2
+3
+�
+w∈Fp3\Fp
+(µ(w) + µ(wp) + µ(wp2))
+�
+.
+• µ |F×
+p ̸= 1. m(σµ, ρ) = 0.
+• µ |F×
+p3̸= 1 but µ |F×
+p = 1.
+m(σµ, ρ) = 1
+|Γ|
+�
+(q − 1)2(q + 1)(p − 1) − (q − 1)(p − 1)P1 + (p − 1)P2 − (p − 1)E2
+�
+= (q − p)(q − p2)(q + p2 + p − 1)
+p3(p − 1)2(p + 1)(p2 + p + 1) .
+21
+
+• µ |F×
+p3= 1.
+m(σµ, ρ) = 1
+|Γ|
+�
+(q − 1)2(q + 1)(p − 1) − (q − 1)(p − 1)P1 + (p − 1)P2 + (p3 − p)E2
+�
+= q(q2 − q − p4 − p3 + p) + p4(p4 − p2 + 1)
+p3(p − 1)2(p + 1)(p2 + p + 1)
+.
+References
+[1] G. Ahumada. Fonctions périodiques et formule des traces de selberg sur les arbres. C.R. Acad. Sci. Paris,
+305:709–712, 1987.
+[2] M. I. al Ali, C.H. Hering, and J. Schäeffer. On the conjugacy classes of the general linear group GL(3, q). Bulletin
+mathématique de la Société des Sciences Mathématiques de Roumanie, 38 (86)(3/4):105–111, 1994.
+[3] J. Arthur. An introduction to the trace formula, Clay Mathematics Proceedings, Vol 4, 2005.
+[4] M. G. Martinez. The finite upper half space and related hypergraphs. Journal of Number Theory, 84:342–360,
+10 2000.
+[5] M. R. Murty. Ramanujan graphs. J. Ramanujan Math. Soc., 18, No.1:1–20, 2003.
+[6] P. Sarnak. Class numbers of indefinite binary quadratic forms. Journal of Number Theory, 15(2):229–247, 1982.
+[7] A. Selberg. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applica-
+tions to Dirichlet series. J. Indian Math. Soc., 20:47–87, 1956.
+[8] R. Steinberg. The representations of gl(3,q), gl(4,q), pgl(3,q), and pgl(4,q). Canadian Journal of Mathematics,
+3:225 - 235, 1951.
+[9] A. Terras. Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts.
+Cambridge University Press, 1999.
+[10] D. I. Wallace. The Selberg trace formula for SL(3, Z)\SL(3, R)/SO(3, R). Transactions of the American Mathe-
+matical Society, 345(1):1–36, 1994.
+22
+
+10. Appendix A: Fundamental Domains and Orbital Sums
+In this appendix, we give proofs of fundamental domains as well as some computations of orbital
+sums.
+• Proof of Lemma 2.2
+Proof. Observe that if α, β ∈ Fq
+�
+δ1/3�
+then (α, β) ∈ Hq if and only if {α, β, 1} form a basis
+of Fq
+�
+δ1/3�
+over Fq.
+Suppose (α, β) ∈ Hq and x, y, z ∈ Fq are such that:
+x · aα + bβ + c
+rα + sβ + t + y · dα + eβ + f
+rα + sβ + t + z · 1 = 0.
+The denominator rα+sβ +t could only be nonzero if r = s = t = 0 (as α, β, 1 are linearly
+independent) but that cannot happen as
+������
+a
+b
+c
+d
+e
+f
+r
+s
+t
+������
+̸= 0. Thus we find:
+x(aα + bβ + c) + y(dα + eβ + f) + z(rα + sβ + t) = 0
+In matrix form:
+�
+x
+y
+z
+�
+
+
+a
+b
+c
+d
+e
+f
+r
+s
+t
+
+
+
+
+α
+β
+1
+
+ = 0.
+This implies that
+�
+x
+y
+z
+�
+
+
+a
+b
+c
+d
+e
+f
+r
+s
+t
+
+ =
+�
+0
+0
+0
+�
+because α, β, 1 are linearly indepen-
+dent over Fq.
+Since our matrix is nonsingular, it follows that x = y = z = 0 and
+�
+aα+bβ+c
+rα+sβ+t, dα+eβ+f
+rα+sβ+t , 1
+�
+is a basis of Fq
+�
+δ1/3�
+over Fq, proving the claim that GL3(Fq) has
+an action on Hq of the above form.
+□
+23
+
+• Second Hyperbolic Term.
+IG(f, γ) =
+�
+t∈Gγ\G/K
+�
+u∈K/{aI}
+f((tu)−1γ(tu))
+=
+�
+t∈Gγ\Hq
+|K|
+|{aI}|f(t−1γt)
+= q3 − 1
+q − 1
+�
+x,y,r,s∈Fq
+ys̸=1
+f
+
+
+
+
+s
+sy−1
+1
+1−sy
+r−sx
+sy−1
+1
+1−sy
+y
+sy−1
+x−ry
+sy−1
+0
+0
+1
+
+ γ
+
+
+y
+1
+x
+1
+s
+r
+0
+0
+1
+
+ p0
+
+ +
++ q3 − 1
+q − 1
+�
+x,r,s∈Fq
+s̸=0
+f
+
+
+
+
+1
+0
+−x
+−s
+1
+xs − r
+0
+0
+1
+
+ γ
+
+
+1
+0
+x
+s
+1
+r
+0
+0
+1
+
+ p0
+
+ +
++ q3 − 1
+q − 1
+�
+x,y,r∈Fq
+y̸=0
+f
+
+
+
+
+1
+−y
+−x + yr
+0
+1
+−r
+0
+0
+1
+
+ γ
+
+
+1
+y
+x
+0
+1
+r
+0
+0
+1
+
+ p0
+
+
++ q3 − 1
+q − 1
+�
+x,r∈Fq
+f
+
+
+
+
+1
+0
+−x
+0
+1
+−r
+0
+0
+1
+
+ γ
+
+
+1
+0
+x
+0
+1
+r
+0
+0
+1
+
+ p0
+
+
+= (q2 + q + 1)
+�
+x,y,r,s∈Fq
+ys̸=1
+f
+
+
+
+
+
+b−asy
+1−sy
+s(a−b)
+sy−1
+asx−br+c(r−sx)
+sy−1
+y(b−a)
+sy−1
+a−bsy
+1−sy
+−ax+bry+c(x−ry)
+sy−1
+0
+0
+c
+
+ p0
+
+
+ +
++ (q2 + q + 1)
+�
+x,r,s∈Fq
+s̸=0
+f
+
+
+
+
+a
+0
+ax − cx
+bs − as
+b
+−axs + br + c (xs − r)
+0
+0
+c
+
+ p0
+
+ +
++ (q2 + q + 1)
+�
+x,y,r∈Fq
+y̸=0
+f
+
+
+
+
+a
+ay − by
+ax − byr + c (yr − x)
+0
+b
+br − cr
+0
+0
+c
+
+ p0
+
+ +
++ (q2 + q + 1)
+�
+x,y∈Fq
+f
+
+
+
+
+a
+0
+ax − cx
+0
+b
+br − cr
+0
+0
+c
+
+ p0
+
+ = (q2 + q + 1)(S1 + S2 + S3 + S4)
+We simplify each of the four terms separately. Observe that the substitution:
+�
+˜x
+˜y
+�
+=
+� asx−br+c(r−sx)
+sy−1
+−ax+bry+c(x−ry)
+sy−1
+�
+=
+1
+sy − 1
+�
+(a − c)s
+c − b
+c − a
+(b − c)y
+� �
+x
+r
+�
+is invertible as
+����
+(a − c)s
+c − b
+c − a
+(b − c)y
+���� = (a − c)(b − c)(sy − 1) ̸= 0. Therefore:
+S1 =
+�
+x,r∈Fq
+�
+y,s∈Fq
+ys̸=1
+f
+
+
+
+
+
+b−asy
+1−sy
+s(a−b)
+sy−1
+x
+y(b−a)
+sy−1
+a−bsy
+1−sy
+r
+0
+0
+c
+
+ p0
+
+
+ =
+�
+x,r∈Fq
+f
+
+
+
+
+b
+0
+x
+0
+a
+r
+0
+0
+c
+
+ p0
+
+ +
+24
+
++
+�
+x,y,r∈Fq
+y̸=0
+f
+
+
+
+
+b
+0
+x
+y(a − b)
+a
+r
+0
+0
+c
+
+ p0
+
++
+�
+x,r,s∈Fq
+s̸=0
+f
+
+
+
+
+b
+s(b − a)
+x
+0
+a
+r
+0
+0
+c
+
+ p0
+
++
+�
+x,r∈Fq
+�
+y,s∈F×
+q
+ys̸=1
+f
+
+
+
+
+
+b−asy
+1−sy
+s(a−b)
+sy−1
+x
+y(b−a)
+sy−1
+a−bsy
+1−sy
+r
+0
+0
+c
+
+ p0
+
+
+
+We can make similar changes of variables
+�
+˜x
+˜r
+�
+as above and find that:
+S2 =
+�
+x,r∈Fq
+s∈F×
+q
+f
+
+
+
+
+a
+0
+x
+(b − a)s
+b
+r
+0
+0
+c
+
+ p0
+
+
+S3 =
+�
+x,r∈Fq
+y∈F×
+q
+f
+
+
+
+
+a
+(a − b)y
+x
+0
+b
+r
+0
+0
+c
+
+ p0
+
+
+S4 =
+�
+x,r∈Fq
+f
+
+
+
+
+a
+0
+x
+0
+b
+r
+0
+0
+c
+
+ p0
+
+
+Putting together all the above we get that S1 + S2 + S3 + S4 is the following sum:
+S1 + S2 + S3 + S4 = Hf
+��
+a/c
+0
+0
+b/c
+��
+This is because each conjugate of
+� a 0
+0 b
+�
+appears exactly once in the upper 2 × 2 block in
+the sum S1 + S2 + S3 + S4.
+Thus, the total contribution is:
+1
+6(q2 + q + 1)(q − 1)3
+(p − 1)3
+�
+a,b,c∈F×
+p
+a̸=b̸=c
+Hf
+��
+a/c
+0
+0
+b/c
+��
+= 1
+6(q2 + q + 1)(q − 1)4
+(p − 1)3
+�
+a,b∈F×
+p \{1}
+a̸=b
+Hf
+��
+a
+0
+0
+b
+��
+=
+= 1
+3(q3 − 1)(q − 1)3
+(p − 1)3
+�
+{a,b}⊆F×
+p \{1}
+Hf
+��
+a
+0
+0
+b
+��
+• Proof of Proposition 5.1
+Proof. First to show the uniqueness of each representative, suppose there exists m ∈ Gγ
+such that
+m · (u1δ1/3, v1δ1/3 + δ2/3) = (u2δ1/3, v2δ1/3 + δ2/3).
+Then we deduce that y = 0 and so u1 = u2. We also see c = d, implying v1 = v2. Next,
+suppose
+m · (u1δ1/3, v1δ1/3 + δ2/3) = (δ1/3 + u2δ2/3, δ1/3).
+But this immediately forces c = 0, a contradiction. Similarly,
+m · (δ1/3 + u1δ2/3, δ1/3) = (δ1/3 + u2δ2/3, δ1/3),
+directly implies u1 = u2.
+25
+
+To show the completeness of this fundamental domain, take an arbitrary element (α, β) ∈
+Hq. If β3 ̸= 0, then observe that
+
+
+1
+α3
+α1
+0
+β3
+β1
+0
+0
+1
+
+
+�
+(α2 − α3
+β2
+β3
+)δ1/3, β2
+β3
+δ1/3 + δ2/3
+�
+= (α, β).
+Otherwise, we must have β2 ̸= 0, in which case
+
+
+1
+α2 − 1
+α1
+0
+β2
+β1
+0
+0
+1
+
+
+�
+δ1/3 + α3δ2/3, δ1/3�
+= (α, β).
+□
+• First Parabolic Term.
+IG(f, γ) =
+�
+t∈Gγ\G/K
+�
+u∈K/{aI}
+f((tu)−1γ(tu))
+=
+�
+t∈Gγ\Hq
+|K|
+|{aI}|f(t−1γt)
+= q3 − 1
+q − 1
+
+
+
+
+�
+u,v∈Fq
+u̸=0
+f
+
+
+
+
+− v
+u
+1
+0
+1
+u
+0
+0
+0
+0
+1
+
+ γ
+
+
+0
+u
+0
+1
+v
+0
+0
+0
+1
+
+ p0
+
+ +
+�
+u∈F×
+q
+f
+
+
+
+
+1
+u
+− 1
+u
+0
+0
+1
+0
+0
+0
+1
+
+ γ
+
+
+u
+1
+0
+0
+1
+0
+0
+0
+1
+
+ p0
+
+
+
+
+
+
+= (q2 + q + 1)
+
+
+
+
+�
+u,v∈Fq
+u̸=0
+f
+
+
+
+
+a
+0
+− av
+u
+0
+a
+a
+u
+0
+0
+a
+
+ p0
+
+ +
+�
+u∈F×
+q
+f
+
+
+
+
+a
+0
+a
+u
+0
+a
+0
+0
+0
+a
+
+ p0
+
+
+
+
+
+
+= (q2 + q + 1)
+
+
+
+
+�
+u,v∈Fq
+u̸=0
+f
+�
+−v
+u + δ2/3, 1
+u + δ1/3
+�
++
+�
+u∈F×
+q
+f
+�
+δ2/3 + 1
+u, δ1/3
+�
+
+
+
+
+= (q2 + q + 1) (Hf(I) − f(p0))
+• Proof of Proposition 5.2. First, we check that if m ∈ B such that mx = y for x, y ∈
+B\Hq, then x = y. Suppose there exists m ∈ B such that
+m(δ1/3, v1δ1/3 + δ2/3) = (δ1/3, v2δ1/3 + δ2/3).
+Then we right away see that c/a = 1 and so v1 = v2. Finally, it is also easy to see that we
+cannot use B to move elements between the two sets in the disjoint union of B\Hq.
+Next, we check that for any arbitrary element (α, β) ∈ Hq (where α = α1+α2δ1/3+α3δ2/3
+and β = β1 + β2δ1/3 + β3δ2/3), there exists m ∈ B and x ∈ B\Hq such that mx = (α, β).
+First suppose that β3 ̸= 0.
+
+
+α2 − α3
+β2
+β3
+α3
+α1
+0
+β3
+β1
+0
+0
+1
+
+
+�
+δ1/3, β2
+β3
+δ1/3 + δ2/3
+�
+= (α, β)
+26
+
+If β3 = 0, then
+
+
+α3
+α2
+α1
+0
+β2
+β1
+0
+0
+1
+
+
+�
+δ2/3, δ1/3�
+= (α, β)
+• Second Parabolic Term.
+IG(f, γ) =
+�
+t∈Gγ\G/K
+�
+u∈K/{aI}
+f((tu)−1γ(tu))
+=
+�
+t∈Gγ\Hq
+|K|
+|aI|f(t−1γt)
+= q3 − 1
+q − 1
+
+f
+
+
+
+
+1
+0
+0
+0
+1
+0
+0
+0
+1
+
+ γ
+
+
+1
+0
+0
+0
+1
+0
+0
+0
+1
+
+ p0
+
+ +
+�
+v∈Fq
+f
+
+
+
+
+−v
+1
+0
+1
+0
+0
+0
+0
+1
+
+ γ
+
+
+0
+1
+0
+1
+v
+0
+0
+0
+1
+
+ p0
+
+
+
+
+= (q2 + q + 1)
+
+f
+
+
+
+
+a
+a
+0
+0
+a
+a
+0
+0
+a
+
+ p0
+
+ +
+�
+v∈Fq
+f
+
+
+
+
+−av + a
+−av2
+a
+a
+av + a
+0
+0
+0
+a
+
+ p0
+
+
+
+
+= (q2 + q + 1)
+
+f
+�
+δ2/3 + δ1/3, δ1/3 + 1
+�
++
+�
+v∈Fq
+f
+�
+(1 − v)δ2/3 − v2δ1/3 + 1, δ2/3 + (v + 1)δ1/3�
+
+
+• Third Parabolic Term.
+Gγ\Hq = {(δ1/3 + uδ2/3 + v, rδ2/3 + s) : r ̸= 0} ⊔ {(δ2/3 + v, rδ1/3 + s) : r ̸= 0}.
+IG(f, γ) =
+�
+t∈Gγ\G/K
+�
+u∈K/{aI}
+f((tu)−1γ(tu))
+=
+�
+t∈Gγ\Hq
+|K|
+|{aI}|f(t−1γt)
+= q3 − 1
+q − 1
+
+
+
+
+�
+u,v,r,s∈Fq
+r̸=0
+f
+
+
+
+
+0
+1
+r
+− s
+r
+1
+− u
+r
+su
+r − v
+0
+0
+1
+
+ γ
+
+
+u
+1
+v
+r
+0
+s
+0
+0
+1
+
+ p0
+
+ +
+�
+v,r,s∈Fq
+r̸=0
+f
+
+
+
+
+1
+0
+−v
+0
+1
+r
+− s
+r
+0
+0
+1
+
+ γ
+
+
+1
+0
+v
+0
+r
+s
+0
+0
+1
+
+ p0
+
+
+
+
+
+
+= (q2 + q + 1)
+�
+�
+u,v,r,s∈Fq
+r̸=0
+f
+
+
+
+
+a(r+u)
+r
+a
+r
+a(s+v)−bs
+r
+− au2
+r
+a(r−u)
+r
+− −arv+asu+auv+brv−bsu
+r
+0
+0
+b
+
+ p0
+
+
++
+�
+v,r,s∈Fq
+r̸=0
+f
+
+
+
+
+a
+0
+v(a − b)
+a
+r
+a
+a(s+v)−bs
+r
+0
+0
+b
+
+ p0
+
+
+�
+.
+Employ the following change of variables in the two sums:
+�
+˜v
+˜s
+�
+= 1
+r
+�
+a
+(a − b)
+(a − b)r − au
+(b − a)u
+� �
+v
+s
+�
+�
+v∗
+s∗
+�
+=
+�
+a − b
+0
+a/r
+(a − b)/r
+� �
+v
+s
+�
+27
+
+The determinants are both (a − b)2/r ̸= 0 hence the transformations are invertible. It
+follows that the above sums reduce to horocycle transforms of parabolic conjugacy classes
+in GL2(Fq).
+IG(f, γ) = (q2 + q + 1)Hf
+��
+a/b
+1
+0
+a/b
+��
+• Proof of Proposition 6.1.
+We first show that every Gγ-orbit on Hq contains at least one element of the above form.
+Let z ∈ Hq and write:
+z =
+
+
+a
+b
+c
+d
+e
+f
+0
+0
+1
+
+ p0
+We want to find r, s ∈ Fq not both zero and t ∈ F×
+q such that:
+
+
+r
+sξ
+0
+s
+r
+0
+0
+0
+t
+
+
+
+
+a
+b
+c
+d
+e
+f
+0
+0
+1
+
+ =
+
+
+v
+u
+x
+0
+1
+y
+0
+0
+1
+
+
+for some v ∈ F×
+q and u, x, y ∈ Fq. This amounts to solving the following set of equations
+for s, r, t over Fq:
+sa + rd = 0
+sb + re = 1
+t = 1
+By assumption, the determinant
+����
+a
+b
+d
+e
+���� is nonzero, therefore the above system admits a
+unique solution for s, r, t over Fq, moreover, s and r cannot be simultaneously zero.
+We now need to show that the domain does not contain any orbit repetitions. Suppose
+that:
+
+
+r
+sξ
+0
+s
+r
+0
+0
+0
+t
+
+
+
+
+v
+u
+x
+0
+1
+y
+0
+0
+1
+
+ =
+
+
+v′
+u′
+x′
+0
+1
+y′
+0
+0
+1
+
+
+Writing up the above equations again we get:
+sv = 0
+su + r = 1
+t = 1
+As v ̸= 0 we find that s = 0, r = 1 and t = 1 so that the first matrix on the left-hand side
+is the identity, which implies that x = x′, y = y′, u = u′ and v = v′, concluding the proof.
+• Second Elliptic Term.
+IG(f, γ) = (q − 1)(q2 − 1)
+(p − 1)(p2 − 1) · q3 − 1
+q − 1
+�
+x,y,u∈Fq
+v∈F×
+q
+f
+
+
+
+
+v
+u
+x
+0
+1
+y
+0
+0
+1
+
+
+−1 
+
+k
+lξ
+0
+l
+k
+0
+0
+0
+m
+
+
+
+
+v
+u
+x
+0
+1
+y
+0
+0
+1
+
+ p0
+
+ =
+28
+
+= (q3 − 1)(q2 − 1)
+(p − 1)(p2 − 1)
+�
+u∈Fq
+v∈F×
+q
+�
+x,y∈Fq
+f
+
+
+
+
+−lu + k
+lξ−lu2
+v
+x(−lu+k)+y(lξ−ku)+m(uy−x)
+v
+vl
+lu + k
+lx + ky − my
+0
+0
+m
+
+ p0
+
+
+Set:
+˜x = (k − m − lu)
+v
+· x + (m − k)u + lξ
+v
+· y
+˜y = lx + (k − m)y
+The determinant corresponding to this change of variables is
+�
+(k − m)2 − l2ξ
+�
+/v. It could
+only be zero if k − m = 0 and l = 0 since ξ is not a square, but then γ would be similar to a
+diagonal matrix, contradicting our assumption. Thus the orbital sum can be simplified as:
+IG(f, γ) = (q3 − 1)(q2 − 1)
+(p − 1)(p2 − 1)
+�
+u∈Fq
+v∈F×
+q
+�
+˜x,˜y∈Fq
+f
+
+
+
+
+−lu + k
+lξ−lu2
+v
+˜x
+vl
+lu + k
+˜y
+0
+0
+m
+
+ p0
+
+
+Finally, notice that the upper-left 2 × 2 block is similar to the elliptic matrix
+�
+k lξ
+l k
+�
+over
+GL2(Fq). Hence we obtain
+IG(f, γ) = (q3 − 1)(q2 − 1)
+(p − 1)(p2 − 1) Hf
+��
+k/m
+lξ/m
+l/m
+k/m
+��
+.
+Department of Mathematics, Brown University, Providence, RI, USA, 02912
+Email address: daksh_aggarwal@brown.edu
+Department of Mathematics, University of Western Ontario, London, ON, Canada, N6A 5B7
+Email address: aghorba@uwo.ca
+Department of Mathematics, University of Western Ontario, London, ON, Canada, N6A 5B7
+Email address: masoud@uwo.ca
+Department of Mathematics, University of Toronto, Toronto, ON, Canada, M5S 2E4
+Email address: jiyuan.lu@mail.utoronto.ca
+DAMTP, University of Cambridge, Cambridge, UK, CB3 0WA
+Email address: bn273@cam.ac.uk
+Department of Mathematics, University of Toronto, Toronto, ON, Canada, M5S 2E4
+Email address: cyu@math.toronto.edu
+29
+
diff --git a/t9AzT4oBgHgl3EQfr_3p/content/tmp_files/load_file.txt b/t9AzT4oBgHgl3EQfr_3p/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..34cb22e8e5fb170b91606e91d83f15456a9ca7b6
--- /dev/null
+++ b/t9AzT4oBgHgl3EQfr_3p/content/tmp_files/load_file.txt
@@ -0,0 +1,809 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf,len=808
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='01654v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='NT]  4 Jan 2023  A SELBERG TRACE FORMULA FOR GL3(Fp)\\ GL3(Fq)/K  DAKSH AGGARWAL, ASGHAR GHORBANPOUR, MASOUD KHALKHALI, JIYUAN LU, BALÁZS NÉMETH,  AND C SHIJIA YU  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In this paper, we prove a discrete analog of the Selberg Trace Formula for the group  GL3(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' By considering a cubic extension of the finite field Fq, we define an analogue of the upper  half space and an action of GL3(Fq) on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' To compute the orbital sums we explicitly identify the  double coset spaces and fundamental domains in our upper half space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' To understand the spectral  side of the trace formula we decompose the induced representation ρ = IndG  Γ 1 for G = GL3(Fq)  and Γ = GL3(Fp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Introduction  1  Acknowledgments  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  A finite upper half-space  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Trace formula  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  The pre-trace formula  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Double cosets and fundamental domains  6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Central and Hyperbolic Terms  7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Central terms  7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Hyperbolic terms of the first kind  7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Hyperbolic terms of the second kind  8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Parabolic Terms  9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Parabolic terms of the first kind  9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Parabolic terms of the second kind  9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Parabolic terms of the third kind  10  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Elliptic Terms  10  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Elliptic terms of the first kind  10  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Elliptic terms of the second kind  12  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Examples  13  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Character of the representation ρ = IndG  Γ 1  14  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Decomposition of ρ when 2, 3 ∤ n  15  References  22  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Appendix A: Fundamental Domains and Orbital Sums  23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Introduction  The Selberg trace formula [7] is one of the most celebrated mathematical results of the past  century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In its original form, it relates the spectrum of the Laplacian on a hyperbolic surface to  lengths of closed geodesics on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' It can be viewed as a non-abelian generalization of the  well-known Poisson summation formula in Fourier analysis and has found numerous applications in  many fields, for example in number theory and the theory of automorphic forms (see Sarnak [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  1  The Selberg trace formula deals with surfaces which can be represented as double cosets of the  form Γ\\ SL2(R)/ SO2(R), where Γ is a discrete subgroup of SL2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' It is a natural question to  work out the trace formula for higher-dimensional matrix groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' To quote Dennis Hejhal: “Of  course, we won’t really understand the trace formula until it is written down for SL4(Z).” Dorothy  Wallace [10] has worked out the explicit details of the trace formula for SL3(Z)\\ SL3(R)/ SO3(R)  but the four-dimensional case still remains mysterious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' A further generalization is the so-called  Arthur-Selberg trace formula [3], which plays an important role in the Langlands program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Selberg type trace formulae have been derived for discrete spaces like graphs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' By realizing  a k-regular graph as a quotient of the infinite k-regular tree, Ahumada [1] found such a formula for  k-regular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Audrey Terras [9] introduced the concept of finite upper half-planes as a finite  analogue of the Poincaré upper half-plane model where the complex number field is replaced by a  quadratic extension of a finite field Fq, where q = pr is a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' She constructed a family of  Ramanujan graphs [5] using these upper-half planes and developed a representation-theoretic trace  formula on GL2(Fp)\\ GL2(Fq)/K, where K is the finite analogue of SO2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  The aim of this paper is to present an analogous trace formula for GL3(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Our main result in  this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Consider the pre-trace formula  �  π∈ ˆG  m(π, ρ) Tr( ˆf(π)) =  �  {γ}∈CΓ  |Gγ|  |Γγ| IG(f, γ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1)  where G is a finite group, Γ is a subgroup of G, ρ = IndG  Γ 1 is the induced representation of the  trivial representation of Γ, and IG(f, γ) is the orbital sum of f at γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' These terms are defined in  Section 3 and a proof of the pre-trace formula is recalled there as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Note that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1) can be  viewed as a representation-theoretic analogue of Selberg’s trace formula [7, 9]: the left hand side  can be thought of as the spectral side, summing over the irreducible representations of G and their  multiplicities in ρ, while the right hand side can be interpreted as the geometric side, summing over  conjugacy classes in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In this paper, to obtain a trace formula for the finite upper half-space we  take G = GL3(Fq) and Γ = GL3(Fp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Also, f : G → C is chosen to be a K−bi-invariant function  (K is the stabilizer of p0 as above), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' ∀k, h ∈ K and ∀x ∈ G we have f(kxh) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' One can  also think of f as a function on G/K ≃ Hq which is invariant under left multiplication by elements  in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  In Section 2, we define an analogue, denoted Hq, of the  upper half plane for the group GL3(Fq) by considering a cubic extension of the finite field Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  We define an action of GL3(Fq) on Hq and this sets the stage for our discrete trace formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In  Section 3, we recall a standard pre-trace formula for finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In Sections 4-6, we compute each  term on the right hand side of the pre-trace formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='4) (the so called geometric side) using the  explicit description of conjugacy classes of GL3(Fq) given in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The conjugacy classes are separated  into central, hyperbolic, parabolic, and elliptic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Each of these types of terms are explicitly  identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' It would be helpful to have a more conceptual understanding of these computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In  Section 7, we give some simple examples of the application of our formula in which the left side of  the pre-trace formula is known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In Sections 8-9, we calculate the left side of the pre-trace  formula by computing the character of the induced representation ρ = IndG  Γ 1 and then using that to  compute its decomposition into irreducible representations in the case that 2, 3 ∤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The calculation  of the decomposition in the other cases is similar but involve more cases and hence they are not  incorporated in the arXiv version of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' They are available on demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' We would like to thank the Fields Institue for organizing the Fields Under-  graduate Summer Research Program in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Daksh Aggarwal, Jiyuan Lu, Balazs Németh, and C  Yu were participants in this program while working on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The project was proposed and  supervised by Masoud Khalkhali and Asghar Ghorbanpour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' A finite upper half-space  To generalize Terras’s results to the GL3(Fq) case, we must first discuss the generalization of the  finite upper half-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let p be a prime number and q = pn for a positive integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The definition  of the upper half space Hq comes from considering the cubic extension of Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The construction was  first given by Martinez in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Here we describe it explicitly for our special case of GL3(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Fq has a cubic nonresidue if and only if q ≡ 1 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let q = pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Consider the cube map x �→ x3 : F×  q → F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' This map has as kernel the subgroup  µ3 consisting of the cube roots of unity in F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let α be a generator of F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Then if q ≡ 1 (mod 3),  we have µ3 = {1, α(q−1)/3, α2(q−1)/3}, in which case there are (q − 1)/3 cubic residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Otherwise,  µ3 = {1}, and so each element of F×  q is a cubic residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  □  From now on, we assume that q ≡ 1 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Take a cubic nonresidue δ ∈ Fq and consider the  cubic extension Fq  �  3√  δ  �  ≃ Fq3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' A basis for Fq  �  3√  δ  �  /Fq is {1, δ1/3, δ2/3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' For an element α, we  will write (α1, α2, α3) to denote its components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=',  α = α1 + α2δ1/3 + α3δ2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  We define  Hq =  �  (α, β) ∈ Fq(  3√  δ) × Fq(  3√  δ) :  ����  α2  α3  β2  β3  ���� ̸= 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The following map defines an action of GL3(Fq) on Hq :  \uf8ee  \uf8f0  a  b  c  d  e  f  r  s  t  \uf8f9  \uf8fb (α, β) =  �aα + bβ + c  rα + sβ + t, dα + eβ + f  rα + sβ + t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The compatibility of the action with multiplication in GL3(Fq) follows from the fact that  GL3(Fq) has an action on the projective space P2(Fq3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' So, we need to check that the action is  closed in Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' This is a routine verification, which can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  □  Our distinguished point will be p0 = (δ2/3, δ1/3) ∈ Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The action of Aff3(Fq) ≤ GL3(Fq) is  transitive on Hq, where  Aff3(Fq) :=  \uf8f1  \uf8f2  \uf8f3  \uf8ee  \uf8f0  a  b  c  d  e  f  0  0  1  \uf8f9  \uf8fb  \uf8fc  \uf8fd  \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  With this action, we calculate that  K := stab(p0) =  \uf8f1  \uf8f2  \uf8f3  \uf8ee  \uf8f0  a  cδ  bδ  b  a  cδ  c  b  a  \uf8f9  \uf8fb  \uf8fc  \uf8fd  \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Indeed, we find that  K  ∼  −→ Fq(δ1/3)× :  \uf8ee  \uf8f0  a  cδ  bδ  b  a  cδ  c  b  a  \uf8f9  \uf8fb �→ a + bδ1/3 + cδ2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Also, we have the homogeneous space Hq ≃ GL3(Fq)/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  As a quick check note that we have  | GL3(Fq)| = (q3 − 1)(q3 − q)(q3 − q2) and |K| = q3 − 1 (from the canonical isomorphism), so we  should have |Hq| equal to  (q3 − 1)(q3 − q)(q3 − q2)  q3 − 1  = (q2 − 1)(q − 1)q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  3  Indeed, from the definition we have  |Hq| = | GL2(Fq)| · q2 = (q2 − 1)(q2 − q)q2 = (q2 − 1)(q − 1)q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Observe that the above construction is a natural generalization of the finite upper  half-plane defined by Terras [9] (here δ is a nonsquare in the finite field Fq):  Hq =  �  x + y  √  δ : x, y ∈ Fq, y ̸= 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  The condition y ̸= 0 is equivalent to 1 and x + y  √  δ being linearly independent elements of Fq  �√  δ  �  over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Trace formula  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The pre-trace formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Our trace formula relies on interpretation of a known result, the  pre-trace formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' We now derive the pre-trace formula and contextualize it in our setting of the  group G = GL3(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The idea behind the pre-trace formula is to compute the trace of a specific  operator in two different bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' We may construct the relevant information for this operator as  follows:  Suppose that G is a finite group and Γ ≤ G is a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let f : G → C be a complex-valued  function on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' For a representation κ : G → GL(V ) of G (where V is a finite-dimensional complex  vector space endowed with a linear G-action), define the Fourier transform of f at κ by:  ˆf(κ) =  �  g∈G  f(g)κ  �  g−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Observe that the Fourier transform ˆf(κ) is End V -valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The pre-trace formula is now obtained  by computing the trace of ˆf(ρ) in two different bases with ρ = IndG  Γ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' First, we can decompose ρ  as a direct sum of irreducible representations of G to get:  Tr  �  ˆf(ρ)  �  =  �  π∈ ˆG  m(π, ρ) Tr  �  ˆf(π)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1)  Here ˆG is the set of irreducible representations of G and m(π, ρ) is the multiplicity of the irre-  ducible representation π ∈ ˆG in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let  V = {f : G → C | f(γg) = f(g)  ∀γ ∈ Γ, g ∈ G} = L2(Γ\\G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  As G acts on linearly on V by right multiplication, the corresponding representation is precisely  ρ = IndG  Γ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' More concretely, for φ ∈ V , g, x ∈ G we have  [ρ(g)φ] (x) = φ(xg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Thus:  �  ˆf(ρ)φ  �  (x) =  �  y∈G  f(y)φ  �  xy−1�  =  �  u∈G  f  �  u−1x  �  φ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Further, as φ is Γ-invariant,  �  ˆf(ρ)φ  �  (x) =  �  y∈Γ\\G  �  γ∈Γ  f  �  y−1γx  �  φ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  4  Now we are ready to compute the trace of ˆf(ρ) in another way: consider the indicator basis  {δy}y∈Γ\\G of V 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' with respect to this basis, ˆf(ρ) is represented by a matrix where the entry corre-  sponding to x, y ∈ Γ\\G is �  γ∈Γ f  �  y−1γx  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Therefore,  Tr  �  ˆf(ρ)  �  =  �  x∈Γ\\G  �  γ∈Γ  f  �  x−1γx  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2)  We can rewrite the right-hand side as a sum over conjugacy classes in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let:  Γγ =  �  x ∈ Γ | x−1γx = γ  �  = centralizer of γ in Γ  Gγ =  �  x ∈ G | x−1γx = γ  �  = centralizer of γ in G  {γ} =  �  x−1γx | x ∈ G  �  = conjugacy class of γ in G  CΓ = set of conjugacy classes in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Note that there is a 1-to-1 correspondence between right cosets Γγ\\Γ and elements of {γ} given  by Γγx �→ x−1γx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Since conjugacy classes partition Γ we can rewrite the inner sum as:  �  x∈Γ\\G  �  γ∈Γ  f(x−1γx) =  �  x∈Γ\\G  �  {γ}∈CΓ  �  u∈Γγ\\Γ  f  �  x−1u−1γux  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Set y = ux, then we can rewrite this as a sum over right cosets y ∈ Γγ\\G:  �  x∈Γ\\G  �  γ∈Γ  f(x−1γx) =  �  y∈Γγ\\G  �  {γ}∈CΓ  f(y−1γy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Observe that Γγ ≤ Gγ hence we can write y = ts for t ∈ Γγ\\Gγ and s ∈ Gγ\\G:  �  x∈Γ\\G  �  γ∈Γ  f(x−1γx) =  �  t∈Γγ\\Gγ  �  s∈Gγ\\G  �  {γ}∈CΓ  f  �  s−1t−1γts  �  =  �  {γ}∈CΓ  |Gγ|  |Γγ|  �  s∈Gγ\\G  f(s−1γs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='3)  Define the orbital sum IG(f, γ) of f at γ to be:  IG(f, γ) :=  �  s∈Gγ\\G  f(s−1γs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  The pre-trace formula now follows from combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1)−(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='3):  �  π∈ ˆG  m(π, ρ) Tr( ˆf(π)) =  �  {γ}∈CΓ  |Gγ|  |Γγ| IG(f, γ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='4)  where ρ = IndG  Γ 1 and IG(f, γ) is the orbital sum of f at γ as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='4) can  be viewed as a representation-theoretic analogue of Selberg’s trace formula [7, 9]: the left hand side  can be thought of as the spectral side, summing over the irreducible representations of G and their  multiplicities in ρ, while the right hand side can be interpreted as the geometric side, summing over  conjugacy classes in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' For further reference see [9, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  To obtain the trace formula for the finite upper half-space we take G = GL3(Fq) and Γ = GL3(Fp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Also, f : G → C is chosen to be a K−bi-invariant function (K is the stabilizer of p0 as above), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  ∀k, h ∈ K and ∀x ∈ G we have f(kxh) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' One can also think of f as a function on G/K ≃ Hq  which is invariant under left multiplication by elements in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  1For y ∈ Γ\\G, δy is the function on Γ\\G which outputs 1 if the input is y and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  5  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='4) can be expressed solely in terms of the characters χπ of the  group G:  Tr  �  ˆf(π)  �  = Tr  \uf8ee  \uf8f0�  g∈G  f(g)π  �  g−1�  \uf8f9  \uf8fb =  �  g∈G  f(g) Tr  �  π  �  g−1��  =  =  �  g∈G  f(g)χπ  �  g−1�  =  �  g∈G  f(g)χπ(g) = ⟨f, χπ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Here ⟨x, y⟩ denotes the usual Hermitian inner product on the complex vector space L2(G):  ⟨x, y⟩ =  �  g∈G  x(g)y(g)  We will, in Sections 4-6, compute each term in the sum on the right hand side of the pre-  trace formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='4) using the explicit description of conjugacy classes of GL3(Fp) given in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The  conjugacy classes are separated into the cases of central, hyperbolic, parabolic, and elliptic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  It is helpful to introduce one more tool that will help us simplify these computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Double cosets and fundamental domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' To calculate the orbital sums it will be con-  venient to identify Gγ\\G with Gγ\\G/K × K/Z (when possible), where Z is the center of G, the  subgroup of diagonal matrices {aI : a ∈ F×  q }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let {ti}l  i=1 be representatives of double cosets in  Gγ\\G/K and consider the map:  Gγ\\G/K × K/Z → Gγ\\G  (GγtiK, kZ) �→ Gγtik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  We first show that this map is well-defined and onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' If z ∈ Z then Gγtikz = Gγztik = Gγtik  because z ∈ Gγ as z commutes with every element of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Now let Gγx ∈ G\\Gγ, then since double  cosets GγtiK partition G and they are unions of right cosets Gγy we can find a representative ti ∈ G  and k ∈ K such that Gγx = Gγtik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  As for injectivity, let ti, tj be representatives of two double cosets in Gγ\\G/K and k, k′ be  representatives of two left cosets in K/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Suppose that Gγtik = Gγtjk′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Since double cosets are  either disjoint or identical this is only possible if ti = tj, let t = ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' So, we have,  Gγ = Gγtk′k−1t−1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='6)  which happens if and only if tk′k−1t−1 ∈ Gγ, equivalently, if k′k−1 ∈ tGγt−1 ∩ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' This can also be  stated as kH = k′H where H = tGγt−1 ∩ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Note that Z ≤ K and Z ≤ tGγt−1 = Gtγt−1 as Z  is the intersection of all centralizers in G, hence we always have Z ≤ K ∩ tGγt−1 = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' If Z = H,  then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='6) is equivalent to kZ = k′Z, which implies that the above map is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In our specific  example, the centralizers Gγ contain no non-central elements conjugate to some matrix in K unless  γ itself is conjugate to a member of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' (This can be seen e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' by taking simple representatives γ  of conjugacy classes in G and looking at the characteristic polynomials and eigenvalues of elements  in Gγ, which are preserved by conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=')  Thus we have two cases:  Case 1: γ is not conjugate to an element of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Then we have:  IG(f, γ) =  �  s∈Gγ\\G  f(s−1γs) =  �  t∈Gγ\\G/K  �  k∈K/Z  f((tk)−1γtk) = |K/Z|  �  t∈Gγ\\G/K  f(t−1γt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  6  (The last equality follows from the fact that f is K−bi-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=') We can identify Gγ\\G/K  with Gγ\\Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In order to calculate orbital sums we will find fundamental domains for the action of  Gγ on Hq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' subsets of Hq which contain exactly one element from each Gγ-orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Case 2: γ is conjugate to an element of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In this case, the above map is not necessarily injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  However, Gγ = G or K depending on whether γ is conjugate to a central element of K or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Then  either Gγ\\G is trivial or it can be identified with Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Central and Hyperbolic Terms  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Central terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let γ =  \uf8ee  \uf8f0  a  0  0  0  a  0  0  0  a  \uf8f9  \uf8fb for a ∈ F×  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In this case, Gγ = G and Γγ = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' So,  IG(f, γ) = f(p0) and the total contribution from such γ is  |G|  |Γ| f(p0) · (p − 1) = (q3 − 1)(q2 − 1)(q − 1)q3  (p3 − 1)(p2 − 1)p3  f(p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Hyperbolic terms of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let γ =  \uf8ee  \uf8f0  a  0  0  0  a  0  0  0  b  \uf8f9  \uf8fb , a ̸= b ∈ F×  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Here  Gγ =  \uf8f1  \uf8f2  \uf8f3  \uf8ee  \uf8f0  x  y  0  w  z  0  0  0  t  \uf8f9  \uf8fb  \uf8fc  \uf8fd  \uf8fe ≃ GL2(Fq) × F×  q  and similarly for Γγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  A fundamental domain for the action of Gγ is given by  Gγ\\Hq = {(u + δ1/3, v + δ2/3) : u, v ∈ Fq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1)  We verify this in the next proposition for the sake of illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In the next sections, the verifici-  ation of the fundamental domains is relegated to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' A fundamental domain for Gγ is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' First to show the uniqueness of each representative, suppose there exists m ∈ Gγ such that  m · (u1 + δ1/3, v1 + δ2/3) = (u2 + δ1/3, v2 + δ2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Then we deduce that y = 0 = w, and x = t = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' So, u1 = u2 and v1 = v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Next, to show the completeness of this fundamental domain, take an arbitrary element (α, β) ∈  Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' We wish to find m ∈ Gγ, u, v ∈ Fq such that m · (u + δ1/3, v + δ2/3) = (α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Set x = α2, y =  α3, w = β2, z = β3, and t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Then we see u, v is a solution to  �  α2  α3  β2  β3  � �  u  v  �  =  �  α1  β1  �  ,  which has a solution since the left matrix is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  □  7  Thus,  IG(f, γ) =  �  t∈Gγ\\G/K  �  u∈K/{aI}  f((tu)−1γ(tu))  =  �  t∈Gγ\\Hq  |K|  |{aI}|f(t−1γt)  = q3 − 1  q − 1  �  x,y∈Fq  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  0  1  −y  1  0  −x  0  0  1  \uf8f9  \uf8fb γ  \uf8ee  \uf8f0  0  1  x  1  0  y  0  0  1  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  = (q2 + q + 1)  �  x,y∈Fq  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  a  0  (a − b)y  0  a  (a − b)x  0  0  b  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  = (q2 + q + 1)  �  x,y∈Fq  f  �  (x + a  b δ2/3, y + a  b δ1/3)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Let us define the horocycle transform Hf : GL2(Fq) → C of f ∈ L2(K\\G/K) by  Hf(κ) :=  �  x,y∈Fq  �  ξ∈{κ}  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  ξ11  ξ12  x  ξ21  ξ22  y  0  0  1  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8 ,  where the inner sum runs over ξ ∈ GL2(Fq) in the conjugacy class of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' So, this hyperbolic term  is equal to  (q2 + q + 1)  �  a,b∈F×  p  a̸=b  (q2 − 1)(q2 − q)(q − 1)  (p2 − 1)(p2 − p)(p − 1) · Hf  �a  b I  �  = (q3 − 1)(q2 − 1)(q2 − q)  (p2 − 1)(p2 − p)(p − 1)  �  a∈F×  p  a̸=1  (p − 1) · Hf(aI)  = (q3 − 1)(q2 − 1)(q2 − q)  (p2 − 1)(p2 − p)  �  a∈F×  p  a̸=1  Hf(aI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  We have shown the full computation for the orbital sums for this case both to demonstrate the  computational technique and to motivate and introduce the horocycle transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  For the next  cases we omit detailed computations for orbital sums, but included some in Appendix A for the  curious reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Hyperbolic terms of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let γ =  \uf8ee  \uf8f0  a  0  0  0  b  0  0  0  c  \uf8f9  \uf8fb with a, b, c ∈ F×  p distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Here  Gγ is the set of diagonal matrices and:  Gγ\\Hq = {(x + δ1/3 + yδ2/3, r + sδ1/3 + δ2/3) : ys ̸= 1} ⊔ {(x + δ2/3, r + δ1/3 + sδ2/3) : s ̸= 0}  ⊔ {(x + yδ1/3 + δ2/3, r + δ1/3) : y ̸= 0} ⊔ {(x + δ2/3, y + δ1/3)}  8  We have  IG(f, γ) = (q2 + q + 1) · Hf  ��  a/c  0  0  b/c  ��  ,  where a, b, c ∈ F×  p comes from a change of variables outlined in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  The total contribution from the second hyperbolic term is:  1  6(q2 + q + 1)(q − 1)3  (p − 1)3  �  a,b,c∈F×  p  a̸=b̸=c  Hf  ��  a/c  0  0  b/c  ��  = 1  6(q2 + q + 1)(q − 1)3  (p − 1)2  �  a,b∈F×  p \\{1}  a̸=b  Hf  ��  a  0  0  b  ��  =  = 1  3(q3 − 1)(q − 1)2  (p − 1)2  �  {a,b}⊆F×  p \\{1}  Hf  ��  a  0  0  b  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Parabolic Terms  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Parabolic terms of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let γ =  \uf8ee  \uf8f0  a  0  a  0  a  0  0  0  a  \uf8f9  \uf8fb , a ∈ F×  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In this case,  Gγ =  \uf8f1  \uf8f2  \uf8f3  \uf8ee  \uf8f0  d  y  x  0  c  b  0  0  d  \uf8f9  \uf8fb : c, d ∈ F×  q , b, x, y ∈ Fq  \uf8fc  \uf8fd  \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' A fundamental domain for Gγ is  Gγ\\Hq = {(uδ1/3, vδ1/3 + δ2/3) : u, v ∈ Fq, u ̸= 0} ⊔ {(δ1/3 + uδ2/3, δ1/3) : u ∈ F×  q }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  □  Following the computations outlined in Appendix A, the orbital sum in this case is  IG(f, γ) = (q2 + q + 1) (Hf(I) − f(p0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  And in total, the contribution from the parabolic terms of the first kind is  q3(q − 1)2  p3(p − 1)2 (q2 + q + 1) (Hf(I) − f(p0)) · (p − 1) = q3(q3 − 1)(q − 1)  p3(p − 1)  (Hf(I) − f(p0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Parabolic terms of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let γ =  \uf8ee  \uf8f0  a  a  0  0  a  a  0  0  a  \uf8f9  \uf8fb , a ∈ F×  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' We find that  Gγ = B =  \uf8f1  \uf8f2  \uf8f3  \uf8ee  \uf8f0  z  y  x  0  c  b  0  0  d  \uf8f9  \uf8fb : d, c, z ∈ F×  q , b, x, y ∈ Fq  \uf8fc  \uf8fd  \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Here B is the Borel subgroup of upper-triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' A fundamental domain for B is  B\\Hq = {(δ1/3, vδ1/3 + δ2/3) : v ∈ Fq} ⊔ {(δ2/3, δ1/3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  □  9  We find the orbital sum (see Appendix A) is  IG(f, γ) = (q2+q+1)  \uf8eb  \uf8edf  �  δ2/3 + δ1/3, δ1/3 + 1  �  +  �  v∈Fq  f  �  (1 − v)δ2/3 − v2δ1/3 + 1, δ2/3 + (v + 1)δ1/3�  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Note that the orbital sum is independent of a, and so we denote each IG(f, γ) by ˜IG(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Thus, in  total, the contribution of these parabolic terms is  q3(q − 1)2  p3(p − 1)2 ˜IG(f) · (p − 1) = q3(q − 1)2  p3(p − 1)  ˜IG(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Parabolic terms of the third kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let γ =  \uf8ee  \uf8f0  a  0  0  a  a  0  0  0  b  \uf8f9  \uf8fb, with a ̸= b ∈ F×  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Here  Gγ =  \uf8f1  \uf8f2  \uf8f3  \uf8ee  \uf8f0  x  0  0  y  x  0  0  0  z  \uf8f9  \uf8fb : x, z ∈ F×  q , y ∈ Fq  \uf8fc  \uf8fd  \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Again, via a change of variables, we find the orbital sum (see Appendix A)  IG(f, γ) = (q2 + q + 1) · Hf  ��  a/b  1  0  a/b  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  The overall contribution of these terms is  �  a,b∈F×  p  a̸=b  q(q − 1)2  p(p − 1)2 (q2 + q + 1) · Hf  ��  a/b  1  0  a/b  ��  = q(q − 1)2  p(p − 1)2 (q2 + q + 1)  �  c∈F×  p  c̸=1  (p − 1) · Hf  ��  c  1  0  c  ��  = q(q − 1)(q3 − 1)  p(p − 1)  �  c∈F×  p  c̸=1  Hf  ��  c  1  0  c  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Elliptic Terms  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Elliptic terms of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Irreducible γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The characteristic polynomial of such a γ is  irreducible in Fp[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Recall that q = pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Depending on whether n ≡ 0 (mod 3), there are two cases  to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  n ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In this case, we have the tower of field extensions Fq/Fp3/Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' So, γ is  similar to a diagonal matrix in GL3(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' But that is not the case in GL3(Fp), which is  why this case is different from the second hyperbolic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Therefore Gγ is the subgroup of  diagonal matrices over Fq while Γγ is K with entries in Fp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' |Γγ| = p3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Since Gγ is the  same, we can reuse the fundamental domain for Gγ from earlier, in the second hyperbolic  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Suppose γ is similar to  � ξ1 0  0  0 ξ2 0  0  0 ξ3  �  in GL3(Fq), where ξ1, ξ2, ξ3 ∈ Fp3\\Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Similarly as for  the second hyperbolic term the orbital sum will be:  IG(f, γ) = |K/Z| · Hf  ��  ξ1/ξ3  0  0  ξ2/ξ3  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Since ξ1, ξ2, ξ3 are Galois conjugates without loss of generality ξ1 = ξp  3 and ξ2 = ξp2  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The  total contribution of the first elliptic terms is therefore:  10  1  3 · (q − 1)3  p3 − 1 · q3 − 1  q − 1  �  ξ∈F×  p3\\F×  p  Hf  ��ξp−1  0  0  ξp2−1  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  (The factor of 1  3 comes from the fact that each γ is counted three times in the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=')  n ̸≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In this case, there is no cubic extension intermediate to Fq/Fp, and so  Fq(δ1/3) is the minimal field (of course, in the sense of containment, not size) over which the  characteristic polynomial of γ has a root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Suppose an eigenvalue of γ is α1 +α2δ1/3 +α3δ2/3  with eigenvector x1 + x2δ1/3 + x3δ2/3, where αi ∈ Fq and xi ∈ F3  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Then,  γx1 = α1x1 + α3δx2 + α2δx3,  γx2 = α2x1 + α1x2 + α3δx3,  γx3 = α3x1 + α2x2 + α1x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Thus, viewed in GL3(Fq), γ is similar to a matrix in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Conversely, non-diagonal matrices  in K give us irreducible elements in GL3(Fp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Further, in this case Gγ is precisely K, and  so Gγ\\G = Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' After identifying elements y ∈ Hq with elements y =  \uf8ee  \uf8f0  a  b  c  d  e  f  0  0  1  \uf8f9  \uf8fb ∈ Aff3(Fq)  and writing γ =  \uf8ee  \uf8f0  α  εδ  βδ  β  α  εδ  ε  β  α  \uf8f9  \uf8fb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' we can compute:  IG(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' γ) =  �  y∈Hq  f(y−1γy)  =  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='f∈Fq  f  \uf8eb  \uf8ed  1  bd − ae  \uf8ee  \uf8f0  −e  b  ce − bf  d  −a  af − cd  0  0  bd − ae  \uf8f9  \uf8fb  \uf8ee  \uf8f0  α  εδ  βδ  β  α  εδ  ε  β  α  \uf8f9  \uf8fb  \uf8ee  \uf8f0  a  b  c  d  e  f  0  0  1  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  =  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='f∈Fq  f  \uf8eb  \uf8ed  1  bd − ae  \uf8ee  \uf8f0  x11  x12  x13  x21  x22  x23  x31  x32  x33  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  where the entries of the matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' by column,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' are as follows:  ⋄ x11 = α(bd − ae) + β(ab + cde − bdf) + ε(ace − abf − deδ)  ⋄ x21 = β(adf − a2 − cd2) + ε(a2f − acd + d2δ)  ⋄ x31 = βd + εa  ⋄ x12 = β(ce2 − bef + b2) + ε(bce − b2f − e2δ)  ⋄ x22 = α(bd − ae) + β(aef − cde − ab) + ε(abf − bcd + deδ)  ⋄ x32 = βe + εb  ⋄ x13 = β(cef + bc − bf 2 − eδ)  ⋄ x23 = β(af 2 − cdf − ac + dδ) + ε(acf − ac2 + dfδ − aδ)  ⋄ x33 = α + βf + εc  Unfortunately,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' there does not seem to be an obvious way to simplify this expression further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  When computing this case for a specific field Fq, it seems that applying this formula may  not be more helpful than directly computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  11  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Elliptic terms of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' γ =  \uf8ee  \uf8f0  a  b  0  c  d  0  0  0  e  \uf8f9  \uf8fb, where g =  �  a  b  c  d  �  is irreducible, and  e ∈ F×  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Also,  Γγ =  \uf8f1  \uf8f2  \uf8f3  \uf8ee  \uf8f0  a  bξ  0  b  a  0  0  0  c  \uf8f9  \uf8fb : c(a2 − b2ξ) ̸= 0  \uf8fc  \uf8fd  \uf8fe ,  where ξ is a nonsquare in Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Once again, depending on whether n ≡ 0 (mod 2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' whether n is even or odd, there are two  cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In this case, Fp ⊂ Fp2 ⊆ Fq and γ is diagonalizable in GL3(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Gγ is a subgroup  of diagonal matrices over Fq, so the fundamental domain Gγ\\Hq is the same as for the  second hyperbolic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Suppose γ is similar to  � η1 0 0  0 η2 0  0  0 e  �  in G, where η1, η2 ∈ Fp2\\Fp and e ∈ F×  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The orbital  sum will be (analogously to the second hyperbolic case):  IG(f, γ) = |K/Z| · Hf  ��  η1/e  0  0  η2/e  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  As η1, η2 are Galois conjugates without loss of generality we may assume that η1 = ηp  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Therefore the total contribution of the second elliptic terms will be:  1  2 ·  (q − 1)3  (p − 1)(p2 − 1) · q3 − 1  q − 1  �  e∈F×  p  �  η∈F×  p2\\F×  p  Hf  ��  η/e  0  0  ηp/e  ��  =  = 1  2 · (q − 1)3  (p2 − 1) · q3 − 1  q − 1  �  η∈F×  p2\\F×  p  Hf  ��  η  0  0  ηp  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In this case γ would not be diagonalizable in GL3(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' In general, it will be  conjugate to an element of the form:  \uf8ee  \uf8f0  k  lξ  0  l  k  0  0  0  m  \uf8f9  \uf8fb ,  where k ∈ Fp, m, l, ξ ∈ F×  p such that ξ is a nonsquare in Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  (This ensures that the  determinant m(k2 − l2ξ) is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=') In this case we have:  Gγ =  \uf8f1  \uf8f2  \uf8f3  \uf8ee  \uf8f0  a  bξ  0  b  a  0  0  0  c  \uf8f9  \uf8fb : a, b, c ∈ Fq, c(a2 − b2ξ) ̸= 0  \uf8fc  \uf8fd  \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  We seek a fundamental domain for this specific Gγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' A fundamental domain for Gγ is given by:  Gγ\\Hq =  ��  x + uδ1/3 + vδ2/3, y + δ1/3�  : v ∈ F×  q , x, y, u ∈ Fq  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  The corresponding orbital sum is  IG(f, γ) = (q3 − 1)(q2 − 1)  (p − 1)(p2 − 1) Hf  ��  k/m  lξ/m  l/m  k/m  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  12  The total contribution from the second elliptic terms in this case is  (q3 − 1)(q2 − 1)  (p2 − 1)  �  s∈Fp  t∈F×  p  Hf  ��  s  tξ  t  s  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  A detailed proof of both the proposition and the orbital sum calculation can be found in  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Examples  Throughout this section, subscripts under multiplicities indicate which group is the domain of  the corresponding representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Set ρ = IndG  Γ 1 and κ = IndG  K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let f : G → C be the constant one function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The right-hand side of the trace  formula will be |G| while the left-hand side becomes (by orthogonality of characters):  �  π∈ ˆG  m(π, ρ)G⟨f, χπ⟩ =  �  π∈ ˆG  m(π, ρ)G⟨χ1, χπ⟩ = |G| · m(1, ρ)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  So the trace formula says |G| · m(1, ρ) = |G| which is a manifestation of Frobenius reciprocity:  m(1, ρ)G =  1  |G|⟨1, IndG  Γ 1⟩G = 1  |Γ|⟨ResG  Γ 1, 1⟩Γ = m(1, 1)Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Let f : Hq → C be the indicator function δp0 which is 1 on p0 and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  It is K−bi-invariant because stab(p0) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' On the left-hand side of the trace formula we have:  �  π∈ ˆG  m(π, ρ)G⟨f, χπ⟩ =  �  π∈ ˆG  m(π, ρ)G  �  g∈G  f(g · p0)χπ  �  g−1�  =  �  π∈ ˆG  m(π, ρ)G  �  k∈K  χπ  �  k−1�  =  =  �  π∈ ˆG  m(π, ρ)G⟨1, χπ⟩K = |K| ·  �  π∈ ˆG  m(π, ρ)G · m(1, ResG  K π)K =  = |K| ·  �  π∈ ˆG  m(π, ρ)G · m(π, κ)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  On the right-hand side, the only nonzero terms are the central terms and the first elliptic terms  when n is not divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' If that is the case, we claim that for a non-central γ in K we have  IG(f, γ) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Suppose γ corresponds to multiplication by ξ ∈ Fq  �  δ1/3�× under the isomorphism K ≃ Fq  �  δ1/3�×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  When viewed as an element of GL3(Fq), the eigenvalues of γ are the Galois conjugates of ξ, which  are ξ, ξq and ξq2 in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' This means that the conjugates of γ lying in K are precisely γq and  γq2, hence there will be exactly three non-zero terms in each orbital sum IG(f, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  The total contribution of the first elliptic terms will be:  1  3 · q3 − 1  p3 − 1  �  γ∈K  γ̸∈Z  IG(f, γ) = (q3 − 1)(q3 − q)  p3 − 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Thus we get:  Case 1: n is divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Then we have  �  π∈ ˆG  m(π, ρ)G · m(π, κ)G = q3(q2 − 1)(q − 1)  p3(p2 − 1)(p3 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  13  Case 2: n is not divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Then we have  �  π∈ ˆG  m(π, ρ)G · m(π, κ)G = q3(q2 − 1)(q − 1)  p3(p2 − 1)(p3 − 1) + q3 − q  p3 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Note that the sum on the left-hand side can also be written as:  �  π∈ ˆG  m(π, ρ)G · m(π, κ)G =  1  |G|⟨χρ, χκ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Character of the representation ρ = IndG  Γ 1  Having effectively computed the right-hand side of the pre-trace formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='4), we now turn to  the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' For this, we require the character χρ of ρ, which we now calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' We use the  usual formula for the character of an induced representation (see [9, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' 16]):  χρ(γ) = 1  |Γ|  �  x∈G  1Γ(xγx−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  We treat each conjugacy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' By H1,q, H2,q, P1,q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' , E2,q we denote the size of various types of  conjugacy classes in GL3(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Given a set S, we use  1S to denote the indicator function for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Central class: γ =  \uf8ee  \uf8f0  a  0  0  0  a  0  0  0  a  \uf8f9  \uf8fb , a ∈ F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Since γ ∈ Z(G), we have xγx−1 = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' So,  χρ(γ) = |G|  |Γ|  1Fp(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  First hyperbolic class: γ =  \uf8ee  \uf8f0  a  0  0  0  a  0  0  0  b  \uf8f9  \uf8fb with a ̸= b ∈ F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Now, γ is similar to an element of  Γ only when a, b are roots of a polynomial over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' This occurs only when a, b ∈ F×  p ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' for  instance, this can be seen by considering the determinant Det(γ) = a2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Being the constant  coefficient of charγ ∈ Fp[x], we have Det(γ) ∈ Fp, which implies Det(γ) is invariant under  the Frobenius automorphism (as Gal(Fq/Fp) is generated by the Frobenius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Since Fq/Fp  is Galois, if a, b ̸∈ Fp, they must be Galois conjugates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=', ap = b and bp = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' But this  forces a2b = (a2b)p = b2a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=', a = b, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' So, by a simple application of the  orbit-stabilizer theorem, it follows that  χρ(γ) = |G|  |Γ|  H1,p  H1,q  1Γ(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Second hyperbolic class: γ =  \uf8ee  \uf8f0  a  0  0  0  b  0  0  0  c  \uf8f9  \uf8fb with a, b, c ∈ F×  q distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' There are three cases  which allow γ to be similar to an element of Γ: either all three are Galois conjugates in  Fp3/Fp (only possible if 3 | n), or a, b are Galois conjugates in Fp2/Fp and c ∈ Fp (only  possible if 2 | n), or all three belong to Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' These cases give rise to the three terms in the  sum:  χρ(γ) =  |G|  |Γ| · H2,q  �  E2,p ·  1Fp3\\Fp(a)δb(ap)δc(ap2) + E1,p ·  1Fp2\\Fp(a)δb(ap)1Fp(c) + H2,p ·  1Γ(γ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  14  First parabolic class: γ =  \uf8ee  \uf8f0  a  1  0  0  a  0  0  0  a  \uf8f9  \uf8fb , a ∈ F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' If a ̸∈ Fp was the root of a polynomial over  Fp, its Galois conjugate ap ̸= a would also be a root, which is clearly not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' So,  χρ(γ) = |G|  |Γ|  P1,p  P1,q  1Fp(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Second parabolic class: γ =  \uf8ee  \uf8f0  a  1  0  0  a  1  0  0  a  \uf8f9  \uf8fb , a ∈ F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Similar to the previous class,  χρ(γ) = |G|  |Γ|  P2,p  P2,q  1Fp(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Third parabolic class: γ =  \uf8ee  \uf8f0  a  1  0  0  a  0  0  0  b  \uf8f9  \uf8fb with a ̸= b ∈ F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' By a similar argument as for the  first hyperbolic class, we have  χρ(g) = |G|  |Γ|  P3,p  P3,q  1Γ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  First elliptic class: γ =  \uf8ee  \uf8f0  w  0  0  0  wq  0  0  0  r  \uf8f9  \uf8fb with w ∈ Fq2 \\ Fq and r ∈ F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Here γ is similar to  an element of Γ if and only if w, wq are roots of a degree-2 irreducible polynomial over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  However, if 2 | n, then Fp2 is an intermediate extension in Fq/Fp and so roots of any degree-2  polynomial over Fp are included in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Thus, χρ(γ) = 0 if n is divisible by 2, otherwise  χρ(γ) = |G|  |Γ|  E1,p  E1,q  1Fp2\\Fp(w)1Fp(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Second elliptic class: γ =  \uf8ee  \uf8f0  w  0  0  0  wq  0  0  0  wq2  \uf8f9  \uf8fb , w ∈ Fq(δ1/3) \\ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' As for the previous class,  χρ(γ) = 0 if n is divisible by 3, otherwise  χρ(γ) = |G|  |Γ|  E2,p  E2,q  1Fp3\\Fp(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Decomposition of ρ when 2, 3 ∤ n  In this section, we compute the decomposition of ρ when neither 2 nor 3 divide n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' thus elliptic  elements in GL3(Fp) remain elliptic in GL3(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  We use the character table for GL3(Fq) in Terras [9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' 383-4] and Steinberg [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' As usual, to  calculate multiplicity m(π, ρ) of an irreducible representation π present in ρ, we compute the inner  product of their respective characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' By H1, H2, P1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' , E2 we denote the size of different types of  conjugacy classes in GL3(Fp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' By NmE/F, we denote the norm of the field extension E/F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  15  (1) α, a character of F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Then,  m(α, ρ) = 1  |Γ|  � �  a∈F×  p  α(a3) + H1  �  a,b∈F×  p  a̸=b  α(a2b) + H2  6  �  a,b,c∈F×  p  a,b,c distinct  α(abc) + P1  �  a∈F×  p  α(a3) + P2  �  a∈F×  p  α(a3)  + P3  �  a,b∈F×  p  a̸=b  α(a2b) + E1  2  �  w∈Fp2\\Fp  r∈F×  p  α(r NmFq2/Fq(w)) + E2  3  �  w∈Fp3\\Fp  α(NmFq3/Fq(w))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  We can simplify these sums on the basis of three cases:  α3 |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Each of the sums is zero and so m(α, ρ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α3 |F×  p = 1 but α |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' A short calculation yields  m(α, ρ) = 1  |Γ|  �  1 − H1 + H2  3 + P1 + P2 − P3 − E2  3  �  (p − 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α |F×  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Then simply  m(α, ρ) = 1  |Γ|  �  {g}∈CΓ  |{g}| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  (2) πα, where α is a character of F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα, ρ) = 1  |Γ|  �  (q2 + q)  �  a∈F×  p  α(a3) + (q + 1)H1  �  a,b∈F×  p  a̸=b  α(a2b) + H2  3  �  a,b,c∈F×  p  a,b,c distinct  α(abc) + qP1  �  a∈F×  p  α(a3)  + 0 + P3  �  a,b∈F×  p  a̸=b  α(a2b) + 0 − E2  3  �  w∈Fp3\\Fp  α(NmFq3/Fq(w))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α3 |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Here again all the sums are zero, giving m(πα, ρ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α3 |F×  p = 1 but α |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα, ρ) = 1  |Γ|  �  (q2 + q) − (q + 1)H1 + 2  3H2 + qP1 − P3 + E2  3  �  (p − 1)  =  (q − p)(q − p2)  p3(p − 1)2(p + 1)(p2 + p + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α |F×  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα, ρ) = 1  |Γ|  �  (q2 + q)(p − 1) + (q + 1)H1(p − 1)(p − 2) + (p − 1)(p − 2)(p − 3)H2  3  + qP1(p − 1) + (p − 1)(p − 2)P3 − E2  3 (p3 − p)  �  =  (q − p)(q + p5 − 2p2)  p3(p − 1)2(p + 1)(p2 + p + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  As a sanity check, showing this is actually an integer is not very hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  16  (3) π′  α, where α is a character of F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(π′  α, ρ) = 1  |Γ|  �  q3 �  a∈F×  p  α(a3) + qH1  �  a,b∈F×  p  a̸=b  α(a2b) + H2  6  �  a,b,c∈F×  p  a,b,c distinct  α(abc) + 0 + 0 + 0  − E1  2  �  w∈Fp2\\Fp  r∈F×  p  α(r NmFq2/Fq(w)) + E2  3  �  w∈Fp3\\Fp  α(NmFq3/Fq(w))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α3 |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' m(πα, ρ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α3 |F×  p = 1 but α |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(π′  α, ρ) = 1  |Γ|  �  q3 − qH1 + 1  3H2 − E2  3  �  (p − 1) =  (q − p)(q − p2)(q + p2 + p)  p3(p − 1)2(p + 1)(p2 + p + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α |F×  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(π′  α, ρ) = 1  |Γ|  �  q3(p − 1) + q(p − 1)(p − 2)H1 + (p − 1)(p − 2)(p − 3)H2  6  − (p2 − p)(p − 1)E1  2 + (p3 − p)E2  3  �  = (q − p)(q2 + pq + p5 − p4 − p3 − p2)  p3(p − 1)2(p + 1)(p2 + p + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  (4) πα,β for distinct characters α and β of F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα,β, ρ) = 1  |Γ|  �  (q2 + q + 1)  �  a∈F×  p  α(a2)β(a) + H1  �  a,b∈F×  p  a̸=b  ((q + 1)α(ab)β(a) + α(a2)β(b))  + H2  6  �  a,b,c∈F×  p  a,b,c distinct  (β(a)α(bc) + β(b)α(ac) + β(c)α(ab)) + (q + 1)P1  �  a∈F×  p  α(a2)β(a)  + P2  �  a∈F×  p  α(a2)β(a) + P3  �  a,b∈F×  p  a̸=b  (α(ab)β(a) + α(a2)β(b))  + E1  2  �  w∈Fp2\\Fp  r∈F×  p  α(NmFq2/Fq(w))β(r) + 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α2β |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' m(πα,β, ρ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  17  α2β |F×  p = 1 but α2 |F×  p ̸= 1 and β |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα,β, ρ) = 1  |Γ|  �  (q2 + q + 1)(p − 1) − (q + 2)(p − 1)H1 + (p − 1)H2  + (q + 1)(p − 1)P1 + (p − 1)P2 − 2(p − 1)P3  �  =  (q − p)(q − p2)  p3(p − 1)2(p + 1)(p2 + p + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α2 |F×  p = 1 and β |F×  p = 1 but α |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα,β, ρ) = 1  |Γ|  �  (q2 + q + 1)(p − 1) − (p − 1)(q − p + 3)H1 − (p − 1)(p − 3)H2  2  + (q + 1)(p − 1)P1 + (p − 1)P2 + (p − 1)(p − 3)P3 − (p − 1)2 E1  2  �  =  (q − p)(q − p2)  p3(p − 1)2(p + 1)(p2 + p + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α |F×  p = 1 and β |F×  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα,β, ρ) = 1  |Γ|  �  (q2 + q + 1)(p − 1) + (q + 2)(p − 1)(p − 2)H1 + (p − 1)(p − 2)(p − 3)H2  2  + (q + 1)(p − 1)P1 + (p − 1)P2 + 2(p − 1)(p − 2)P3 + (p − 1)(p2 − p)E1  2  �  = q(q + p5 − 2p2 − p) + p3(p5 − 2p3 − p2 + 3)  p3(p − 1)2(p + 1)(p2 + p + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  (5) π′  α,β for distinct characters α and β of F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(π′  α,β, ρ) = 1  |Γ|  �  q(q2 + q + 1)  �  a∈F×  p  α(a2)β(a) + H1  �  a,b∈F×  p  a̸=b  ((q + 1)α(ab)β(a) + qα(a2)β(b))  + H2  6  �  a,b,c∈F×  p  a,b,c distinct  (β(a)α(bc) + β(b)α(ac) + β(c)α(ab)) + qP1  �  a∈F×  p  α(a2)β(a)  + 0 + P3  �  a,b∈F×  p  a̸=b  α(ab)β(a) − E1  2  �  w∈Fp2\\Fp  r∈F×  p  α(NmFq2/Fq(w))β(r) + 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α2β |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' m(π′  α,β, ρ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  18  α2β |F×  p = 1 but α2 |F×  p ̸= 1 and β |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(π′  α,β, ρ) = 1  |Γ|  �  q(q2 + q + 1)(p − 1) − (2q + 1)(p − 1)H1 + (p − 1)H2  + q(p − 1)P1 − (p − 1)P3  �  = (q − p)(q − p2)(q + p2 + p + 1)  p3(p − 1)2(p + 1)(p2 + p + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α2 |F×  p = 1 and β |F×  p = 1 but α |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα,β, ρ) = 1  |Γ|  �  q(q2 + q + 1)(p − 1) + (p − 1)(qp − 3q − 1)H1 − (p − 1)(p − 3)H2  2  + q(p − 1)P1 − (p − 1)P3 + (p − 1)2 E1  2  �  = (q − p)(q(q + p + 1) + p2(p3 − p2 − p − 2))  p3(p − 1)2(p + 1)(p2 + p + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α |F×  p = 1 and β |F×  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα,β, ρ) = 1  |Γ|  �  q(q2 + q + 1)(p − 1) + (2q + 1)(p − 1)(p − 2)H1 + (p − 1)(p − 2)(p − 3)H2  2  + q(p − 1)P1 + (p − 1)(p − 2)P3 − (p − 1)(p2 − p)E1  2  �  = (q − p)(q(q + p + 1) + p2(2p − 3)(p2 + p + 1))  p3(p − 1)2(p + 1)(p2 + p + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  (6) πα,β,γ for distinct characters α, β, γ of F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα,β,γ, ρ) = 1  |Γ|  �  (q + 1)(q2 + q + 1)  �  a∈F×  p  α(a)β(a)γ(a) + H2  �  a,b,c∈F×  p  a,b,c distinct  α(a)β(b)γ(c)  + (q + 1)H1  �  a,b∈F×  p  a̸=b  (α(b)β(a)γ(a) + β(b)γ(a)α(a) + γ(b)α(a)β(a))  + (2q + 1)P1  �  a∈F×  p  α(a)β(a)γ(a) + P2  �  a∈F×  p  α(a)β(a)γ(a)  + P3  �  a,b∈F×  p  a̸=b  (α(b)β(a)γ(a) + β(b)γ(a)α(a) + γ(b)α(a)β(a)) + 0 + 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  αβγ |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' m(πα,β,γ, ρ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  19  αβγ |F×  p = 1 but none of α, β, or γ restrict to the trivial character on F×  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα,β,γ, ρ) = 1  |Γ|  �  (q + 1)(q2 + q + 1)(p − 1) + 2(p − 1)H2 − 3(q + 1)(p − 1)H1  + (2q + 1)(p − 1)P1 + (p − 1)P2 − 3(p − 1)P3  �  = (q − p)(q − p2)(q + p2 + p + 2)  p3(p − 1)2(p + 1)(p2 + p + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  αβ |F×  p = 1 and γ |F×  p = 1 but α and β are nontrivial on F×  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Note that since m(πα,β,γ, ρ)  is symmetric in α, β, γ, the calculation here also holds for the other two permutations  of this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα,β,γ, ρ) = 1  |Γ|  �  (q + 1)(q2 + q + 1)(p − 1) − (p − 1)(p − 3)H2 + (q + 1)(p − 1)(p − 4)H1  + (2q + 1)(p − 1)P1 + (p − 1)P2 + (p − 1)(p − 4)P3  �  = (q − p)(q(q + p + 2) + p2(p3 − p2 − p − 3))  p3(p − 1)2(p + 1)(p2 + p + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  All three α, β, γ restrict to the trivial character on F×  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(πα,β,γ, ρ) = 1  |Γ|  �  (q + 1)(q2 + q + 1)(p − 1) + (p − 1)(p − 2)(p − 3)H2 + 3(q + 1)(p − 1)(p − 2)H1  + (2q + 1)(p − 1)P1 + (p − 1)P2 + 3(p − 1)(p − 2)P3  �  = (q(q2 + 2q + 3p5 − p4 − p3 − 6p2 − 2p) + p3(p5 − 4p3 + p + 6))  p3(p − 1)2(p + 1)(p2 + p + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  (7) ρα,ν for α a character of F×  q and ν a character of F×  q2 such that νq ̸= ν (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=', it is nondecom-  posable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(ρα,ν, ρ) = 1  |Γ|  �  (q − 1)(q2 + q + 1)  �  a∈F×  p  α(a)ν(a) + (q − 1)H1  �  a,b∈F×  p  a̸=b  α(b)ν(a) + 0  − P1  �  a∈F×  p  α(a)ν(a) − P2  �  a∈F×  p  α(a)ν(a) − P3  �  a,b∈F×  p  a̸=b  α(b)ν(a)  − E1  2  �  w∈Fp2\\Fp  r∈F×  p  α(r)(ν(w) + ν(wp)) + 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Note that the last nonzero sum can be written as −E1  �  w∈Fp2\\Fp  r∈F×  p  α(r)ν(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  αν |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' m(ρα,ν, ρ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  20  αν |F×  p = 1 but α |F×  p ̸= 1 and ν |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(ρα,ν, ρ) = 1  |Γ|  �  (q − 1)(q2 + q + 1)(p − 1) − (q − 1)(p − 1)H1  − (p − 1)P1 − (p − 1)P2 + (p − 1)P3  �  =  (q − p)(q − p2)(q + p2 + p)  p3(p − 1)2(p + 1)(p2 + p + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α |F×  p = 1, ν |F×  p3̸= 1 but ν |F×  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(ρα,ν, ρ) = 1  |Γ|  �  (q − 1)(q2 + q + 1)(p − 1) + (q − 1)(p − 1)(p − 2)H1  − (p − 1)P1 − (p − 1)P2 − (p − 1)(p − 2)P3 + (p − 1)2E1  �  = (q − p)(q(q + p) + p2(p3 − p2 − p − 1))  (p − 1)2p3(p + 1)(p2 + p + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  α |F×  p = 1 and ν |F×  p3= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(ρα,ν, ρ) = 1  |Γ|  �  (q − 1)(q2 + q + 1)(p − 1) + (q − 1)(p − 1)(p − 2)H1  − (p − 1)P1 − (p − 1)P2 − (p − 1)(p − 2)P3 − (p − 1)(p2 − p)E1  �  = q(q2 + p5 − p4 − p3 − 2p2) + p4(−p4 + 2p + 1)  p3(p − 1)2(p + 1)(p2 + p + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  (8) σµ for µ a character of F×  q3 such that µq ̸= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(σµ, ρ) = 1  |Γ|  �  (q − 1)2(q + 1)  �  a∈F×  p  µ(a) + 0 + 0 − (q − 1)P1  �  a∈F×  p  µ(a) + P2  �  a∈F×  p  µ(a) + 0  + 0 + E2  3  �  w∈Fp3\\Fp  (µ(w) + µ(wp) + µ(wp2))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  µ |F×  p ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' m(σµ, ρ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  µ |F×  p3̸= 1 but µ |F×  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(σµ, ρ) = 1  |Γ|  �  (q − 1)2(q + 1)(p − 1) − (q − 1)(p − 1)P1 + (p − 1)P2 − (p − 1)E2  �  = (q − p)(q − p2)(q + p2 + p − 1)  p3(p − 1)2(p + 1)(p2 + p + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  21  µ |F×  p3= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  m(σµ, ρ) = 1  |Γ|  �  (q − 1)2(q + 1)(p − 1) − (q − 1)(p − 1)P1 + (p − 1)P2 + (p3 − p)E2  �  = q(q2 − q − p4 − p3 + p) + p4(p4 − p2 + 1)  p3(p − 1)2(p + 1)(p2 + p + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  References  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
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+page_content=' Martinez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The finite upper half space and related hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Journal of Number Theory, 84:342–360,  10 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Murty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Ramanujan graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Ramanujan Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=', 18, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1:1–20, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Sarnak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Class numbers of indefinite binary quadratic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Journal of Number Theory, 15(2):229–247, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Selberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applica-  tions to Dirichlet series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Indian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=', 20:47–87, 1956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Steinberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The representations of gl(3,q), gl(4,q), pgl(3,q), and pgl(4,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Canadian Journal of Mathematics,  3:225 - 235, 1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Terras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Fourier Analysis on Finite Groups and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' London Mathematical Society Student Texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Cambridge University Press, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Wallace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' The Selberg trace formula for SL(3, Z)\\SL(3, R)/SO(3, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Transactions of the American Mathe-  matical Society, 345(1):1–36, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  22  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Appendix A: Fundamental Domains and Orbital Sums  In this appendix, we give proofs of fundamental domains as well as some computations of orbital  sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Observe that if α, β ∈ Fq  �  δ1/3�  then (α, β) ∈ Hq if and only if {α, β, 1} form a basis  of Fq  �  δ1/3�  over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Suppose (α, β) ∈ Hq and x, y, z ∈ Fq are such that:  x · aα + bβ + c  rα + sβ + t + y · dα + eβ + f  rα + sβ + t + z · 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  The denominator rα+sβ +t could only be nonzero if r = s = t = 0 (as α, β, 1 are linearly  independent) but that cannot happen as  ������  a  b  c  d  e  f  r  s  t  ������  ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Thus we find:  x(aα + bβ + c) + y(dα + eβ + f) + z(rα + sβ + t) = 0  In matrix form:  �  x  y  z  �  \uf8ee  \uf8f0  a  b  c  d  e  f  r  s  t  \uf8f9  \uf8fb  \uf8ee  \uf8f0  α  β  1  \uf8f9  \uf8fb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  This implies that  �  x  y  z  �  \uf8ee  \uf8f0  a  b  c  d  e  f  r  s  t  \uf8f9  \uf8fb =  �  0  0  0  �  because α, β, 1 are linearly indepen-  dent over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Since our matrix is nonsingular, it follows that x = y = z = 0 and  �  aα+bβ+c  rα+sβ+t, dα+eβ+f  rα+sβ+t , 1  �  is a basis of Fq  �  δ1/3�  over Fq, proving the claim that GL3(Fq) has  an action on Hq of the above form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  □  23  Second Hyperbolic Term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  IG(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' γ) =  �  t∈Gγ\\G/K  �  u∈K/{aI}  f((tu)−1γ(tu))  =  �  t∈Gγ\\Hq  |K|  |{aI}|f(t−1γt)  = q3 − 1  q − 1  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='s∈Fq  ys̸=1  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  s  sy−1  1  1−sy  r−sx  sy−1  1  1−sy  y  sy−1  x−ry  sy−1  0  0  1  \uf8f9  \uf8fb γ  \uf8ee  \uf8f0  y  1  x  1  s  r  0  0  1  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8 +  + q3 − 1  q − 1  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='s∈Fq  s̸=0  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  1  0  −x  −s  1  xs − r  0  0  1  \uf8f9  \uf8fb γ  \uf8ee  \uf8f0  1  0  x  s  1  r  0  0  1  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8 +  + q3 − 1  q − 1  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r∈Fq  y̸=0  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  1  −y  −x + yr  0  1  −r  0  0  1  \uf8f9  \uf8fb γ  \uf8ee  \uf8f0  1  y  x  0  1  r  0  0  1  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  + q3 − 1  q − 1  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r∈Fq  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  1  0  −x  0  1  −r  0  0  1  \uf8f9  \uf8fb γ  \uf8ee  \uf8f0  1  0  x  0  1  r  0  0  1  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  = (q2 + q + 1)  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='s∈Fq  ys̸=1  f  \uf8eb  \uf8ec  \uf8ed  \uf8ee  \uf8ef\uf8f0  b−asy  1−sy  s(a−b)  sy−1  asx−br+c(r−sx)  sy−1  y(b−a)  sy−1  a−bsy  1−sy  −ax+bry+c(x−ry)  sy−1  0  0  c  \uf8f9  \uf8fa\uf8fb p0  \uf8f6  \uf8f7  \uf8f8 +  + (q2 + q + 1)  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='s∈Fq  s̸=0  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  a  0  ax − cx  bs − as  b  −axs + br + c (xs − r)  0  0  c  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8 +  + (q2 + q + 1)  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r∈Fq  y̸=0  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  a  ay − by  ax − byr + c (yr − x)  0  b  br − cr  0  0  c  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8 +  + (q2 + q + 1)  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='y∈Fq  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  a  0  ax − cx  0  b  br − cr  0  0  c  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8 = (q2 + q + 1)(S1 + S2 + S3 + S4)  We simplify each of the four terms separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Observe that the substitution:  �  ˜x  ˜y  �  =  � asx−br+c(r−sx)  sy−1  −ax+bry+c(x−ry)  sy−1  �  =  1  sy − 1  �  (a − c)s  c − b  c − a  (b − c)y  � �  x  r  �  is invertible as  ����  (a − c)s  c − b  c − a  (b − c)y  ���� = (a − c)(b − c)(sy − 1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Therefore:  S1 =  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r∈Fq  �  y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='s∈Fq  ys̸=1  f  \uf8eb  \uf8ec  \uf8ed  \uf8ee  \uf8ef\uf8f0  b−asy  1−sy  s(a−b)  sy−1  x  y(b−a)  sy−1  a−bsy  1−sy  r  0  0  c  \uf8f9  \uf8fa\uf8fb p0  \uf8f6  \uf8f7  \uf8f8 =  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r∈Fq  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  b  0  x  0  a  r  0  0  c  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8 +  24  +  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r∈Fq  y̸=0  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  b  0  x  y(a − b)  a  r  0  0  c  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8+  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='s∈Fq  s̸=0  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  b  s(b − a)  x  0  a  r  0  0  c  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8+  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r∈Fq  �  y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='s∈F×  q  ys̸=1  f  \uf8eb  \uf8ec  \uf8ed  \uf8ee  \uf8ef\uf8f0  b−asy  1−sy  s(a−b)  sy−1  x  y(b−a)  sy−1  a−bsy  1−sy  r  0  0  c  \uf8f9  \uf8fa\uf8fb p0  \uf8f6  \uf8f7  \uf8f8  We can make similar changes of variables  �  ˜x  ˜r  �  as above and find that:  S2 =  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r∈Fq  s∈F×  q  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  a  0  x  (b − a)s  b  r  0  0  c  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  S3 =  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r∈Fq  y∈F×  q  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  a  (a − b)y  x  0  b  r  0  0  c  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  S4 =  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r∈Fq  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  a  0  x  0  b  r  0  0  c  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  Putting together all the above we get that S1 + S2 + S3 + S4 is the following sum:  S1 + S2 + S3 + S4 = Hf  ��  a/c  0  0  b/c  ��  This is because each conjugate of  � a 0  0 b  �  appears exactly once in the upper 2 × 2 block in  the sum S1 + S2 + S3 + S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Thus, the total contribution is:  1  6(q2 + q + 1)(q − 1)3  (p − 1)3  �  a,b,c∈F×  p  a̸=b̸=c  Hf  ��  a/c  0  0  b/c  ��  = 1  6(q2 + q + 1)(q − 1)4  (p − 1)3  �  a,b∈F×  p \\{1}  a̸=b  Hf  ��  a  0  0  b  ��  =  = 1  3(q3 − 1)(q − 1)3  (p − 1)3  �  {a,b}⊆F×  p \\{1}  Hf  ��  a  0  0  b  ��  Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' First to show the uniqueness of each representative, suppose there exists m ∈ Gγ  such that  m · (u1δ1/3, v1δ1/3 + δ2/3) = (u2δ1/3, v2δ1/3 + δ2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Then we deduce that y = 0 and so u1 = u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' We also see c = d, implying v1 = v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Next,  suppose  m · (u1δ1/3, v1δ1/3 + δ2/3) = (δ1/3 + u2δ2/3, δ1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  But this immediately forces c = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Similarly,  m · (δ1/3 + u1δ2/3, δ1/3) = (δ1/3 + u2δ2/3, δ1/3),  directly implies u1 = u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  25  To show the completeness of this fundamental domain, take an arbitrary element (α, β) ∈  Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' If β3 ̸= 0, then observe that  \uf8ee  \uf8f0  1  α3  α1  0  β3  β1  0  0  1  \uf8f9  \uf8fb  �  (α2 − α3  β2  β3  )δ1/3, β2  β3  δ1/3 + δ2/3  �  = (α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Otherwise, we must have β2 ̸= 0, in which case  \uf8ee  \uf8f0  1  α2 − 1  α1  0  β2  β1  0  0  1  \uf8f9  \uf8fb  �  δ1/3 + α3δ2/3, δ1/3�  = (α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  □  First Parabolic Term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  IG(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' γ) =  �  t∈Gγ\\G/K  �  u∈K/{aI}  f((tu)−1γ(tu))  =  �  t∈Gγ\\Hq  |K|  |{aI}|f(t−1γt)  = q3 − 1  q − 1  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='v∈Fq  u̸=0  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  − v  u  1  0  1  u  0  0  0  0  1  \uf8f9  \uf8fb γ  \uf8ee  \uf8f0  0  u  0  1  v  0  0  0  1  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8 +  �  u∈F×  q  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  1  u  − 1  u  0  0  1  0  0  0  1  \uf8f9  \uf8fb γ  \uf8ee  \uf8f0  u  1  0  0  1  0  0  0  1  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  \uf8f6  \uf8f7  \uf8f7  \uf8f8  = (q2 + q + 1)  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='v∈Fq  u̸=0  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  a  0  − av  u  0  a  a  u  0  0  a  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8 +  �  u∈F×  q  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  a  0  a  u  0  a  0  0  0  a  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  \uf8f6  \uf8f7  \uf8f7  \uf8f8  = (q2 + q + 1)  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='v∈Fq  u̸=0  f  �  −v  u + δ2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' 1  u + δ1/3  �  +  �  u∈F×  q  f  �  δ2/3 + 1  u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' δ1/3  �  \uf8f6  \uf8f7  \uf8f7  \uf8f8  = (q2 + q + 1) (Hf(I) − f(p0))  Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' First, we check that if m ∈ B such that mx = y for x, y ∈  B\\Hq, then x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Suppose there exists m ∈ B such that  m(δ1/3, v1δ1/3 + δ2/3) = (δ1/3, v2δ1/3 + δ2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Then we right away see that c/a = 1 and so v1 = v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Finally, it is also easy to see that we  cannot use B to move elements between the two sets in the disjoint union of B\\Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Next, we check that for any arbitrary element (α, β) ∈ Hq (where α = α1+α2δ1/3+α3δ2/3  and β = β1 + β2δ1/3 + β3δ2/3), there exists m ∈ B and x ∈ B\\Hq such that mx = (α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  First suppose that β3 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  \uf8ee  \uf8f0  α2 − α3  β2  β3  α3  α1  0  β3  β1  0  0  1  \uf8f9  \uf8fb  �  δ1/3, β2  β3  δ1/3 + δ2/3  �  = (α, β)  26  If β3 = 0, then  \uf8ee  \uf8f0  α3  α2  α1  0  β2  β1  0  0  1  \uf8f9  \uf8fb  �  δ2/3, δ1/3�  = (α, β)  Second Parabolic Term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  IG(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' γ) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='t∈Gγ\\G/K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='u∈K/{aI}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='f((tu)−1γ(tu))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='t∈Gγ\\Hq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='|K|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='|aI|f(t−1γt)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='= q3 − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='q − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='\uf8edf  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='\uf8ee  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
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+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='= (q2 + q + 1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='\uf8edf  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='δ2/3 + δ1/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' δ1/3 + 1  �  +  �  v∈Fq  f  �  (1 − v)δ2/3 − v2δ1/3 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' δ2/3 + (v + 1)δ1/3�  \uf8f6  \uf8f8  Third Parabolic Term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Gγ\\Hq = {(δ1/3 + uδ2/3 + v, rδ2/3 + s) : r ̸= 0} ⊔ {(δ2/3 + v, rδ1/3 + s) : r ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  IG(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' γ) =  �  t∈Gγ\\G/K  �  u∈K/{aI}  f((tu)−1γ(tu))  =  �  t∈Gγ\\Hq  |K|  |{aI}|f(t−1γt)  = q3 − 1  q − 1  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='s∈Fq  r̸=0  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  0  1  r  − s  r  1  − u  r  su  r − v  0  0  1  \uf8f9  \uf8fb γ  \uf8ee  \uf8f0  u  1  v  r  0  s  0  0  1  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8 +  �  v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='s∈Fq  r̸=0  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  1  0  −v  0  1  r  − s  r  0  0  1  \uf8f9  \uf8fb γ  \uf8ee  \uf8f0  1  0  v  0  r  s  0  0  1  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  \uf8f6  \uf8f7  \uf8f7  \uf8f8  = (q2 + q + 1)  �  �  u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='s∈Fq  r̸=0  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  a(r+u)  r  a  r  a(s+v)−bs  r  − au2  r  a(r−u)  r  − −arv+asu+auv+brv−bsu  r  0  0  b  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  +  �  v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='s∈Fq  r̸=0  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  a  0  v(a − b)  a  r  a  a(s+v)−bs  r  0  0  b  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Employ the following change of variables in the two sums:  �  ˜v  ˜s  �  = 1  r  �  a  (a − b)  (a − b)r − au  (b − a)u  � �  v  s  �  �  v∗  s∗  �  =  �  a − b  0  a/r  (a − b)/r  � �  v  s  �  27  The determinants are both (a − b)2/r ̸= 0 hence the transformations are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' It  follows that the above sums reduce to horocycle transforms of parabolic conjugacy classes  in GL2(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  IG(f, γ) = (q2 + q + 1)Hf  ��  a/b  1  0  a/b  ��  Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  We first show that every Gγ-orbit on Hq contains at least one element of the above form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Let z ∈ Hq and write:  z =  \uf8ee  \uf8f0  a  b  c  d  e  f  0  0  1  \uf8f9  \uf8fb p0  We want to find r, s ∈ Fq not both zero and t ∈ F×  q such that:  \uf8ee  \uf8f0  r  sξ  0  s  r  0  0  0  t  \uf8f9  \uf8fb  \uf8ee  \uf8f0  a  b  c  d  e  f  0  0  1  \uf8f9  \uf8fb =  \uf8ee  \uf8f0  v  u  x  0  1  y  0  0  1  \uf8f9  \uf8fb  for some v ∈ F×  q and u, x, y ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' This amounts to solving the following set of equations  for s, r, t over Fq:  sa + rd = 0  sb + re = 1  t = 1  By assumption, the determinant  ����  a  b  d  e  ���� is nonzero, therefore the above system admits a  unique solution for s, r, t over Fq, moreover, s and r cannot be simultaneously zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  We now need to show that the domain does not contain any orbit repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Suppose  that:  \uf8ee  \uf8f0  r  sξ  0  s  r  0  0  0  t  \uf8f9  \uf8fb  \uf8ee  \uf8f0  v  u  x  0  1  y  0  0  1  \uf8f9  \uf8fb =  \uf8ee  \uf8f0  v′  u′  x′  0  1  y′  0  0  1  \uf8f9  \uf8fb  Writing up the above equations again we get:  sv = 0  su + r = 1  t = 1  As v ̸= 0 we find that s = 0, r = 1 and t = 1 so that the first matrix on the left-hand side  is the identity, which implies that x = x′, y = y′, u = u′ and v = v′, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Second Elliptic Term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  IG(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' γ) = (q − 1)(q2 − 1)  (p − 1)(p2 − 1) · q3 − 1  q − 1  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='u∈Fq  v∈F×  q  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  v  u  x  0  1  y  0  0  1  \uf8f9  \uf8fb  −1 \uf8ee  \uf8f0  k  lξ  0  l  k  0  0  0  m  \uf8f9  \uf8fb  \uf8ee  \uf8f0  v  u  x  0  1  y  0  0  1  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8 =  28  = (q3 − 1)(q2 − 1)  (p − 1)(p2 − 1)  �  u∈Fq  v∈F×  q  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='y∈Fq  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  −lu + k  lξ−lu2  v  x(−lu+k)+y(lξ−ku)+m(uy−x)  v  vl  lu + k  lx + ky − my  0  0  m  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  Set:  ˜x = (k − m − lu)  v  x + (m − k)u + lξ  v  y  ˜y = lx + (k − m)y  The determinant corresponding to this change of variables is  �  (k − m)2 − l2ξ  �  /v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' It could  only be zero if k − m = 0 and l = 0 since ξ is not a square, but then γ would be similar to a  diagonal matrix, contradicting our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Thus the orbital sum can be simplified as:  IG(f, γ) = (q3 − 1)(q2 − 1)  (p − 1)(p2 − 1)  �  u∈Fq  v∈F×  q  �  ˜x,˜y∈Fq  f  \uf8eb  \uf8ed  \uf8ee  \uf8f0  −lu + k  lξ−lu2  v  ˜x  vl  lu + k  ˜y  0  0  m  \uf8f9  \uf8fb p0  \uf8f6  \uf8f8  Finally, notice that the upper-left 2 × 2 block is similar to the elliptic matrix  �  k lξ  l k  �  over  GL2(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content=' Hence we obtain  IG(f, γ) = (q3 − 1)(q2 − 1)  (p − 1)(p2 − 1) Hf  ��  k/m  lξ/m  l/m  k/m  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='  Department of Mathematics, Brown University, Providence, RI, USA, 02912  Email address: daksh_aggarwal@brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='edu  Department of Mathematics, University of Western Ontario, London, ON, Canada, N6A 5B7  Email address: aghorba@uwo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='ca  Department of Mathematics, University of Western Ontario, London, ON, Canada, N6A 5B7  Email address: masoud@uwo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='ca  Department of Mathematics, University of Toronto, Toronto, ON, Canada, M5S 2E4  Email address: jiyuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='lu@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='ca  DAMTP, University of Cambridge, Cambridge, UK, CB3 0WA  Email address: bn273@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
+page_content='uk  Department of Mathematics, University of Toronto, Toronto, ON, Canada, M5S 2E4  Email address: cyu@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfr_3p/content/2301.01654v1.pdf'}
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+SKEWED BERNSTEIN–VON MISES THEOREM
+AND SKEW–MODAL APPROXIMATIONS
+BY DANIELE DURANTE1,*, FRANCESCO POZZA2,‡ AND BOTOND SZABO1,*,†
+1Department of Decision Sciences and Institute for Data Science and Analytics, Bocconi University,
+*daniele.durante@unibocconi.it; †botond.szabo@unibocconi.it
+2Department of Statistical Sciences, University of Padova, ‡francesco.pozza.2@phd.unipd.it
+Deterministic Gaussian approximations of intractable posterior distributions are com-
+mon in Bayesian inference. From an asymptotic perspective, a theoretical justification in
+regular parametric settings is provided by the Bernstein–von Mises theorem. However, such
+a limiting behavior may require a large sample size before becoming visible in practice. In
+fact, in situations with small–to–moderate sample size, even simple parametric models often
+yield posterior distributions which are far from resembling a Gaussian shape, mainly due to
+skewness. In this article, we provide rigorous theoretical arguments for such a behavior by
+deriving a novel limiting law that coincides with a closed–form and tractable skewed gener-
+alization of Gaussian densities, and yields a total variation distance from the exact posterior
+whose convergence rate can be shown to be of order 1/n, up to a logarithmic factor. Such a
+result provides a substantial accuracy improvement over the classical Bernstein–von Mises
+theorem whose convergence rate, under similar conditions, is of order 1/√n, again up to
+a logarithmic factor. In contrast to higher–order approximations based on, e.g., Edgeworth
+expansions, which require finite truncations for inference, possibly leading to even negative
+densities, our theory characterizes the limiting behavior of Bayesian posteriors with respect
+to a sequence of valid and tractable densities. These advancements further motivate a prac-
+tical plug–in version which replaces the unknown model parameters with the corresponding
+maximum–a–posteriori estimate to obtain a novel skew–modal approximation achieving the
+same improved rate of convergence of our skewed Bernstein–von Mises theorem. Exten-
+sive simulations and a real–data application confirm that our new theory closely matches the
+empirical behavior observed in practice even in finite, possibly small, sample regimes. The
+proposed skew–modal approximation further exhibits improved accuracy not only relative to
+classical Laplace approximation, but also with respect to state–of–the–art Gaussian and non–
+Gaussian approximations from mean–field variational Bayes and expectation–propagation.
+1. Introduction. Modern Bayesian statistics relies extensively on deterministic approx-
+imations of intractable posterior distributions to ease inference in challenging contexts (e.g.,
+Tierney and Kadane, 1986; Minka, 2001; Rue, Martino and Chopin, 2009; Blei, Kucukelbir
+and McAuliffe, 2017). A natural option to enforce the desired tractability is to constrain the
+approximating distribution within a suitable family which facilitates the evaluation of func-
+tionals of interest for inference. To this end, both classical solutions, such as the approxima-
+tion of posterior distributions induced by the Laplace method (e.g., Bishop, 2006, Ch. 4.4),
+and state–of–the–art strategies, including, for example, the standard versions of expectation–
+*Co–funded by the European Union (ERC, BigBayesUQ, project number: 101041064). Views and opinions expressed are however
+those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the
+European Union nor the granting authority can be held responsible for them.
+MSC2020 subject classifications: Primary 62F15; secondary 62F03, 62E17.
+Keywords and phrases: Bernstein–von Mises theorem, Deterministic approximation, Generalized skew–normal distribution.
+1
+arXiv:2301.03038v1  [math.ST]  8 Jan 2023
+
+2
+propagation (Minka, 2001), employ Gaussian approximations. These further appear, either
+as the final solution or as a recurring building–block, also in several routinely–implemented
+alternatives, such as mean–field variational Bayes (Blei, Kucukelbir and McAuliffe, 2017)
+and integrated nested Laplace approximation (INLA) (Rue, Martino and Chopin, 2009). See
+also Opper and Archambeau (2009); Wang and Blei (2013); Chopin and Ridgway (2017);
+Durante and Rigon (2019); Ray and Szabó (2022) and Vehtari et al. (2020), among others,
+for further references and examples illustrating the relevance of Gaussian approximations.
+From a theoretical perspective, the above Gaussian approximations are typically justified,
+in asymptotic regimes, by deriving Bernstein–von Mises type results. In its original for-
+mulation (Doob, 1949; Le Cam, 2012; Van der Vaart, 2000), the celebrated Bernstein–von
+Mises theorem states that, in regular parametric models, the posterior distribution converges
+in total variation distance, with probability tending to one under the law of the data, to a
+Gaussian distribution. The mean of such a limiting Gaussian is a known function of the true
+data–generative parameter, or any efficient estimator of this quantity, such as the maximum
+likelihood estimator, while the variance is the inverse of the Fisher information. Besides
+supporting deterministic Gaussian approximations of intractable posterior distributions, this
+central theorem provides also guarantees for the frequentist coverage of suitable Bayesian
+credible regions (Van der Vaart, 2000). The relevance of these results has motivated several
+extensions of the Bernstein–von Mises theorem to more complex settings. Relevant contri-
+butions along these directions include, among others, generalizations to high-dimensional
+regimes (Boucheron and Gassiat, 2009; Bontemps, 2011; Spokoiny and Panov, 2021), along
+with in–depth treatments of misspecified (Kleijn and van der Vaart, 2012) and irregular
+(Bochkina and Green, 2014) models. Semiparametric (Bickel and Kleijn, 2012; Castillo and
+Rousseau, 2015) and nonparametric (Castillo and Nickl, 2014) settings have also been ad-
+dressed. In the nonparametric context, Bernstein–von Mises type results do not hold in gen-
+eral, but asymptotic Gaussianity can be still proved for weak Sobolev spaces via multiscale
+analysis (Castillo and Nickl, 2014). See also Dehaene and Barthelmé (2018), Wang and Blei
+(2019) and Kasprzak, Giordano and Broderick (2022) for recent Bernstein–von Mises type
+results with a specific focus on scalable deterministic approximations from expectation–
+propagation (EP), variational Bayes (VB) and Laplace method, respectively.
+Albeit considering different regimes, the above results share a common focus on asymp-
+totic Gaussianity of the posterior distribution. Although this perspective has led to funda-
+mental advancements in Bayesian asymptotics and deterministic approximations, as illus-
+trated in Figure 1, even basic parametric models may require, in practice, moderate–to–large
+sample size before yielding posterior distributions that closely resemble Gaussian shapes.
+In fact, in small–to–moderate sample size settings, general posterior distributions induced
+by routinely–implemented statistical models, such as Bayesian generalized linear models,
+display a non–Gaussian behavior, mainly due to skewness (e.g., Kuss, Rasmussen and Her-
+brich, 2005; Challis and Barber, 2012; Durante, 2019). For example, Durante (2019), Fasano
+and Durante (2022) and Anceschi et al. (2023) have recently proved that, under a broad class
+of priors which includes multivariate normals, the posterior distribution induced by probit,
+multinomial probit and tobit models belongs to a skewed generalization of the Gaussian dis-
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+3
+n = 10
+BvM
+Skewed BvM
+0.0
+0.1
+0.2
+-8
+-4
+0
+4
+8
+n = 20
+BvM
+Skewed BvM
+0.0
+0.1
+0.2
+-8
+-4
+0
+4
+8
+n = 30
+BvM
+Skewed BvM
+0.0
+0.1
+0.2
+-8
+-4
+0
+4
+8
+Fig 1: For varying sample size n ∈ {10,20,30}, graphical comparison between the approximations of the
+exact posterior density (grey area) provided by the classical Gaussian Bernstein–von Mises theorem (BvM) and
+the newly–developed skewed version relying on generalized skew–normal approximating densities (Skewed
+BvM). Results refer to an exponential EXP(θ) model with EXP(1) prior on the rate parameter θ.
+tribution known as unified skew–normal (SUN) (Arellano-Valle and Azzalini, 2006). More
+generally, available extensions of Gaussian deterministic approximations which account, ei-
+ther explicitly or implicitly, for skewness (e.g., Rue, Martino and Chopin, 2009; Challis and
+Barber, 2012; Fasano, Durante and Zanella, 2022) have shown evidence of improved empir-
+ical accuracy relative to the Gaussian counterparts. Nonetheless, these approximations are
+often model–specific and a general justification similar to the Bernstein–von Mises theorem
+is not available yet. In fact, current theory on skewed approximations is either lacking or is
+tailored to the specific models and priors studied (e.g., Fasano, Durante and Zanella, 2022).
+In this article, we cover the aforementioned gaps by deriving a novel limiting law for pos-
+terior distributions, in regular parametric models, which belongs to the class of generalized
+skew–normal distributions (Ma and Genton, 2004; Genton and Loperfido, 2005) and yields
+noticeable improvements over the convergence rate of the classical Bernstein–von Mises
+theorem based on limiting Gaussian distributions. More specifically, in Section 2.3 we prove,
+under standard assumptions, that a suitably–derived sequence of generalized skew–normal
+distributions yields a total variation distance from the exact posterior of order 1/n, up to
+a logarithmic factor, with probability tending to one under the law of the data. This results
+in a noticeable improvement over the 1/√n order, still up to a logarithmic factor, obtained
+under the classical Gaussian approximating sequences. As clarified in Section 2.1, we derive
+such a class of skewed approximations from a novel treatment of a third–order extension of
+the Laplace method. This provides a valid and tractable generalized skew–normal density
+that perturbs the one of a multivariate Gaussian with the cumulative distribution function
+of a univariate normal evaluated at a cubic function of the parameter. As a consequence,
+our results also yield theoretical and practical advancements relative to higher–order studies
+relying on Edgeworth or other types of expansions (see e.g., Johnson, 1970; Weng, 2010;
+Kolassa and Kuffner, 2020, and the references therein). In fact, these contributions pro-
+vide supporting theory for arbitrarily truncated versions of infinite expansions which do not
+necessarily correspond to closed–form valid densities, even after normalization — e.g., the
+density approximation is not guaranteed to be non–negative (see e.g., Kolassa and Kuffner,
+
+4
+2020, Remark 11). This limits the methodological implications and the practical relevance
+of such theoretical results. In contrast, the skewed Bernstein–von Mises results derived in
+Section 2.3, establish convergence to a valid and interpretable class of densities which are
+as tractable as the Gaussian counterpart, and arise from a perturbation of such a symmetric
+density mainly regulated by higher–order terms which progressively disappear as the sample
+size increases. This implies that in the limit the novel skewed approximation approaches the
+Gaussian derived under the classical Bernstein–von Mises theorem. However, the total vari-
+ation distance from the exact posterior distribution is substantially reduced, thus clarifying
+the role of skewness in achieving remarkable improvements in approximation accuracy.
+Besides refining classical Bernstein–von Mises type results and yielding improvements
+over available higher–order theoretical studies, the above advancements naturally motivate
+the development of a novel practical class of skew–modal approximations. This can be ob-
+tained by replacing the true unknown data–generative model parameters in the statement of
+the skewed Bernstein–von Mises theorem with the corresponding maximum–a–posteriori
+estimates. This novel plug–in version is derived in Section 3, where we also verify that, un-
+der minimal additional conditions, the total variation distance between such a skew–modal
+approximation and the exact posterior distribution achieves the same 1/n order, up to a log-
+arithmic factor, as its population counterpart. This outperforms the classical Gaussian modal
+approximation induced by the standard Laplace method for which the rate can be proved to
+be of order 1/√n, again up to a logarithmic factor. The practical performance of our new re-
+sults is illustrated through extensive simulations and a real–data application in Sections 2.4
+and 3.2. The numerical analysis demonstrates that our theoretical findings closely match the
+empirical behavior observed in practice even in finite, possibly small, sample regimes. Our
+skew–modal approximation further exhibits improved accuracy not only relative to classi-
+cal Gaussian modal approximations, but also with respect to state–of–the–art Gaussian and
+non–Gaussian approximations from mean–field variational Bayes and expectation propa-
+gation (Minka, 2001; Blei, Kucukelbir and McAuliffe, 2017; Chopin and Ridgway, 2017;
+Durante and Rigon, 2019; Fasano, Durante and Zanella, 2022).
+Concluding remarks in Section 4 clarify that the scope and impact of our theoretical re-
+sults is not limited to providing support on the improved accuracy of skewed approximations
+of posterior distributions. In fact, this article can contribute more generally in refining the
+extensive theory based on Bernstein–von Mises type results, and in further improving the
+quality of more sophisticated methods, such as INLA (Rue, Martino and Chopin, 2009), by
+replacing the building–block Gaussian approximations with the proposed skew–modal ones.
+The proof of the skewed Bernstein–von Mises theorem is detailed in Section 2.3. Additional
+theoretical results and the corresponding technical lemmas are proved in the Appendix.
+Such lemmas guarantee a careful control over the order of the terms involved in the expan-
+sions underlying the proof of the new skewed Bernstein–von Mises theorem. In fact, unlike
+for theory on classical Bernstein–von Mises type results and higher–order expansions, the
+proofs of Theorems 2.1 and 3.1 not only seek to establish a polynomially higher approxi-
+mation accuracy relative to the classical Gaussian counterparts, but also require that such an
+improvement is achieved by a novel limiting law yielding valid and closed–form densities.
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+5
+1.1. Notations. Let {Xi}i≥1 denote a sequence of independent random variables, mod-
+eled by a parametric family {P ∞
+θ ,θ ∈ Θ}, with Θ ⊆ Rd. In the following, we assume that
+the joint probability measure P n
+θ of the random vector Xn = (X1,...,Xn) is dominated by
+a common σ–finite measure µn and we denote by pn
+θ the density of P n
+θ with respect to µn.
+The true unknown data–generating mechanism is indicated by P ∞
+θ0 , with θ0 ∈ Θ, whereas
+the log–likelihood of the model is ℓ(θ) = ℓ(θ,Xn) = log pn
+θ (Xn).
+As mentioned in Section 1, our novel results rely on higher–order expansions and deriva-
+tives. To this end, we characterize operations among vectors, matrices and arrays in a com-
+pact manner by adopting the index notation along with the Einstein’s summation convention
+(e.g., Pace and Salvan, 1997, page 335). More specifically, the inner product Z⊺a between
+the generic random vector Z ∈ Rd, with components Zs for s = 1,...,d, and the vector of
+coefficients a ∈ Rd having elements as for s = 1,...,d, is expressed as asZs, with the sum
+being implicit in the repetition of the indexes. Similarly, if B is a d × d matrix with entries
+bst for s,t = 1,...,d, the quadratic form Z⊺BZ is expressed as bstZsZt. The generalization
+to operations involving arrays with higher dimensions is obtained under the same reasoning.
+Leveraging the above notation, the score vector evaluated at θ0 is defined as
+ℓ(1)
+θ0 = [ℓ(1)
+s (θ)]|θ=θ0 = [(∂/∂θs)ℓ(θ)]|θ=θ0 ∈ Rd,
+whereas, the second, third and fourth order derivatives of ℓ(θ), still evaluated at θ0, are
+ℓ(2)
+θ0 = [ℓ(2)
+st (θ)]|θ=θ0 = [∂/(∂θs∂θt)ℓ(θ)]|θ=θ0 ∈ Rd×d,
+ℓ(3)
+θ0 = [ℓ(3)
+stl(θ)]|θ=θ0 = [∂/(∂θs∂θt∂θl)ℓ(θ)]|θ=θ0 ∈ Rd×d×d,
+ℓ(4)
+θ0 = [ℓ(4)
+stlk(θ)]|θ=θ0 = [∂/(∂θs∂θt∂θl∂θk)ℓ(θ)]|θ=θ0 ∈ Rd×d×d×d,
+where all the indexes in the above definitions and in the subsequent ones go from 1 to d.
+The observed and expected Fisher information are denoted by Jθ0 = [jst] = −[ℓ(2)
+θ0,st] ∈ Rd×d
+and Iθ0 = [ist] = [En
+θ0jst] ∈ Rd×d, where En
+θ0 is the expectation with respect to P n
+θ0. Prior and
+posterior distributions are denoted by Π(·) and Π(· | Xn), respectively. In addition,
+log π(1)
+θ0 =[log π(θ)(1)
+s ]|θ=θ0 = [∂/(∂θs)log π(θ)]|θ=θ0 ∈ Rd,
+log π(2)
+θ0 =[log π(θ)(2)
+st ]|θ=θ0 = [∂/(∂θs∂θt)log π(θ)]|θ=θ0 ∈ Rd×d,
+represent the first two derivatives of the log–prior density, evaluated at θ0.
+The Euclidean norm of a vector a ∈ Rd is denoted by ∥a∥, whereas, for a generic d × d
+matrix B, the notation |B| indicates its determinant. Furthermore, u∧v and u∨v correspond
+to min{u,v} and max{u,v}, respectively. For two positive sequences un,vn we employ
+un ≲ vn if there exists a universal positive constant C such that un ≤ Cvn. When un ≲ vn
+and vn ≲ un are satisfied simultaneously, we write un ≍ vn. Finally, KL(pn
+θ0∥pn
+θ) denotes the
+Kullback–Leibler divergence between the two densities pn
+θ0 and pn
+θ.
+
+6
+2. The skewed Bernstein–von Mises theorem. This section presents the main theoret-
+ical contribution of the article. More specifically, we first show that a suitable treatment of
+a third–order version of the Laplace method can yield a novel limiting law of known densi-
+ties corresponding to distributions within the class of generalized skew–normals (GSN) (Ma
+and Genton, 2004; Genton and Loperfido, 2005). Focusing on this alternative sequence of
+approximating densities, we then prove that the convergence rate of its total variation dis-
+tance from the exact posterior distribution improves by a √n factor the one provided by the
+classical Bernstein–von Mises theorem under limiting Gaussians (Van der Vaart, 2000).
+Consistent with standard Bernstein–von Mises type theory (Van der Vaart, 2000), let us
+consider the re–parametrization h = √n(θ − θ0) ∈ Rd. Then, as illustrated below in Sec-
+tion 2.1, the newly–derived class of approximating GSN densities pn
+GSN has the general form
+(1)
+pn
+GSN (h) = 2φd (h;ξ,Ω)Φ(α(h)).
+In equation (1), φd (·;ξ,Ω) is the density of a d–variate Gaussian variable with mean vector
+ξ and covariance matrix Ω defined as
+ξ = [ξs] = [n(J−1
+θ0 )stwt] ∈ Rd,
+with wt = (ℓ(1)
+θ0 + log π(1)
+θ0 )t/√n, t = 1,...,d,
+Ω = V −1 ∈ Rd×d,
+with V = [vst] = [jst/n − (ξlℓ(3)
+θ0,stl)/n3/2] ∈ Rd×d.
+The term Φ(α(h)) ∈ (0,1) is instead responsible for inducing skewness, and corresponds
+to the cumulative distribution function of a standard normal variable evaluated at
+α(h) = {
+√
+2π/(12√n)}(ℓ(3)
+θ0,stl/n){(h − ξ)s(h − ξ)t(h − ξ)l + 3(h − ξ)sξtξl} ∈ R,
+where α(h) is an odd function of (h − ξ).
+Interestingly, the first factor in the right hand side of (1) closely resembles the limiting
+Gaussian density with mean vector ℓ(1)
+θ0 /√n and covariance matrix (Iθ0/n)−1 from the clas-
+sical Bernstein–von Mises theorem which, however, fails to incorporate skewness. To this
+end, the symmetric density factor in (1) is further perturbed via a skewness–inducing mech-
+anism regulated by Φ(α(h)) to obtain a valid and known asymmetric density with tractable
+normalizing constant. In fact, since α(h) is an odd function of (h − ξ), and φd (·;ξ,Ω) is
+symmetric about ξ, it follows that pn
+GSN (h) in (1) is the density of a GSN variable (Genton
+and Loperfido, 2005, Equation 4.1). This further belongs to the even more general family
+of skew–symmetric distributions (Azzalini and Capitanio, 2003; Wang, Boyer and Genton,
+2004); see also Ma and Genton (2004). Hence, unlike for higher–order studies of posterior
+distributions based on Edgeworth or other expansions (e.g., Johnson, 1970; Weng, 2010; Ko-
+lassa and Kuffner, 2020), our theoretical results focus on a valid and interpretable limiting
+distribution that is as tractable as multivariate Gaussians, both in terms of evaluation of the
+corresponding density, and i.i.d. sampling. More specifically, let z ∈ Rd and u ∈ [0,1] de-
+note samples from a d–variate Gaussian with density φd(z;0,Ω) and a uniform with support
+[0,1], respectively. Then, adapting, e.g., Wang, Boyer and Genton (2004), a sample from the
+GSN distribution with density in (1) can be readily obtained via ξ + sgn(Φ(α(z)) − u)z.
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+7
+The aforementioned discussion clarifies the practical and methodological impact of the
+skewed Bernstein–von Mises theorem. More specifically, our result points toward the possi-
+bility of employing a more refined family of valid approximating distributions arising from
+a simple perturbation of multivariate Gaussian densities that still allows tractable inference,
+while yielding provable, and noticeable, improvements in approximation accuracy.
+2.1. Derivation of the generalized skew–normal approximating distribution. Prior to state
+and prove the skewed Bernstein–von Mises theorem in Section 2.3, let us focus on provid-
+ing a constructive derivation of the generalized skew–normal approximating density in (1)
+via a third–order version of the Laplace method. To simplify the notation, we consider the
+univariate case d = 1. The extension to d > 1 yielding the general expression for pn
+GSN (h)
+in (1) follows as a direct adaptation of the univariate case.
+As a first step towards deriving the GSN density pn
+GSN (h) in (1), notice that the posterior
+for h = √n(θ − θ0) can be expressed as
+(2)
+π(h | Xn) ∝
+pn
+θ0+h/√n
+pn
+θ0
+(Xn)π(θ0 + h/√n)
+π(θ0)
+,
+since pn
+θ0(Xn) and π(θ0) do not depend on h, and θ = θ0 + h/√n.
+Under suitable regularity conditions discussed in Section 2.2 below, the third–order Tay-
+lor’s expansion for the logarithm of the likelihood ratio in Equation (2) is
+(3)
+log
+pn
+θ0+h/√n
+pn
+θ0
+(Xn) = ℓ(1)
+θ0
+√nh − 1
+2
+jθ0
+n h2 +
+1
+6√n
+ℓ(3)
+θ0
+n h3 + OP n
+θ0
+�n−1�,
+whereas the first order Taylor’s expansion of the log–prior ratio is
+(4)
+log π(θ0 + h/√n)
+π(θ0)
+= log π(1)
+θ0
+√n
+h + O
+�n−1�.
+Combining (3) and (4) it is possible to reformulate the right end side of Equation (2) as
+(5)
+pn
+θ0+h/√n
+pn
+θ0
+(Xn)π(θ0 + h/√n)
+π(θ0)
+= exp
+�
+wh − 1
+2
+jθ0
+n h2 +
+1
+6√n
+ℓ(3)
+θ0
+n h3�
++ OP n
+θ0(n−1),
+where w = (ℓ(1)
+θ0 + log π(1)
+θ0 )/√n.
+Notice that, up to a multiplicative constant, the Gaussian density arising from the classical
+Bernstein–von Mises theorem can be obtained by neglecting all terms in (3)–(4) which con-
+verge to zero in probability. These correspond to the contribution of the prior, the difference
+between the observed and expected Fisher information, and the term associated to the third–
+order log–likelihood derivative. Maintaining these quantities would surely yield improved
+accuracy, but it is unclear whether a valid and similarly–tractable density can be still iden-
+tified. In fact, current solutions (e.g., Johnson, 1970) consider approximations based on the
+
+8
+sum among a Gaussian density and additional terms in the Taylor’s expansion. However, as
+for related alternatives arising from Edgeworth–type expansions (e.g., Weng, 2010; Kolassa
+and Kuffner, 2020), there is no guarantee that such constructions provide valid densities.
+As a first key contribution we prove below that a valid and tractable approximating density
+can be, in fact, derived from the above expansions and belongs to the GSN class. To this end,
+let ω = 1/v with v = jθ0/n − ξℓ(3)
+θ0 /n3/2 and ξ = n(jθ0)−1w, and note that, by replacing h3
+in the right hand side of Equation (5) with (h + ξ − ξ)3, the exponential term in (5) can be
+rewritten as proportional to
+(6)
+φ(h;ξ,ω)exp({1/(6√n)}(ℓ(3)
+θ0 /n)
+�(h − ξ)3 + 3(h − ξ)ξ2�).
+Next, recall that, for x = O(1), we can write exp(x) = 1 + x + O(x2) and 2Φ(x) = 1 +
+(2/π)1/2x + O(x3). Therefore, leveraging the similarity among these two expansions and
+the fact that the exponent in Equation (6) is an odd function of (h − ξ) about 0, and its order
+is OP n
+θ0(n−1/2), it follows that (6) is equal to
+2φ(h;ξ,ω)Φ(α(h)) + OP n
+θ0(n−1),
+with α(h) defined below Equation (1), thereby yielding the density of the generalized skew–
+normal in (1), for the univariate case, up to an OP n
+θ0(n−1) term. The direct extension of the
+above derivations to the multivariate case provides the general form of pn
+GSN (h) in (1).
+In Sections 2.2–2.3 we further formalize the above results by proving that the total varia-
+tion distance between the exact posterior and the approximating distribution with density as
+in (1) is OP n
+θ0((log n)d/2+3/n). This improves the classical Bernstein–von Mises type results
+by a √n factor, up to a logarithmic term.
+2.2. Assumptions. The skewed Bernstein–von Mises theorem in Section 2.3 requires as-
+sumptions A0, A1, A2, A3 and A4 stated below. For convenience, let us also introduce the
+notation Mn = √c0 log n, where c0 is a positive constant to be specified later.
+A0.0 The log–likelihood of the model is four times differentiable with
+ℓ(1)
+θ0,s = OP n
+θ0(n1/2),
+ℓ(2)
+θ0,st = OP n
+θ0(n),
+ℓ(3)
+θ0,stl = OP n
+θ0(n),
+for s,t,l = 1,...,d,
+and sup∥h∥≤Mn |ℓ(4)
+θ0,stlk(h)| = OP n
+θ0(n), for s,t,l,k = 1,...,d, with ℓ(4)
+θ0,stlk(h) := ℓ(4)
+stlk(θ0+
+h/√n). Moreover, let ℓ(1)
+i,θ0 be the first derivative of the log–likelihood ℓ(θ,Xi) evaluated
+at θ0, and assume that Eθ0∥ℓ(1)
+i,θ0/√n∥3 = O(1) for each i = 1,...,n.
+A0.1 The entries ist of the Fisher information matrix Iθ0 satisfy ist = O(n) and jst/n −
+ist/n = OP n
+θ0(n−1/2), for s,t = 1,...,d. Moreover, i−1
+st = O(n−1), where i−1
+st = (I−1
+θ0 )st
+denote the entries of the inverse Fisher information matrix, for s,t = 1,...,d.
+A1 The log–prior density log π(θ) is two times continuously differentiable in a neighbor-
+hood of θ0, and π(θ0) > 0.
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+9
+A2 For some compact neighborhood Θ0 ⊂ Θ of θ0 there exists a sequence of tests φn such
+that En
+θ0φn → 0 and supθ/∈Θ0 En
+θ(1 − φn) → 0.
+A3 For every constant C1 > 0 there exist C2 > 0 and C3 > 0 such that Π(θ : KL(pn
+θ0∥pn
+θ) <
+n−C1) ≥ C3n−C2.
+A4 There exists a constant η > 0 such that |Iθ0/n| > η.
+Assumptions A0–A4 provide a slightly more restrictive, yet still mild, counterpart of the
+standard conditions commonly employed in state–of–the–art versions of the Bernstein–von
+Mises result (e.g., Van der Vaart, 2000, Chapter 10). More specifically, A0 and A4 are readily
+verifiable for a wide range of routinely–implemented models, including for instance expo-
+nential families, and are required to carefully control the error between the log–likelihood
+ratio and its third order Taylor’s expansion. In principle, these conditions can be relaxed, e.g.,
+by considering the locally asymptotically normal assumption as in Chapter 10 of Van der
+Vaart (2000). However, these results only derive asymptotically vanishing total variance
+distance between the posterior and its Gaussian approximation, whereas we aim to derive a
+sharp rate, up to a logarithmic factor, and hence slightly stronger assumptions are required.
+As for A0 and A4, also condition A1 on the prior is mild and standard, and is similarly
+required to quantify the approximation accuracy. Condition A3 is the usual prior small ball
+assumption used routinely in Bayesian nonparametrics to derive contraction rates. Together
+with the standard testing assumption A2, required in the classical Bernstein–von Mises the-
+orem as well, it guarantees that the posterior mass decreases fast enough in the area where
+the skewed approximation does not hold, so that its impact on the overall rate is negligible.
+2.3. Theorem. Theorem 2.1 states the main theoretical result of this article. It establishes
+improved rates of convergence for the total variation distance between the exact posterior
+and the generalized skew–normal approximating distribution derived in Section 2.1, under
+assumptions A0–A4. Our result clarifies that a substantially more accurate representation
+for the asymptotic behavior of the exact posterior distribution, relative to the one obtained
+under limiting multivariate Gaussians, can be achieved via a similarly–tractable class of gen-
+eralized skew–normals. As illustrated in Sections 2.4 and 3, such a theory has fundamental
+implications in the development of novel rigorous approximations with noticeable empirical
+improvements in accuracy.
+THEOREM 2.1.
+Under assumptions A0–A4, if the map θ → P n
+θ is one–to–one and θ0 is
+an inner point of Θ, then
+(7)
+∥Π(· | Xn) − P n
+GSN(·)∥TV = OP n
+θ0
+�(log n)d/2+3/n
+�,
+where ∥ · ∥TV is the total variation distance, and P n
+GSN(S) =
+�
+S pn
+GSN(h)dh for S ⊂ Rd with
+pn
+GSN(h) defined as in Equation (1).
+
+10
+REMARK 2.2.
+Under related conditions and a simpler proof, it is possible to show that
+the order of convergence for the usual Bernstein–von Mises theorem based on Gaussian
+limiting distributions is OP n
+θ0((log n)γ/√n), for some known constant γ ≥ 0, possibly de-
+pending on d. Hence, Theorem 2.1 guarantees that by relying on GSN approximating distri-
+butions with density defined as in (1) and derived in Section 2.1, it is possible to improve the
+classical Bernstein–von Mises result by a √n factor, up a logarithmic term.
+The proof of Theorem 2.1 is reported below and follows the general reasoning behind
+the classical Bernstein–von Mises type derivations (e.g., Szabo and Van der Vaart, 2021),
+extended to GSN distributions. Nonetheless, as mentioned before, the need to derive a sharp
+rate which establishes a polynomially higher approximation accuracy, relative to Gaussian
+limiting distributions, requires a number of additional technical lemmas and refined argu-
+ments ensuring a tight control of the error terms in the expansions underlying Theorem 2.1.
+PROOF. To prove Theorem 2.1, we first show that the posterior mass outside the ball
+Θn = {θ ∈ Θ : ∥θ − θ0∥ < Mn/√n},
+(8)
+is asymptotically negligible, and then prove that, when restricted to Θn, the posterior distri-
+bution can be accurately approximated by the generalized skew–normal in (1) with an error
+of order OPθ0(M d+6
+n
+/n). Setting Mn = √c0 log n for some constant c0 > 0 to be specified
+later, yields the statement of Theorem 2.1. As outlined in Section 2.1, this improvement fol-
+lows from the fact that A0.0 and A1 allow to combine a third order Taylor’s expansion of
+the log–likelihood ratio with a first order Taylor’s expansion of the log–prior ratio to obtain
+an approximation for the exact posterior density whose kernel closely matches the one of a
+generalized skew–normal. To simplify notation, we drop below the n from P n
+θ , pn
+θ and the
+corresponding expectation.
+To start the proof, let us first denote the posterior restricted to Θn as
+(9)
+ΠΘn(θ ∈ B | Xn) =
+�
+θ∈B∩Θn pθ (Xn)dΠ(θ)
+�
+θ∈Θn pθ (Xn)dΠ(θ) ,
+B ⊆ Θ,
+and recall that, by a standard inequality for the TV norm, it follows that
+(10)
+supB|Π(θ ∈ B | Xn) − ΠΘn(θ ∈ B | Xn)| < 2Π(Θc
+n | Xn).
+To prove that Π(Θc
+n | Xn) is asymptotically negligible, we show that this quantity converges
+to zero in mean and, therefore, in probability. To this end, let An,1 be the event defined as
+An,1 = {
+� pθ/pθ0(Xn)dΠ(θ) > Π(θ : KL(pθ0∥pθ) < n−C1)}, for some C1 > 0. Then,
+Eθ0Π(Θc
+n | Xn) ≤ Eθ0Π(Θc
+n | Xn)1An,1 + Pθ0(Ac
+n,1).
+since Π(Θc
+n | Xn) can be expressed as Π(Θc
+n | Xn)1An,1 + Π(Θc
+n | Xn)1Ac
+n,1.
+At this point, note that the probability Pθ0(Ac
+n,1) in the above bound can be neglected since
+it converges to zero at any polynomial rate by Lemma B.1. As for Eθ0Π(Θc
+n | Xn)1An,1, let
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+11
+us re–write such a quantity as Eθ0Π(Θc
+n | Xn)1An,1(1−φn)+Eθ0Π(Θc
+n | Xn)1An,1φn, where
+φn is a sequence of tests, and notice that Π(Θc
+n | Xn)1An,1φn ≤ φn, since all the involved
+quantities are less than one. In addition, recall that the posterior density π(θ | Xn) can be
+expressed as (pθ/pθ0)(Xn)π(θ)/
+� (pθ/pθ0)(Xn)dΠ(θ), where the denominator is greater
+than Π(θ : KL(pθ0∥pθ) < n−C1), when restricting to An,1. Therefore, Π(Θc
+n | Xn)1An,1(1 −
+φn) can be bounded above by
+�
+Θcn(pθ/pθ0)(Xn)(1 − φn)dΠ(θ)/Π(θ : KL(pθ0∥pθ) < n−C1).
+Combining these results with Lemma B.2, it follows that there exists a test φn, such that
+Eθ0Π(Θc
+n | Xn)1An,1 ≤
+Eθ0
+�
+Θcn(pθ/pθ0)(Xn)(1 − φn)dΠ(θ)
+Π(θ : KL(pθ0∥pθ) < n−C1)
++ exp(−ctM 2
+n),
+for some positive constant ct > 0. As a direct consequence of assumptions A3 and Fubini’s
+theorem, the first term on the right hand side of the above inequality can be bounded above
+by a multiple of nC2 �
+Θcn Eθ(1 − φn)dΠ(θ), after noticing that Eθ0(pθ/pθ0)(Xn)(1 − φn) =
+� (1 − φn)pθ(Xn)dXn = Eθ(1 − φn). Leveraging now assumption A2 along with the fact
+that, as proved within Lemma B.2, Eθ (1 − φn) ≤ exp(−ctn(∥θ −θ0∥2 ∧1)), it additionally
+follows that nC2 �
+Θcn Eθ(1 − φn)dΠ(θ) can be further bounded above by
+nC2e−ctM 2
+n �
+{Mn/√n<∥θ−θ0∥<1} dΠ(θ) + nC2e−nct �
+{∥θ−θ0∥≥1} dΠ(θ).
+Therefore, by setting Mn as in Section 2.2, with c0 > (C2 + 1)/ct, we have that Eθ0Π(Θc
+n |
+Xn) = O(n−1), and hence, leveraging the inequality for the TV norm in Equation (10), it
+follows that supB|Π(θ ∈ B | Xn) − ΠΘn(θ ∈ B | Xn)| = OPθ0(n−1). Since the total varia-
+tion distance is invariant to scale and location transformations, this implies the same bound
+for the posterior on h. Namely
+(11)
+supS|Π(h ∈ S | Xn) − ΠSMn(h ∈ S | Xn)| = OPθ0(n−1)
+where SMn = {h : ∥h∥ < Mn}.
+Let us now move to the second part of the proof which shows that, when restricted to Θn,
+the posterior distribution for the re–parameterized quantity h = √n(θ − θ0) is accurately
+approximated by the generalized skew–normal with density as in (1). To this end, define
+Yn(h) = 2e
+hsws − hsht
+2
+jst
+n +
+1
+2√n
+ℓ(3)
+θ0,stl
+n
+(h − ξ)s(h − ξ)tξl +
+1
+6√n
+ℓ(3)
+θ0,stl
+n
+ξsξtξlΦ(α(h)),
+Ln(h) = pθ0+h/√n
+pθ0
+(Xn)π(θ0 + h/√n)
+π(θ0)
+,
+(12)
+where Φ(α(h)) denotes the skewness–inducing factor defined below Equation (1). In (12),
+the quantity Yn(h) coincides, up a normalizing constant, with the generalized skew–normal
+approximating density pGSN(h) in (1). In fact, notice that the exponential term is quadratic in
+h and, hence, can be suitably re–written as proportional to the Gaussian density factor in (1).
+
+12
+The quantity Ln(h) is instead the exact posterior–density ratio that we seek to approximate
+via Yn(h); see also Equation (2) and the constructive derivation of the GSN density pGSN(h)
+in Section 2.1.
+In view of A0 and A1 let us now consider the third order Taylor’s expansion of log Ln(h)
+around h = 0, satisfying log Ln(h) = (ℓ(1)
+θ0,shs)/√n − (isthsht)/2n + rn,h + qn,h,, with rn,h
+and qn,h defined as
+rn,h =
+hs log π(1)
+θ0,s
+√n
+− hsht
+2
+�jst
+n − ist
+n
+�
++
+ℓ(3)
+θ0,stl
+n
+hshthl
+6√n = OPθ0({∥h∥3 ∨ 1}/√n),
+qn,h =
+ℓ(4)
+θ0,stlk(˜h)
+n
+hshthlhk
+24n
++ log π(θ0 + ¯h/√n)(2)
+st
+n
+hsht,
+(13)
+where ˜h,¯h ∈ Rd satisfy ∥˜h∥ ∨ ∥¯h∥ ≤ ∥h∥. Then, by triangle inequality and in view of the
+second order Taylor’s expansion of eqn,h = 1 + qn,h + eβqn,hq2
+n,h/2, for a β ∈ (0,1), we have
+(14)
+|Ln(h) − Yn(h)| ≤ |exp(un,h) − Yn(h)| + |exp(un,h)(qn,h + eβqn,hq2
+n,h/2)|,
+where un,h = (ℓ(1)
+θ0,shs)/√n − (hshtist)/2n + rn,h.
+Since the statement of Theorem 2.1 is in probability, let us replace some of the above
+random variables with truncated versions which converge in probability to the original ones.
+This ensures the existence of the moments for the truncated variables and facilitates a tight
+control of the corresponding limiting behavior. Hence, define
+˜ℓ(1)
+θ0 /√n =
+�ℓ(1)
+θ0 /√n
+�1∥ℓ(1)
+θ0 /√n∥<Cn,
+with Cn = M 1/α
+n
+for α > 2,
+˜rn,h = rn,h1|rn,h|<Kn,h,
+with Kn,h = log(n){∥h∥3 ∨ 1}/√n,
+and consider the event An,2 = �
+h:∥h∥≤Mn(|rn,h| < Kn,h) ∩ (∥ℓ(1)
+θ0 /√n∥ < Cn). Notice that,
+by assumption A0 and (13) it follows that Pθ0(Ac
+n,2) = O(1). Furthermore, let us define the
+quantity λn,h = (1/6√n)(ℓ(3)
+θ0,stl/n){ξsξtξl + 3(h − ξ)s(h − ξ)tξl − hshthl}, and set
+˜ℓ(3)
+θ0,stl = ℓ(3)
+θ0,stl1An,3,
+and
+˜λn,h = λn,h1An,3,
+where An,3 is the event defined as An,3 = �
+h:∥h∥≤Mn{|λn,h| ≤ τn} with τn = FnM 3
+n/√n,
+for any Fn that goes to infinity arbitrary slow. Notice that by A0–A1 also Pθ0(Ac
+n,3) = O(1).
+Finally, set also ˜ℓ(4)
+θ0,stlk(h) = ℓ(4)
+θ0,stlk(h)1An,4 where An,4 = �
+h:∥h∥≤Mn{|ℓ(4)
+θ0,stlk(h)| < Fnn},
+and note that A0 further implies Pθ0(Ac
+n,4) = O(1).
+Combining all the above truncated quantities, let us also denote with ˜Ln(h) and ˜Yn(h) the
+alternative versions of Ln(h) and Yn(h), given in (12), where all the stochastic quantities
+have been replaced by their truncated counterparts. Since the statement of our theorem is in
+probability, it is sufficient to restrict ourselves to the event An,1:4 = An,1∩An,2∩An,3∩An,4,
+provided that Pθ0(Ac
+n,1:4) = O(1).
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+13
+In view of Hölder’s inequality with ρn = (1 + M 2
+n)/M2
+n and ˜ρn = 1 + M 2
+n, the truncated
+version of the second term in the right hand side of (14) is bounded from above by
+suph∈SMnE1/ρn
+θ0
+exp(ρn˜un,h) · E1/˜ρn
+θ0
+(˜qn,h + eβ˜qn,h˜q2
+n,h/2)˜ρn,
+(15)
+where ˜un,h and ˜qn,h denote the truncated versions of un,h and qn,h, respectively. Hence, since
+˜qn,h is a function of ˜ℓ(4)
+θ0 (h) whose entries are truncated above by Fnn, it follows that
+sup
+h∈SMn
+˜qn,h + eβ˜qn,h˜q2
+n,h
+2
+≲
+�Fn(∥h∥4 ∨ 1)
+24n
++ e
+Fn(∥h∥4 ∨ 1)
+24n
+�Fn(∥h∥4 ∨ 1)
+24n
+�2�
+= O
+�M 4
+n
+n
+�
+.
+Therefore, together with Lemma B.3, which guarantees control on the limiting behavior of
+the first expectation in Equation (15), it follows that the expression in (15) is O
+�M 4
+n/n
+�
+.
+This implies, together with the truncated version of (14) and Lemma B.5 on the limiting
+behavior of the first term in the right hand side of (14), that
+(16)
+suph∈SMnEθ0 |˜Ln(h) − ˜Yn(h)| = O
+�M 6
+n/n
+�.
+Combining (16) with Fubini’s theorem and Jensen’s inequality it follows that
+Eθ0 supS |
+�
+S∩SMn ˜Ln(h)dh −
+�
+S∩SMn ˜Yn(h)dh|
+≤
+�
+SMn Eθ0|˜Ln(h) − ˜Yn(h)|dh ≲
+�
+SMn(M 6
+n/n)dh = O
+�M d+6
+n
+/n
+�,
+for any S ⊆ Rd. Let us now focus on obtaining a similar result for a normalized version of
+the above quantities. To this end, replace ˜Ln(h) and ˜Yn(h) with the corresponding normal-
+ized counterparts, and note that by summing and subtracting
+�
+S∩SMn ˜Ln(h)dh/
+�
+SMn ˜Yn(h)dh
+within the above absolute difference it follows, as a direct consequence of the triangle and
+Jensen’s inequalities, that
+Eθ0 supS |
+�
+S∩SMn ˜Ln(h)dh/
+�
+SMn ˜Ln(h)dh −
+�
+S∩SMn ˜Yn(h)dh/
+�
+SMn ˜Yn(h)dh|
+≤ 2Eθ0(
+�
+SMn |˜Yn(h) − ˜Ln(h)|dh/
+�
+SMn ˜Yn(h)dh) = O
+�M d+6
+n
+/n
+�,
+since the denominator is bounded away from zero with probability one. Leveraging the fact
+that Pθ0(Ac
+n,1:4) = O(1) and the connection between convergence in mean and convergence
+in probability, this translates into a similar statement for the un–truncated quantities. Hence,
+since, by (12),
+�
+S∩SMn Ln(h)dh/
+�
+SMn Ln(h)dh = ΠSMn(h ∈ S | Xn), it follows that
+(17)
+supS
+��ΠSMn(h ∈ S | Xn) − PGSN,SMn(S)
+�� = OPθ0(M d+6
+n
+/n).
+where PGSN,SMn(S) =
+�
+S∩SMn Yn(h)dh/
+�
+SMn Yn(h)dh. To conclude the proof, it is now nec-
+essary to study the behavior of the absolute difference between PGSN,SMn(S) and PGSN(S)
+
+14
+defined in Theorem 2.1. To accomplish this goal, first notice that by the definition of Yn(h)
+in (12), the normalizing factor
+�
+SMn Yn(h)dh can be reformulated as
+e
+ξsξt
+2
+jst
+n +
+1
+6√n
+ℓ(3)
+θ0,stl
+n
+ξsξtξl�
+SMn
+2e
+−1
+2(h − ξ)s(h − ξ)t
+�jst
+n − ξl
+√n
+ℓ(3)
+θ0,stl
+n
+�
+Φ(α(h))dh,
+where the term inside the integral coincides with pGSN(h) in (1), up to the normalizing con-
+stant of the Gaussian density in (1). Therefore,
+�
+SMn Yn(h)dh can be expressed as a constant
+times (1 − PGSN(Sc
+Mn)). This also implies that PGSN,SMn(S) is equal to PGSN(S ∩ SMn)/(1−
+PGSN(Sc
+Mn)). As a consequence, since PGSN(S) = PGSN(S ∩ SMn) + PGSN(S ∩ Sc
+Mn), by ap-
+plying the triangle inequality, we have that
+��PGSN,SMn(S) − PGSN(S)
+�� ≤ {PGSN(Sc
+Mn)/(1 − PGSN(Sc
+Mn))}PGSN
+�S ∩ SMn
+� + PGSN(Sc
+Mn).
+Now, notice that, since the density in (1) has the same tail behavior of a Gaussian, it holds
+PGSN(Sc
+Mn) = OPθ0(n−D), D > 1 for sufficiently large choice of c0 in Mn = √c0 log n. Thus
+(18)
+supS
+��PGSN,SMn(S) − PGSN(S)
+�� = OPθ0(n−D).
+Recall also that assumption A4 together with Lemma B.7 guarantee that V = Ω−1 is positive
+definite with determinant bounded away from zero with Pθ0–probability tending to one.
+To conclude the proof notice that, by triangle inequality, the total variation distance ∥Π(· |
+Xn) − P n
+GSN(·)∥TV in Theorem 2.1, can be bounded above by the sum of the left end side
+quantities in (11), (17), and (18). The leading term in such a summation is OPθ0(M d+6
+n
+/n)
+with Mn = √c0 log n. Hence, ∥Π(· | Xn) − P n
+GSN(·)∥TV = OP n
+θ0((log n)d/2+3/n).
+2.4. Empirical results. Sections 2.4.1–2.4.3 illustrate the applicability of the asymptotic
+results in Sections 2.1–2.3. More specifically, the focus is on providing comprehensive em-
+pirical evidence of the improved accuracy achieved by the GSN limiting approximation in (1)
+relative to its Gaussian counterpart arising from the classical Bernstein–von Mises theorem.
+In fact, our approach demonstrates improved performance and noticeable accuracy already
+for small sample sizes n; see e.g., Figure 1. Through extensive examples, we further demon-
+strate that conditions A0–A4 underlying Theorem 2.1 are mild and readily fulfilled in broad
+settings, including those considered by alternative higher–order versions of the Bernstein–
+von Mises theorem (e.g., Johnson, 1970; Weng, 2010; Kolassa and Kuffner, 2020).
+Notice that, as for the classical Bernstein–von Mises theorem, also our results in Sections
+2.1–2.3 require knowledge of the true data–generating parameter θ0, which is clearly un-
+known in practical implementations. In Sections 3–3.1, we address this aspect via a plug–in
+version of the generalized skew–normal limiting approximation, which replaces θ0 with its
+maximum–a–posteriori estimate, to obtain similar theoretical and empirical support; see the
+additional simulations and real–data applications in Section 3.2.
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+15
+2.4.1. Exponential model. Let Xi
+iid
+∼ EXP(θ0), for i = 1,...,n, where EXP(θ0) denotes
+the exponential distribution with rate parameter θ0 > 0. Moreover, consider an EXP(1) prior
+for θ. To obtain the generalized skew–normal approximation for the posterior distribution
+induced by such a Bayesian model, let us first verify that all conditions of Theorem 2.1 hold.
+To address this goal first notice that, under the above model, the first four derivates of the
+log–likelihood at θ are n/θ − �n
+i=1 xi, −n/θ2, 2n/θ3 and −6n/θ4, respectively. Hence,
+conditions A0 and A4 are both satisfied, even around a small neighborhood of θ0. In addi-
+tion, the KL divergence between pn
+θ0 and pn
+θ is KL(pn
+θ0∥pn
+θ) = n{θ/θ0 − 1 + log(θ0/θ)}. This
+implies, in view of
+{θ : KL
+�pn
+θ0∥pn
+θ
+� < n−C1} ⊃ {θ > θ0 : (θ/θ0 − 1) < n−C1−1},
+that also assumption A3 is fulfilled for any prior density π bounded away from zero around
+θ0. This condition, along with A1, is met by a broad class of routinely–implemented priors.
+For instance, EXP(1) can be considered in this case. Finally, since the model is parametric
+and it satisfies the usual regularity conditions, the existence of the test required by assump-
+tion A2 is guaranteed (see e.g., Van der Vaart, 2000; Szabo and Van der Vaart, 2021).
+The above derivations ensure that Theorem 2.1 holds. Hence, let us derive the parameters
+of the generalized skew–normal approximating density in (1) under this exponential exam-
+ple. To this end, first notice that, since the prior is an EXP(1), then log π(1)
+θ0 = −1. Therefore,
+ξ = θ2
+0(n/θ0 − �n
+i=1 xi − 1)/√n and Ω = 1/(θ−2
+0
+− 2θ−1
+0 {1/θ0 − (�n
+i=1 xi)/n − 1/n}),
+while the cubic function appearing in the skewness–inducing mechanism regulated by Φ(·)
+is defined as α(h) = {
+√
+2π/(6√nθ3
+0)}{(h − ξ)3 + 3(h − ξ)ξ2}.
+Table 1 compares the accuracy of the generalized skew–normal (S–BVM) and the Gaus-
+sian (BVM) approximations corresponding to the newly–derived and classical Bernstein–
+von Mises theorems, respectively, under growing sample size and replicated experiments.
+More specifically, we consider 50 different simulated datasets with sample size n∗ = 5000.
+Then, within each of these 50 experiments, we derive the exact posterior under several sub-
+sets of data x1,...,xn with a growing sample size n ∈ {1,10,50,100,500,1000,5000},
+and compute the total variation (TV) distances between such a posterior and the two approx-
+imations provided by the classical and skewed Bernstein–von Mises theorems. Since both
+TABLE 1
+In the exponential example, average, over 50 replicated studies, of the log–total–variation (TV) distances
+corresponding to the classical (BVM) and skewed (S–BVM) Bernstein–von Mises theorem. Summaries for
+the ratios between the two TV distances are also reported. Results are displayed for different sample sizes
+from n = 1 to n = 5000. The standard deviations of the ratios, computed over the 50 replicated studies, are
+displayed within round brackets. The bold values indicate the best performance for each sample size.
+n = 1
+n = 10
+n = 50
+n = 100
+n = 500
+n = 1000
+n = 5000
+log n
+0.00
+2.30
+3.91
+4.61
+6.21
+6.91
+8.52
+log TVBVM
+−1.52
+−1.74
+−2.23
+−2.49
+−3.37
+−3.54
+−4.41
+log TVS–BVM
+− 2.39
+−2.78
+−3.69
+−4.26
+−5.94
+−6.35
+−8.07
+TVS–BVM/TVBVM
+0.45
+0.39
+0.26
+0.19
+0.08
+0.06
+0.03
+(0.19)
+(0.18)
+(0.13)
+(0.08)
+(0.03)
+(0.02)
+(0.01)
+
+16
+the exact posterior and the two approximating densities are available in closed–form, the to-
+tal variation distances TVn
+BVM = 2
+�
+R |π(h | Xn) − pn
+GAUSS(h)|dh and TVn
+S–BVM = 2
+�
+R |π(h |
+Xn) − pn
+GSN(h)|dh can be evaluated numerically, for each n, via standard routines in R. Ta-
+ble 1 displays, for each n, the summaries of the logarithm of such distances along with the
+corresponding ratios, computed across the 50 replicated experiments.
+In line with Theorem 2.1 and Remark 2.2, the results in Table 1 clarify that the gener-
+alized skew–normal approximation consistently yields substantial accuracy improvements
+relative to the Gaussian counterpart for any n. Moreover, on average log TVn
+S–BVM ≈ −log n,
+whereas log TVn
+BVM ≈ −0.5log n, which are exactly the results stated for the asymptotic
+regime in Theorem 2.1 and Remark 2.2. This empirical finding clarifies that our theoretical
+results for the limiting case are, in fact, visible also in finite, even small, sample size set-
+tings, thus making our theory of direct practical impact, while motivating the adoption of
+the generalized skew–normal approximation in place of the Gaussian one. In fact, as is clear
+from the ratios in Table 1, S–BVM yields TV distances from the exact posterior which are
+orders of magnitude lower than those obtained under BVM.
+2.4.2. Bernoulli model. Let us now consider a Bernoulli model with a focus on the natu-
+ral parameter θ0 ∈ R defined as θ0 = logit(g(θ0)), with g(θ0) = pr(Xi = 1|θ0), i = 1,...,n.
+Hence, Xi
+iid
+∼ Bern(g(θ0)), for every i = 1,...,n, where g(θ0) = eθ0/(1 + eθ0). As for Sec-
+tion 2.4.1, let us first verify that all conditions underlying Theorem 2.1 hold, thus implying
+the skewed version of the Bernstein–von Mises result.
+Conditions A0 and A4 are easily met, since the first four derivatives of the log–likelihood,
+evaluated at θ, under the above Bernoulli model, are �n
+i=1 xi − ng(θ), −ng(θ)/(1 + eθ),
+ng(θ)(eθ − 1)/(1 + eθ)2, and −n(e3θ − 4e2θ + eθ)/(1 + eθ)4, respectively, and hence sat-
+isfy the required bounds uniformly around θ0 in probability. To verify condition A3, notice
+that the Kullback–Leibler divergence between pn
+θ0 and pn
+θ can be derived analytically also
+in this example and is given by KL(pn
+θ0∥pn
+θ) = n(θ0 − θ)g(θ0) + nlog((1 + eθ)/(1 + eθ0)).
+Hence, for C1 > 0, we have that
+{θ : KL(pn
+θ0∥pn
+θ) < n−C1} ⊃ {θ < θ0 : (θ0 − θ)g(θ0) < n−C1−1},
+where the volume of the set in the right hand side is (1 + eθ0)/eθ0n−C1+1, proving condition
+A3 for any prior density π bounded away from zero about θ0. As discussed in Section 2.4.1,
+also assumption A2 is readily verifiable under this parametric model. Similarly, A1 holds
+for a broad class of priors, including the Gaussian one often employed in Bayesian inference
+under such a Bernoulli model. Therefore, for any prior density π which is positive and differ-
+entiable about θ0, Theorem 2.1 holds for the generalized skew–normal approximating den-
+sity in (1) with location ξ = e−θ0(1 + eθ0)2(�n
+i=1 xi − ng(θ0) + log π(1)
+θ0 )/√n, and variance
+Ω = [g(θ0)/(1 + eθ0) − (eθ0 − 1)(�n
+i=1 xi − ng(θ0) + log π(1)
+θ0 )/n(1 + eθ0)]−1. The func-
+tion α(h) in the skewness–inducing mechanism regulated by the perturbing factor Φ(·) is
+instead equal to α(h) = (
+√
+2πeθ0(eθ0 − 1)/{12√n(1 + eθ0)3}){(h − ξ)3 + 3(h − ξ)ξ2}.
+Figure 2 provides a visual assessment of the accuracy improvements provided by such a
+novel skewed approximation relative to the Gaussian one from the classical Bernstein–von
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+17
+n = 10
+BvM
+Skewed BvM
+0.0
+0.1
+0.2
+-10
+-6
+-2
+2
+6
+10
+n = 20
+BvM
+Skewed BvM
+0.0
+0.1
+0.2
+-10
+-6
+-2
+2
+6
+10
+n = 30
+BvM
+Skewed BvM
+0.0
+0.1
+-10
+-6
+-2
+2
+6
+10
+Fig 2: For varying sample size n ∈ {10,20,30}, graphical comparison between the approximations of the
+exact posterior density (grey area) provided by the classical Gaussian Bernstein–von Mises result (BvM),
+and the newly–developed skewed version (Skewed BvM) relying on generalized skew–normal approximating
+densities. Results refer to the Bernoulli model with N(0,4) prior for θ = logit(pr(Xi = 1 | θ)).
+Mises theorem. The analysis is performed for three different sample sizes n ∈ {10,20,30}
+under a Bernoulli data–generating model with θ0 = logit(0.2) and prior N(0,4) for θ. Con-
+sistent with the empirical evidence form Section 2.4.1, and with the statements in Theorem
+2.1 and Remark 2.2, the skewed Bernstein–von Mises approximation is more accurate than
+its Gaussian counterpart also in this study.
+2.4.3. Gamma model. We conclude our empirical analyses of the skewed Bernstein–von
+Mises result with a bivariate example focusing on the parameters θ0 = (α0,β0) of a Gamma
+model. Hence, for data Xi
+iid
+∼ Gamma(θ0), the first derivative of the log–likelihood at θ, is
+(log β + �n
+i=1(log xi)/n − ψ(0)(α),α/β − �n
+i=1 xi/n), where ψ(0)(·) is the polygamma
+function of order 0. Similarly, letting ψ(1)(·) denote the polygamma function of order 1, the
+2 × 2 symmetric matrix of the second order derivatives at θ has diagonal entries −ψ(1)(α)
+and −α/β2, while the off–diagonal is given by 1/β. Finally, the 2 × 2 × 2 array comprising
+the third order derivatives at θ, is
+ℓ(3)
+θ,··1 = n
+�
+−ψ(2)(α)
+0
+0
+−1/β2
+�
+,
+ℓ(3)
+θ,··2 = n
+�
+0
+−1/β2
+−1/β2
+2α/β3
+�
+,
+where ψ(2)(·) denotes the polygamma function of order 2. Moreover, notice that Jθ = Iθ and
+|Iθ/n| = (αψ(1)(α) − 1)/β2 > 0. Hence, for any suitable choice of the prior distribution,
+conditions A0, A1 and A4 are satisfied. Notice also that the fourth order derivatives yield a
+2 × 2 × 2 × 2 array which is not reported here for simplicity of exposition. Nonetheless, it
+is straightforward to verify that the condition A0 on the fourth order derivative is verified in
+this context. Additionally, since we are dealing again with a regular parametric model also
+condition A2 is fulfilled.
+We are left to prove the validity of assumption A3. To this end, notice that KL(pn
+θ0∥pn
+θ) =
+n{log Γ(α0)(α0 − α) + log(Γ(α)/Γ(α0)) + αlog(β0/β) + (α0/β0)(β − β0)}. Hence the
+
+18
+set {θ = (α,β) : KL(pn
+θ0∥pn
+θ) < n−C1} contains
+(19)
+{α < α0, β > β0 : log Γ(α0)(α0 − α) + (α0/β0)(β − β0) < n−(C1+1)}.
+Note that, if log Γ(α0) ≤ 0 then log Γ(α0)(α0 −α) ≤ 0 which implies that the set defined in
+(19) contains {α < α0, β0 < β < (β0/α0)n−(C1+1)}. This latter set has prior mass bounded
+at least constant times n−(C1+1) for prior densities π bounded away from zero in a neigh-
+bourhood of θ0. For log Γ(α0) > 0, the set {θ = (α,β) : KL(pn
+θ0∥pn
+θ) < n−C1} contains
+(20)
+{α0 − n−(C1+1)/
+�2log Γ(α0)
+� < α < α0, β0 < β < β0 + n−(C1+1)β0/(2α0)},
+which has prior mass proportional to n−(C1+1) for any prior density π bounded away from
+zero about θ0, thereby verifying condition A3. Therefore, since A0–A4 are met, the exact
+posterior admits the generalized skew–normal approximation in Equation (1) with ξ, Ω and
+α(h) that can be computed analytically using the derivatives of the log–likelihood function
+given above. For clarity of presentation we refrain from presenting these formulas here.
+Figure 3 compares graphically the classical and skewed Bernstein–von Mises approxima-
+tions for the posterior of h = √n(θ−θ0) in a simulation study with sample size n = 15, true
+data–generative parameter θ0 = (α0,β0) = (1,0.5) and prior chosen as the product of two
+BvM
+Skewed BvM
+0.0
+0.2
+-6
+-2
+2
+6
+n(α − α0)
+-1
+0
+1
+2
+-2
+0
+2
+n(α − α0)
+n(β − β0)
+-1
+0
+1
+2
+-2
+0
+2
+n(α − α0)
+n(β − β0)
+BvM
+Skewed BvM
+0.0
+0.2
+0.4
+-3
+-1
+1
+3
+5
+n(β − β0)
+Fig 3: Graphical comparison between the approximations of the marginal and bivariate exact posterior den-
+sities (grey areas and lines) provided by the classical Gaussian Bernstein–von Mises result (BvM) and the
+newly–developed skewed version (Skewed BvM) based on generalized skew–normal approximating densities.
+Results refer to the Gamma(α,β) model with independent exponential priors on the parameters α and β.
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+19
+independent exponential distributions with rate parameter equal to 2 in both cases. As for
+the examples in Sections 2.4.1–2.4.2, the generalized skew–normal approximation is con-
+siderably more accurate than the Gaussian counterpart for both the joint and the marginal
+posterior densities.
+3. Skew–modal approximation. As mentioned in Section 2.4, Bernstein–von Mises
+type results, including our Theorem 2.1, describe the asymptotic behavior of the exact pos-
+terior with respect to a limiting distribution whose parameters depend on the true data–
+generating θ0. Since this quantity is not available in practice, it is necessary to replace θ0
+with a sensible estimate in order to ensure that the approximation derived is of practical rel-
+evance. To this end, in Section 3.1 we consider a simple, yet effective, plug–in version of the
+generalized skew–normal density in (1) which replaces θ0 with its maximum–a–posteriori
+(MAP) estimator, without losing theoretical accuracy guarantees. Notice that, in general, θ0
+can be replaced by any efficient estimator, but by relying on the MAP several quantities in
+(1) simplify, giving rise to a highly tractable and accurate approximation, as further shown
+in Section 3.2.
+3.1. Derivation and theoretical guarantees. Consistent with the above discussion, we
+consider the plug–in version ˆpn
+GSN of pn
+GSN in Equation (1), where the unknown θ0 is replaced
+by the MAP ˆθ = argmaxθ∈Θ{ℓ(θ) + log π(θ)}. As a consequence, this yields the generalized
+skew–normal density, for the rescaled parameter ˆh = √n(θ − ˆθ) ∈ Rd, defined as
+(21)
+ˆpn
+GSN(ˆh) = 2φd(ˆh;0, ˆΩ)Φ(α(ˆh)),
+where ˆΩ = ˆV −1 with ˆV = [ˆvst] = [jˆθ,st/n] ∈ Rd×d, while the skewness function entering the
+cumulative distribution Φ(·) is given by ˆα(h) = {
+√
+2π/(12√n)}(ℓ(3)
+ˆθ,stl/n)ˆhsˆhtˆhl ∈ R.
+Notice that, relative to the expression for pn
+GSN(·) in Equation (1), the location parameter
+ξ is zero in (21), since ˆξ is a function of the quantity (ℓ(1)
+ˆθ + log π(1)
+ˆθ )/√n which is zero by
+definition when ˆθ is the MAP. For the same reason, in the expression for the precision matrix
+of the Gaussian density factor in (21), the term −(ˆξlℓ(3)
+ˆθ,stl)/n3/2 disappears.
+Equation (21) provides a practical skewed approximation of the exact posterior leveraging
+its local behavior at the mode. For this reason, such a solution is referred to as skew–modal
+approximation. In order to provide theoretical guarantees for this practical version, similar
+to those in Theorem 2.1, let us introduce two mild assumptions in addition to those outlined
+in Section 2.2.
+M1 The MAP estimator ˆθ = argmaxθ∈Θ{ℓ(θ)+log π(θ)}, satisfies Eθ0∥ˆθ−θ0∥2 = O(n−1).
+M2 Let ˆV = Jˆθ/n. With P n
+θ0–probability tending to one, ˆV is positive definite and | ˆV | > ˆη
+for some ˆη > 0. Moreover, there exist two positive constants δ and L such that the in-
+equalities | ℓ(3)
+stl(θ)/n | < L, | ℓ(4)
+stlk(θ)/n | < L, and |log π(2)
+st (θ)| < L hold uniformly over
+θ ∈ Bδ(ˆθ) = {θ ∈ Θ : ∥θ − ˆθ∥ < δ}, with P n
+θ0–probability tending to one.
+Condition M1 is mild and holds generally in regular parametric models. This assumption
+ensures us that the MAP is in a small neighborhood of θ0, where the centering took place in
+
+20
+Theorem 2.1. Condition M2 is a weaker version of the analytical assumption for Laplace’s
+method described in Kass, Tierney and Kadane (1990). Note also that assumption M1 im-
+plies that M2 is a stronger version of A0.0 and A4, requiring the upper bound to hold in a
+neighborhood of θ0. These conditions ensure uniform control on the difference between the
+log–likelihood ratio and its third order Taylor’s expansion. Finally, we remark that even the
+more restrictive analytical assumption for Laplace’s methods introduced in Kass, Tierney
+and Kadane (1990) are satisfied in several examples of practical interest.
+Based on the above additional assumptions we state a Bernstein–von Mises type result for
+the skew–modal approximation in (21), similar to Theorem 2.1. The proof of the theorem
+follows closely the reasoning and derivations considered to prove Theorem 2.1. Hence, such
+a proof, along with those of the technical lemmas, is deferred to the Appendix.
+THEOREM 3.1.
+Let ˆh = √n(θ − ˆθ). If the assumptions of Theorem 2.1 and conditions
+M1–M2 are satisfied, then the posterior for ˆh satisfies
+(22)
+∥Π(· | Xn) − ˆP n
+GSN(·)∥TV = OPθ0
+�(log n)3+d/2/n
+�,
+where ˆP n
+GSN(S) =
+�
+S ˆpn
+GSN(ˆh)dˆh for S ⊂ Rd with ˆpn
+GSN(ˆh) defined as in (21).
+REMARK 3.2.
+Since the total variation distance is invariant to scale and location trans-
+formations, the above result can be stated also for the original parametrization θ of interest.
+This means that the generalized skew–normal density
+(23)
+ˆpn
+GSN(θ) = 2φd(θ; ˆθ,J−1
+ˆθ )Φ((
+√
+2π/12)ℓ(3)
+ˆθ,stl(θ − ˆθ)s(θ − ˆθ)t(θ − ˆθ)l),
+approximates the posterior density for θ with the rate derived in Theorem 3.1.
+The skew–modal approximation in (23) provides, therefore, a similarly tractable, yet sub-
+stantially more accurate, alternative to the classical Gaussian counterpart arising from the
+Laplace method. As discussed in Section 2, the closed–form density in (23) can be evalu-
+ated essentially at the same cost of a Gaussian density and further admits a straightforward
+i.i.d. sampling scheme that facilitates Monte Carlo estimation of any functional of interest.
+Such a scheme has the same cost of sampling from a d–variate Gaussian and, hence, can be
+readily implemented via standard R packages for simulating from these variables.
+3.2. Empirical results. Sections 3.2.1–3.2.2 demonstrate the practical applicability of the
+skew–modal approximation derived in Section 3.1 on both a synthetic simulation and a real
+data application on Cushing’s syndrome (Venables and Ripley, 2002). These empirical anal-
+yses provide consistent evidence of substantial accuracy improvements not only relative to
+the classical Gaussian approximation induced by the Laplace method, but also with respect
+to state–of–the–art implementations of mean–field variational Bayes (e.g., Blei, Kucukel-
+bir and McAuliffe, 2017) and expectation–propagation (e.g., Vehtari et al., 2020), including
+also recent skewed extensions (e.g., Fasano, Durante and Zanella, 2022).
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+21
+TABLE 2
+In the exponential example introduced in Section 2.4.1, average, over 50 replicated studies, of the
+log–total–variation (TV) distances corresponding to the Gaussian modal approximation induced by the
+Laplace method (GM) and the newly–developed skew–modal (SKEW–M) solution in Section 3. Summaries for
+the ratios between the two TV distances are also reported. Results are displayed for different sample sizes
+from n = 1 to n = 5000. The standard deviations of the ratios, computed over the 50 replicated studies, are
+displayed within round brackets. The bold values indicate the best performance for each sample size.
+n = 1
+n = 10
+n = 50
+n = 100
+n = 500
+n = 1000
+n = 5000
+log n
+0.00
+2.30
+3.91
+4.61
+6.21
+6.91
+8.52
+log TVGM
+−1.08
+−1.43
+−1.98
+−2.29
+−3.05
+−3.40
+−4.20
+log TVSKEW–M
+−2.32
+−3.01
+−4.13
+−4.74
+−6.28
+−6.97
+−8.56
+TVSKEW–M/TVGM
+0.29
+0.21
+0.12
+0.09
+0.04
+0.03
+0.01
+(10−5)
+(10−5)
+(10−5)
+(10−5)
+(10−5)
+(10−5)
+(10−5)
+3.2.1. Exponential model revisited. Let us first replicate the simulation study on the ex-
+ponential model under the same settings and comparisons in Section 2.4.1, but with a focus
+on the plug–in skew–modal approximation in (23), rather than its population version which
+assumes knowledge of θ0. Consistent with this focus, performance is compared against the
+Gaussian approximation N(ˆθ,J−1
+ˆθ ) arising from the Laplace method (e.g., Gelman et al.,
+2013, p. 318). Notice that such a model meets the additional assumptions M1–M2 required
+by Theorem 3.1 and Remark 3.2. In fact, ˆθ is asymptotically equivalent to the maximum
+likelihood estimator which implies that condition M1 is fulfilled. Moreover, in view of the
+expressions for the first three log–likelihood derivatives in Section 2.4.1 also M2 is satisfied.
+Table 2 displays the same summaries of Table 1, but now the two total variation distances
+from the exact posterior are computed with respect to the skew–modal approximation in
+(23) and the Gaussian N(ˆθ,J−1
+ˆθ ), respectively. Similarly to the population version assessed
+in Section 2.4.1, also the practical skew–modal approximation yields considerable accuracy
+gains relative to the Gaussian counterpart. In addition, also in this context, the asymptotic
+theory in Theorem 3.1 and Remark 3.2 closely matches the empirical behavior observed in
+practice even in finite, possibly small, sample size settings. Notice also that, when compared
+to the results in Table 1, the variability across experiments in Table 2 is substantially lower.
+This is due to the fact that, unlike for the population counterpart, both approximations are
+centered at the MAP of the actual posterior rather than at θ0, which is not necessarily the
+maximum–a–posteriori in finite sample size settings. This allows for a more accurate fit to
+the posterior in practice and less variability across experiments in the TV distances.
+3.2.2. Probit and logistic regression model. We conclude with a real–data application on
+the Cushings dataset (Venables and Ripley, 2002) which is openly–available in the R library
+Mass. In this case the true data–generative model is not known and, therefore, such a final
+assessment is useful to evaluate performance in situations which are not guaranteed to meet
+the assumptions underlying our theory in Sections 2.3 and 3.1.
+The data are obtained from a medical study, on n = 27 subjects, aimed at investigating
+the relationship between four different sub–types of Cushing’s syndrome and two steroid
+
+22
+metabolites, Tetrahydrocortisone and Pregnanetriol respectively. To simplify the analysis,
+we consider the binary response Xi ∈ {0,1}, i = 1,...,n taking value 1 if the patient is
+affected by bilateral hyperplasia, and 0 otherwise, for i = 1...,n. The observed covariates
+are zi1 = “urinary excretion rate (mg/24hr) of Tetrahydrocortisone for patient i" and zi2 =
+“urinary excretion rate (mg/24hr) of Pregnanetriol for patient i". For inference, we focus
+on the two most widely–implemented regression models for binary data, namely:
+Probit regression. Xi ∼ Bern(Φ(θ0 + θ1zi1 + θ2zi2)), independently for i = 1,...,n, with
+regression coefficients (θ0,θ1,θ2) = θ ∈ R3.
+Logistic regression. Xi ∼ Bern(g(θ0+θ1zi1+θ2zi2)), independently for i = 1,...,n, with
+(θ0,θ1,θ2) = θ ∈ R3, and g(·) the inverse logit function defined in Section 2.4.2.
+Under both models, Bayesian inference proceeds via routinely–implemented weakly in-
+formative Gaussian priors N(0,25) for the three regression coefficients. Recalling Durante
+(2019), such a choice yields a closed–form unified skew–normal (Arellano-Valle and Az-
+zalini, 2006) posterior for θ under probit regression, which admits tractable i.i.d. sampling
+schemes. Extending the results in Durante (2019), a recent contribution by Onorati and
+Liseo (2022) has proved that similar findings, for an extension of the unified skew–normal,
+can be obtained also under logistic models. Although yielding important advancements in
+the field of Bayesian inference for binary regression models, the derived closed–form pos-
+teriors require evaluation of n–dimensional Gaussian cumulative distribution functions, and
+hence, inference under the exact posterior becomes intractable as n grows. This motivates
+alternative solutions within the class of deterministic approximations to the exact posterior.
+State–of–the–art methods under these regression models are mean–field variational Bayes
+(MF–VB) (Consonni and Marin, 2007; Durante and Rigon, 2019), partially–factorized varia-
+tional Bayes (PFM–VB) (Fasano, Durante and Zanella, 2022), expectation–propagation (EP)
+(Chopin and Ridgway, 2017), and the classical Gaussian modal approximation (GM) based
+on the Laplace method (Tierney and Kadane, 1986).
+Table 3 and Figures 4–5, compare the approximation accuracy of the newly–developed
+skew–modal approximation in (23) relative to the aforementioned alternatives in both probit
+and logit models. To evaluate also the i.i.d. sampling scheme for generalized skew–normal
+distributions introduced in Section 2, the marginals, bivariates and joint total variation dis-
+tances are computed via Monte Carlo methods leveraging 105 samples from both the exact
+posteriors for θ and from the corresponding approximations provided by the different meth-
+ods analyzed. Under both probit and logistic regression, the skew–modal approximation and
+the classical Gaussian modal approximation (GM) can be readily derived from the deriva-
+tives of the log–likelihood and log–prior which are available in closed–form. Moreover,
+since the prior is Gaussian, the MAP under both models coincides with the ridge–regression
+solution and hence can be computed via basic R functions. MF–VB and PFM–VB for probit
+regression leverage the implementation in the GitHub repository Probit-PFMVB (Fasano,
+Durante and Zanella, 2022), whereas in the logistic setting we rely on the codes in the repos-
+itory logisticVB (Durante and Rigon, 2019). Notice that PFM–VB has been developed
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+23
+TABLE 3
+For probit and logistic regression, estimated joint, bivariate and marginal total variation distances between
+the exact posterior and the deterministic approximations under analysis in the Cushings application. The
+bold values indicate the best performance for each subset of parameters.
+TVθ
+TVθ01
+TVθ02
+TVθ12
+TVθ0
+TVθ1
+TVθ2
+Probit
+SKEW–M
+0.42
+0.19
+0.25
+0.35
+0.13
+0.16
+0.20
+GM
+0.75
+0.40
+0.55
+0.72
+0.37
+0.31
+0.44
+EP
+0.51
+0.30
+0.38
+0.45
+0.01
+0.29
+0.36
+MF–VB
+1.96
+1.28
+1.65
+1.84
+0.71
+1.06
+1.40
+PFM–VB
+0.97
+0.49
+0.87
+0.89
+0.25
+0.34
+0.78
+Logit
+SKEW–M
+0.57
+0.32
+0.38
+0.52
+0.22
+0.24
+0.29
+GM
+0.92
+0.52
+0.69
+0.88
+0.45
+0.39
+0.57
+EP
+2.03
+2.04
+2.18
+2.02
+2.52
+2.02
+2.04
+MF–VB
+1.01
+0.52
+1.05
+0.95
+0.31
+0.42
+0.73
+only for probit regression. Finally, EP is implemented, under both models, via the R library
+EPGLM (Chopin and Ridgway, 2017). The approximations provided by GM, MF–VB and EP
+are all Gaussian and, hence, admit straightforward i.i.d. sampling, while PFM–VB yields a
+computationally–tractable unified skew–normal approximation from which i.i.d. values can
+be simulated via Algorithm 3 in Fasano, Durante and Zanella (2022). Sampling from the
+exact probit posterior can be instead performed via Algorithm 1 in Durante (2019), whereas
+for the logistic regression we implement Hamiltonian Monte Carlo via the rstan library.
+Table 3 displays the TV distance estimates from the exact posteriors of θ for the differ-
+ent approximations, under both probit and logistic regression. In both models, the proposed
+skew–modal (SKEW–M) solution generally yields remarkable accuracy improvements rela-
+tive to all the state–of–the–art competitors. More specifically, SKEW–M halves, on average,
+the TV distance associated with GM, while yielding noticeable improvements relative to MF–
+VB and PFM–VB. The gains over PFM–VB in the probit model are remarkable since also such
+a strategy leverages a skewed approximation of the exact posterior distribution. This yields
+higher accuracy than MF–VB, but the improvements are not as noticeable as SKEW–M. A rea-
+son for this result is that PFM–VB has been originally developed to provide high accuracy in
+high–dimensional p > n settings (Fasano, Durante and Zanella, 2022), whereas in this study
+p = 3 and n = 27. Except for one marginal in the probit model, SKEW–M also outperforms
+state–of–the–art EP methods, with the accuracy improvements being particularly remark-
+able in the logit model. This is mainly due to the fact that, unlike for the proposed SKEW–M
+approximation, EP relies on Gaussian approximations obtained via iterative algorithms with
+no guarantees of convergence and reduced theoretical support. Hence, the poor EP perfor-
+mance in logistic regression might be due to convergence issues of the R function EPlogit
+(Ridgway, 2016). Conversely, our computationally–tractable SKEW–M approximation only
+requires derivation and evaluation of log–likelihood and log–prior derivatives. Furthermore,
+it has also strong theoretical support from Theorem 3.1 and Remark 3.2, thereby yielding
+
+24
+-0.1
+-0.1
+0.0
+-1.0
+-0.2
+0.7
+1.5
+θ0
+θ1
+-0.8
+-0.4
+-0.1
+0.3
+-1.0
+-0.2
+0.7
+1.5
+θ0
+θ2
+-0.8
+-0.4
+-0.1
+0.3
+-0.1
+-0.1
+0.0
+θ1
+θ2
+-0.1
+-0.1
+0.0
+-1.0
+-0.2
+0.7
+1.5
+θ0
+θ1
+-0.8
+-0.4
+-0.1
+0.3
+-1.0
+-0.2
+0.7
+1.5
+θ0
+θ2
+-0.8
+-0.4
+-0.1
+0.3
+-0.1
+-0.1
+0.0
+θ1
+θ2
+Fig 4: Visual comparison between skew–modal (blue) and Gaussian (orange) approximations of the exact
+bivariate posteriors (grey) for the three coefficients of the probit regression model in the Cushings application.
+-0.2
+-0.1
+0.0
+0.1
+-1.5
+-0.3
+0.8
+2.0
+θ0
+θ1
+-1.1
+-0.6
+0.0
+0.5
+-1.5
+-0.3
+0.8
+2.0
+θ0
+θ2
+-1.1
+-0.6
+0.0
+0.5
+-0.2
+-0.1
+0.0
+0.1
+θ1
+θ2
+-0.2
+-0.1
+0.0
+0.1
+-1.5
+-0.3
+0.8
+2.0
+θ0
+θ1
+-1.1
+-0.6
+0.0
+0.5
+-1.5
+-0.3
+0.8
+2.0
+θ0
+θ2
+-1.1
+-0.6
+0.0
+0.5
+-0.2
+-0.1
+0.0
+0.1
+θ1
+θ2
+Fig 5: Visual comparison between skew–modal (blue) and Gaussian (orange) approximations of the exact
+bivariate posteriors (grey) for the three coefficients of the logistic regression model in the Cushings application.
+gains relative to all state–of–the–art competitors. Clearly, as the exact posterior approaches
+a Gaussian shape, such improvements become less noticeable. Nonetheless, as illustrated in
+Figures 4–5, even relatively–minor deviations from Gaussianity in the posterior for θ, can
+yield remarkable gains when approximated via SKEW–M.
+4. Discussion. This article proves that substantial theoretical and practical accuracy im-
+provements can be obtained by replacing limiting Gaussian approximations of exact poste-
+rior distributions with generalized skew–normals constructively derived form a third–order
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+25
+version of the Laplace method. Beside clarifying the role of skewness in Bayesian asymp-
+totics, our contribution has also a broad theoretical scope and remarkable methodological
+impact. It can directly motivate refinements of several theoretical results relying on the clas-
+sical Gaussian Bernstein–von Mises theorem. The tractability of generalized skew–normals
+and the corresponding skew–modal practical counterparts, along with the theoretical and
+empirical improvements presented in the article, also open new avenues to stimulate exten-
+sive advancements in the field of Bayesian inference based on deterministic approximations.
+More specifically, our contribution motivates future efforts toward developing skewed ver-
+sions of state–of–the–art variational Bayes and expectation propagation. These more effec-
+tive skew–modal approximations, which can readily replace the classical Gaussian one, can
+be further included in more refined strategies, such as INLA, which consider these Gaussian
+approximations as building–blocks.
+Future research directions of particular interest include the extension of our theory and
+methods beyond parametric settings, with a specific focus on semiparametric (Bickel and
+Kleijn, 2012; Castillo and Rousseau, 2015) and high–dimensional (Boucheron and Gassiat,
+2009; Bontemps, 2011; Spokoiny and Panov, 2021) contexts. Moreover, although the in-
+clusion of skewness is arguably sufficient to yield an accurate approximation of posterior
+distributions in small–to–moderate sample size regimes, accounting for kurtosis might yield
+additional improvements both in theory and in practice. To this end, a relevant research di-
+rection is to seek for an alternative to generalized skew–normals, possibly obtained from a
+fourth–order version of the Laplace method that yields an improved characterization of the
+tails. In principle, this extension is expected to provide further improvements in the rate of
+convergence, while maintaining good sampling properties. Our conjecture is that such an
+alternative limiting law can be found in the class of skew–symmetric distributions (Azzalini
+and Capitanio, 2003; Wang, Boyer and Genton, 2004) and relies on the perturbation of a
+density with heavier tails than the Gaussian one.
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+
+28
+APPENDIX A: PROOF OF THEOREM 3.1
+To prove Theorem 3.1 we follow a similar line of reasoning as in Theorem 2.1, and high-
+light the main differences between the two proofs. To this end, let us first consider the sieve
+ˆΘn = {θ ∈ Θ : ∥θ − ˆθ∥ < 2Mn/√n}. We show below that Π(ˆΘc
+n | Xn) converges to zero
+in probability. To this end let us consider the event Bn = {∥ˆθ − θ0∥ ≤ Mn/√n}, and recall
+the definition of Θn in (8). Note that
+Π(ˆΘc
+n | Xn) ≤ Π(Θc
+n | Xn) + 1Bcn.
+The first term on the right hand side is OPθ0(n−1) in view of Markov’s inequality and the
+fact that, as shown in Theorem 2.1, Eθ0Π(Θc
+n | Xn) = O(n−1). The second term has the
+same order in view of assumption M1 and applying again Markov’s inequality. Therefore,
+following (10), it is sufficient to work with the restricted posterior ΠˆΘn(·|Xn) defined in (9),
+with Θn replaced by ˆΘn. As for the proof of Theorem 2.1, let us introduce the quantities
+Yn(ˆh) = 2exp
+�
+− 1
+2
+jˆθ,st
+n
+ˆhsˆht
+�
+Φ
+� √
+2π
+12√n
+ℓ(3)
+ˆθ,stl
+n
+ˆhsˆhtˆhl
+�
+,
+Ln(ˆh) =
+ˆpˆθ+ˆh/√n
+ˆpˆθ
+(Xn)π(ˆθ + ˆh/√n)
+π(ˆθ)
+.
+In view of assumption M2 we can express log Ln(ˆh) as log Ln(ˆh) = −(1/2)(jˆθ,st/n)ˆhsˆht+
+(1/6√n)(ℓ(3)
+ˆθ,stl/n)ˆhsˆhtˆhl + qn,ˆh, where
+qn,ˆh =
+ℓ(4)
+ˆθ,stlk(˜h)
+n
+ˆhsˆhtˆhlˆhk + 1
+2
+log π(ˆθ + ¯h/√n)(2)
+st
+n
+ˆhsˆht = OPθ0
+�∥ˆh∥4 ∨ 1
+n
+�
+,
+and ˜h,¯h are some points between zero and ˆh. Now, from triangle inequality and in view of
+the second order Taylor expansion of eqn,ˆh = 1 + qn,ˆh + eβqn,ˆhq2
+n,ˆh/2, for some β ∈ (0,1), it
+follows that
+(24) |Ln(ˆh) − Yn(ˆh)| ≤|exp(un,ˆh)(qn,ˆh + exp(βqn,ˆh)q2
+n,ˆh/2)| + |exp(un,ˆh) − Yn(ˆh)|,
+with un,ˆh = −(1/2)(jˆθ,st/n)ˆhsˆht + (1/6√n)(ℓ(3)
+ˆθ,stl/n)ˆhsˆhtˆhl.
+As for the proof of Theorem 2.1, let us replace the stochastic quantities involved in the
+above expressions with their truncated counterparts. In particular, let ˜ℓ(3)
+ˆθ,stl = (ℓ(3)
+ˆθ,stl ∨ −nL)∧
+nL, ˜ℓ(4)
+ˆθ,stlk(˜h) = (ℓ(4)
+ˆθ,stlk(˜h) ∨ −nL) ∧ nL and log ˜π(ˆθ+¯h/√n)(2)
+st = (log π(ˆθ + ¯h/√n)st(2)∨
+−L) ∧ L, and denote with ˜qˆh,n, ˜Ln(ˆh) and ˜Yn(ˆh) the versions of qˆh,n, Ln(ˆh) and Yn(ˆh) in
+which the different quantities appearing within the corresponding expressions have been
+replaced by the truncated counterparts. Notice that the Pθ0–probability that any of these
+truncations takes place is O(1). Therefore, by applying in (24) the thresholds introduced by
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+29
+the truncation, we directly get that the first term on the right hand side is O(M 4
+n/n), while
+in view of Lemma B.6 the second term is O(M 6
+n/n). This implies that
+(25)
+supˆh∈ ˆSMn |˜Ln(ˆh) − ˜Yn(ˆh)| = O(M 6
+n/n).
+where ˆSMn = {ˆh : ||ˆh|| < 2Mn}. Hence, combining (25) with Fubini’s theorem and Jensen’s
+inequality, it holds
+Eθ0 supS |
+�
+S∩ ˆSMn ˜Ln(ˆh)dˆh −
+�
+S∩ ˆSMn ˜Yn(ˆh)dˆh|
+≤
+�
+ˆSMn Eθ|˜Ln(ˆh) − ˜Yn(ˆh)|dˆh ≲
+�
+ˆSMn(M 6
+n/n)dˆh = O
+�M d+6
+n
+/n
+�.
+Then, by applying the triangle inequality and noticing that Pθ0(
+�
+ˆSMn ˜Yn(ˆh)dˆh > 0) = 1 it
+follows from the above result that
+Eθ0 supS |
+�
+S∩ ˆSMn ˜Ln(ˆh)dˆh/
+�
+ˆSMn ˜Ln(ˆh)dˆh−
+�
+S∩ ˆSMn ˜Yn(ˆh)dˆh/
+�
+ˆSMn ˜Yn(ˆh)dˆh| = O
+�M d+6
+n
+/n
+�,
+which in turn implies the same upper bound, in Pθ0–probability, for the non–truncated ver-
+sion. More specifically, recalling the proof of Theorem 2.1 and considering a similar nota-
+tion, it follows that
+(26)
+supS | Π ˆSMn(ˆh ∈ S | Xn) − ˆPGSN, ˆSMn(S)| = OPθ0(M d+6
+n
+/n).
+where ˆPGSN, ˆSMn(S) =
+�
+S∩ ˆSMn Yn(ˆh)dˆh/
+�
+ˆSMn Yn(ˆh)dˆh. Moreover, as for the proof of Theo-
+rem 2.1, notice that the denominator in the expression for ˆPGSN, ˆSMn(S) is equal to a constant
+times (1 − ˆPGSN( ˆSc
+Mn)), where ˆPGSN( ˆSc
+Mn) = OPθ0(n−D), D > 1 for a sufficiently large c0 in
+the definition of Mn. Hence, since
+| ˆPGSN, ˆSMn(S) − ˆPGSN(S)| ≤ { ˆPGSN( ˆSc
+Mn)/(1 − ˆPGSN( ˆSc
+Mn))} ˆPGSN
+�S ∩ ˆSMn
+� + ˆPGSN( ˆSc
+Mn),
+we have that
+(27)
+supS| ˆPGSN, ˆSMn(S) − ˆPGSN(S)| = OPθ0(n−D).
+Combining (26) and (27) with the first part of the proof, and applying the triangle equality
+similarly to what done for Theorem 2.1, concludes the proof of Theorem 3.1.
+APPENDIX B: TECHNICAL LEMMAS
+In this section we collect the technical lemmas used in the proofs of our two main theo-
+rems. Also in this case we omit for convenience the n from the notation of the probability
+measure Pθ and the corresponding expectation and densities.
+The first lemma is a refined version of Lemma 6.26 in Ghosal and Van der Vaart (2017),
+with (possibly) a polynomial instead of logarithmic Pθ0–convergence rate, used together
+with Lemma B.2, to bound the Type II error.
+
+30
+LEMMA B.1.
+For any prior Π on Θ, sequence Dn > 1 and ϵ > 0, it holds
+(28)
+�
+(pθ/pθ0)(Xn)dΠ(θ) ≥ Π
+�θ : KL(pθ0∥pθ) < ϵ2�exp{−2Dn(
+�
+nϵ2/2 ∨ nϵ2)}
+with Pθ0–probability at least 1 − D−1
+n .
+PROOF. First note that for B = {θ : KL(pθ0∥pθ) < ϵ2}, in view of Jensen’s inequality
+log
+� (pθ/pθ0)(Xn)dΠ(θ)
+Π(B)
+≥ log
+�
+(pθ/pθ0)(Xn)dΠ(θ | B)
+≥
+�
+log(pθ/pθ0)(Xn)dΠ(θ | B) =: Z.
+Now, note that by Markov’s and Jensen’s inequality, the probability of the complement of
+the event in (28) can be bounded as
+Pr(Z < −2Dn(
+�
+nϵ2/2 ∨ nϵ2)) ≤
+Eθ0(−Z)+
+2Dn(
+�
+nϵ2/2 ∨ nϵ2)
+≤ Eθ0
+� (−log(pθ/pθ0)(Xn))+ dΠ(θ | B)
+2Dn(
+�
+nϵ2/2 ∨ nϵ2)
+,
+where the generic (Z)+ is defined as Z ∨0. Then, by Fubini’s theorem and Pinsker’s second
+inequality (e.g. Lemma B.13 of Ghosal and Van der Vaart (2017)) the right hand side of the
+above inequality is further bounded by
+�
+KL(pθ0∥pθ) +
+�
+KL(pθ0∥pθ)/2dΠ(θ|B)
+2Dn(
+�
+nϵ2/2 ∨ nϵ2)
+≤
+2(
+�
+nϵ2/2 ∨ nϵ2)
+2Dn(
+�
+nϵ2/2 ∨ nϵ2)
+= 1
+Dn
+,
+concluding the proof of Lemma B.1.
+The next lemma adapts Lemma 10.3 in Van der Vaart (2000) to ensure a tighter control on
+the Type I error. In the proofs of our main theorem, it is required to control the probability
+mass assigned by the posterior distribution outside the ball of radius Mn/√n.
+LEMMA B.2.
+Under the conditions of Theorem 2.1 there exists, for every Mn → ∞, a
+sequence of tests φn and a constant ct such that, for every sufficiently large n and every
+∥θ − θ0∥ > Mn/√n, we have
+Eθ0φn ≤ exp
+�−ctM 2
+n
+�
+and
+Eθ (1 − φn) ≤ exp(−ctn(∥θ − θ0∥2 ∧ 1)).
+PROOF. Condition A0.0 allows us to consider the truncated version of the score function
+˜ℓ(1)
+θ0 ∈ [−L,L], with L > 0 taken such that Eθ0˜ℓ(1)
+θ0 (˜ℓ(1)
+θ0 )⊺ is nonsingular. Define now the test
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+31
+ωn = 1{∥(En − Eθ0)˜ℓ(1)
+θ0 ∥ > c∗Mn/√n}, where the constant c∗ > 0 is determined later and
+En is the expectation with respect to the empirical measure. By Hoeffding’s inequality,
+Eθ0ωn ≤ exp
+�−ctM 2
+n
+�,
+for some constant ct depending on L.
+To conclude the proof, it remains to study the Type II error. By the triangle inequality
+∥(Eθ − Eθ0)˜ℓ(1)
+θ0 ∥ ≤ ∥(En − Eθ))˜ℓ(1)
+θ0 ∥ + ∥(En − Eθ0))˜ℓ(1)
+θ0 ∥.
+Furthermore, in view of the differentiability of the likelihood function there exists a constant
+c > 0 for which ∥(Eθ − Eθ0)˜ℓ(1)
+θ0 ∥ > c∥θ − θ0∥ for all θ such that ∥θ − θ0∥ < ϵ; see Lemma
+10.3 in Van der Vaart (2000). Therefore, by choosing c∗ = c/2 and applying the preceding
+two displays together with Hoeffding’s inequality we get that
+Eθ(1 − ωn) ≤ Pθ
+�
+∥(En − Eθ)˜ℓ(1)
+θ0 ∥ > ∥(Eθ − Eθ0)˜ℓ(1)
+θ0 ∥ − c∗Mn/√n
+�
+.
+≤ Pθ
+�
+∥(En − Eθ)˜ℓ(1)
+θ0 ∥ > c∥θ − θ0∥/2
+�
+≤ exp
+�−ctn∥θ − θ0∥2�,
+for a sufficiently small constant ct depending on L. For ∥θ − θ0∥ > ϵ it is possible to con-
+struct a test with Type I and Type II errors that decrease exponentially in n by proceeding
+as in Lemma 10.3 of Van der Vaart (2000). Combining such tests, concludes the proof.
+The next lemma gives an upper bound for the first term in (15).
+LEMMA B.3.
+Let ˜∆n,θ = ˜ℓ(1)
+θ /√n =
+�ℓ(1)
+θ /√n
+�1∥ℓ(1)
+θ /√n∥<Cn where Cn = M 1/α
+n
+for α >
+2. Under the notations and conditions of Theorem 2.1, we have
+(29)
+limn→∞ suph∈SMn |Eθ expγ˜un,h − 1| → 0,
+for any γ > 0, where ˜un,h = hs{ ˜∆n,θ}s − (hshtist)/2n + ˜rn,h.
+PROOF. To prove Lemma B.3, let us first notice that, for ˜∆θ = ∆θ1∥∆θ∥<Cn with ∆θ ∼
+N(0,Iθ/n), it holds Eθ exp{γ
+�hs{ ˜∆θ}s − (hshtist)/2n
+�} − 1 = O(1). Therefore, by trian-
+gle inequality if follows that the left hand side of (29) is bounded from above by
+suph∈SMn |Eθ expγ˜un,h − Eθ expγ(hs{ ˜∆θ}s − (hshtist)/2n)| + O(1).
+Theorem 2.7 in Van der Vaart (2000) implies that ( ˜∆n,θ, ˜rn,h) converges in distribution to
+( ˜∆θ,0). Since Iθ is a deterministic positive definite matrix we can concentrate only on
+limn→∞ suph∈SMn |Eθ exp[γhs{ ˜∆n,θ}s + γ˜rn,h] − Eθ exp[γhs{ ˜∆θ}s]|.
+Let us now define J = Jϵ = {x ∈ Rp:∥x∥ < Rϵ}, with Rϵ > 0 such that Pθ (∆θ∈J) > 1−ϵ.
+Clearly, for all δ > 0, Pθ ((∆θ,0) ∈ J × (−δ,δ)) > 1 − ϵ. Since the functions fh(x,y) =
+
+32
+exp(γhsxs + γy) are uniformly continuous on I = J × (−δ,δ), for all h, one can cover I
+with finitely many subsets such that fh varies at most (constant times) ϵ in each of them. In
+particular, let us define the following sets
+(30)
+J0 ={x : ∥x∥ > (Cn ∧ Rϵ)},
+Jj ={x : log[(j − 1)ϵ1] < γhsxs ≤ log(jϵ1)} ∩ {x : ∥x∥ ≤ (Cn ∧ Rϵ)},
+Ek ={y : log[(k − 1)ϵ2] < γy ≤ log(kϵ2)} ∩ {y : −δ < y < δ},
+with ϵ1 = ϵexp(−γδ), ϵ2 = ϵexp(−γMnCn), and j,k ∈ {1,...,Nn} for Nn such that Nn ≤
+exp{γ(MnCn + δ)}/ϵ. Furthermore, note that there exists a unique k∗ ∈ {1,...,Nn} such
+that 0 ∈ Ek∗. Let us now merge the sets Ek∗−1, Ek∗, Ek∗+1 into Ek∗ for technical reasons to
+be seen later. Then, let us take Ij,k = Jj ×Ek ⊂ I, j ∈ {0,...,Nn},k ∈ {1,...,Nn}\{k∗ −
+1,k∗+1} and note that following from the triangle inequality the function fh(x1(∥x∥<Cn),y)
+varies at most 4ϵ in each Ij,k. Moreover, notice that for j > 0 each Jj is convex since it is
+the intersection of two convex sets. Therefore, the multivariate Berry–Esseen’s theorem, see
+Lemma B.8, can be directly applied to Pθ(∆n,θ ∈ Jj). On the contrary, J0 is not convex but
+we can compute Pθ(∆n,θ ∈ J0) in terms of its complement which is convex.
+We use the sets Ij,k to approximate fh with a step function. Let us take an arbitrary point
+(xj,yk) ∈ Ij,k for j ∈ {0,...,Nn},k ∈ {1,...,Nn}\{k∗ − 1,k∗ + 1} and define the step
+function
+fϵ,h(x,y) =
+�Nn
+j=0
+�Nn
+k=1 fh(xj,yk)1(x,y)∈Ij,k.
+Notice that, for notational convenience, we consider the sum over k = 1,...,Nn, but in fact
+the k∗ −1 and k∗ +1 cells are merged with the k∗ cell. Then, by triangle inequality, we have
+(31)
+��Eθ fh( ˜∆n,θ, ˜rn,h) − Eθfh( ˜∆θ,0)
+�� ≤
+��Eθ fh( ˜∆n,θ, ˜rn,h) − Eθ fϵ,h( ˜∆n,θ, ˜rn,h)
+��
++
+��Eθ fϵ,h( ˜∆n,θ, ˜rn,h) − Eθfϵ,h( ˜∆θ,0)
+��
++
+��Eθ fh( ˜∆θ,0) − Eθfϵ,h( ˜∆θ,0)
+��.
+We now study the three terms on the right hand side of (31) separately.
+For the second term, we can exploit the fact that fϵ,h is a step function and that it is zero
+outside of I. This result, in combination with Jensen’s inequality and the bound fh(xj,yk) ≤
+exp(γMnCn +γδ) for j,k ∈ {1,...,Nn}, ensures that
+��Eθ fϵ,h( ˜∆n,θ, ˜rn,h)−Eθfϵ,h( ˜∆θ,0)
+��
+is bounded from above by
+�Nn
+j=1
+�Nn
+k=1 eγ(MnCn+δ)|Pθ(( ˜∆n,θ, ˜rn,h) ∈ Ij,k) − Pθ (( ˜∆θ,0) ∈ Ij,k)|.
+The asymptotic behavior of the above expression depends on the difference between the
+probability measure of ( ˜∆n,θ, ˜rn,h) and the one of ˜∆θ. First, consider the case 0 /∈ Ek (i.e.
+k ̸= k∗) where Pθ
+�( ˜∆θ,0) ∈ Ij,k
+� = 0 holds. In view of |log(1 + y)| ≥ |y|/2 for |y| ≤ 2.5
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+33
+we have that Ek ⊂ {|˜rn,h| > ϵ2}, since |k − k∗| ≥ 2. Therefore, following by (the truncated
+version of) assertion (13)
+(32)
+|Pθ(( ˜∆n,θ, ˜rn,h) ∈ Ij,k
+� − Pθ
+�( ˜∆θ,0) ∈ Ij,k)|
+≤ Pθ(˜rn,h ∈ Ek) ≤ Pθ(|˜rn,h| > ϵ2) = O(log n{∥h∥3 ∨ 1}eγMnCn/(√nϵ)).
+Similarly, if 0 ∈ Ek, i.e. k = k∗, by applying the triangle inequality
+|Pθ(( ˜∆n,θ, ˜rn,h) ∈ Ij,k) − Pθ ( ˜∆θ ∈ Jj)| ≤ |Pθ( ˜∆n,θ ∈ Jj) − Pθ ( ˜∆θ ∈ Jj)| + Pθ(˜rn,h /∈ Ek)
+= O
+�log n{∥h∥3 ∨ 1}eγMnCn/(√nϵ)
+�,
+where the last step follows from the multivariate Berry–Esseen’s theorem; see Lemma B.8
+below (with Yi = ℓ(1)(θ0,Xi)/√n), and (the truncated version of) assertion (13).
+For the first term on the right hand side of (31), note that
+(33)
+��Eθfh( ˜∆n,θ, ˜rn,h) − Eθ fϵ,h( ˜∆n,θ, ˜rn,h)
+��
+≤ 4ϵPθ
+�( ˜∆n,θ, ˜rn,h) ∈ I
+� + eγ(MnCn+δ)Pθ
+�( ˜∆n,θ, ˜rn,h) /∈ I
+�.
+Hence, as before
+Pθ
+�( ˜∆n,θ, ˜rn,h) /∈ I
+� ≤ |Pθ
+� ˜∆n,θ /∈ J
+� − Pθ
+� ˜∆θ /∈ J
+�| + Pθ( ˜∆θ /∈ J) + Pθ(|˜rn,h| > δ)
+= O(log n{∥h∥3 ∨ 1}/(√nϵ)) + ϵ.
+(34)
+Finally, the third term on the right hand side of (31) can be bounded similarly as
+(35)
+��Eθ fh( ˜∆θ,0) − Eθfϵ,h( ˜∆θ,0)
+��
+≤ 4ϵPθ( ˜∆θ ∈ J) + eγMnCnPθ( ˜∆θ /∈ J) ≤ 4ϵ + eγMnCnϵ.
+We conclude the proof by applying the bounds (32), (33), (34) and (35) in (31), by choosing
+ϵ = ϵn ≍ n−β for β ∈ (0,1/6) and by recalling that Cn = M 1/α
+n
+with α > 2 to obtain
+limn→∞ suph∈SMn |Eθ fh( ˜∆n,θ, ˜rn,h) − Eθfh( ˜∆θ,0)| = 0.
+LEMMA B.4.
+Define λn,h = (1/6√n)(ℓ(3)
+θ0,stl/n){ξsξtξl + 3(h − ξ)s(h − ξ)tξl − hshthl}.
+Then, under the conditions of Theorem 2.1, it follows that
+(36)
+Yn(h) = exp(un,h)(1 + λn,h + eβ∗λn,hλ2
+n,h/2)
+× (1 − λn,h + (1/12)e−π(˜βλn,h)2/4{1 − (˜βλn,h
+�
+π/2)2}λ3
+n,hπ),
+for some β∗ ∈ (0,1) and ˜β ∈ (0,1), with λn,h = OPθ0((∥h∥3 ∨ 1)/√n).
+
+34
+PROOF. Notice first that assumption A0 implies that λn,h = OPθ0({∥h∥3 ∨ 1}/√n). Fur-
+thermore, −λn,h = (1/6√n)(ℓ(3)
+θ0,stl/n)((h − ξ)s(h − ξ)t(h − ξ)l + 3(h − ξ)sξtξl). There-
+fore, by taking a Taylor’s expansion of Φ(·) around 0, i.e Φ(x) = 1/2 + x/
+√
+2π + O(x3),
+for x = O(1), we obtain for some ˜β ∈ (0,1) that
+Φ(−
+�
+π/2λn,h) = 1/2 − λn,h/2 + (1/24)e−π(˜βλn,h)2/4{1 − (˜βλn,h
+�
+π/2)2}λ3
+n,hπ.
+To conclude the proof, recall that in (12), the variable Yn(h) can be alternatively defined as
+Yn(h) = 2exp(un,h)exp(λn,h)Φ(−
+�
+π/2λn,h).
+Combining the two equations above with the second order Taylor’s expansion ex = 1 + x +
+eβxx2/2 for some β ∈ (0,1), concludes the proof.
+LEMMA B.5.
+Under the notations and conditions of Theorem 2.1, we have
+(37)
+suph∈SMn Eθ|exp(˜un,h) − ˜Yn(h)| = O(M 6
+n/n).
+PROOF. In view of Lemma B.4 and by Hölder’s inequality we have that Eθ|exp(˜un,h) −
+˜Yn(h)| is bounded from above by the product between E1/ρn
+θ
+exp(ρn˜un,h) and E1/˜ρn
+θ
+|1−(1+
+˜λn,h + eβ∗˜λn,h˜λ2
+n,h/2)(1 − λn,h + (1/12)e−π(˜βλn,h)2/4{1 − (˜βλn,h
+�
+π/2)2}λ3
+n,hπ)|˜ρn. Now,
+notice that Lemma B.3 implies E1/ρn
+θ
+exp(ρn˜un,h) = O(1) for ρn = (1 + M 2
+n)/M2
+n. Further-
+more, in view of the truncation |˜λn,h| ≤ τn, the second factor is O(M 6
+n/n), concluding the
+proof of the statement.
+The following lemma is the counterpart Lemma B.5 for the skew–modal approximation.
+LEMMA B.6.
+Under the conditions of Theorem 3.1, it holds
+supˆh∈ ˆSMn |exp(˜un,ˆh) − ˜Yn(ˆh)| = O(M 6
+n/n),
+where ˜un,ˆh = −(1/2)(jˆθ,st/n)ˆhsˆht + (1/6√n)(˜ℓ(3)
+ˆθ,stl/n)ˆhsˆhtˆhl.
+PROOF. First note that ˜λn,ˆh := (˜ℓ(3)
+ˆθ,stl/n)ˆhsˆhtˆhl ≤ CL,d(∥ˆh∥3 ∨ 1), for some constant CL,d
+depending on L and d. Moreover, by the Taylor’s expansion of Φ(·) around 0, we have
+Φ((
+√
+2π/12√n)˜λn,ˆh) =1/2 + ˜λn,ˆh/(12√n) + O((∥˜h∥9 ∨ 1)/n3/2).
+These results, together with the positive definiteness of Jˆθ, and the fact that exp(x) = 1 +
+x + exp(βx)x2/2 for some β ∈ (0,1) lead to
+supˆh∈ ˆSMn |exp(˜un,ˆh) − ˜Yn(ˆh)|
+≤ supˆh∈ ˆSMn |exp((1/6√n)˜λn,ˆh) − 2Φ((
+√
+2π/12√n)˜λn,ˆh)| = O((∥˜h∥6 ∨ 1)/n),
+concluding the proof of the lemma.
+
+SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS
+35
+Next we show that both the empirical Fisher information matrix and the covariance matrix
+of the Gaussian density in the generalized skew–normal approximation are positive definite
+with determinant bounded away from zero.
+LEMMA B.7.
+Under A0 and A4, with Pθ0–probability tending to 1, V = Ω−1 and Jθ0/n
+are positive definite matrices and there exists ˜η > 0 such that |V | > ˜η and |Jθ0/n| > ˜η.
+PROOF. Leveraging the assumptions in A0, let us first notice that Jθ0/n = Iθ0/n + RJ
+and V = Iθ0/n + RV , with RJ and RV having entries of order OPθ0(n−1/2). In the follow-
+ing, we focus on RJ since the statement for RV can be obtained under the same reasoning
+and steps. Notice that condition A0 implies the existence of a sequence Fn ≍ n−1/κ, with
+κ > 2 such that
+Pθ0(|RJ,st| > Fn) = O(1),
+for each s,t = 1,...,d. Define now the matrix M with entries Mst = |RJ,st| ∧ Fn for s,t =
+1,...,d. From Wielandt’s theorem (Zwillinger and Jeffrey, 2007), with probability 1−O(1),
+the spectral radius of M is an upper bound of the spectral radius of RJ. Moreover, since M
+is a non–negative matrix, the Perron–Frobenius theorem (Perron, 1907; Frobenius, 1912)
+implies that the largest eigenvalue in absolute value is bounded by constant times Fn. Since
+both Iθ0/n and RJ are real symmetric matrices, in view of the Weyl’s inequalities (e.g., Tao,
+2011, equation (1.54)), the eigenvalues of Jθ0/n and Iθ0/n can differ at most by constant
+times Fn with probability 1−O(1). Since the entries of Iθ0/n are O(1) this implies that, with
+Pθ0–probability tending to one, |Jθ0/n| is bounded away from 0 and that Jθ0 is a positive–
+definite matrix.
+Finally, for completeness, we recall Berry–Esseen’s theorem; see for instance Theorem 1
+of Bentkus (2005).
+LEMMA B.8.
+Let Y1,...,Yn be independent random variables on Rd with EYi = 0,i =
+1,...,n. Write Z = Y1 + ··· + Yn, Σ = Cov Z and take Z0 ∼ Nd(0,Σ). Assume that Σ is
+invertible. Furthermore, define η = η1 + ··· + ηn, with ηi = E∥Σ−1/2Yi∥3. Then, there exist
+C > 0 such that for every convex set A ⊂ Rd, it holds |P(Z ∈ A) − P(Z0 ∈ A)| ≤ Cd1/4η.
+
diff --git a/tNE1T4oBgHgl3EQfQQOd/content/tmp_files/load_file.txt b/tNE1T4oBgHgl3EQfQQOd/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1474 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf,len=1473
+page_content='SKEWED BERNSTEIN–VON MISES THEOREM  AND SKEW–MODAL APPROXIMATIONS  BY DANIELE DURANTE1,*, FRANCESCO POZZA2,‡ AND BOTOND SZABO1,*,†  1Department of Decision Sciences and Institute for Data Science and Analytics, Bocconi University,  daniele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='durante@unibocconi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' †botond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='szabo@unibocconi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='it  2Department of Statistical Sciences, University of Padova, ‡francesco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='pozza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2@phd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='it  Deterministic Gaussian approximations of intractable posterior distributions are com-  mon in Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' From an asymptotic perspective, a theoretical justification in  regular parametric settings is provided by the Bernstein–von Mises theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' However, such  a limiting behavior may require a large sample size before becoming visible in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In  fact, in situations with small–to–moderate sample size, even simple parametric models often  yield posterior distributions which are far from resembling a Gaussian shape, mainly due to  skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In this article, we provide rigorous theoretical arguments for such a behavior by  deriving a novel limiting law that coincides with a closed–form and tractable skewed gener-  alization of Gaussian densities, and yields a total variation distance from the exact posterior  whose convergence rate can be shown to be of order 1/n, up to a logarithmic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Such a  result provides a substantial accuracy improvement over the classical Bernstein–von Mises  theorem whose convergence rate, under similar conditions, is of order 1/√n, again up to  a logarithmic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In contrast to higher–order approximations based on, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Edgeworth  expansions, which require finite truncations for inference, possibly leading to even negative  densities, our theory characterizes the limiting behavior of Bayesian posteriors with respect  to a sequence of valid and tractable densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' These advancements further motivate a prac-  tical plug–in version which replaces the unknown model parameters with the corresponding  maximum–a–posteriori estimate to obtain a novel skew–modal approximation achieving the  same improved rate of convergence of our skewed Bernstein–von Mises theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Exten-  sive simulations and a real–data application confirm that our new theory closely matches the  empirical behavior observed in practice even in finite, possibly small, sample regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The  proposed skew–modal approximation further exhibits improved accuracy not only relative to  classical Laplace approximation, but also with respect to state–of–the–art Gaussian and non–  Gaussian approximations from mean–field variational Bayes and expectation–propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Modern Bayesian statistics relies extensively on deterministic approx-  imations of intractable posterior distributions to ease inference in challenging contexts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',  Tierney and Kadane, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Minka, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Rue, Martino and Chopin, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Blei, Kucukelbir  and McAuliffe, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' A natural option to enforce the desired tractability is to constrain the  approximating distribution within a suitable family which facilitates the evaluation of func-  tionals of interest for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this end, both classical solutions, such as the approxima-  tion of posterior distributions induced by the Laplace method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Bishop, 2006, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4),  and state–of–the–art strategies, including, for example, the standard versions of expectation–  Co–funded by the European Union (ERC, BigBayesUQ, project number: 101041064).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Views and opinions expressed are however  those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Neither the  European Union nor the granting authority can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  MSC2020 subject classifications: Primary 62F15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' secondary 62F03, 62E17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Keywords and phrases: Bernstein–von Mises theorem, Deterministic approximation, Generalized skew–normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='03038v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='ST]  8 Jan 2023  2  propagation (Minka, 2001), employ Gaussian approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' These further appear, either  as the final solution or as a recurring building–block, also in several routinely–implemented  alternatives, such as mean–field variational Bayes (Blei, Kucukelbir and McAuliffe, 2017)  and integrated nested Laplace approximation (INLA) (Rue, Martino and Chopin, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' See  also Opper and Archambeau (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Wang and Blei (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Chopin and Ridgway (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Durante and Rigon (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Ray and Szabó (2022) and Vehtari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' (2020), among others,  for further references and examples illustrating the relevance of Gaussian approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  From a theoretical perspective, the above Gaussian approximations are typically justified,  in asymptotic regimes, by deriving Bernstein–von Mises type results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In its original for-  mulation (Doob, 1949;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Le Cam, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Van der Vaart, 2000), the celebrated Bernstein–von  Mises theorem states that, in regular parametric models, the posterior distribution converges  in total variation distance, with probability tending to one under the law of the data, to a  Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The mean of such a limiting Gaussian is a known function of the true  data–generative parameter, or any efficient estimator of this quantity, such as the maximum  likelihood estimator, while the variance is the inverse of the Fisher information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Besides  supporting deterministic Gaussian approximations of intractable posterior distributions, this  central theorem provides also guarantees for the frequentist coverage of suitable Bayesian  credible regions (Van der Vaart, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The relevance of these results has motivated several  extensions of the Bernstein–von Mises theorem to more complex settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Relevant contri-  butions along these directions include, among others, generalizations to high-dimensional  regimes (Boucheron and Gassiat, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Bontemps, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Spokoiny and Panov, 2021), along  with in–depth treatments of misspecified (Kleijn and van der Vaart, 2012) and irregular  (Bochkina and Green, 2014) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Semiparametric (Bickel and Kleijn, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Castillo and  Rousseau, 2015) and nonparametric (Castillo and Nickl, 2014) settings have also been ad-  dressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In the nonparametric context, Bernstein–von Mises type results do not hold in gen-  eral, but asymptotic Gaussianity can be still proved for weak Sobolev spaces via multiscale  analysis (Castillo and Nickl, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' See also Dehaene and Barthelmé (2018), Wang and Blei  (2019) and Kasprzak, Giordano and Broderick (2022) for recent Bernstein–von Mises type  results with a specific focus on scalable deterministic approximations from expectation–  propagation (EP), variational Bayes (VB) and Laplace method, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Albeit considering different regimes, the above results share a common focus on asymp-  totic Gaussianity of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Although this perspective has led to funda-  mental advancements in Bayesian asymptotics and deterministic approximations, as illus-  trated in Figure 1, even basic parametric models may require, in practice, moderate–to–large  sample size before yielding posterior distributions that closely resemble Gaussian shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  In fact, in small–to–moderate sample size settings, general posterior distributions induced  by routinely–implemented statistical models, such as Bayesian generalized linear models,  display a non–Gaussian behavior, mainly due to skewness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Kuss, Rasmussen and Her-  brich, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Challis and Barber, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Durante, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' For example, Durante (2019), Fasano  and Durante (2022) and Anceschi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' (2023) have recently proved that, under a broad class  of priors which includes multivariate normals, the posterior distribution induced by probit,  multinomial probit and tobit models belongs to a skewed generalization of the Gaussian dis-  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  3  n = 10  BvM  Skewed BvM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2  8  4  0  4  8  n = 20  BvM  Skewed BvM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2  8  4  0  4  8  n = 30  BvM  Skewed BvM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2  8  4  0  4  8  Fig 1: For varying sample size n ∈ {10,20,30}, graphical comparison between the approximations of the  exact posterior density (grey area) provided by the classical Gaussian Bernstein–von Mises theorem (BvM) and  the newly–developed skewed version relying on generalized skew–normal approximating densities (Skewed  BvM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Results refer to an exponential EXP(θ) model with EXP(1) prior on the rate parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  tribution known as unified skew–normal (SUN) (Arellano-Valle and Azzalini, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' More  generally, available extensions of Gaussian deterministic approximations which account, ei-  ther explicitly or implicitly, for skewness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Rue, Martino and Chopin, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Challis and  Barber, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Fasano, Durante and Zanella, 2022) have shown evidence of improved empir-  ical accuracy relative to the Gaussian counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Nonetheless, these approximations are  often model–specific and a general justification similar to the Bernstein–von Mises theorem  is not available yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In fact, current theory on skewed approximations is either lacking or is  tailored to the specific models and priors studied (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Fasano, Durante and Zanella, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  In this article, we cover the aforementioned gaps by deriving a novel limiting law for pos-  terior distributions, in regular parametric models, which belongs to the class of generalized  skew–normal distributions (Ma and Genton, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Genton and Loperfido, 2005) and yields  noticeable improvements over the convergence rate of the classical Bernstein–von Mises  theorem based on limiting Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' More specifically, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3 we prove,  under standard assumptions, that a suitably–derived sequence of generalized skew–normal  distributions yields a total variation distance from the exact posterior of order 1/n, up to  a logarithmic factor, with probability tending to one under the law of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This results  in a noticeable improvement over the 1/√n order, still up to a logarithmic factor, obtained  under the classical Gaussian approximating sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As clarified in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, we derive  such a class of skewed approximations from a novel treatment of a third–order extension of  the Laplace method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This provides a valid and tractable generalized skew–normal density  that perturbs the one of a multivariate Gaussian with the cumulative distribution function  of a univariate normal evaluated at a cubic function of the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As a consequence,  our results also yield theoretical and practical advancements relative to higher–order studies  relying on Edgeworth or other types of expansions (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Johnson, 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Weng, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Kolassa and Kuffner, 2020, and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In fact, these contributions pro-  vide supporting theory for arbitrarily truncated versions of infinite expansions which do not  necessarily correspond to closed–form valid densities, even after normalization — e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', the  density approximation is not guaranteed to be non–negative (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Kolassa and Kuffner,  4  2020, Remark 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This limits the methodological implications and the practical relevance  of such theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In contrast, the skewed Bernstein–von Mises results derived in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3, establish convergence to a valid and interpretable class of densities which are  as tractable as the Gaussian counterpart, and arise from a perturbation of such a symmetric  density mainly regulated by higher–order terms which progressively disappear as the sample  size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This implies that in the limit the novel skewed approximation approaches the  Gaussian derived under the classical Bernstein–von Mises theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' However, the total vari-  ation distance from the exact posterior distribution is substantially reduced, thus clarifying  the role of skewness in achieving remarkable improvements in approximation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Besides refining classical Bernstein–von Mises type results and yielding improvements  over available higher–order theoretical studies, the above advancements naturally motivate  the development of a novel practical class of skew–modal approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This can be ob-  tained by replacing the true unknown data–generative model parameters in the statement of  the skewed Bernstein–von Mises theorem with the corresponding maximum–a–posteriori  estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This novel plug–in version is derived in Section 3, where we also verify that, un-  der minimal additional conditions, the total variation distance between such a skew–modal  approximation and the exact posterior distribution achieves the same 1/n order, up to a log-  arithmic factor, as its population counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This outperforms the classical Gaussian modal  approximation induced by the standard Laplace method for which the rate can be proved to  be of order 1/√n, again up to a logarithmic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The practical performance of our new re-  sults is illustrated through extensive simulations and a real–data application in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4  and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The numerical analysis demonstrates that our theoretical findings closely match the  empirical behavior observed in practice even in finite, possibly small, sample regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Our  skew–modal approximation further exhibits improved accuracy not only relative to classi-  cal Gaussian modal approximations, but also with respect to state–of–the–art Gaussian and  non–Gaussian approximations from mean–field variational Bayes and expectation propa-  gation (Minka, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Blei, Kucukelbir and McAuliffe, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Chopin and Ridgway, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Durante and Rigon, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Fasano, Durante and Zanella, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Concluding remarks in Section 4 clarify that the scope and impact of our theoretical re-  sults is not limited to providing support on the improved accuracy of skewed approximations  of posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In fact, this article can contribute more generally in refining the  extensive theory based on Bernstein–von Mises type results, and in further improving the  quality of more sophisticated methods, such as INLA (Rue, Martino and Chopin, 2009), by  replacing the building–block Gaussian approximations with the proposed skew–modal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The proof of the skewed Bernstein–von Mises theorem is detailed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Additional  theoretical results and the corresponding technical lemmas are proved in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Such lemmas guarantee a careful control over the order of the terms involved in the expan-  sions underlying the proof of the new skewed Bernstein–von Mises theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In fact, unlike  for theory on classical Bernstein–von Mises type results and higher–order expansions, the  proofs of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 not only seek to establish a polynomially higher approxi-  mation accuracy relative to the classical Gaussian counterparts, but also require that such an  improvement is achieved by a novel limiting law yielding valid and closed–form densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Let {Xi}i≥1 denote a sequence of independent random variables, mod-  eled by a parametric family {P ∞  θ ,θ ∈ Θ}, with Θ ⊆ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In the following, we assume that  the joint probability measure P n  θ of the random vector Xn = (X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',Xn) is dominated by  a common σ–finite measure µn and we denote by pn  θ the density of P n  θ with respect to µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The true unknown data–generating mechanism is indicated by P ∞  θ0 , with θ0 ∈ Θ, whereas  the log–likelihood of the model is ℓ(θ) = ℓ(θ,Xn) = log pn  θ (Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  As mentioned in Section 1, our novel results rely on higher–order expansions and deriva-  tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this end, we characterize operations among vectors, matrices and arrays in a com-  pact manner by adopting the index notation along with the Einstein’s summation convention  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Pace and Salvan, 1997, page 335).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' More specifically, the inner product Z⊺a between  the generic random vector Z ∈ Rd, with components Zs for s = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',d, and the vector of  coefficients a ∈ Rd having elements as for s = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',d, is expressed as asZs, with the sum  being implicit in the repetition of the indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Similarly, if B is a d × d matrix with entries  bst for s,t = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',d, the quadratic form Z⊺BZ is expressed as bstZsZt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The generalization  to operations involving arrays with higher dimensions is obtained under the same reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Leveraging the above notation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' the score vector evaluated at θ0 is defined as  ℓ(1)  θ0 = [ℓ(1)  s (θ)]|θ=θ0 = [(∂/∂θs)ℓ(θ)]|θ=θ0 ∈ Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  whereas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' the second,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' third and fourth order derivatives of ℓ(θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' still evaluated at θ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' are  ℓ(2)  θ0 = [ℓ(2)  st (θ)]|θ=θ0 = [∂/(∂θs∂θt)ℓ(θ)]|θ=θ0 ∈ Rd×d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  ℓ(3)  θ0 = [ℓ(3)  stl(θ)]|θ=θ0 = [∂/(∂θs∂θt∂θl)ℓ(θ)]|θ=θ0 ∈ Rd×d×d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  ℓ(4)  θ0 = [ℓ(4)  stlk(θ)]|θ=θ0 = [∂/(∂θs∂θt∂θl∂θk)ℓ(θ)]|θ=θ0 ∈ Rd×d×d×d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  where all the indexes in the above definitions and in the subsequent ones go from 1 to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The observed and expected Fisher information are denoted by Jθ0 = [jst] = −[ℓ(2)  θ0,st] ∈ Rd×d  and Iθ0 = [ist] = [En  θ0jst] ∈ Rd×d, where En  θ0 is the expectation with respect to P n  θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Prior and  posterior distributions are denoted by Π(·) and Π(· | Xn), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In addition,  log π(1)  θ0 =[log π(θ)(1)  s ]|θ=θ0 = [∂/(∂θs)log π(θ)]|θ=θ0 ∈ Rd,  log π(2)  θ0 =[log π(θ)(2)  st ]|θ=θ0 = [∂/(∂θs∂θt)log π(θ)]|θ=θ0 ∈ Rd×d,  represent the first two derivatives of the log–prior density, evaluated at θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The Euclidean norm of a vector a ∈ Rd is denoted by ∥a∥, whereas, for a generic d × d  matrix B, the notation |B| indicates its determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Furthermore, u∧v and u∨v correspond  to min{u,v} and max{u,v}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' For two positive sequences un,vn we employ  un ≲ vn if there exists a universal positive constant C such that un ≤ Cvn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' When un ≲ vn  and vn ≲ un are satisfied simultaneously, we write un ≍ vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Finally, KL(pn  θ0∥pn  θ) denotes the  Kullback–Leibler divergence between the two densities pn  θ0 and pn  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The skewed Bernstein–von Mises theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This section presents the main theoret-  ical contribution of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' More specifically, we first show that a suitable treatment of  a third–order version of the Laplace method can yield a novel limiting law of known densi-  ties corresponding to distributions within the class of generalized skew–normals (GSN) (Ma  and Genton, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Genton and Loperfido, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Focusing on this alternative sequence of  approximating densities, we then prove that the convergence rate of its total variation dis-  tance from the exact posterior distribution improves by a √n factor the one provided by the  classical Bernstein–von Mises theorem under limiting Gaussians (Van der Vaart, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Consistent with standard Bernstein–von Mises type theory (Van der Vaart, 2000), let us  consider the re–parametrization h = √n(θ − θ0) ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Then, as illustrated below in Sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, the newly–derived class of approximating GSN densities pn  GSN has the general form  (1)  pn  GSN (h) = 2φd (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='ξ,Ω)Φ(α(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  In equation (1), φd (·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='ξ,Ω) is the density of a d–variate Gaussian variable with mean vector  ξ and covariance matrix Ω defined as  ξ = [ξs] = [n(J−1  θ0 )stwt] ∈ Rd,  with wt = (ℓ(1)  θ0 + log π(1)  θ0 )t/√n, t = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',d,  Ω = V −1 ∈ Rd×d,  with V = [vst] = [jst/n − (ξlℓ(3)  θ0,stl)/n3/2] ∈ Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The term Φ(α(h)) ∈ (0,1) is instead responsible for inducing skewness, and corresponds  to the cumulative distribution function of a standard normal variable evaluated at  α(h) = {  √  2π/(12√n)}(ℓ(3)  θ0,stl/n){(h − ξ)s(h − ξ)t(h − ξ)l + 3(h − ξ)sξtξl} ∈ R,  where α(h) is an odd function of (h − ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Interestingly, the first factor in the right hand side of (1) closely resembles the limiting  Gaussian density with mean vector ℓ(1)  θ0 /√n and covariance matrix (Iθ0/n)−1 from the clas-  sical Bernstein–von Mises theorem which, however, fails to incorporate skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this  end, the symmetric density factor in (1) is further perturbed via a skewness–inducing mech-  anism regulated by Φ(α(h)) to obtain a valid and known asymmetric density with tractable  normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In fact, since α(h) is an odd function of (h − ξ), and φd (·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='ξ,Ω) is  symmetric about ξ, it follows that pn  GSN (h) in (1) is the density of a GSN variable (Genton  and Loperfido, 2005, Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This further belongs to the even more general family  of skew–symmetric distributions (Azzalini and Capitanio, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Wang, Boyer and Genton,  2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' see also Ma and Genton (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence, unlike for higher–order studies of posterior  distributions based on Edgeworth or other expansions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Johnson, 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Weng, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Ko-  lassa and Kuffner, 2020), our theoretical results focus on a valid and interpretable limiting  distribution that is as tractable as multivariate Gaussians, both in terms of evaluation of the  corresponding density, and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' More specifically, let z ∈ Rd and u ∈ [0,1] de-  note samples from a d–variate Gaussian with density φd(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0,Ω) and a uniform with support  [0,1], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Then, adapting, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Wang, Boyer and Genton (2004), a sample from the  GSN distribution with density in (1) can be readily obtained via ξ + sgn(Φ(α(z)) − u)z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  7  The aforementioned discussion clarifies the practical and methodological impact of the  skewed Bernstein–von Mises theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' More specifically, our result points toward the possi-  bility of employing a more refined family of valid approximating distributions arising from  a simple perturbation of multivariate Gaussian densities that still allows tractable inference,  while yielding provable, and noticeable, improvements in approximation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Derivation of the generalized skew–normal approximating distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Prior to state  and prove the skewed Bernstein–von Mises theorem in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3, let us focus on provid-  ing a constructive derivation of the generalized skew–normal approximating density in (1)  via a third–order version of the Laplace method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To simplify the notation, we consider the  univariate case d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The extension to d > 1 yielding the general expression for pn  GSN (h)  in (1) follows as a direct adaptation of the univariate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  As a first step towards deriving the GSN density pn  GSN (h) in (1), notice that the posterior  for h = √n(θ − θ0) can be expressed as  (2)  π(h | Xn) ∝  pn  θ0+h/√n  pn  θ0  (Xn)π(θ0 + h/√n)  π(θ0)  ,  since pn  θ0(Xn) and π(θ0) do not depend on h, and θ = θ0 + h/√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Under suitable regularity conditions discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2 below, the third–order Tay-  lor’s expansion for the logarithm of the likelihood ratio in Equation (2) is  (3)  log  pn  θ0+h/√n  pn  θ0  (Xn) = ℓ(1)  θ0  √nh − 1  2  jθ0  n h2 +  1  6√n  ℓ(3)  θ0  n h3 + OP n  θ0  �n−1�,  whereas the first order Taylor’s expansion of the log–prior ratio is  (4)  log π(θ0 + h/√n)  π(θ0)  = log π(1)  θ0  √n  h + O  �n−1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Combining (3) and (4) it is possible to reformulate the right end side of Equation (2) as  (5)  pn  θ0+h/√n  pn  θ0  (Xn)π(θ0 + h/√n)  π(θ0)  = exp  �  wh − 1  2  jθ0  n h2 +  1  6√n  ℓ(3)  θ0  n h3�  + OP n  θ0(n−1),  where w = (ℓ(1)  θ0 + log π(1)  θ0 )/√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Notice that, up to a multiplicative constant, the Gaussian density arising from the classical  Bernstein–von Mises theorem can be obtained by neglecting all terms in (3)–(4) which con-  verge to zero in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' These correspond to the contribution of the prior, the difference  between the observed and expected Fisher information, and the term associated to the third–  order log–likelihood derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Maintaining these quantities would surely yield improved  accuracy, but it is unclear whether a valid and similarly–tractable density can be still iden-  tified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In fact, current solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Johnson, 1970) consider approximations based on the  8  sum among a Gaussian density and additional terms in the Taylor’s expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' However, as  for related alternatives arising from Edgeworth–type expansions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Weng, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Kolassa  and Kuffner, 2020), there is no guarantee that such constructions provide valid densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  As a first key contribution we prove below that a valid and tractable approximating density  can be, in fact, derived from the above expansions and belongs to the GSN class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this end,  let ω = 1/v with v = jθ0/n − ξℓ(3)  θ0 /n3/2 and ξ = n(jθ0)−1w, and note that, by replacing h3  in the right hand side of Equation (5) with (h + ξ − ξ)3, the exponential term in (5) can be  rewritten as proportional to  (6)  φ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='ξ,ω)exp({1/(6√n)}(ℓ(3)  θ0 /n)  �(h − ξ)3 + 3(h − ξ)ξ2�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Next, recall that, for x = O(1), we can write exp(x) = 1 + x + O(x2) and 2Φ(x) = 1 +  (2/π)1/2x + O(x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Therefore, leveraging the similarity among these two expansions and  the fact that the exponent in Equation (6) is an odd function of (h − ξ) about 0, and its order  is OP n  θ0(n−1/2), it follows that (6) is equal to  2φ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='ξ,ω)Φ(α(h)) + OP n  θ0(n−1),  with α(h) defined below Equation (1), thereby yielding the density of the generalized skew–  normal in (1), for the univariate case, up to an OP n  θ0(n−1) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The direct extension of the  above derivations to the multivariate case provides the general form of pn  GSN (h) in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  In Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3 we further formalize the above results by proving that the total varia-  tion distance between the exact posterior and the approximating distribution with density as  in (1) is OP n  θ0((log n)d/2+3/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This improves the classical Bernstein–von Mises type results  by a √n factor, up to a logarithmic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The skewed Bernstein–von Mises theorem in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3 requires as-  sumptions A0, A1, A2, A3 and A4 stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' For convenience, let us also introduce the  notation Mn = √c0 log n, where c0 is a positive constant to be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0 The log–likelihood of the model is four times differentiable with  ℓ(1)  θ0,s = OP n  θ0(n1/2),  ℓ(2)  θ0,st = OP n  θ0(n),  ℓ(3)  θ0,stl = OP n  θ0(n),  for s,t,l = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',d,  and sup∥h∥≤Mn |ℓ(4)  θ0,stlk(h)| = OP n  θ0(n), for s,t,l,k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',d, with ℓ(4)  θ0,stlk(h) := ℓ(4)  stlk(θ0+  h/√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover, let ℓ(1)  i,θ0 be the first derivative of the log–likelihood ℓ(θ,Xi) evaluated  at θ0, and assume that Eθ0∥ℓ(1)  i,θ0/√n∥3 = O(1) for each i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 The entries ist of the Fisher information matrix Iθ0 satisfy ist = O(n) and jst/n −  ist/n = OP n  θ0(n−1/2), for s,t = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover, i−1  st = O(n−1), where i−1  st = (I−1  θ0 )st  denote the entries of the inverse Fisher information matrix, for s,t = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  A1 The log–prior density log π(θ) is two times continuously differentiable in a neighbor-  hood of θ0, and π(θ0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  9  A2 For some compact neighborhood Θ0 ⊂ Θ of θ0 there exists a sequence of tests φn such  that En  θ0φn → 0 and supθ/∈Θ0 En  θ(1 − φn) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  A3 For every constant C1 > 0 there exist C2 > 0 and C3 > 0 such that Π(θ : KL(pn  θ0∥pn  θ) <  n−C1) ≥ C3n−C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  A4 There exists a constant η > 0 such that |Iθ0/n| > η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Assumptions A0–A4 provide a slightly more restrictive, yet still mild, counterpart of the  standard conditions commonly employed in state–of–the–art versions of the Bernstein–von  Mises result (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Van der Vaart, 2000, Chapter 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' More specifically, A0 and A4 are readily  verifiable for a wide range of routinely–implemented models, including for instance expo-  nential families, and are required to carefully control the error between the log–likelihood  ratio and its third order Taylor’s expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In principle, these conditions can be relaxed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',  by considering the locally asymptotically normal assumption as in Chapter 10 of Van der  Vaart (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' However, these results only derive asymptotically vanishing total variance  distance between the posterior and its Gaussian approximation, whereas we aim to derive a  sharp rate, up to a logarithmic factor, and hence slightly stronger assumptions are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  As for A0 and A4, also condition A1 on the prior is mild and standard, and is similarly  required to quantify the approximation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Condition A3 is the usual prior small ball  assumption used routinely in Bayesian nonparametrics to derive contraction rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Together  with the standard testing assumption A2, required in the classical Bernstein–von Mises the-  orem as well, it guarantees that the posterior mass decreases fast enough in the area where  the skewed approximation does not hold, so that its impact on the overall rate is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 states the main theoretical result of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' It establishes  improved rates of convergence for the total variation distance between the exact posterior  and the generalized skew–normal approximating distribution derived in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, under  assumptions A0–A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Our result clarifies that a substantially more accurate representation  for the asymptotic behavior of the exact posterior distribution, relative to the one obtained  under limiting multivariate Gaussians, can be achieved via a similarly–tractable class of gen-  eralized skew–normals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As illustrated in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4 and 3, such a theory has fundamental  implications in the development of novel rigorous approximations with noticeable empirical  improvements in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  THEOREM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Under assumptions A0–A4, if the map θ → P n  θ is one–to–one and θ0 is  an inner point of Θ, then  (7)  ∥Π(· | Xn) − P n  GSN(·)∥TV = OP n  θ0  �(log n)d/2+3/n  �,  where ∥ · ∥TV is the total variation distance, and P n  GSN(S) =  �  S pn  GSN(h)dh for S ⊂ Rd with  pn  GSN(h) defined as in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  10  REMARK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Under related conditions and a simpler proof, it is possible to show that  the order of convergence for the usual Bernstein–von Mises theorem based on Gaussian  limiting distributions is OP n  θ0((log n)γ/√n), for some known constant γ ≥ 0, possibly de-  pending on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 guarantees that by relying on GSN approximating distri-  butions with density defined as in (1) and derived in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, it is possible to improve the  classical Bernstein–von Mises result by a √n factor, up a logarithmic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 is reported below and follows the general reasoning behind  the classical Bernstein–von Mises type derivations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Szabo and Van der Vaart, 2021),  extended to GSN distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Nonetheless, as mentioned before, the need to derive a sharp  rate which establishes a polynomially higher approximation accuracy, relative to Gaussian  limiting distributions, requires a number of additional technical lemmas and refined argu-  ments ensuring a tight control of the error terms in the expansions underlying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, we first show that the posterior mass outside the ball  Θn = {θ ∈ Θ : ∥θ − θ0∥ < Mn/√n},  (8)  is asymptotically negligible, and then prove that, when restricted to Θn, the posterior distri-  bution can be accurately approximated by the generalized skew–normal in (1) with an error  of order OPθ0(M d+6  n  /n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Setting Mn = √c0 log n for some constant c0 > 0 to be specified  later, yields the statement of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, this improvement fol-  lows from the fact that A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0 and A1 allow to combine a third order Taylor’s expansion of  the log–likelihood ratio with a first order Taylor’s expansion of the log–prior ratio to obtain  an approximation for the exact posterior density whose kernel closely matches the one of a  generalized skew–normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To simplify notation, we drop below the n from P n  θ , pn  θ and the  corresponding expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  To start the proof, let us first denote the posterior restricted to Θn as  (9)  ΠΘn(θ ∈ B | Xn) =  �  θ∈B∩Θn pθ (Xn)dΠ(θ)  �  θ∈Θn pθ (Xn)dΠ(θ) ,  B ⊆ Θ,  and recall that, by a standard inequality for the TV norm, it follows that  (10)  supB|Π(θ ∈ B | Xn) − ΠΘn(θ ∈ B | Xn)| < 2Π(Θc  n | Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  To prove that Π(Θc  n | Xn) is asymptotically negligible, we show that this quantity converges  to zero in mean and, therefore, in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this end, let An,1 be the event defined as  An,1 = {  � pθ/pθ0(Xn)dΠ(θ) > Π(θ : KL(pθ0∥pθ) < n−C1)}, for some C1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Then,  Eθ0Π(Θc  n | Xn) ≤ Eθ0Π(Θc  n | Xn)1An,1 + Pθ0(Ac  n,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  since Π(Θc  n | Xn) can be expressed as Π(Θc  n | Xn)1An,1 + Π(Θc  n | Xn)1Ac  n,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  At this point, note that the probability Pθ0(Ac  n,1) in the above bound can be neglected since  it converges to zero at any polynomial rate by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As for Eθ0Π(Θc  n | Xn)1An,1, let  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  11  us re–write such a quantity as Eθ0Π(Θc  n | Xn)1An,1(1−φn)+Eθ0Π(Θc  n | Xn)1An,1φn, where  φn is a sequence of tests, and notice that Π(Θc  n | Xn)1An,1φn ≤ φn, since all the involved  quantities are less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In addition, recall that the posterior density π(θ | Xn) can be  expressed as (pθ/pθ0)(Xn)π(θ)/  � (pθ/pθ0)(Xn)dΠ(θ), where the denominator is greater  than Π(θ : KL(pθ0∥pθ) < n−C1), when restricting to An,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Therefore, Π(Θc  n | Xn)1An,1(1 −  φn) can be bounded above by  �  Θcn(pθ/pθ0)(Xn)(1 − φn)dΠ(θ)/Π(θ : KL(pθ0∥pθ) < n−C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Combining these results with Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2, it follows that there exists a test φn, such that  Eθ0Π(Θc  n | Xn)1An,1 ≤  Eθ0  �  Θcn(pθ/pθ0)(Xn)(1 − φn)dΠ(θ)  Π(θ : KL(pθ0∥pθ) < n−C1)  + exp(−ctM 2  n),  for some positive constant ct > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As a direct consequence of assumptions A3 and Fubini’s  theorem, the first term on the right hand side of the above inequality can be bounded above  by a multiple of nC2 �  Θcn Eθ(1 − φn)dΠ(θ), after noticing that Eθ0(pθ/pθ0)(Xn)(1 − φn) =  � (1 − φn)pθ(Xn)dXn = Eθ(1 − φn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Leveraging now assumption A2 along with the fact  that, as proved within Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2, Eθ (1 − φn) ≤ exp(−ctn(∥θ −θ0∥2 ∧1)), it additionally  follows that nC2 �  Θcn Eθ(1 − φn)dΠ(θ) can be further bounded above by  nC2e−ctM 2  n �  {Mn/√n<∥θ−θ0∥<1} dΠ(θ) + nC2e−nct �  {∥θ−θ0∥≥1} dΠ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Therefore, by setting Mn as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2, with c0 > (C2 + 1)/ct, we have that Eθ0Π(Θc  n |  Xn) = O(n−1), and hence, leveraging the inequality for the TV norm in Equation (10), it  follows that supB|Π(θ ∈ B | Xn) − ΠΘn(θ ∈ B | Xn)| = OPθ0(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Since the total varia-  tion distance is invariant to scale and location transformations, this implies the same bound  for the posterior on h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Namely  (11)  supS|Π(h ∈ S | Xn) − ΠSMn(h ∈ S | Xn)| = OPθ0(n−1)  where SMn = {h : ∥h∥ < Mn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Let us now move to the second part of the proof which shows that, when restricted to Θn,  the posterior distribution for the re–parameterized quantity h = √n(θ − θ0) is accurately  approximated by the generalized skew–normal with density as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this end, define  Yn(h) = 2e  hsws − hsht  2  jst  n +  1  2√n  ℓ(3)  θ0,stl  n  (h − ξ)s(h − ξ)tξl +  1  6√n  ℓ(3)  θ0,stl  n  ξsξtξlΦ(α(h)),  Ln(h) = pθ0+h/√n  pθ0  (Xn)π(θ0 + h/√n)  π(θ0)  ,  (12)  where Φ(α(h)) denotes the skewness–inducing factor defined below Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In (12),  the quantity Yn(h) coincides, up a normalizing constant, with the generalized skew–normal  approximating density pGSN(h) in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In fact, notice that the exponential term is quadratic in  h and, hence, can be suitably re–written as proportional to the Gaussian density factor in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  12  The quantity Ln(h) is instead the exact posterior–density ratio that we seek to approximate  via Yn(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' see also Equation (2) and the constructive derivation of the GSN density pGSN(h)  in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  In view of A0 and A1 let us now consider the third order Taylor’s expansion of log Ln(h)  around h = 0, satisfying log Ln(h) = (ℓ(1)  θ0,shs)/√n − (isthsht)/2n + rn,h + qn,h,, with rn,h  and qn,h defined as  rn,h =  hs log π(1)  θ0,s  √n  − hsht  2  �jst  n − ist  n  �  +  ℓ(3)  θ0,stl  n  hshthl  6√n = OPθ0({∥h∥3 ∨ 1}/√n),  qn,h =  ℓ(4)  θ0,stlk(˜h)  n  hshthlhk  24n  + log π(θ0 + ¯h/√n)(2)  st  n  hsht,  (13)  where ˜h,¯h ∈ Rd satisfy ∥˜h∥ ∨ ∥¯h∥ ≤ ∥h∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Then, by triangle inequality and in view of the  second order Taylor’s expansion of eqn,h = 1 + qn,h + eβqn,hq2  n,h/2, for a β ∈ (0,1), we have  (14)  |Ln(h) − Yn(h)| ≤ |exp(un,h) − Yn(h)| + |exp(un,h)(qn,h + eβqn,hq2  n,h/2)|,  where un,h = (ℓ(1)  θ0,shs)/√n − (hshtist)/2n + rn,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Since the statement of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 is in probability, let us replace some of the above  random variables with truncated versions which converge in probability to the original ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  This ensures the existence of the moments for the truncated variables and facilitates a tight  control of the corresponding limiting behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence, define  ˜ℓ(1)  θ0 /√n =  �ℓ(1)  θ0 /√n  �1∥ℓ(1)  θ0 /√n∥<Cn,  with Cn = M 1/α  n  for α > 2,  ˜rn,h = rn,h1|rn,h|<Kn,h,  with Kn,h = log(n){∥h∥3 ∨ 1}/√n,  and consider the event An,2 = �  h:∥h∥≤Mn(|rn,h| < Kn,h) ∩ (∥ℓ(1)  θ0 /√n∥ < Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Notice that,  by assumption A0 and (13) it follows that Pθ0(Ac  n,2) = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Furthermore, let us define the  quantity λn,h = (1/6√n)(ℓ(3)  θ0,stl/n){ξsξtξl + 3(h − ξ)s(h − ξ)tξl − hshthl}, and set  ˜ℓ(3)  θ0,stl = ℓ(3)  θ0,stl1An,3,  and  ˜λn,h = λn,h1An,3,  where An,3 is the event defined as An,3 = �  h:∥h∥≤Mn{|λn,h| ≤ τn} with τn = FnM 3  n/√n,  for any Fn that goes to infinity arbitrary slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Notice that by A0–A1 also Pθ0(Ac  n,3) = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Finally, set also ˜ℓ(4)  θ0,stlk(h) = ℓ(4)  θ0,stlk(h)1An,4 where An,4 = �  h:∥h∥≤Mn{|ℓ(4)  θ0,stlk(h)| < Fnn},  and note that A0 further implies Pθ0(Ac  n,4) = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Combining all the above truncated quantities, let us also denote with ˜Ln(h) and ˜Yn(h) the  alternative versions of Ln(h) and Yn(h), given in (12), where all the stochastic quantities  have been replaced by their truncated counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Since the statement of our theorem is in  probability, it is sufficient to restrict ourselves to the event An,1:4 = An,1∩An,2∩An,3∩An,4,  provided that Pθ0(Ac  n,1:4) = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  13  In view of Hölder’s inequality with ρn = (1 + M 2  n)/M2  n and ˜ρn = 1 + M 2  n, the truncated  version of the second term in the right hand side of (14) is bounded from above by  suph∈SMnE1/ρn  θ0  exp(ρn˜un,h) · E1/˜ρn  θ0  (˜qn,h + eβ˜qn,h˜q2  n,h/2)˜ρn,  (15)  where ˜un,h and ˜qn,h denote the truncated versions of un,h and qn,h, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence, since  ˜qn,h is a function of ˜ℓ(4)  θ0 (h) whose entries are truncated above by Fnn, it follows that  sup  h∈SMn  ˜qn,h + eβ˜qn,h˜q2  n,h  2  ≲  �Fn(∥h∥4 ∨ 1)  24n  + e  Fn(∥h∥4 ∨ 1)  24n  �Fn(∥h∥4 ∨ 1)  24n  �2�  = O  �M 4  n  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Therefore, together with Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3, which guarantees control on the limiting behavior of  the first expectation in Equation (15), it follows that the expression in (15) is O  �M 4  n/n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  This implies, together with the truncated version of (14) and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='5 on the limiting  behavior of the first term in the right hand side of (14), that  (16)  suph∈SMnEθ0 |˜Ln(h) − ˜Yn(h)| = O  �M 6  n/n  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Combining (16) with Fubini’s theorem and Jensen’s inequality it follows that  Eθ0 supS |  �  S∩SMn ˜Ln(h)dh −  �  S∩SMn ˜Yn(h)dh|  ≤  �  SMn Eθ0|˜Ln(h) − ˜Yn(h)|dh ≲  �  SMn(M 6  n/n)dh = O  �M d+6  n  /n  �,  for any S ⊆ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Let us now focus on obtaining a similar result for a normalized version of  the above quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' replace ˜Ln(h) and ˜Yn(h) with the corresponding normal-  ized counterparts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' and note that by summing and subtracting  �  S∩SMn ˜Ln(h)dh/  �  SMn ˜Yn(h)dh  within the above absolute difference it follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' as a direct consequence of the triangle and  Jensen’s inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' that  Eθ0 supS |  �  S∩SMn ˜Ln(h)dh/  �  SMn ˜Ln(h)dh −  �  S∩SMn ˜Yn(h)dh/  �  SMn ˜Yn(h)dh|  ≤ 2Eθ0(  �  SMn |˜Yn(h) − ˜Ln(h)|dh/  �  SMn ˜Yn(h)dh) = O  �M d+6  n  /n  �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  since the denominator is bounded away from zero with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Leveraging the fact  that Pθ0(Ac  n,1:4) = O(1) and the connection between convergence in mean and convergence  in probability, this translates into a similar statement for the un–truncated quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence,  since, by (12),  �  S∩SMn Ln(h)dh/  �  SMn Ln(h)dh = ΠSMn(h ∈ S | Xn), it follows that  (17)  supS  ��ΠSMn(h ∈ S | Xn) − PGSN,SMn(S)  �� = OPθ0(M d+6  n  /n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  where PGSN,SMn(S) =  �  S∩SMn Yn(h)dh/  �  SMn Yn(h)dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To conclude the proof, it is now nec-  essary to study the behavior of the absolute difference between PGSN,SMn(S) and PGSN(S)  14  defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To accomplish this goal, first notice that by the definition of Yn(h)  in (12), the normalizing factor  �  SMn Yn(h)dh can be reformulated as  e  ξsξt  2  jst  n +  1  6√n  ℓ(3)  θ0,stl  n  ξsξtξl�  SMn  2e  −1  2(h − ξ)s(h − ξ)t  �jst  n − ξl  √n  ℓ(3)  θ0,stl  n  �  Φ(α(h))dh,  where the term inside the integral coincides with pGSN(h) in (1), up to the normalizing con-  stant of the Gaussian density in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Therefore,  �  SMn Yn(h)dh can be expressed as a constant  times (1 − PGSN(Sc  Mn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This also implies that PGSN,SMn(S) is equal to PGSN(S ∩ SMn)/(1−  PGSN(Sc  Mn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As a consequence, since PGSN(S) = PGSN(S ∩ SMn) + PGSN(S ∩ Sc  Mn), by ap-  plying the triangle inequality, we have that  ��PGSN,SMn(S) − PGSN(S)  �� ≤ {PGSN(Sc  Mn)/(1 − PGSN(Sc  Mn))}PGSN  �S ∩ SMn  � + PGSN(Sc  Mn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Now, notice that, since the density in (1) has the same tail behavior of a Gaussian, it holds  PGSN(Sc  Mn) = OPθ0(n−D), D > 1 for sufficiently large choice of c0 in Mn = √c0 log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Thus  (18)  supS  ��PGSN,SMn(S) − PGSN(S)  �� = OPθ0(n−D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Recall also that assumption A4 together with Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='7 guarantee that V = Ω−1 is positive  definite with determinant bounded away from zero with Pθ0–probability tending to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  To conclude the proof notice that, by triangle inequality, the total variation distance ∥Π(· |  Xn) − P n  GSN(·)∥TV in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, can be bounded above by the sum of the left end side  quantities in (11), (17), and (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The leading term in such a summation is OPθ0(M d+6  n  /n)  with Mn = √c0 log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence, ∥Π(· | Xn) − P n  GSN(·)∥TV = OP n  θ0((log n)d/2+3/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Empirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3 illustrate the applicability of the asymptotic  results in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' More specifically, the focus is on providing comprehensive em-  pirical evidence of the improved accuracy achieved by the GSN limiting approximation in (1)  relative to its Gaussian counterpart arising from the classical Bernstein–von Mises theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  In fact, our approach demonstrates improved performance and noticeable accuracy already  for small sample sizes n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Through extensive examples, we further demon-  strate that conditions A0–A4 underlying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 are mild and readily fulfilled in broad  settings, including those considered by alternative higher–order versions of the Bernstein–  von Mises theorem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Johnson, 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Weng, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Kolassa and Kuffner, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Notice that, as for the classical Bernstein–von Mises theorem, also our results in Sections  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3 require knowledge of the true data–generating parameter θ0, which is clearly un-  known in practical implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In Sections 3–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, we address this aspect via a plug–in  version of the generalized skew–normal limiting approximation, which replaces θ0 with its  maximum–a–posteriori estimate, to obtain similar theoretical and empirical support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' see the  additional simulations and real–data applications in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Exponential model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Let Xi  iid  ∼ EXP(θ0), for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',n, where EXP(θ0) denotes  the exponential distribution with rate parameter θ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover, consider an EXP(1) prior  for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To obtain the generalized skew–normal approximation for the posterior distribution  induced by such a Bayesian model, let us first verify that all conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  To address this goal first notice that, under the above model, the first four derivates of the  log–likelihood at θ are n/θ − �n  i=1 xi, −n/θ2, 2n/θ3 and −6n/θ4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence,  conditions A0 and A4 are both satisfied, even around a small neighborhood of θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In addi-  tion, the KL divergence between pn  θ0 and pn  θ is KL(pn  θ0∥pn  θ) = n{θ/θ0 − 1 + log(θ0/θ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This  implies, in view of  {θ : KL  �pn  θ0∥pn  θ  � < n−C1} ⊃ {θ > θ0 : (θ/θ0 − 1) < n−C1−1},  that also assumption A3 is fulfilled for any prior density π bounded away from zero around  θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This condition, along with A1, is met by a broad class of routinely–implemented priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  For instance, EXP(1) can be considered in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Finally, since the model is parametric  and it satisfies the usual regularity conditions, the existence of the test required by assump-  tion A2 is guaranteed (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Van der Vaart, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Szabo and Van der Vaart, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The above derivations ensure that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence, let us derive the parameters  of the generalized skew–normal approximating density in (1) under this exponential exam-  ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this end, first notice that, since the prior is an EXP(1), then log π(1)  θ0 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Therefore,  ξ = θ2  0(n/θ0 − �n  i=1 xi − 1)/√n and Ω = 1/(θ−2  0  − 2θ−1  0 {1/θ0 − (�n  i=1 xi)/n − 1/n}),  while the cubic function appearing in the skewness–inducing mechanism regulated by Φ(·)  is defined as α(h) = {  √  2π/(6√nθ3  0)}{(h − ξ)3 + 3(h − ξ)ξ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Table 1 compares the accuracy of the generalized skew–normal (S–BVM) and the Gaus-  sian (BVM) approximations corresponding to the newly–derived and classical Bernstein–  von Mises theorems, respectively, under growing sample size and replicated experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  More specifically, we consider 50 different simulated datasets with sample size n∗ = 5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Then, within each of these 50 experiments, we derive the exact posterior under several sub-  sets of data x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',xn with a growing sample size n ∈ {1,10,50,100,500,1000,5000},  and compute the total variation (TV) distances between such a posterior and the two approx-  imations provided by the classical and skewed Bernstein–von Mises theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Since both  TABLE 1  In the exponential example, average, over 50 replicated studies, of the log–total–variation (TV) distances  corresponding to the classical (BVM) and skewed (S–BVM) Bernstein–von Mises theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Summaries for  the ratios between the two TV distances are also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Results are displayed for different sample sizes  from n = 1 to n = 5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The standard deviations of the ratios, computed over the 50 replicated studies, are  displayed within round brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The bold values indicate the best performance for each sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  n = 1  n = 10  n = 50  n = 100  n = 500  n = 1000  n = 5000  log n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='91  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='61  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='21  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='91  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='52  log TVBVM  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='52  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='74  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='23  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='49  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='37  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='54  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='41  log TVS–BVM  − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='39  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='78  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='69  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='26  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='94  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='35  −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='07  TVS–BVM/TVBVM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='03  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='19)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='18)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='13)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='08)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='03)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='02)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='01)  16  the exact posterior and the two approximating densities are available in closed–form, the to-  tal variation distances TVn  BVM = 2  �  R |π(h | Xn) − pn  GAUSS(h)|dh and TVn  S–BVM = 2  �  R |π(h |  Xn) − pn  GSN(h)|dh can be evaluated numerically, for each n, via standard routines in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Ta-  ble 1 displays, for each n, the summaries of the logarithm of such distances along with the  corresponding ratios, computed across the 50 replicated experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  In line with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2, the results in Table 1 clarify that the gener-  alized skew–normal approximation consistently yields substantial accuracy improvements  relative to the Gaussian counterpart for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover, on average log TVn  S–BVM ≈ −log n,  whereas log TVn  BVM ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='5log n, which are exactly the results stated for the asymptotic  regime in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This empirical finding clarifies that our theoretical  results for the limiting case are, in fact, visible also in finite, even small, sample size set-  tings, thus making our theory of direct practical impact, while motivating the adoption of  the generalized skew–normal approximation in place of the Gaussian one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In fact, as is clear  from the ratios in Table 1, S–BVM yields TV distances from the exact posterior which are  orders of magnitude lower than those obtained under BVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Bernoulli model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Let us now consider a Bernoulli model with a focus on the natu-  ral parameter θ0 ∈ R defined as θ0 = logit(g(θ0)), with g(θ0) = pr(Xi = 1|θ0), i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Hence, Xi  iid  ∼ Bern(g(θ0)), for every i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',n, where g(θ0) = eθ0/(1 + eθ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As for Sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, let us first verify that all conditions underlying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 hold, thus implying  the skewed version of the Bernstein–von Mises result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Conditions A0 and A4 are easily met, since the first four derivatives of the log–likelihood,  evaluated at θ, under the above Bernoulli model, are �n  i=1 xi − ng(θ), −ng(θ)/(1 + eθ),  ng(θ)(eθ − 1)/(1 + eθ)2, and −n(e3θ − 4e2θ + eθ)/(1 + eθ)4, respectively, and hence sat-  isfy the required bounds uniformly around θ0 in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To verify condition A3, notice  that the Kullback–Leibler divergence between pn  θ0 and pn  θ can be derived analytically also  in this example and is given by KL(pn  θ0∥pn  θ) = n(θ0 − θ)g(θ0) + nlog((1 + eθ)/(1 + eθ0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Hence, for C1 > 0, we have that  {θ : KL(pn  θ0∥pn  θ) < n−C1} ⊃ {θ < θ0 : (θ0 − θ)g(θ0) < n−C1−1},  where the volume of the set in the right hand side is (1 + eθ0)/eθ0n−C1+1, proving condition  A3 for any prior density π bounded away from zero about θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1,  also assumption A2 is readily verifiable under this parametric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Similarly, A1 holds  for a broad class of priors, including the Gaussian one often employed in Bayesian inference  under such a Bernoulli model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Therefore, for any prior density π which is positive and differ-  entiable about θ0, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 holds for the generalized skew–normal approximating den-  sity in (1) with location ξ = e−θ0(1 + eθ0)2(�n  i=1 xi − ng(θ0) + log π(1)  θ0 )/√n, and variance  Ω = [g(θ0)/(1 + eθ0) − (eθ0 − 1)(�n  i=1 xi − ng(θ0) + log π(1)  θ0 )/n(1 + eθ0)]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The func-  tion α(h) in the skewness–inducing mechanism regulated by the perturbing factor Φ(·) is  instead equal to α(h) = (  √  2πeθ0(eθ0 − 1)/{12√n(1 + eθ0)3}){(h − ξ)3 + 3(h − ξ)ξ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Figure 2 provides a visual assessment of the accuracy improvements provided by such a  novel skewed approximation relative to the Gaussian one from the classical Bernstein–von  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  17  n = 10  BvM  Skewed BvM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2  10  6  2  2  6  10  n = 20  BvM  Skewed BvM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2  10  6  2  2  6  10  n = 30  BvM  Skewed BvM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1  10  6  2  2  6  10  Fig 2: For varying sample size n ∈ {10,20,30}, graphical comparison between the approximations of the  exact posterior density (grey area) provided by the classical Gaussian Bernstein–von Mises result (BvM),  and the newly–developed skewed version (Skewed BvM) relying on generalized skew–normal approximating  densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Results refer to the Bernoulli model with N(0,4) prior for θ = logit(pr(Xi = 1 | θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Mises theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The analysis is performed for three different sample sizes n ∈ {10,20,30}  under a Bernoulli data–generating model with θ0 = logit(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2) and prior N(0,4) for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Con-  sistent with the empirical evidence form Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, and with the statements in Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2, the skewed Bernstein–von Mises approximation is more accurate than  its Gaussian counterpart also in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Gamma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' We conclude our empirical analyses of the skewed Bernstein–von  Mises result with a bivariate example focusing on the parameters θ0 = (α0,β0) of a Gamma  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence, for data Xi  iid  ∼ Gamma(θ0), the first derivative of the log–likelihood at θ, is  (log β + �n  i=1(log xi)/n − ψ(0)(α),α/β − �n  i=1 xi/n), where ψ(0)(·) is the polygamma  function of order 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Similarly, letting ψ(1)(·) denote the polygamma function of order 1, the  2 × 2 symmetric matrix of the second order derivatives at θ has diagonal entries −ψ(1)(α)  and −α/β2, while the off–diagonal is given by 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Finally, the 2 × 2 × 2 array comprising  the third order derivatives at θ, is  ℓ(3)  θ,··1 = n  �  −ψ(2)(α)  0  0  −1/β2  �  ,  ℓ(3)  θ,··2 = n  �  0  −1/β2  −1/β2  2α/β3  �  ,  where ψ(2)(·) denotes the polygamma function of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover, notice that Jθ = Iθ and  |Iθ/n| = (αψ(1)(α) − 1)/β2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence, for any suitable choice of the prior distribution,  conditions A0, A1 and A4 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Notice also that the fourth order derivatives yield a  2 × 2 × 2 × 2 array which is not reported here for simplicity of exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Nonetheless, it  is straightforward to verify that the condition A0 on the fourth order derivative is verified in  this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Additionally, since we are dealing again with a regular parametric model also  condition A2 is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  We are left to prove the validity of assumption A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this end, notice that KL(pn  θ0∥pn  θ) =  n{log Γ(α0)(α0 − α) + log(Γ(α)/Γ(α0)) + αlog(β0/β) + (α0/β0)(β − β0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence the  18  set {θ = (α,β) : KL(pn  θ0∥pn  θ) < n−C1} contains  (19)  {α < α0, β > β0 : log Γ(α0)(α0 − α) + (α0/β0)(β − β0) < n−(C1+1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Note that, if log Γ(α0) ≤ 0 then log Γ(α0)(α0 −α) ≤ 0 which implies that the set defined in  (19) contains {α < α0, β0 < β < (β0/α0)n−(C1+1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This latter set has prior mass bounded  at least constant times n−(C1+1) for prior densities π bounded away from zero in a neigh-  bourhood of θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' For log Γ(α0) > 0, the set {θ = (α,β) : KL(pn  θ0∥pn  θ) < n−C1} contains  (20)  {α0 − n−(C1+1)/  �2log Γ(α0)  � < α < α0, β0 < β < β0 + n−(C1+1)β0/(2α0)},  which has prior mass proportional to n−(C1+1) for any prior density π bounded away from  zero about θ0, thereby verifying condition A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Therefore, since A0–A4 are met, the exact  posterior admits the generalized skew–normal approximation in Equation (1) with ξ, Ω and  α(h) that can be computed analytically using the derivatives of the log–likelihood function  given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' For clarity of presentation we refrain from presenting these formulas here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Figure 3 compares graphically the classical and skewed Bernstein–von Mises approxima-  tions for the posterior of h = √n(θ−θ0) in a simulation study with sample size n = 15, true  data–generative parameter θ0 = (α0,β0) = (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='5) and prior chosen as the product of two  BvM  Skewed BvM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2  6  2  2  6  n(α − α0)  1  0  1  2  2  0  2  n(α − α0)  n(β − β0)  1  0  1  2  2  0  2  n(α − α0)  n(β − β0)  BvM  Skewed BvM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4  3  1  1  3  5  n(β − β0)  Fig 3: Graphical comparison between the approximations of the marginal and bivariate exact posterior den-  sities (grey areas and lines) provided by the classical Gaussian Bernstein–von Mises result (BvM) and the  newly–developed skewed version (Skewed BvM) based on generalized skew–normal approximating densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Results refer to the Gamma(α,β) model with independent exponential priors on the parameters α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  19  independent exponential distributions with rate parameter equal to 2 in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As for  the examples in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2, the generalized skew–normal approximation is con-  siderably more accurate than the Gaussian counterpart for both the joint and the marginal  posterior densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Skew–modal approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4, Bernstein–von Mises  type results, including our Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, describe the asymptotic behavior of the exact pos-  terior with respect to a limiting distribution whose parameters depend on the true data–  generating θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Since this quantity is not available in practice, it is necessary to replace θ0  with a sensible estimate in order to ensure that the approximation derived is of practical rel-  evance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this end, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 we consider a simple, yet effective, plug–in version of the  generalized skew–normal density in (1) which replaces θ0 with its maximum–a–posteriori  (MAP) estimator, without losing theoretical accuracy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Notice that, in general, θ0  can be replaced by any efficient estimator, but by relying on the MAP several quantities in  (1) simplify, giving rise to a highly tractable and accurate approximation, as further shown  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Derivation and theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Consistent with the above discussion, we  consider the plug–in version ˆpn  GSN of pn  GSN in Equation (1), where the unknown θ0 is replaced  by the MAP ˆθ = argmaxθ∈Θ{ℓ(θ) + log π(θ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As a consequence, this yields the generalized  skew–normal density, for the rescaled parameter ˆh = √n(θ − ˆθ) ∈ Rd, defined as  (21)  ˆpn  GSN(ˆh) = 2φd(ˆh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0, ˆΩ)Φ(α(ˆh)),  where ˆΩ = ˆV −1 with ˆV = [ˆvst] = [jˆθ,st/n] ∈ Rd×d, while the skewness function entering the  cumulative distribution Φ(·) is given by ˆα(h) = {  √  2π/(12√n)}(ℓ(3)  ˆθ,stl/n)ˆhsˆhtˆhl ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Notice that, relative to the expression for pn  GSN(·) in Equation (1), the location parameter  ξ is zero in (21), since ˆξ is a function of the quantity (ℓ(1)  ˆθ + log π(1)  ˆθ )/√n which is zero by  definition when ˆθ is the MAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' For the same reason, in the expression for the precision matrix  of the Gaussian density factor in (21), the term −(ˆξlℓ(3)  ˆθ,stl)/n3/2 disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Equation (21) provides a practical skewed approximation of the exact posterior leveraging  its local behavior at the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' For this reason, such a solution is referred to as skew–modal  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In order to provide theoretical guarantees for this practical version, similar  to those in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, let us introduce two mild assumptions in addition to those outlined  in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  M1 The MAP estimator ˆθ = argmaxθ∈Θ{ℓ(θ)+log π(θ)}, satisfies Eθ0∥ˆθ−θ0∥2 = O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  M2 Let ˆV = Jˆθ/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' With P n  θ0–probability tending to one, ˆV is positive definite and | ˆV | > ˆη  for some ˆη > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover, there exist two positive constants δ and L such that the in-  equalities | ℓ(3)  stl(θ)/n | < L, | ℓ(4)  stlk(θ)/n | < L, and |log π(2)  st (θ)| < L hold uniformly over  θ ∈ Bδ(ˆθ) = {θ ∈ Θ : ∥θ − ˆθ∥ < δ}, with P n  θ0–probability tending to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Condition M1 is mild and holds generally in regular parametric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This assumption  ensures us that the MAP is in a small neighborhood of θ0, where the centering took place in  20  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Condition M2 is a weaker version of the analytical assumption for Laplace’s  method described in Kass, Tierney and Kadane (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Note also that assumption M1 im-  plies that M2 is a stronger version of A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0 and A4, requiring the upper bound to hold in a  neighborhood of θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' These conditions ensure uniform control on the difference between the  log–likelihood ratio and its third order Taylor’s expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Finally, we remark that even the  more restrictive analytical assumption for Laplace’s methods introduced in Kass, Tierney  and Kadane (1990) are satisfied in several examples of practical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Based on the above additional assumptions we state a Bernstein–von Mises type result for  the skew–modal approximation in (21), similar to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The proof of the theorem  follows closely the reasoning and derivations considered to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence, such  a proof, along with those of the technical lemmas, is deferred to the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Let ˆh = √n(θ − ˆθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' If the assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 and conditions  M1–M2 are satisfied, then the posterior for ˆh satisfies  (22)  ∥Π(· | Xn) − ˆP n  GSN(·)∥TV = OPθ0  �(log n)3+d/2/n  �,  where ˆP n  GSN(S) =  �  S ˆpn  GSN(ˆh)dˆh for S ⊂ Rd with ˆpn  GSN(ˆh) defined as in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  REMARK 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Since the total variation distance is invariant to scale and location trans-  formations, the above result can be stated also for the original parametrization θ of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  This means that the generalized skew–normal density  (23)  ˆpn  GSN(θ) = 2φd(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' ˆθ,J−1  ˆθ )Φ((  √  2π/12)ℓ(3)  ˆθ,stl(θ − ˆθ)s(θ − ˆθ)t(θ − ˆθ)l),  approximates the posterior density for θ with the rate derived in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The skew–modal approximation in (23) provides, therefore, a similarly tractable, yet sub-  stantially more accurate, alternative to the classical Gaussian counterpart arising from the  Laplace method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As discussed in Section 2, the closed–form density in (23) can be evalu-  ated essentially at the same cost of a Gaussian density and further admits a straightforward  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' sampling scheme that facilitates Monte Carlo estimation of any functional of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Such a scheme has the same cost of sampling from a d–variate Gaussian and, hence, can be  readily implemented via standard R packages for simulating from these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Empirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2 demonstrate the practical applicability of the  skew–modal approximation derived in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 on both a synthetic simulation and a real  data application on Cushing’s syndrome (Venables and Ripley, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' These empirical anal-  yses provide consistent evidence of substantial accuracy improvements not only relative to  the classical Gaussian approximation induced by the Laplace method, but also with respect  to state–of–the–art implementations of mean–field variational Bayes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Blei, Kucukel-  bir and McAuliffe, 2017) and expectation–propagation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Vehtari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', 2020), including  also recent skewed extensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Fasano, Durante and Zanella, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  21  TABLE 2  In the exponential example introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, average, over 50 replicated studies, of the  log–total–variation (TV) distances corresponding to the Gaussian modal approximation induced by the  Laplace method (GM) and the newly–developed skew–modal (SKEW–M) solution in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Summaries for  the ratios between the two TV distances are also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Results are displayed for different sample sizes  from n = 1 to n = 5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The standard deviations of the ratios, computed over the 50 replicated studies, are  displayed within round brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The bold values indicate the best performance for each sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  n = 1  n = 10  n = 50  n = 100  n = 500  n = 1000  n = 5000  log n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='91  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='61  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='21  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='91  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='52  log TVGM  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='08  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='43  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='98  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='29  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='05  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='40  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='20  log TVSKEW–M  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='32  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='01  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='13  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='74  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='28  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='97  −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='56  TVSKEW–M/TVGM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='01  (10−5)  (10−5)  (10−5)  (10−5)  (10−5)  (10−5)  (10−5)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Exponential model revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Let us first replicate the simulation study on the ex-  ponential model under the same settings and comparisons in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, but with a focus  on the plug–in skew–modal approximation in (23), rather than its population version which  assumes knowledge of θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Consistent with this focus, performance is compared against the  Gaussian approximation N(ˆθ,J−1  ˆθ ) arising from the Laplace method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',  2013, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' 318).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Notice that such a model meets the additional assumptions M1–M2 required  by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In fact, ˆθ is asymptotically equivalent to the maximum  likelihood estimator which implies that condition M1 is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover, in view of the  expressions for the first three log–likelihood derivatives in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 also M2 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Table 2 displays the same summaries of Table 1, but now the two total variation distances  from the exact posterior are computed with respect to the skew–modal approximation in  (23) and the Gaussian N(ˆθ,J−1  ˆθ ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Similarly to the population version assessed  in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, also the practical skew–modal approximation yields considerable accuracy  gains relative to the Gaussian counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In addition, also in this context, the asymptotic  theory in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2 closely matches the empirical behavior observed in  practice even in finite, possibly small, sample size settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Notice also that, when compared  to the results in Table 1, the variability across experiments in Table 2 is substantially lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  This is due to the fact that, unlike for the population counterpart, both approximations are  centered at the MAP of the actual posterior rather than at θ0, which is not necessarily the  maximum–a–posteriori in finite sample size settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This allows for a more accurate fit to  the posterior in practice and less variability across experiments in the TV distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Probit and logistic regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' We conclude with a real–data application on  the Cushings dataset (Venables and Ripley, 2002) which is openly–available in the R library  Mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In this case the true data–generative model is not known and, therefore, such a final  assessment is useful to evaluate performance in situations which are not guaranteed to meet  the assumptions underlying our theory in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The data are obtained from a medical study, on n = 27 subjects, aimed at investigating  the relationship between four different sub–types of Cushing’s syndrome and two steroid  22  metabolites, Tetrahydrocortisone and Pregnanetriol respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To simplify the analysis,  we consider the binary response Xi ∈ {0,1}, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',n taking value 1 if the patient is  affected by bilateral hyperplasia, and 0 otherwise, for i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The observed covariates  are zi1 = “urinary excretion rate (mg/24hr) of Tetrahydrocortisone for patient i" and zi2 =  “urinary excretion rate (mg/24hr) of Pregnanetriol for patient i".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' For inference, we focus  on the two most widely–implemented regression models for binary data, namely:  Probit regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Xi ∼ Bern(Φ(θ0 + θ1zi1 + θ2zi2)), independently for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',n, with  regression coefficients (θ0,θ1,θ2) = θ ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Xi ∼ Bern(g(θ0+θ1zi1+θ2zi2)), independently for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',n, with  (θ0,θ1,θ2) = θ ∈ R3, and g(·) the inverse logit function defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Under both models, Bayesian inference proceeds via routinely–implemented weakly in-  formative Gaussian priors N(0,25) for the three regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Recalling Durante  (2019), such a choice yields a closed–form unified skew–normal (Arellano-Valle and Az-  zalini, 2006) posterior for θ under probit regression, which admits tractable i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' sampling  schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Extending the results in Durante (2019), a recent contribution by Onorati and  Liseo (2022) has proved that similar findings, for an extension of the unified skew–normal,  can be obtained also under logistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Although yielding important advancements in  the field of Bayesian inference for binary regression models, the derived closed–form pos-  teriors require evaluation of n–dimensional Gaussian cumulative distribution functions, and  hence, inference under the exact posterior becomes intractable as n grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This motivates  alternative solutions within the class of deterministic approximations to the exact posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  State–of–the–art methods under these regression models are mean–field variational Bayes  (MF–VB) (Consonni and Marin, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Durante and Rigon, 2019), partially–factorized varia-  tional Bayes (PFM–VB) (Fasano, Durante and Zanella, 2022), expectation–propagation (EP)  (Chopin and Ridgway, 2017), and the classical Gaussian modal approximation (GM) based  on the Laplace method (Tierney and Kadane, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Table 3 and Figures 4–5, compare the approximation accuracy of the newly–developed  skew–modal approximation in (23) relative to the aforementioned alternatives in both probit  and logit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To evaluate also the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' sampling scheme for generalized skew–normal  distributions introduced in Section 2, the marginals, bivariates and joint total variation dis-  tances are computed via Monte Carlo methods leveraging 105 samples from both the exact  posteriors for θ and from the corresponding approximations provided by the different meth-  ods analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Under both probit and logistic regression, the skew–modal approximation and  the classical Gaussian modal approximation (GM) can be readily derived from the deriva-  tives of the log–likelihood and log–prior which are available in closed–form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover,  since the prior is Gaussian, the MAP under both models coincides with the ridge–regression  solution and hence can be computed via basic R functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' MF–VB and PFM–VB for probit  regression leverage the implementation in the GitHub repository Probit-PFMVB (Fasano,  Durante and Zanella, 2022), whereas in the logistic setting we rely on the codes in the repos-  itory logisticVB (Durante and Rigon, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Notice that PFM–VB has been developed  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  23  TABLE 3  For probit and logistic regression, estimated joint, bivariate and marginal total variation distances between  the exact posterior and the deterministic approximations under analysis in the Cushings application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The  bold values indicate the best performance for each subset of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  TVθ  TVθ01  TVθ02  TVθ12  TVθ0  TVθ1  TVθ2  Probit  SKEW–M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='20  GM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='44  EP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='36  MF–VB  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='96  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='71  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='40  PFM–VB  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='78  Logit  SKEW–M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='29  GM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='57  EP  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='03  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='04  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='18  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='02  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='52  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='02  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='04  MF–VB  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='73  only for probit regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Finally, EP is implemented, under both models, via the R library  EPGLM (Chopin and Ridgway, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The approximations provided by GM, MF–VB and EP  are all Gaussian and, hence, admit straightforward i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' sampling, while PFM–VB yields a  computationally–tractable unified skew–normal approximation from which i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' values can  be simulated via Algorithm 3 in Fasano, Durante and Zanella (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Sampling from the  exact probit posterior can be instead performed via Algorithm 1 in Durante (2019), whereas  for the logistic regression we implement Hamiltonian Monte Carlo via the rstan library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Table 3 displays the TV distance estimates from the exact posteriors of θ for the differ-  ent approximations, under both probit and logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In both models, the proposed  skew–modal (SKEW–M) solution generally yields remarkable accuracy improvements rela-  tive to all the state–of–the–art competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' More specifically, SKEW–M halves, on average,  the TV distance associated with GM, while yielding noticeable improvements relative to MF–  VB and PFM–VB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The gains over PFM–VB in the probit model are remarkable since also such  a strategy leverages a skewed approximation of the exact posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This yields  higher accuracy than MF–VB, but the improvements are not as noticeable as SKEW–M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' A rea-  son for this result is that PFM–VB has been originally developed to provide high accuracy in  high–dimensional p > n settings (Fasano, Durante and Zanella, 2022), whereas in this study  p = 3 and n = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Except for one marginal in the probit model, SKEW–M also outperforms  state–of–the–art EP methods, with the accuracy improvements being particularly remark-  able in the logit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This is mainly due to the fact that, unlike for the proposed SKEW–M  approximation, EP relies on Gaussian approximations obtained via iterative algorithms with  no guarantees of convergence and reduced theoretical support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence, the poor EP perfor-  mance in logistic regression might be due to convergence issues of the R function EPlogit  (Ridgway, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Conversely, our computationally–tractable SKEW–M approximation only  requires derivation and evaluation of log–likelihood and log–prior derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Furthermore,  it has also strong theoretical support from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2, thereby yielding  24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='5  θ0  θ1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
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+page_content='1  θ1  θ2  Fig 5: Visual comparison between skew–modal (blue) and Gaussian (orange) approximations of the exact  bivariate posteriors (grey) for the three coefficients of the logistic regression model in the Cushings application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  gains relative to all state–of–the–art competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Clearly, as the exact posterior approaches  a Gaussian shape, such improvements become less noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Nonetheless, as illustrated in  Figures 4–5, even relatively–minor deviations from Gaussianity in the posterior for θ, can  yield remarkable gains when approximated via SKEW–M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This article proves that substantial theoretical and practical accuracy im-  provements can be obtained by replacing limiting Gaussian approximations of exact poste-  rior distributions with generalized skew–normals constructively derived form a third–order  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  25  version of the Laplace method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Beside clarifying the role of skewness in Bayesian asymp-  totics, our contribution has also a broad theoretical scope and remarkable methodological  impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' It can directly motivate refinements of several theoretical results relying on the clas-  sical Gaussian Bernstein–von Mises theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The tractability of generalized skew–normals  and the corresponding skew–modal practical counterparts, along with the theoretical and  empirical improvements presented in the article, also open new avenues to stimulate exten-  sive advancements in the field of Bayesian inference based on deterministic approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  More specifically, our contribution motivates future efforts toward developing skewed ver-  sions of state–of–the–art variational Bayes and expectation propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' These more effec-  tive skew–modal approximations, which can readily replace the classical Gaussian one, can  be further included in more refined strategies, such as INLA, which consider these Gaussian  approximations as building–blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Future research directions of particular interest include the extension of our theory and  methods beyond parametric settings, with a specific focus on semiparametric (Bickel and  Kleijn, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Castillo and Rousseau, 2015) and high–dimensional (Boucheron and Gassiat,  2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Bontemps, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Spokoiny and Panov, 2021) contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover, although the in-  clusion of skewness is arguably sufficient to yield an accurate approximation of posterior  distributions in small–to–moderate sample size regimes, accounting for kurtosis might yield  additional improvements both in theory and in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this end, a relevant research di-  rection is to seek for an alternative to generalized skew–normals, possibly obtained from a  fourth–order version of the Laplace method that yields an improved characterization of the  tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In principle, this extension is expected to provide further improvements in the rate of  convergence, while maintaining good sampling properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Our conjecture is that such an  alternative limiting law can be found in the class of skew–symmetric distributions (Azzalini  and Capitanio, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Wang, Boyer and Genton, 2004) and relies on the perturbation of a  density with heavier tails than the Gaussian one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  REFERENCES  ANCESCHI, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', FASANO, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', DURANTE, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' and ZANELLA, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Bayesian conjugacy in probit, tobit,  multinomial probit and extensions: A review and new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Journal of the American Statistical Association  In press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  ARELLANO-VALLE, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' and AZZALINI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' On the unification of families of skew-normal distri-  butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Scandinavian Journal of Statistics 33 561–574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  AZZALINI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' and CAPITANIO, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Distributions generated by perturbation of symmetry with em-  phasis on a multivariate skew t-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
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+page_content=' Expectation propagation as a way of life: A  framework for Bayesian inference on partitioned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
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+page_content=' Springer, New  York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' ISBN 0-387-95457-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  WANG, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
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+page_content=' Variational inference in nonconjugate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Journal of Machine Learn-  ing Research 14 1005–1031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  WANG, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
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+page_content=' Frequentist consistency of variational Bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Journal of the American  Statistical Association 114 1147–1161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  WANG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', BOYER, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' and GENTON, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
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+page_content=' A skew-symmetric representation of multivariate distri-  butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Statistica Sinica 1259–1270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  WENG, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' A Bayesian Edgeworth expansion by Stein’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Bayesian Analysis 5 741–763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  ZWILLINGER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' and JEFFREY, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Table of integrals, series, and products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  28  APPENDIX A: PROOF OF THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1  To prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 we follow a similar line of reasoning as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, and high-  light the main differences between the two proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this end, let us first consider the sieve  ˆΘn = {θ ∈ Θ : ∥θ − ˆθ∥ < 2Mn/√n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' We show below that Π(ˆΘc  n | Xn) converges to zero  in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To this end let us consider the event Bn = {∥ˆθ − θ0∥ ≤ Mn/√n}, and recall  the definition of Θn in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Note that  Π(ˆΘc  n | Xn) ≤ Π(Θc  n | Xn) + 1Bcn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The first term on the right hand side is OPθ0(n−1) in view of Markov’s inequality and the  fact that, as shown in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, Eθ0Π(Θc  n | Xn) = O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' The second term has the  same order in view of assumption M1 and applying again Markov’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Therefore,  following (10), it is sufficient to work with the restricted posterior ΠˆΘn(·|Xn) defined in (9),  with Θn replaced by ˆΘn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' As for the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, let us introduce the quantities  Yn(ˆh) = 2exp  �  − 1  2  jˆθ,st  n  ˆhsˆht  �  Φ  � √  2π  12√n  ℓ(3)  ˆθ,stl  n  ˆhsˆhtˆhl  �  ,  Ln(ˆh) =  ˆpˆθ+ˆh/√n  ˆpˆθ  (Xn)π(ˆθ + ˆh/√n)  π(ˆθ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  In view of assumption M2 we can express log Ln(ˆh) as log Ln(ˆh) = −(1/2)(jˆθ,st/n)ˆhsˆht+  (1/6√n)(ℓ(3)  ˆθ,stl/n)ˆhsˆhtˆhl + qn,ˆh, where  qn,ˆh =  ℓ(4)  ˆθ,stlk(˜h)  n  ˆhsˆhtˆhlˆhk + 1  2  log π(ˆθ + ¯h/√n)(2)  st  n  ˆhsˆht = OPθ0  �∥ˆh∥4 ∨ 1  n  �  ,  and ˜h,¯h are some points between zero and ˆh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Now, from triangle inequality and in view of  the second order Taylor expansion of eqn,ˆh = 1 + qn,ˆh + eβqn,ˆhq2  n,ˆh/2, for some β ∈ (0,1), it  follows that  (24) |Ln(ˆh) − Yn(ˆh)| ≤|exp(un,ˆh)(qn,ˆh + exp(βqn,ˆh)q2  n,ˆh/2)| + |exp(un,ˆh) − Yn(ˆh)|,  with un,ˆh = −(1/2)(jˆθ,st/n)ˆhsˆht + (1/6√n)(ℓ(3)  ˆθ,stl/n)ˆhsˆhtˆhl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  As for the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, let us replace the stochastic quantities involved in the  above expressions with their truncated counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In particular, let ˜ℓ(3)  ˆθ,stl = (ℓ(3)  ˆθ,stl ∨ −nL)∧  nL, ˜ℓ(4)  ˆθ,stlk(˜h) = (ℓ(4)  ˆθ,stlk(˜h) ∨ −nL) ∧ nL and log ˜π(ˆθ+¯h/√n)(2)  st = (log π(ˆθ + ¯h/√n)st(2)∨  −L) ∧ L, and denote with ˜qˆh,n, ˜Ln(ˆh) and ˜Yn(ˆh) the versions of qˆh,n, Ln(ˆh) and Yn(ˆh) in  which the different quantities appearing within the corresponding expressions have been  replaced by the truncated counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Notice that the Pθ0–probability that any of these  truncations takes place is O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Therefore, by applying in (24) the thresholds introduced by  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  29  the truncation, we directly get that the first term on the right hand side is O(M 4  n/n), while  in view of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='6 the second term is O(M 6  n/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This implies that  (25)  supˆh∈ ˆSMn |˜Ln(ˆh) − ˜Yn(ˆh)| = O(M 6  n/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  where ˆSMn = {ˆh : ||ˆh|| < 2Mn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence, combining (25) with Fubini’s theorem and Jensen’s  inequality, it holds  Eθ0 supS |  �  S∩ ˆSMn ˜Ln(ˆh)dˆh −  �  S∩ ˆSMn ˜Yn(ˆh)dˆh|  ≤  �  ˆSMn Eθ|˜Ln(ˆh) − ˜Yn(ˆh)|dˆh ≲  �  ˆSMn(M 6  n/n)dˆh = O  �M d+6  n  /n  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Then, by applying the triangle inequality and noticing that Pθ0(  �  ˆSMn ˜Yn(ˆh)dˆh > 0) = 1 it  follows from the above result that  Eθ0 supS |  �  S∩ ˆSMn ˜Ln(ˆh)dˆh/  �  ˆSMn ˜Ln(ˆh)dˆh−  �  S∩ ˆSMn ˜Yn(ˆh)dˆh/  �  ˆSMn ˜Yn(ˆh)dˆh| = O  �M d+6  n  /n  �,  which in turn implies the same upper bound, in Pθ0–probability, for the non–truncated ver-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' More specifically, recalling the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 and considering a similar nota-  tion, it follows that  (26)  supS | Π ˆSMn(ˆh ∈ S | Xn) − ˆPGSN, ˆSMn(S)| = OPθ0(M d+6  n  /n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  where ˆPGSN, ˆSMn(S) =  �  S∩ ˆSMn Yn(ˆh)dˆh/  �  ˆSMn Yn(ˆh)dˆh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover, as for the proof of Theo-  rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, notice that the denominator in the expression for ˆPGSN, ˆSMn(S) is equal to a constant  times (1 − ˆPGSN( ˆSc  Mn)), where ˆPGSN( ˆSc  Mn) = OPθ0(n−D), D > 1 for a sufficiently large c0 in  the definition of Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Hence, since  | ˆPGSN, ˆSMn(S) − ˆPGSN(S)| ≤ { ˆPGSN( ˆSc  Mn)/(1 − ˆPGSN( ˆSc  Mn))} ˆPGSN  �S ∩ ˆSMn  � + ˆPGSN( ˆSc  Mn),  we have that  (27)  supS| ˆPGSN, ˆSMn(S) − ˆPGSN(S)| = OPθ0(n−D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Combining (26) and (27) with the first part of the proof, and applying the triangle equality  similarly to what done for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, concludes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  APPENDIX B: TECHNICAL LEMMAS  In this section we collect the technical lemmas used in the proofs of our two main theo-  rems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Also in this case we omit for convenience the n from the notation of the probability  measure Pθ and the corresponding expectation and densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The first lemma is a refined version of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='26 in Ghosal and Van der Vaart (2017),  with (possibly) a polynomial instead of logarithmic Pθ0–convergence rate, used together  with Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2, to bound the Type II error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  30  LEMMA B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  For any prior Π on Θ, sequence Dn > 1 and ϵ > 0, it holds  (28)  �  (pθ/pθ0)(Xn)dΠ(θ) ≥ Π  �θ : KL(pθ0∥pθ) < ϵ2�exp{−2Dn(  �  nϵ2/2 ∨ nϵ2)}  with Pθ0–probability at least 1 − D−1  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' First note that for B = {θ : KL(pθ0∥pθ) < ϵ2}, in view of Jensen’s inequality  log  � (pθ/pθ0)(Xn)dΠ(θ)  Π(B)  ≥ log  �  (pθ/pθ0)(Xn)dΠ(θ | B)  ≥  �  log(pθ/pθ0)(Xn)dΠ(θ | B) =: Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Now, note that by Markov’s and Jensen’s inequality, the probability of the complement of  the event in (28) can be bounded as  Pr(Z < −2Dn(  �  nϵ2/2 ∨ nϵ2)) ≤  Eθ0(−Z)+  2Dn(  �  nϵ2/2 ∨ nϵ2)  ≤ Eθ0  � (−log(pθ/pθ0)(Xn))+ dΠ(θ | B)  2Dn(  �  nϵ2/2 ∨ nϵ2)  ,  where the generic (Z)+ is defined as Z ∨0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Then, by Fubini’s theorem and Pinsker’s second  inequality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='13 of Ghosal and Van der Vaart (2017)) the right hand side of the  above inequality is further bounded by  �  KL(pθ0∥pθ) +  �  KL(pθ0∥pθ)/2dΠ(θ|B)  2Dn(  �  nϵ2/2 ∨ nϵ2)  ≤  2(  �  nϵ2/2 ∨ nϵ2)  2Dn(  �  nϵ2/2 ∨ nϵ2)  = 1  Dn  ,  concluding the proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The next lemma adapts Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3 in Van der Vaart (2000) to ensure a tighter control on  the Type I error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In the proofs of our main theorem, it is required to control the probability  mass assigned by the posterior distribution outside the ball of radius Mn/√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  LEMMA B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Under the conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1 there exists, for every Mn → ∞, a  sequence of tests φn and a constant ct such that, for every sufficiently large n and every  ∥θ − θ0∥ > Mn/√n, we have  Eθ0φn ≤ exp  �−ctM 2  n  �  and  Eθ (1 − φn) ≤ exp(−ctn(∥θ − θ0∥2 ∧ 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Condition A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='0 allows us to consider the truncated version of the score function  ˜ℓ(1)  θ0 ∈ [−L,L], with L > 0 taken such that Eθ0˜ℓ(1)  θ0 (˜ℓ(1)  θ0 )⊺ is nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Define now the test  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  31  ωn = 1{∥(En − Eθ0)˜ℓ(1)  θ0 ∥ > c∗Mn/√n}, where the constant c∗ > 0 is determined later and  En is the expectation with respect to the empirical measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' By Hoeffding’s inequality,  Eθ0ωn ≤ exp  �−ctM 2  n  �,  for some constant ct depending on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  To conclude the proof, it remains to study the Type II error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' By the triangle inequality  ∥(Eθ − Eθ0)˜ℓ(1)  θ0 ∥ ≤ ∥(En − Eθ))˜ℓ(1)  θ0 ∥ + ∥(En − Eθ0))˜ℓ(1)  θ0 ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Furthermore, in view of the differentiability of the likelihood function there exists a constant  c > 0 for which ∥(Eθ − Eθ0)˜ℓ(1)  θ0 ∥ > c∥θ − θ0∥ for all θ such that ∥θ − θ0∥ < ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' see Lemma  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3 in Van der Vaart (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Therefore, by choosing c∗ = c/2 and applying the preceding  two displays together with Hoeffding’s inequality we get that  Eθ(1 − ωn) ≤ Pθ  �  ∥(En − Eθ)˜ℓ(1)  θ0 ∥ > ∥(Eθ − Eθ0)˜ℓ(1)  θ0 ∥ − c∗Mn/√n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  ≤ Pθ  �  ∥(En − Eθ)˜ℓ(1)  θ0 ∥ > c∥θ − θ0∥/2  �  ≤ exp  �−ctn∥θ − θ0∥2�,  for a sufficiently small constant ct depending on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' For ∥θ − θ0∥ > ϵ it is possible to con-  struct a test with Type I and Type II errors that decrease exponentially in n by proceeding  as in Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3 of Van der Vaart (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Combining such tests, concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The next lemma gives an upper bound for the first term in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  LEMMA B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Let ˜∆n,θ = ˜ℓ(1)  θ /√n =  �ℓ(1)  θ /√n  �1∥ℓ(1)  θ /√n∥<Cn where Cn = M 1/α  n  for α >  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Under the notations and conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, we have  (29)  limn→∞ suph∈SMn |Eθ expγ˜un,h − 1| → 0,  for any γ > 0, where ˜un,h = hs{ ˜∆n,θ}s − (hshtist)/2n + ˜rn,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' To prove Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3, let us first notice that, for ˜∆θ = ∆θ1∥∆θ∥<Cn with ∆θ ∼  N(0,Iθ/n), it holds Eθ exp{γ  �hs{ ˜∆θ}s − (hshtist)/2n  �} − 1 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Therefore, by trian-  gle inequality if follows that the left hand side of (29) is bounded from above by  suph∈SMn |Eθ expγ˜un,h − Eθ expγ(hs{ ˜∆θ}s − (hshtist)/2n)| + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='7 in Van der Vaart (2000) implies that ( ˜∆n,θ, ˜rn,h) converges in distribution to  ( ˜∆θ,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Since Iθ is a deterministic positive definite matrix we can concentrate only on  limn→∞ suph∈SMn |Eθ exp[γhs{ ˜∆n,θ}s + γ˜rn,h] − Eθ exp[γhs{ ˜∆θ}s]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Let us now define J = Jϵ = {x ∈ Rp:∥x∥ < Rϵ}, with Rϵ > 0 such that Pθ (∆θ∈J) > 1−ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Clearly, for all δ > 0, Pθ ((∆θ,0) ∈ J × (−δ,δ)) > 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Since the functions fh(x,y) =  32  exp(γhsxs + γy) are uniformly continuous on I = J × (−δ,δ), for all h, one can cover I  with finitely many subsets such that fh varies at most (constant times) ϵ in each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In  particular, let us define the following sets  (30)  J0 ={x : ∥x∥ > (Cn ∧ Rϵ)},  Jj ={x : log[(j − 1)ϵ1] < γhsxs ≤ log(jϵ1)} ∩ {x : ∥x∥ ≤ (Cn ∧ Rϵ)},  Ek ={y : log[(k − 1)ϵ2] < γy ≤ log(kϵ2)} ∩ {y : −δ < y < δ},  with ϵ1 = ϵexp(−γδ), ϵ2 = ϵexp(−γMnCn), and j,k ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',Nn} for Nn such that Nn ≤  exp{γ(MnCn + δ)}/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Furthermore, note that there exists a unique k∗ ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',Nn} such  that 0 ∈ Ek∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Let us now merge the sets Ek∗−1, Ek∗, Ek∗+1 into Ek∗ for technical reasons to  be seen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Then, let us take Ij,k = Jj ×Ek ⊂ I, j ∈ {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',Nn},k ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',Nn}\\{k∗ −  1,k∗+1} and note that following from the triangle inequality the function fh(x1(∥x∥<Cn),y)  varies at most 4ϵ in each Ij,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover, notice that for j > 0 each Jj is convex since it is  the intersection of two convex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Therefore, the multivariate Berry–Esseen’s theorem, see  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='8, can be directly applied to Pθ(∆n,θ ∈ Jj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' On the contrary, J0 is not convex but  we can compute Pθ(∆n,θ ∈ J0) in terms of its complement which is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  We use the sets Ij,k to approximate fh with a step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Let us take an arbitrary point  (xj,yk) ∈ Ij,k for j ∈ {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',Nn},k ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',Nn}\\{k∗ − 1,k∗ + 1} and define the step  function  fϵ,h(x,y) =  �Nn  j=0  �Nn  k=1 fh(xj,yk)1(x,y)∈Ij,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Notice that, for notational convenience, we consider the sum over k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',Nn, but in fact  the k∗ −1 and k∗ +1 cells are merged with the k∗ cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Then, by triangle inequality, we have  (31)  ��Eθ fh( ˜∆n,θ, ˜rn,h) − Eθfh( ˜∆θ,0)  �� ≤  ��Eθ fh( ˜∆n,θ, ˜rn,h) − Eθ fϵ,h( ˜∆n,θ, ˜rn,h)  ��  +  ��Eθ fϵ,h( ˜∆n,θ, ˜rn,h) − Eθfϵ,h( ˜∆θ,0)  ��  +  ��Eθ fh( ˜∆θ,0) − Eθfϵ,h( ˜∆θ,0)  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  We now study the three terms on the right hand side of (31) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  For the second term, we can exploit the fact that fϵ,h is a step function and that it is zero  outside of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' This result, in combination with Jensen’s inequality and the bound fh(xj,yk) ≤  exp(γMnCn +γδ) for j,k ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',Nn}, ensures that  ��Eθ fϵ,h( ˜∆n,θ, ˜rn,h)−Eθfϵ,h( ˜∆θ,0)  ��  is bounded from above by  �Nn  j=1  �Nn  k=1 eγ(MnCn+δ)|Pθ(( ˜∆n,θ, ˜rn,h) ∈ Ij,k) − Pθ (( ˜∆θ,0) ∈ Ij,k)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The asymptotic behavior of the above expression depends on the difference between the  probability measure of ( ˜∆n,θ, ˜rn,h) and the one of ˜∆θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' First, consider the case 0 /∈ Ek (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  k ̸= k∗) where Pθ  �( ˜∆θ,0) ∈ Ij,k  � = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In view of |log(1 + y)| ≥ |y|/2 for |y| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='5  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  33  we have that Ek ⊂ {|˜rn,h| > ϵ2}, since |k − k∗| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Therefore, following by (the truncated  version of) assertion (13)  (32)  |Pθ(( ˜∆n,θ, ˜rn,h) ∈ Ij,k  � − Pθ  �( ˜∆θ,0) ∈ Ij,k)|  ≤ Pθ(˜rn,h ∈ Ek) ≤ Pθ(|˜rn,h| > ϵ2) = O(log n{∥h∥3 ∨ 1}eγMnCn/(√nϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Similarly, if 0 ∈ Ek, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' k = k∗, by applying the triangle inequality  |Pθ(( ˜∆n,θ, ˜rn,h) ∈ Ij,k) − Pθ ( ˜∆θ ∈ Jj)| ≤ |Pθ( ˜∆n,θ ∈ Jj) − Pθ ( ˜∆θ ∈ Jj)| + Pθ(˜rn,h /∈ Ek)  = O  �log n{∥h∥3 ∨ 1}eγMnCn/(√nϵ)  �,  where the last step follows from the multivariate Berry–Esseen’s theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' see Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='8  below (with Yi = ℓ(1)(θ0,Xi)/√n), and (the truncated version of) assertion (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  For the first term on the right hand side of (31), note that  (33)  ��Eθfh( ˜∆n,θ, ˜rn,h) − Eθ fϵ,h( ˜∆n,θ, ˜rn,h)  ��  ≤ 4ϵPθ  �( ˜∆n,θ, ˜rn,h) ∈ I  � + eγ(MnCn+δ)Pθ  �( ˜∆n,θ, ˜rn,h) /∈ I  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Hence, as before  Pθ  �( ˜∆n,θ, ˜rn,h) /∈ I  � ≤ |Pθ  � ˜∆n,θ /∈ J  � − Pθ  � ˜∆θ /∈ J  �| + Pθ( ˜∆θ /∈ J) + Pθ(|˜rn,h| > δ)  = O(log n{∥h∥3 ∨ 1}/(√nϵ)) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  (34)  Finally, the third term on the right hand side of (31) can be bounded similarly as  (35)  ��Eθ fh( ˜∆θ,0) − Eθfϵ,h( ˜∆θ,0)  ��  ≤ 4ϵPθ( ˜∆θ ∈ J) + eγMnCnPθ( ˜∆θ /∈ J) ≤ 4ϵ + eγMnCnϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  We conclude the proof by applying the bounds (32), (33), (34) and (35) in (31), by choosing  ϵ = ϵn ≍ n−β for β ∈ (0,1/6) and by recalling that Cn = M 1/α  n  with α > 2 to obtain  limn→∞ suph∈SMn |Eθ fh( ˜∆n,θ, ˜rn,h) − Eθfh( ˜∆θ,0)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  LEMMA B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Define λn,h = (1/6√n)(ℓ(3)  θ0,stl/n){ξsξtξl + 3(h − ξ)s(h − ξ)tξl − hshthl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Then, under the conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, it follows that  (36)  Yn(h) = exp(un,h)(1 + λn,h + eβ∗λn,hλ2  n,h/2)  × (1 − λn,h + (1/12)e−π(˜βλn,h)2/4{1 − (˜βλn,h  �  π/2)2}λ3  n,hπ),  for some β∗ ∈ (0,1) and ˜β ∈ (0,1), with λn,h = OPθ0((∥h∥3 ∨ 1)/√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  34  PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Notice first that assumption A0 implies that λn,h = OPθ0({∥h∥3 ∨ 1}/√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Fur-  thermore, −λn,h = (1/6√n)(ℓ(3)  θ0,stl/n)((h − ξ)s(h − ξ)t(h − ξ)l + 3(h − ξ)sξtξl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' There-  fore, by taking a Taylor’s expansion of Φ(·) around 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='e Φ(x) = 1/2 + x/  √  2π + O(x3),  for x = O(1), we obtain for some ˜β ∈ (0,1) that  Φ(−  �  π/2λn,h) = 1/2 − λn,h/2 + (1/24)e−π(˜βλn,h)2/4{1 − (˜βλn,h  �  π/2)2}λ3  n,hπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  To conclude the proof, recall that in (12), the variable Yn(h) can be alternatively defined as  Yn(h) = 2exp(un,h)exp(λn,h)Φ(−  �  π/2λn,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Combining the two equations above with the second order Taylor’s expansion ex = 1 + x +  eβxx2/2 for some β ∈ (0,1), concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  LEMMA B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Under the notations and conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, we have  (37)  suph∈SMn Eθ|exp(˜un,h) − ˜Yn(h)| = O(M 6  n/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In view of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='4 and by Hölder’s inequality we have that Eθ|exp(˜un,h) −  ˜Yn(h)| is bounded from above by the product between E1/ρn  θ  exp(ρn˜un,h) and E1/˜ρn  θ  |1−(1+  ˜λn,h + eβ∗˜λn,h˜λ2  n,h/2)(1 − λn,h + (1/12)e−π(˜βλn,h)2/4{1 − (˜βλn,h  �  π/2)2}λ3  n,hπ)|˜ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Now,  notice that Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='3 implies E1/ρn  θ  exp(ρn˜un,h) = O(1) for ρn = (1 + M 2  n)/M2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Further-  more, in view of the truncation |˜λn,h| ≤ τn, the second factor is O(M 6  n/n), concluding the  proof of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  The following lemma is the counterpart Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='5 for the skew–modal approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  LEMMA B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Under the conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='1, it holds  supˆh∈ ˆSMn |exp(˜un,ˆh) − ˜Yn(ˆh)| = O(M 6  n/n),  where ˜un,ˆh = −(1/2)(jˆθ,st/n)ˆhsˆht + (1/6√n)(˜ℓ(3)  ˆθ,stl/n)ˆhsˆhtˆhl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' First note that ˜λn,ˆh := (˜ℓ(3)  ˆθ,stl/n)ˆhsˆhtˆhl ≤ CL,d(∥ˆh∥3 ∨ 1), for some constant CL,d  depending on L and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover, by the Taylor’s expansion of Φ(·) around 0, we have  Φ((  √  2π/12√n)˜λn,ˆh) =1/2 + ˜λn,ˆh/(12√n) + O((∥˜h∥9 ∨ 1)/n3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  These results, together with the positive definiteness of Jˆθ, and the fact that exp(x) = 1 +  x + exp(βx)x2/2 for some β ∈ (0,1) lead to  supˆh∈ ˆSMn |exp(˜un,ˆh) − ˜Yn(ˆh)|  ≤ supˆh∈ ˆSMn |exp((1/6√n)˜λn,ˆh) − 2Φ((  √  2π/12√n)˜λn,ˆh)| = O((∥˜h∥6 ∨ 1)/n),  concluding the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  SKEWED BERNSTEIN–VON MISES THEOREM AND SKEW–MODAL APPROXIMATIONS  35  Next we show that both the empirical Fisher information matrix and the covariance matrix  of the Gaussian density in the generalized skew–normal approximation are positive definite  with determinant bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  LEMMA B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Under A0 and A4, with Pθ0–probability tending to 1, V = Ω−1 and Jθ0/n  are positive definite matrices and there exists ˜η > 0 such that |V | > ˜η and |Jθ0/n| > ˜η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Leveraging the assumptions in A0, let us first notice that Jθ0/n = Iθ0/n + RJ  and V = Iθ0/n + RV , with RJ and RV having entries of order OPθ0(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' In the follow-  ing, we focus on RJ since the statement for RV can be obtained under the same reasoning  and steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Notice that condition A0 implies the existence of a sequence Fn ≍ n−1/κ, with  κ > 2 such that  Pθ0(|RJ,st| > Fn) = O(1),  for each s,t = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Define now the matrix M with entries Mst = |RJ,st| ∧ Fn for s,t =  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' From Wielandt’s theorem (Zwillinger and Jeffrey, 2007), with probability 1−O(1),  the spectral radius of M is an upper bound of the spectral radius of RJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Moreover, since M  is a non–negative matrix, the Perron–Frobenius theorem (Perron, 1907;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Frobenius, 1912)  implies that the largest eigenvalue in absolute value is bounded by constant times Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Since  both Iθ0/n and RJ are real symmetric matrices, in view of the Weyl’s inequalities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=', Tao,  2011, equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='54)), the eigenvalues of Jθ0/n and Iθ0/n can differ at most by constant  times Fn with probability 1−O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Since the entries of Iθ0/n are O(1) this implies that, with  Pθ0–probability tending to one, |Jθ0/n| is bounded away from 0 and that Jθ0 is a positive–  definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Finally, for completeness, we recall Berry–Esseen’s theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' see for instance Theorem 1  of Bentkus (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  LEMMA B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='  Let Y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',Yn be independent random variables on Rd with EYi = 0,i =  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Write Z = Y1 + ··· + Yn, Σ = Cov Z and take Z0 ∼ Nd(0,Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Assume that Σ is  invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Furthermore, define η = η1 + ··· + ηn, with ηi = E∥Σ−1/2Yi∥3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
+page_content=' Then, there exist  C > 0 such that for every convex set A ⊂ Rd, it holds |P(Z ∈ A) − P(Z0 ∈ A)| ≤ Cd1/4η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE1T4oBgHgl3EQfQQOd/content/2301.03038v1.pdf'}
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+arXiv:2301.04380v1  [math.OA]  11 Jan 2023
+Ideals with approximate unit in semicrossed
+products
+Charalampos Magiatis
+January 12, 2023
+Abstract
+We characterize the ideals of the semicrossed product C0(X) ×φ Z+
+with left (resp. right) approximate unit.
+1
+Introduction and Notation
+The semicrossed product is a non-selfadjoint operator algebra which is con-
+structed from a dynamical system. We recall the construction of the semicrossed
+product we will consider in this work. Let X be a locally compact Hausdorff
+space and φ : X → X be a continuous and proper surjection (recall that a map
+φ is proper if the inverse image φ−1(K) is compact for every compact K ⊆ X).
+The pair (X, φ) is called a dynamical system.
+An action of Z+ := N ∪ {0}
+on C0(X) by isometric ∗-automorphisms αn, n ∈ Z+, is obtained by defining
+αn(f) = f◦φn. We write the elements of the Banach space ℓ1(Z+, C0(X)) as for-
+mal series A = �
+n∈Z+ U nfn with the norm given by ∥A∥1 = �
+n∈Z+ ∥fn∥C0(X).
+Multiplication on ℓ1(Z+, C0(X)) is defined by setting
+(U nf)(U mg) = U n+m(αm(f)g) ,
+and extending by linearity and continuity. With this multiplication ℓ1(Z+, C0(X))
+is a Banach algebra.
+The Banach algebra ℓ1(Z+, C0(X)) can be faithfully represented as a (con-
+crete) operator algebra on a Hilbert space. This is achieved by assuming a faith-
+ful action of C0(X) on a Hilbert space H0. Then we can define a faithful contrac-
+tive representation π of ℓ1(Z+, C0(X)) on the Hilbert space H = H0 ⊗ ℓ2(Z+)
+by defining π(U nf) as
+π(U nf)(ξ ⊗ ek) = αk(f)ξ ⊗ ek+n .
+The semicrossed product C0(X)×φZ+ is the closure of the image of ℓ1(Z+, C0(X))
+in B(H) in the representation just defined. We will denote an element π(U nf)
+of C0(X) ×φ Z+ by U nf to simplify the notation.
+1
+
+For A = �
+n∈Z+ U nfn ∈ ℓ1(Z+, C0(X)) we call fn ≡ En(A) the nth Fourier
+coefficient of A. The maps En : ℓ1(Z+, C0(X)) → C0(X) are contractive in the
+(operator) norm of C0(X)×φZ+, and therefore they extend to contractions En :
+C0(X) ×φ Z+ → C0(X). An element A of the semicrossed product C0(X) ×φ
+Z+ is 0 if and only if En(A) = 0, for all n ∈ Z+, and thus A is completely
+determined by its Fourier coefficients. We will denote A by the formal series A =
+�
+n∈Z+ U nfn, where fn = En(A). Note however that the series �
+n∈Z+ U nfn
+does not in general converge to A [6, II.9 p.512]. The kth arithmetic mean of
+A is defined to be ¯Ak =
+1
+k+1
+�k
+l=0 Sl(A), where Sl(A) = �l
+n=0 U nfn. Then,
+the sequence { ¯Ak}k∈Z+ is norm convergent to A [6, Remark p.524]. We refer to
+[6, 4, 3] for more information about the semicrossed product.
+Let {Xn}∞
+n=0 be a sequence of closed subsets of X satisfying
+Xn+1 ∪ φ(Xn+1) ⊆ Xn ,
+(∗)
+for all n ∈ N. Peters proved in [7] that there is a one-to-one correspondence
+between closed two-sided ideals I ⊆ C0(X) ×φ Z+ and sequences {Xn}∞
+n=0 of
+closed subsets of X satisfying (∗), under the additional assumptions that X is
+metrizable and the dynamical system (X, φ) contains no periodic points. In fact,
+the ideal I associated with the sequence {Xn}∞
+n=0 is I = {A ∈ C0(X) ×φ Z+ :
+En(A)(Xn) = {0}}. We will write this as I ∼ {Xn}∞
+n=0.
+Moreover, under
+the above assumptions, he characterizes the maximal and prime ideals of the
+semicrossed product C0(X) ×φ Z+.
+Donsig, Katavolos and Manousos obtained in [4] a characterization of the
+Jacobson radical for the semicrossed product C0(X)×φZ+, where X is a locally
+compact metrisable space and φ : X → X is a continuous and proper surjection.
+Andreolas, Anoussis and the author characterized in [2] the ideal generated by
+the compact elements and in [1] the hypocompact and the scattered radical of
+the semicrossed product C0(X) ×φ Z+, where X is a locally compact Hausdorff
+space and φ : X → X is a homeomorphism. All these ideals are of the form
+I ∼ {Xn}∞
+n=0 for suitable families of closed subsets {Xn}∞
+n=0.
+In the present paper we characterize the closed two-sided ideals I ∼ {Xn}∞
+n=0
+of C0(X) ×φ Z+ with left (resp. right) approximate unit. As a consequence, we
+obtain a complete characterization of ideals with left (resp. right) approximate
+unit under the additional assumptions that X is metrizable and the dynamical
+system (X, φ) contains no periodic points.
+Recall that a left (resp. right) approximate unit of a Banach algebra A is a
+net {uλ}λ∈Λ of elements of A such that:
+1. for some positive number r, ∥uλ∥ ≤ r for all λ ∈ Λ,
+2. lim uλa = a (resp. lim auλ = a), for all a ∈ A, in the norm topology of A.
+A net which is both a left and a right approximate unit of A is called an approx-
+imate unit of A. A left (resp. right) approximate unit {uλ}λ∈Λ that satisfies
+∥uλ∥ ≤ 1 for all λ ∈ Λ is called a contractive left (resp. right) approximate unit.
+We will say that an ideal I of a Banach algebra A has a left (resp. right)
+approximate unit if it has a left (resp. right) approximate unit as an algebra.
+2
+
+2
+Ideals with approximate unit
+In the following theorem the ideals I ∼ {Xn}∞
+n=0 with right approximate unit
+are characterized.
+Theorem 2.1 Let I ∼ {Xn}∞
+n=0 be a non-zero ideal of C0(X) ×φ Z+. The
+following are equivalent:
+1. I has a right approximate unit.
+2. Xn = Xn+1, for all n ∈ Z+.
+Proof.
+We start by proving that (1) ⇒ (2). Let I ∼ {Xn}∞
+n=0 be an ideal
+with right approximate unit {Vλ}λ∈Λ. We suppose that there exists n ∈ Z+
+such that Xn+1 ⊊ Xn. Let
+n0 = min{n ∈ Z+ : Xn+1 ⊊ Xn} ,
+x0 ∈ Xn0 \ Xn0+1 and f ∈ C0(X) such that f(x0) = 1, f(Xn0+1) = {0} and
+∥f∥ = 1. Then, for A = U n0+1f, we have
+∥AVλ − A∥ ≥ ∥En0+1(AVλ − A)∥ = ∥fE0(Vλ) − f∥ ≥ |(fE0(Vλ) − f)(x0)| = 1 ,
+for all λ ∈ Λ, since x0 ∈ Xn0 and E0(Vλ)(Xn0) = 0, which is a contradiction.
+Therefore Xn = Xn+1 for all n ∈ Z+.
+For (2) ⇒ (1), assume that Xn = Xn+1 for all n ∈ Z+. By (∗), we get that
+φ(X0) ⊆ X0. We will show that if {uλ}λ∈Λ is a contractive approximate unit
+of the ideal C0(X \ X0) of C0(X), then {U 0uλ}λ∈Λ is a right approximate unit
+of I. Since ∥uλ∥ ≤ 1, we have ∥U 0uλ∥ ≤ 1.
+Let A ∈ I and ε > 0. Then there exists k ∈ Z+ such that
+∥A − ¯Ak∥ < ε
+4 ,
+where ¯Ak is the kth arithmetic mean of A. Since Xn = X0, En( ¯Ak) ∈ C0(X\X0)
+and {uλ}λ∈Λ is an approximate unit of C0(X \ X0), there exists λ0 ∈ Λ such
+that
+∥El( ¯Ak)uλ − El( ¯Ak)∥ <
+ε
+2(k + 1) ,
+for all l ≤ k and λ > λ0. So, for λ > λ0 we get that
+∥AU 0uλ − A∥
+<
+∥AU 0uλ − ¯AkU 0uλ + ¯AkU 0uλ − ¯Ak + ¯Ak − A∥
+<
+∥AU 0uλ − ¯AkU 0uλ∥ + ∥ ¯AkU 0uλ − ¯Ak∥ + ∥A − ¯Ak∥
+<
+∥ ¯AkU 0uλ − ¯Ak∥ + ε
+2
+≤
+k
+�
+l=0
+∥El( ¯Ak)uλ − El( ¯Ak)∥ + ε
+2
+<
+ε ,
+3
+
+which concludes the proof.
+□
+In the following theorem the ideals I ∼ {Xn}∞
+n=0 with left approximate unit
+are characterized.
+Theorem 2.2 Let I ∼ {Xn}∞
+n=0 be a non-zero ideal of C0(X) ×φ Z+. The
+following are equivalent:
+1. I has a left approximate unit.
+2. X0 ⊊ X and φn(X \ Xn) = X \ X0, for all n ∈ Z+.
+3. φ(X \ X1) = X \ X0 and φ(Xn+1 \ Xn+2) = Xn \ Xn+1, for all n ∈ Z+.
+Proof.
+We start by proving that (1) ⇒ (2). Let I ∼ {Xn}∞
+n=0 be an ideal
+with left approximate unit {Vλ}λ∈Λ.
+First we prove that X0 ⊊ X. We suppose that X0 = X. Then E0(Vλ) = 0,
+for all λ ∈ Λ, and hence for every U nf ∈ I we have
+∥VλU nf − U nf∥ ≥ ∥En(VλU nf − U nf)∥ = ∥E0(Vλ)f − f∥ = ∥f∥ ,
+for all λ ∈ Λ, which is a contradiction. Therefore X0 ⊊ X.
+Now we prove that φn(X \ Xn) = X \ X0, for all n ∈ Z+. We suppose that
+there exists n ∈ Z+ such that φn(X \ Xn) ̸⊆ X \ X0 and let
+n0 = min{n ∈ Z+ : φn(X \ Xn) ̸⊆ X \ X0} .
+Then, there exist x0 ∈ X\Xn0 such that φn0(x0) ∈ X0 and a function f ∈ C0(X)
+such that f(x0) = 1, f(Xn0) = {0} and ∥f∥ = 1. If A = U n0f, we have that
+A ∈ I, ∥A∥ = 1 and
+∥VλA − A∥
+≥
+∥En0(VλA − A)∥
+=
+∥E0(Vλ) ◦ φn0f − f∥
+≥
+|(E0(Vλ) ◦ φn0f − f)(x0)|
+=
+1 ,
+for all λ ∈ Λ, since φn0(x0) ∈ X0 and E0(Vλ)(X0) = {0}, which is a contra-
+diction. Therefore φn(X \ Xn) ⊆ X \ X0. Furthermore, by (∗) we get that
+φn(Xn) ⊆ X0, for all n ∈ Z+, and hence
+X = φn(X) = φn(Xn ∪ (X \ Xn)) = φn(Xn) ∪ φn(X \ Xn) ⊆ X0 ∪ φn(X \ Xn) .
+Since φn(X \ Xn) ⊆ X \ X0 and φ is surjective, φn(X \ Xn) = X \ X0, for all
+n ∈ Z+.
+For (2) ⇒ (1), assume that X0 ⊊ X and φn(X \ Xn) = X \ X0, for all
+n ∈ Z+. We will show that if {uλ}λ∈Λ is a contractive approximate unit of the
+ideal C0(X \ X0) of C0(X), then {U 0uλ}λ∈Λ is a left approximate unit of I.
+Since ∥uλ∥ ≤ 1, we have ∥U 0uλ∥ ≤ 1.
+4
+
+Let A be a norm-one element of I and ε > 0. Then there exists k ∈ Z+ such
+that
+∥A − ¯Ak∥ < ε
+4 ,
+where ¯Ak is the kth arithmetic mean of A. For l ≤ k, let
+Dε(El( ¯Ak)) =
+�
+x ∈ X : |El( ¯Ak)(x)| ≥
+ε
+4(k + 1)
+�
+.
+Since A ∈ I, we have El( ¯Ak)(Xl) = {0} and hence Dε(El( ¯Ak)) ⊆ X \ Xl.
+Furthermore, since φn(X \ Xn) = X \ X0, for all n ∈ Z+, we have that
+φl(Dε(El( ¯Ak))) ⊆ X \ X0. Moreover, the set Dε(El( ¯Ak)) is compact and hence
+the set φl(Dε(El( ¯Ak))) is also compact. By Urysohn’s lemma, there is a norm-
+one function vl ∈ C0(X) such that
+vl(x) =
+�
+1,
+x ∈ φl(Dε(El( ¯Ak)))
+0,
+x ∈ X0
+.
+Then, there exists λ0 ∈ Λ such that
+∥uλvl − vl∥ <
+ε
+2(k + 1) ,
+for all l ≤ k and λ > λ0, and hence
+|uλ(x) − 1| <
+ε
+2(k + 1) ,
+for all x ∈ ∪k
+l=0φl(Dε(El( ¯Ak))) and λ > λ0. Therefore, if x ∈ ∪k
+l=0(Dε(El( ¯Ak)))
+then φl(x) ∈ ∪k
+l=0φl(Dε(El( ¯Ak))) and hence
+∥((uλ ◦ φl)El( ¯Ak) − El( ¯Ak))(x)∥ <
+ε
+2(k + 1) ,
+for all l ≤ k and λ > λ0. On the other hand, if x ̸∈ ∪k
+l=0(Dε(El( ¯Ak))), then
+|El( ¯Ak)(x)| <
+ε
+4(k + 1) ,
+for all l ≤ k, and hence
+∥((uλ ◦ φl)El( ¯Ak) − El( ¯Ak))(x)∥ <
+ε
+2(k + 1) .
+From what we said so far we get that
+∥U 0uλA − A∥
+<
+∥U 0uλ ¯Ak − ¯Ak∥ + ε
+2
+≤
+k
+�
+l=0
+∥(uλ ◦ φl)El( ¯Ak) − El( ¯Ak)∥ + ε
+2
+<
+ε ,
+5
+
+for all λ > λ0.
+Now we show that (2) ⇒ (3).
+We assume that φn(X \ Xn) = X \ X0,
+for all n ∈ Z+. Then φ(X \ Xn+2) ⊆ X \ Xn+1. Indeed, if x ∈ X \ Xn+2 and
+φ(x) ∈ Xn+1 then φn+2(x) ∈ X0, by (∗), which is a contradiction. Furthermore,
+by (∗), we know that φ(Xn+1) ⊆ Xn and hence φ(Xn+1 \ Xn+2) ⊆ Xn \ Xn+1
+for all n ∈ Z+.
+To prove that φ(Xn+1 \ Xn+2) = Xn \ Xn+1 for all n ∈ Z+, we suppose that
+there exists n ∈ Z+ such that φ(Xn+1 \ Xn+2) ⊊ Xn \ Xn+1. If
+n0 = min{n ∈ Z+ : φ(Xn+1 \ Xn+2) ⊊ Xn \ Xn+1} ,
+then
+φ(X)
+=
+φ(Xn0+1 ∪ (X \ Xn0+1))
+=
+φ(Xn0+1) ∪ φ(X \ Xn0+1)
+⊆
+φ(Xn0+1) ∪ (X \ Xn0)
+⊊
+X ,
+which is a contradiction, since φ is surjective. Therefore, φ(Xn+1 \ Xn+2) =
+Xn \ Xn+1 for all n ∈ Z+.
+Finally we show that (3) ⇒ (2). We assume that φ(X \ X1) = X \ X0 and
+φ(Xn+1 \ Xn+2) = Xn \ Xn+1, for all n ∈ Z+. Then, X0 ⊊ X. Indeed, if
+X0 = X, then I ≡ {0} which is a contradiction. If n > 1, we have that
+φ(X \ Xn)
+=
+φ [(X \ X1) ∪ (X1 \ X2) ∪ · · · ∪ (Xn−1 \ Xn)]
+=
+φ(X \ X1) ∪ φ(X1 \ X2) ∪ · · · ∪ φ(Xn−1 \ Xn)
+=
+(X \ X0) ∪ (X0 \ X1) ∪ · · · ∪ (Xn−2 \ Xn−1)
+=
+X \ Xn−1 ,
+and hence, φn(X \ Xn) = X \ X0, for all n ∈ Z+.
+□
+By Theorem 2.2, if I ∼ {Xn}∞
+n=0 is an ideal of C0(X) ×φ Z+ with left
+approximate unit, then Xn+1 = Xn or Xn+1 ⊊ Xn for all n ∈ Z+. If I ∼
+{Xn}∞
+n=0 and Xn+1 = Xn, for all n ∈ Z+, we will write I ∼ {X0}. We obtain
+the following characterization.
+Corollary 2.3 Let I ∼ {X0} be a non-zero ideal of C0(X)×φZ+. The following
+are equivalent:
+1. I has a left approximate unit.
+2. φ(X0) = X0 and φ(X \ X0) = X \ X0.
+Proof.
+By Theorem 2.2 we have φ(X \ X0) = X \ X0.
+By (∗) we have
+φ(X0) ⊆ X0 and since φ is surjective we get φ(X0) = X0.
+□
+In the following proposition the ideals I ∼ {Xn}∞
+n=1 of C0(X) ×φ Z+ with
+left approximate unit are characterized, when φ is a homeomorphism.
+6
+
+Proposition 2.4 Let I ∼ {Xn}∞
+n=1 be a non-zero ideal of C0(X)×φ Z+, where
+φ is a homeomorphism. The following are equivalent:
+1. I has a left approximate unit.
+2. There exist S, W ⊊ X such that S is closed and φ(S) = S, the sets
+φ−1(W), φ−2(W), . . . are pairwise disjoint and φk(W) ∩ S = ∅, for all
+k ∈ Z, and
+Xn = S ∪ (∪∞
+k=nφ−k(W)) ,
+for all n ∈ Z+.
+Proof. The second condition implies the second condition of Theorem 2.2 and
+hence the implication (2) ⇒ (1) is immediate. We will prove the implication (1)
+⇒ (2).
+We set S = ∩∞
+n=0Xn.
+Clearly the set S is closed and, by (∗), we have
+φ(S) ⊆ S. We will prove that φ(S) = S. We suppose φ(S) ⊊ S. Since φ is
+surjective, there exists x ∈ X \ S such that φ(x) ∈ S. Moreover, φn(x) ∈ S for
+all n ≥ 1. However, since x /∈ S there exists n0 such that x /∈ Xn0 and hence
+φn0(x) ∈ X\X0, by Theorem 2.2, which is a contradiction since S∩(X\X0) = ∅.
+By Theorem 2.2, φ(Xn+1 \ Xn+2) = Xn \ Xn+1 for all n ∈ Z+ and hence
+φn(Xn \ Xn+1) = X0 \ X1 or equivalently Xn \ Xn+1 = φ−n(X0 \ X1) since φ is
+a homeomorphism. Furthermore, the sets φ−1(X0 \ X1), φ−2(X0 \ X1), . . . are
+pairwise disjoint.
+We set W = X0 \ X1. Clearly, φk(W) ∩ S = ∅ for all k ∈ Z, since φ(S) = S
+and φ(W) ⊆ X \ X0. Also, X0 = S ∪ (X0 \ X1) ∪ (X1 \ X2) ∪ . . . and hence
+X0 = S ∪ (∪∞
+k=0φ−k(W)) .
+Finally, for all n ∈ Z+ we have that
+X0 = Xn ∪ (∪n
+k=1Xk−1 \ Xk) = Xn ∪ (∪n
+k=1φ−k+1(W)) = Xn ∪ (∪n−1
+k=0φ−k(W)) ,
+and so
+Xn = X0 \ (∪n−1
+k=0φ−k(W)) = S ∪ (∪∞
+k=nφ−k(W)) .
+□
+In the following corollary the ideals with an approximate unit are character-
+ized.
+Corollary 2.5 Let I ∼ {Xn}∞
+n=0 be a non-zero ideal of C0(X) ×φ Z+. The
+following are equivalent:
+1. I has an approximate unit.
+2. Xn = Xn+1, for all n ∈ Z+, and φ(X \ X0) = X \ X0.
+Proof. (1) ⇒ (2) is immediate from Theorem 2.1 and Corollary 2.3.
+We show (2) ⇒ (1). If Xn = Xn+1, by (∗), we have φ(X0) ⊆ X0. Since
+φ(X \ X0) = X \ X0 and φ surjective we have φ(X0) = X0. Theorem 2.1 and
+Corollary 2.3 conclude the proof.
+□
+7
+
+Remark 2.6 If I ∼ {Xn}∞
+n=0 is an ideal of C0(X) ×φ Z+ with a left (resp.
+right) approximate unit, then it has a contractive left (resp.
+right) approxi-
+mate unit. Moreover, the semicrossed product C0(X) ×φ Z+ has a contractive
+approximate unit.
+Let B be a Banach space and C be a subspace of B.
+The set of linear
+functionals that vanish on a subspace C of B is called the annihilator of C. A
+subspace C of a Banach space B is an M-ideal in B if its annihilator is the
+kernel of a projection P on B∗ such that ∥y∥ = ∥P(y)∥ + ∥y − P(y)∥, for all y,
+where B∗ is the dual space of B.
+Effros and Ruan proved that the M-ideals in a unital operator algebra are the
+closed two-sided ideals with an approximate unit, [5, Theorem 2.2]. Therefore,
+we obtain the following corollary about the M-ideals of a semicrossed product.
+Corollary 2.7 Let I ∼ {Xn}∞
+n=0 be a non-zero ideal of C0(X) ×φ Z+, where
+X is compact. The following are equivalent:
+1. I is M-ideal.
+2. I has an approximate unit.
+3. Xn = Xn+1, for all n ∈ Z+, and φ(X \ X0) = X \ X0.
+References
+[1] G. Andreolas, M. Anoussis and C. Magiatis, Topological radicals of semi-
+crossed products, Serdica Math. J. 47 (2021), 81–92.
+[2] G. Andreolas, M. Anoussis and C. Magiatis, Compact multiplication oper-
+ators on semicrossed products, Studia Math, to appear.
+[3] K. R. Davidson, A. H. Fuller and E. T. A. Kakariadis, Semicrossed products
+of operator algebras: a survey, New York J. Math. 24A (2018), 56–86.
+[4] A. Donsig, A. Katavolos and A. Manoussos, The Jacobson radical for an-
+alytic crossed products, J. Funct. Anal. 187 (2001), 129–145.
+[5] E. G. Effros and Z-J. Ruan, On non self-adjoint operator algebras, Proc.
+Amer. Math. Soc. 110 (1990), 915—922.
+[6] J. Peters, Semicrossed products of C∗-algebras, J. Funct. Anal. 59 (1984),
+498–534.
+[7] J. Peters, The ideal structure of certain nonselfadjoint operator algebras,
+Trans. Amer. Math. Soc. 305 (1988), 333–352.
+8
+
diff --git a/tNE3T4oBgHgl3EQfNAnx/content/tmp_files/load_file.txt b/tNE3T4oBgHgl3EQfNAnx/content/tmp_files/load_file.txt
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@@ -0,0 +1,257 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf,len=256
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='04380v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='OA]  11 Jan 2023  Ideals with approximate unit in semicrossed  products  Charalampos Magiatis  January 12, 2023  Abstract  We characterize the ideals of the semicrossed product C0(X) ×φ Z+  with left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' right) approximate unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  1  Introduction and Notation  The semicrossed product is a non-selfadjoint operator algebra which is con-  structed from a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We recall the construction of the semicrossed  product we will consider in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Let X be a locally compact Hausdorff  space and φ : X → X be a continuous and proper surjection (recall that a map  φ is proper if the inverse image φ−1(K) is compact for every compact K ⊆ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  The pair (X, φ) is called a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  An action of Z+ := N ∪ {0}  on C0(X) by isometric ∗-automorphisms αn, n ∈ Z+, is obtained by defining  αn(f) = f◦φn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We write the elements of the Banach space ℓ1(Z+, C0(X)) as for-  mal series A = �  n∈Z+ U nfn with the norm given by ∥A∥1 = �  n∈Z+ ∥fn∥C0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Multiplication on ℓ1(Z+, C0(X)) is defined by setting  (U nf)(U mg) = U n+m(αm(f)g) ,  and extending by linearity and continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' With this multiplication ℓ1(Z+, C0(X))  is a Banach algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  The Banach algebra ℓ1(Z+, C0(X)) can be faithfully represented as a (con-  crete) operator algebra on a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' This is achieved by assuming a faith-  ful action of C0(X) on a Hilbert space H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Then we can define a faithful contrac-  tive representation π of ℓ1(Z+, C0(X)) on the Hilbert space H = H0 ⊗ ℓ2(Z+)  by defining π(U nf) as  π(U nf)(ξ ⊗ ek) = αk(f)ξ ⊗ ek+n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  The semicrossed product C0(X)×φZ+ is the closure of the image of ℓ1(Z+, C0(X))  in B(H) in the representation just defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We will denote an element π(U nf)  of C0(X) ×φ Z+ by U nf to simplify the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  1  For A = �  n∈Z+ U nfn ∈ ℓ1(Z+, C0(X)) we call fn ≡ En(A) the nth Fourier  coefficient of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' The maps En : ℓ1(Z+, C0(X)) → C0(X) are contractive in the  (operator) norm of C0(X)×φZ+, and therefore they extend to contractions En :  C0(X) ×φ Z+ → C0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' An element A of the semicrossed product C0(X) ×φ  Z+ is 0 if and only if En(A) = 0, for all n ∈ Z+, and thus A is completely  determined by its Fourier coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We will denote A by the formal series A =  �  n∈Z+ U nfn, where fn = En(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Note however that the series �  n∈Z+ U nfn  does not in general converge to A [6, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='9 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='512].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' The kth arithmetic mean of  A is defined to be ¯Ak =  1  k+1  �k  l=0 Sl(A), where Sl(A) = �l  n=0 U nfn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Then,  the sequence { ¯Ak}k∈Z+ is norm convergent to A [6, Remark p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='524].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We refer to  [6, 4, 3] for more information about the semicrossed product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Let {Xn}∞  n=0 be a sequence of closed subsets of X satisfying  Xn+1 ∪ φ(Xn+1) ⊆ Xn ,  (∗)  for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Peters proved in [7] that there is a one-to-one correspondence  between closed two-sided ideals I ⊆ C0(X) ×φ Z+ and sequences {Xn}∞  n=0 of  closed subsets of X satisfying (∗), under the additional assumptions that X is  metrizable and the dynamical system (X, φ) contains no periodic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' In fact,  the ideal I associated with the sequence {Xn}∞  n=0 is I = {A ∈ C0(X) ×φ Z+ :  En(A)(Xn) = {0}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We will write this as I ∼ {Xn}∞  n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Moreover, under  the above assumptions, he characterizes the maximal and prime ideals of the  semicrossed product C0(X) ×φ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Donsig, Katavolos and Manousos obtained in [4] a characterization of the  Jacobson radical for the semicrossed product C0(X)×φZ+, where X is a locally  compact metrisable space and φ : X → X is a continuous and proper surjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Andreolas, Anoussis and the author characterized in [2] the ideal generated by  the compact elements and in [1] the hypocompact and the scattered radical of  the semicrossed product C0(X) ×φ Z+, where X is a locally compact Hausdorff  space and φ : X → X is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' All these ideals are of the form  I ∼ {Xn}∞  n=0 for suitable families of closed subsets {Xn}∞  n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  In the present paper we characterize the closed two-sided ideals I ∼ {Xn}∞  n=0  of C0(X) ×φ Z+ with left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' right) approximate unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' As a consequence, we  obtain a complete characterization of ideals with left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' right) approximate  unit under the additional assumptions that X is metrizable and the dynamical  system (X, φ) contains no periodic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Recall that a left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' right) approximate unit of a Banach algebra A is a  net {uλ}λ∈Λ of elements of A such that:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' for some positive number r, ∥uλ∥ ≤ r for all λ ∈ Λ,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' lim uλa = a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' lim auλ = a), for all a ∈ A, in the norm topology of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  A net which is both a left and a right approximate unit of A is called an approx-  imate unit of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' A left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' right) approximate unit {uλ}λ∈Λ that satisfies  ∥uλ∥ ≤ 1 for all λ ∈ Λ is called a contractive left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' right) approximate unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  We will say that an ideal I of a Banach algebra A has a left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' right)  approximate unit if it has a left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' right) approximate unit as an algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  2  2  Ideals with approximate unit  In the following theorem the ideals I ∼ {Xn}∞  n=0 with right approximate unit  are characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='1 Let I ∼ {Xn}∞  n=0 be a non-zero ideal of C0(X) ×φ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' The  following are equivalent:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' I has a right approximate unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Xn = Xn+1, for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  We start by proving that (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Let I ∼ {Xn}∞  n=0 be an ideal  with right approximate unit {Vλ}λ∈Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We suppose that there exists n ∈ Z+  such that Xn+1 ⊊ Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Let  n0 = min{n ∈ Z+ : Xn+1 ⊊ Xn} ,  x0 ∈ Xn0 \\ Xn0+1 and f ∈ C0(X) such that f(x0) = 1, f(Xn0+1) = {0} and  ∥f∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Then, for A = U n0+1f, we have  ∥AVλ − A∥ ≥ ∥En0+1(AVλ − A)∥ = ∥fE0(Vλ) − f∥ ≥ |(fE0(Vλ) − f)(x0)| = 1 ,  for all λ ∈ Λ, since x0 ∈ Xn0 and E0(Vλ)(Xn0) = 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Therefore Xn = Xn+1 for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  For (2) ⇒ (1), assume that Xn = Xn+1 for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' By (∗), we get that  φ(X0) ⊆ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We will show that if {uλ}λ∈Λ is a contractive approximate unit  of the ideal C0(X \\ X0) of C0(X), then {U 0uλ}λ∈Λ is a right approximate unit  of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Since ∥uλ∥ ≤ 1, we have ∥U 0uλ∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Let A ∈ I and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Then there exists k ∈ Z+ such that  ∥A − ¯Ak∥ < ε  4 ,  where ¯Ak is the kth arithmetic mean of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Since Xn = X0, En( ¯Ak) ∈ C0(X\\X0)  and {uλ}λ∈Λ is an approximate unit of C0(X \\ X0), there exists λ0 ∈ Λ such  that  ∥El( ¯Ak)uλ − El( ¯Ak)∥ <  ε  2(k + 1) ,  for all l ≤ k and λ > λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' So, for λ > λ0 we get that  ∥AU 0uλ − A∥  <  ∥AU 0uλ − ¯AkU 0uλ + ¯AkU 0uλ − ¯Ak + ¯Ak − A∥  <  ∥AU 0uλ − ¯AkU 0uλ∥ + ∥ ¯AkU 0uλ − ¯Ak∥ + ∥A − ¯Ak∥  <  ∥ ¯AkU 0uλ − ¯Ak∥ + ε  2  ≤  k  �  l=0  ∥El( ¯Ak)uλ − El( ¯Ak)∥ + ε  2  <  ε ,  3  which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  □  In the following theorem the ideals I ∼ {Xn}∞  n=0 with left approximate unit  are characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='2 Let I ∼ {Xn}∞  n=0 be a non-zero ideal of C0(X) ×φ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' The  following are equivalent:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' I has a left approximate unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' X0 ⊊ X and φn(X \\ Xn) = X \\ X0, for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' φ(X \\ X1) = X \\ X0 and φ(Xn+1 \\ Xn+2) = Xn \\ Xn+1, for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  We start by proving that (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Let I ∼ {Xn}∞  n=0 be an ideal  with left approximate unit {Vλ}λ∈Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  First we prove that X0 ⊊ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We suppose that X0 = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Then E0(Vλ) = 0,  for all λ ∈ Λ, and hence for every U nf ∈ I we have  ∥VλU nf − U nf∥ ≥ ∥En(VλU nf − U nf)∥ = ∥E0(Vλ)f − f∥ = ∥f∥ ,  for all λ ∈ Λ, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Therefore X0 ⊊ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Now we prove that φn(X \\ Xn) = X \\ X0, for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We suppose that  there exists n ∈ Z+ such that φn(X \\ Xn) ̸⊆ X \\ X0 and let  n0 = min{n ∈ Z+ : φn(X \\ Xn) ̸⊆ X \\ X0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Then, there exist x0 ∈ X\\Xn0 such that φn0(x0) ∈ X0 and a function f ∈ C0(X)  such that f(x0) = 1, f(Xn0) = {0} and ∥f∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' If A = U n0f, we have that  A ∈ I, ∥A∥ = 1 and  ∥VλA − A∥  ≥  ∥En0(VλA − A)∥  =  ∥E0(Vλ) ◦ φn0f − f∥  ≥  |(E0(Vλ) ◦ φn0f − f)(x0)|  =  1 ,  for all λ ∈ Λ, since φn0(x0) ∈ X0 and E0(Vλ)(X0) = {0}, which is a contra-  diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Therefore φn(X \\ Xn) ⊆ X \\ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Furthermore, by (∗) we get that  φn(Xn) ⊆ X0, for all n ∈ Z+, and hence  X = φn(X) = φn(Xn ∪ (X \\ Xn)) = φn(Xn) ∪ φn(X \\ Xn) ⊆ X0 ∪ φn(X \\ Xn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Since φn(X \\ Xn) ⊆ X \\ X0 and φ is surjective, φn(X \\ Xn) = X \\ X0, for all  n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  For (2) ⇒ (1), assume that X0 ⊊ X and φn(X \\ Xn) = X \\ X0, for all  n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We will show that if {uλ}λ∈Λ is a contractive approximate unit of the  ideal C0(X \\ X0) of C0(X), then {U 0uλ}λ∈Λ is a left approximate unit of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Since ∥uλ∥ ≤ 1, we have ∥U 0uλ∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  4  Let A be a norm-one element of I and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Then there exists k ∈ Z+ such  that  ∥A − ¯Ak∥ < ε  4 ,  where ¯Ak is the kth arithmetic mean of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' For l ≤ k, let  Dε(El( ¯Ak)) =  �  x ∈ X : |El( ¯Ak)(x)| ≥  ε  4(k + 1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Since A ∈ I, we have El( ¯Ak)(Xl) = {0} and hence Dε(El( ¯Ak)) ⊆ X \\ Xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Furthermore, since φn(X \\ Xn) = X \\ X0, for all n ∈ Z+, we have that  φl(Dε(El( ¯Ak))) ⊆ X \\ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Moreover, the set Dε(El( ¯Ak)) is compact and hence  the set φl(Dε(El( ¯Ak))) is also compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' By Urysohn’s lemma, there is a norm-  one function vl ∈ C0(X) such that  vl(x) =  �  1,  x ∈ φl(Dε(El( ¯Ak)))  0,  x ∈ X0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Then, there exists λ0 ∈ Λ such that  ∥uλvl − vl∥ <  ε  2(k + 1) ,  for all l ≤ k and λ > λ0, and hence  |uλ(x) − 1| <  ε  2(k + 1) ,  for all x ∈ ∪k  l=0φl(Dε(El( ¯Ak))) and λ > λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Therefore, if x ∈ ∪k  l=0(Dε(El( ¯Ak)))  then φl(x) ∈ ∪k  l=0φl(Dε(El( ¯Ak))) and hence  ∥((uλ ◦ φl)El( ¯Ak) − El( ¯Ak))(x)∥ <  ε  2(k + 1) ,  for all l ≤ k and λ > λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' On the other hand, if x ̸∈ ∪k  l=0(Dε(El( ¯Ak))), then  |El( ¯Ak)(x)| <  ε  4(k + 1) ,  for all l ≤ k, and hence  ∥((uλ ◦ φl)El( ¯Ak) − El( ¯Ak))(x)∥ <  ε  2(k + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  From what we said so far we get that  ∥U 0uλA − A∥  <  ∥U 0uλ ¯Ak − ¯Ak∥ + ε  2  ≤  k  �  l=0  ∥(uλ ◦ φl)El( ¯Ak) − El( ¯Ak)∥ + ε  2  <  ε ,  5  for all λ > λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Now we show that (2) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  We assume that φn(X \\ Xn) = X \\ X0,  for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Then φ(X \\ Xn+2) ⊆ X \\ Xn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Indeed, if x ∈ X \\ Xn+2 and  φ(x) ∈ Xn+1 then φn+2(x) ∈ X0, by (∗), which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Furthermore,  by (∗), we know that φ(Xn+1) ⊆ Xn and hence φ(Xn+1 \\ Xn+2) ⊆ Xn \\ Xn+1  for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  To prove that φ(Xn+1 \\ Xn+2) = Xn \\ Xn+1 for all n ∈ Z+, we suppose that  there exists n ∈ Z+ such that φ(Xn+1 \\ Xn+2) ⊊ Xn \\ Xn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' If  n0 = min{n ∈ Z+ : φ(Xn+1 \\ Xn+2) ⊊ Xn \\ Xn+1} ,  then  φ(X)  =  φ(Xn0+1 ∪ (X \\ Xn0+1))  =  φ(Xn0+1) ∪ φ(X \\ Xn0+1)  ⊆  φ(Xn0+1) ∪ (X \\ Xn0)  ⊊  X ,  which is a contradiction, since φ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Therefore, φ(Xn+1 \\ Xn+2) =  Xn \\ Xn+1 for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Finally we show that (3) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We assume that φ(X \\ X1) = X \\ X0 and  φ(Xn+1 \\ Xn+2) = Xn \\ Xn+1, for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Then, X0 ⊊ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Indeed, if  X0 = X, then I ≡ {0} which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' If n > 1, we have that  φ(X \\ Xn)  =  φ [(X \\ X1) ∪ (X1 \\ X2) ∪ · · · ∪ (Xn−1 \\ Xn)]  =  φ(X \\ X1) ∪ φ(X1 \\ X2) ∪ · · · ∪ φ(Xn−1 \\ Xn)  =  (X \\ X0) ∪ (X0 \\ X1) ∪ · · · ∪ (Xn−2 \\ Xn−1)  =  X \\ Xn−1 ,  and hence, φn(X \\ Xn) = X \\ X0, for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  □  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='2, if I ∼ {Xn}∞  n=0 is an ideal of C0(X) ×φ Z+ with left  approximate unit, then Xn+1 = Xn or Xn+1 ⊊ Xn for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' If I ∼  {Xn}∞  n=0 and Xn+1 = Xn, for all n ∈ Z+, we will write I ∼ {X0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We obtain  the following characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='3 Let I ∼ {X0} be a non-zero ideal of C0(X)×φZ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' The following  are equivalent:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' I has a left approximate unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' φ(X0) = X0 and φ(X \\ X0) = X \\ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='2 we have φ(X \\ X0) = X \\ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  By (∗) we have  φ(X0) ⊆ X0 and since φ is surjective we get φ(X0) = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  □  In the following proposition the ideals I ∼ {Xn}∞  n=1 of C0(X) ×φ Z+ with  left approximate unit are characterized, when φ is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  6  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='4 Let I ∼ {Xn}∞  n=1 be a non-zero ideal of C0(X)×φ Z+, where  φ is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' The following are equivalent:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' I has a left approximate unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' There exist S, W ⊊ X such that S is closed and φ(S) = S, the sets  φ−1(W), φ−2(W), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' are pairwise disjoint and φk(W) ∩ S = ∅, for all  k ∈ Z, and  Xn = S ∪ (∪∞  k=nφ−k(W)) ,  for all n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' The second condition implies the second condition of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='2 and  hence the implication (2) ⇒ (1) is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We will prove the implication (1)  ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  We set S = ∩∞  n=0Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Clearly the set S is closed and, by (∗), we have  φ(S) ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We will prove that φ(S) = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' We suppose φ(S) ⊊ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Since φ is  surjective, there exists x ∈ X \\ S such that φ(x) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Moreover, φn(x) ∈ S for  all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' However, since x /∈ S there exists n0 such that x /∈ Xn0 and hence  φn0(x) ∈ X\\X0, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='2, which is a contradiction since S∩(X\\X0) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='2, φ(Xn+1 \\ Xn+2) = Xn \\ Xn+1 for all n ∈ Z+ and hence  φn(Xn \\ Xn+1) = X0 \\ X1 or equivalently Xn \\ Xn+1 = φ−n(X0 \\ X1) since φ is  a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Furthermore, the sets φ−1(X0 \\ X1), φ−2(X0 \\ X1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' are  pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  We set W = X0 \\ X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Clearly, φk(W) ∩ S = ∅ for all k ∈ Z, since φ(S) = S  and φ(W) ⊆ X \\ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Also, X0 = S ∪ (X0 \\ X1) ∪ (X1 \\ X2) ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' and hence  X0 = S ∪ (∪∞  k=0φ−k(W)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Finally, for all n ∈ Z+ we have that  X0 = Xn ∪ (∪n  k=1Xk−1 \\ Xk) = Xn ∪ (∪n  k=1φ−k+1(W)) = Xn ∪ (∪n−1  k=0φ−k(W)) ,  and so  Xn = X0 \\ (∪n−1  k=0φ−k(W)) = S ∪ (∪∞  k=nφ−k(W)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  □  In the following corollary the ideals with an approximate unit are character-  ized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='5 Let I ∼ {Xn}∞  n=0 be a non-zero ideal of C0(X) ×φ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' The  following are equivalent:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' I has an approximate unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Xn = Xn+1, for all n ∈ Z+, and φ(X \\ X0) = X \\ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' (1) ⇒ (2) is immediate from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='1 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  We show (2) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' If Xn = Xn+1, by (∗), we have φ(X0) ⊆ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Since  φ(X \\ X0) = X \\ X0 and φ surjective we have φ(X0) = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='1 and  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='3 conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  □  7  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='6 If I ∼ {Xn}∞  n=0 is an ideal of C0(X) ×φ Z+ with a left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  right) approximate unit, then it has a contractive left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  right) approxi-  mate unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Moreover, the semicrossed product C0(X) ×φ Z+ has a contractive  approximate unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Let B be a Banach space and C be a subspace of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  The set of linear  functionals that vanish on a subspace C of B is called the annihilator of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' A  subspace C of a Banach space B is an M-ideal in B if its annihilator is the  kernel of a projection P on B∗ such that ∥y∥ = ∥P(y)∥ + ∥y − P(y)∥, for all y,  where B∗ is the dual space of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Effros and Ruan proved that the M-ideals in a unital operator algebra are the  closed two-sided ideals with an approximate unit, [5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Therefore,  we obtain the following corollary about the M-ideals of a semicrossed product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='7 Let I ∼ {Xn}∞  n=0 be a non-zero ideal of C0(X) ×φ Z+, where  X is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' The following are equivalent:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' I is M-ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' I has an approximate unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Xn = Xn+1, for all n ∈ Z+, and φ(X \\ X0) = X \\ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  References  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Andreolas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Anoussis and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Magiatis, Topological radicals of semi-  crossed products, Serdica Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' 47 (2021), 81–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Andreolas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Anoussis and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Magiatis, Compact multiplication oper-  ators on semicrossed products, Studia Math, to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Davidson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Fuller and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Kakariadis, Semicrossed products  of operator algebras: a survey, New York J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' 24A (2018), 56–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Donsig, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Katavolos and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Manoussos, The Jacobson radical for an-  alytic crossed products, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' 187 (2001), 129–145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Effros and Z-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Ruan, On non self-adjoint operator algebras, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' 110 (1990), 915—922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Peters, Semicrossed products of C∗-algebras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' 59 (1984),  498–534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Peters, The ideal structure of certain nonselfadjoint operator algebras,  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content=' 305 (1988), 333–352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
+page_content='  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE3T4oBgHgl3EQfNAnx/content/2301.04380v1.pdf'}
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+1 
+ 
+Controlled alignment of supermoiré lattice in double-aligned graphene 
+heterostructures    
+ 
+Junxiong Hu1,2†, Junyou Tan2†, Mohammed M. Al Ezzi1,2†, Udvas Chattopadhyay1,2, 
+Jian Gou1, Yuntian Zheng1, Zihao Wang3, Jiayu Chen1, Reshmi Thottathil1, Jiangbo 
+Luo1, Kenji Watanabe4, Takashi Taniguchi5, Andrew Thye Shen Wee1, Shaffique 
+Adam1,2, Ariando Ariando1* 
+ 
+1Department of Physics, National University of Singapore, Singapore 117542 
+2Centre for Advanced 2D Materials and Graphene Research Centre, National 
+University of Singapore, Singapore 117551 
+3Department of Materials Science and Engineering, National University of Singapore, 
+Singapore 117575, Singapore 
+4Research Center for Functional Materials, National Institute for Materials Science, 
+Tsukuba, Japan 
+5International Center for Materials Nanoarchitectonics, National Institute for 
+Materials Science, Tsukuba, Japan. 
+ 
+†These authors contributed equally to this work. 
+*Email: ariando@nus.edu.sg 
+ 
+ 
+ 
+
+2 
+ 
+The supermoiré lattice, built by stacking two moiré patterns, provides a platform 
+for creating flat mini-bands and studying electron correlations. An ultimate 
+challenge in assembling a graphene supermoiré lattice is in the deterministic 
+control of its rotational alignment, which is made highly probabilistic due to the 
+random nature of the edge chirality and crystal symmetry of each component layer. 
+In this work, we present an experimental strategy to overcome this challenge and 
+realize the controlled alignment of double-aligned hBN/graphene/hBN supermoiré 
+lattice, where graphene is precisely aligned with both top hBN and bottom hBN. 
+Remarkably, we find that the crystallographic edge of neighboring graphite can 
+be used to better guide the stacking alignment, as demonstrated by the controlled 
+production of 20 moiré samples with an accuracy better than ~0.2º. Employing 
+this technique, we are first time to fabricate the perfect double-aligned graphene 
+supermoiré lattice and to observe the sharp resistivity peaks at band filling of 0, -
+4 and -8 electrons per moiré unit cell. Finally, we extend our technique to other 
+strongly correlated electron systems, such as low-angle twisted bilayer graphene 
+and ABC-stacked trilayer graphene, providing a strategy for flat-band 
+engineering in these moiré materials. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+3 
+ 
+The moiré superlattice, created by stacking van der Waals (vdW) heterostructures with 
+a controlled twist angle1-6, enables the engineering of electronic band structures and 
+provides a platform for investigating exotic quantum states, both in the weakly 
+interacting electron systems7-10, as well as recently in the strongly correlated electron 
+systems11-17. Particularly, when two moiré superlattices contact and interface together, 
+the overlay of double moiré will create a new structure called supermoiré lattice, 
+strongly modifying the lattice symmetry and electronic band structure18-22. Compared 
+with the single moiré potential, the double moiré potential will further break the lattice 
+symmetry and create isolated flat moiré minibands, providing a strategy for the band 
+flattening effect23. Recently, evidence of possible correlated states was observed in the 
+double-aligned graphene supermoiré lattice24, which triggered further effort to search 
+for other correlated phenomena, such as superconductivity and ferromagnetic states, as 
+observed in twisted graphene systems25. However, due to the sophisticated stacking and 
+the lack of control of rotational alignment, searching for these correlated phenomena in 
+double-aligned supermoiré lattice remains elusive. 
+ 
+In previous studies of graphene/hexagonal boron nitride (G/hBN) supermoiré lattice, 
+several techniques have been developed to control the rotation alignment, such as in 
+situ 
+rotation 
+mediated 
+by 
+atomic 
+force 
+microscope 
+(AFM) 
+tips 
+and 
+polydimethylsiloxane (PDMS) hemisphere20,21. However, these techniques are 
+restricted by either specialized equipment or complicated sample preparation steps. 
+Consequently, the optical alignment of straight edges is still the most popular and 
+
+4 
+ 
+reliable technique for moiré device fabrication8-10,14-19,22-30. Nevertheless, the 
+conventional optical alignment has two limitations. First, the lack of prior 
+understanding of the crystallographic orientation of graphene or hBN, leading to only 
+a 50% and 25% success rate for alignment of a single and double moiré structure, 
+respectively, because the zigzag (armchair) edge of graphene can be unintentionally 
+aligned to either the zigzag or armchair edge of hBN (Fig. 1a). Moreover, the lattice 
+symmetry of hBN layer also obscures the inversion symmetry of the moiré 
+heterostructure20,21, further decreasing the success rate to 12.5% (Fig. 3b, c). Second, 
+the optical alignment highly depends on the crystal edge of graphene itself and usually 
+it has a short and non-perfect straight edge. This leads to the error in the determination 
+of principle crystallographic axes (PCA) and decreases the accuracy of alignment. 
+Consequently, a perfect double-aligned sample, where the twist angles between 
+graphene and top/bottom hBN are both close to zero, has not been realized so far (See 
+Table S2).   
+ 
+In this work, we overcome the above two limitations and realize the control alignment 
+of double-aligned graphene supermoiré lattice. First, we use a rotation 30° technique to 
+control the alignment of top hBN and graphene (Fig. 1), while we use a flip-over 
+technique to control the alignment of top hBN and bottom hBN (Fig. 3). Based on these 
+two techniques, we can control the lattice symmetry and tune the graphene band 
+structure. In a high-quality perfect double-aligned device with mobility of ~700,000 
+cm2/Vs at 2 K, we observe sharp resistive peaks at band fillings of 0, -4, -8 electrons 
+
+5 
+ 
+per moiré unit cell, consistent with our calculated band structure. Second, we show that 
+the neighboring graphite edge can be used to better guide the alignment, as 
+demonstrated by the controlled production of 20 moiré samples with accuracy better 
+than ~0.2º, significantly improving the productivity and accuracy. Finally, we extend 
+our alignment technique to other correlated systems, such as low-angle twist bilayer 
+graphene and ABC-stacked trilayer graphene, enabling us to examine moiré potential 
+effects in these strongly correlated electron systems. 
+ 
+Fig. 1 | Control alignment of T-hBN and graphene by rotating 30°. a, Alignment of 
+T-hBN, graphene and B-hBN. The Zigzag (ZG) edge of T-hBN is aligned with the ZG 
+edge or Armchair (AR) edge of graphene and is then aligned with the ZG or AR edge 
+of B-hBN, leading to four possible combinations: C1 (0°/0°), C2 (0°/30°), C3 (30°/30°) 
+and C4 (30°/0°). The middle cartoons are two basic moiré patterns of 0° G/hBN and 30° 
+G/hBN. b,c, Calculated interaction energies for G/hBN heterostructure around 0° and 
+30°. Total energy (red circles) contributions from intralayer (elastic energy/blue 
+triangles) and interlayer interactions (adhesive energy/black squares). d-f, Side view 
+and bottom view of 30° rotational alignment. g, Optical image of G/hBN stack on 
+PDMS stamp. The red dash line profiles the outline of 0° G1/hBN and the green dash 
+line profiles the outline of 30° G2/hBN. Scale bar, 20 μm. h, Spatial map of the FWHM 
+of 2D-band for the dashed area in g. The red color map refers to 0° G1/hBN, while the 
+green color map refers to 30° G2/hBN. Scale bar, 5 μm. i, The STM topography image 
+of 0° G1/hBN shows ~14 nm moiré patterns (left, 500mV,15pA), and the topography 
+image of 30° G2/hBN shows the characteristic of quasicrystal (right, 100mV, 100pA). 
+Scale bar, 10 nm. 
+ 
+ 
+30°
+d
+e
+f
+Bottom view
+Bottom view
+Bottom view
+hBN
+Graphene
+SiO2/Si
+Rotate 30°
+PDMS
+Glass handle
+PDMS
+Glass handle
+PDMS
+Glass handle
+i
+g
+b
+c
+PDMS
+T-hBN
+B-hBN
+Graphene
+C1
+(0°/ 0°)
+C2
+(0°/ 30°)
+C3
+(30°/ 30°)
+C4
+(30°/ 0°)
+a
+h
+0° G1/hBN 30° G2/hBN
+ZG
+ZG
+AR
+ZG AR
+ZG AR
+0° G/hBN
+30° G/hBN
+30° G2/hBN
+0° G1/hBN
+40 rel.1/cm
+0 rel.1/cm
+0° G/hBN
+30° G/hBN
+PCA (x)
+PCA (y)
+hBN
+PCA (x)
+PCA (y)
+hBN
+30°
+PCA (x)
+PCA (y)
+hBN
+30°
+Graphene
+Graphene
+G1
+G2
+Graphene
+G1 G2
+0° G1/hBN
+30° G2/hBN
+
+(meV/atom)
+(meV/atom)
+0.01
+0.00
+0.01
+E
+E
+1.0
+-0.5
+0.0
+0.5
+1.0
+29.029.5
+30.030.531.0
+ (°)
+ (°)6 
+ 
+First, we study the controlled alignment between top hBN (T-hBN) and graphene. In 
+the optical alignment technique, the G/hBN moiré superlattice is achieved by aligning 
+the PCA between graphene and hBN. Based on the crystallographic structures, they can 
+form two basic moiré patterns: 0° G/hBN and 30° G/hBN (Fig. 1a, also Fig. S1). When 
+in the assembly of hBN/graphene/hBN sandwich structure, there are four possible 
+configurations: C1 (ZG/ZG/ZG, 0°/0°), C2 (ZG/ZG/AR, 0°/30°), C3 (ZG/AR/ZG, 
+30°/30°), C4 (ZG/AR/AR, 30°/0°) (Fig. 1a). All possible combinations lead to 50% 
+single alignment (C2 and C4), 25% double 30° alignment (C3), and 25% double 0° 
+alignment (C1). Our theoretical calculations investigate the adhesive energies of two 
+basic moiré patterns and find that there are two energy minima at 0° and 30° twist angles, 
+indicating these two moiré patterns are energetically stable states (Fig. 1b, c). These 
+calculations also suggest that graphene and hBN tend to self-rotate to 0° or 30° when 
+stacked together, depending on the initial states31,32. 
+ 
+Based on these calculations, we develop the so-called “30°-rotation technique” to 
+simultaneously obtain these two energy-stable states. The central concept of this 
+technique is illustrated in Fig. 1d-f. Instead of directly picking up the whole graphene 
+as in the conventional optical alignment technique, the hBN is aligned partially with a 
+graphene layer (Fig. 1d). Thanks to a stronger vdW interaction between graphene and 
+hBN, the graphene layer can be torn into two pieces. The graphene area in contact with 
+hBN can be selectively detached (G1), leaving behind a section of the graphene layer 
+(G2) on the silicon wafer. The critical step in our technique is that the left graphene (G2) 
+
+7 
+ 
+is rotated manually by a twist-angle of 30° (Fig. 1e), and G2 is then stacked at another 
+location on the same hBN (Fig. 1f). This process results in two G/hBN structures based 
+on the same hBN layer: G1/hBN and G2/hBN (Fig. 1f). Since G1 and G2 are 30° rotated 
+to each other, one of them must be 0° aligned with hBN, and the other one must be 30° 
+with hBN (Fig. S1). To examine our concept, we investigate the resulting G/hBN 
+structures using optical microscopy, Raman spectroscopy and scanning tunneling 
+microscopy (STM). Figure 1g shows the optical image of G/hBN stacks, showing two 
+sections of graphene rotated 30° to each other as indicated by the dashed lines. Figure 
+1h shows the respective FWHM mapping of Raman 2D peaks. The FWHM mapping 
+of the red area is close to 40 cm-1, indicating the 0° rotation between graphene and 
+hBN33 (Fig. S2). While the FWHM mapping of the green area is close to 20 cm-1, 
+indicating the 30° G/hBN. Moreover, the homogeneous distribution mapping indicates 
+the spatially uniform twist angles. To further confirm the twist angles, we also use STM 
+to directly characterize the moiré patterns of G1/hBN and G2/hBN (Fig. 1i). The 0° 
+G1/hBN has a clear moiré wavelength of ~14 nm34,35, while the 30° G2/hBN shows the 
+character of quasicrystal line, which has 12-fold rotational order but lacks translational 
+symmetry36,37 (see also Supplementary Fig. S3). Moreover, the twist angles are also 
+confirmed by our transport measurements, as discussed later. Therefore, using this 30°-
+rotation technique, we can always obtain the 0° G/hBN without the need to consider 
+the exact chirality edge of each layer. 
+
+8 
+ 
+  
+Fig. 2 | Perfect alignment of T-hBN and graphene using the neighboring graphite 
+edge. a. Optical image of single-layer graphene connecting with a neighboring graphite 
+edge. b, G/hBN stack after alignment using the graphite edge of a. c, Spatial map of the 
+FWHM of Raman 2D-band for the dashed area in b. d, single-layer graphene with one 
+edge has 60° with a neighboring graphite edge. e, G/hBN stack after alignment using 
+the graphite edge of d. f, Spatial map of the FWHM of Raman 2D-band for the dashed 
+area in e. g, single-layer graphene without any connection with an adjacent graphite 
+edge. h, G/hBN stack after alignment using the graphite edge of g. i, Spatial map of the 
+FWHM of Raman 2D-band for the dashed area in h. Scale bars, 20 µm (a, d, g); 5 µm 
+(b, e, h); 2 µm (c, f, i). j, Histogram of the FWHM of Raman 2D-band and twist angle 
+for 20 moiré samples. The FWHM of ~20 cm-1 (green color) corresponds to 30° G/hBN, 
+and the FWHM of ~40 cm-1 (red color) corresponds to 0° G/hBN.  
+Even though we can overcome the uncertainty in the edge chirality, there are still two 
+other challenges for optical alignment. First, it is incredibly challenging to identify a 
+single-layer graphene flake with a straight edge, which usually happens in less than 1% 
+from all exfoliated flakes (Fig. S4). Second, the single-layer graphene usually is not 
+long and straight enough, decreasing the accuracy in the alignment (Fig. S5). To 
+improve productivity and accuracy, we show that the neighboring graphite edge can be 
+better for alignment. The concept is illustrated in Fig. S6. Figure 2 illustrates three 
+typical cases of neighboring graphite edges that can be used for perfect alignment. The 
+first case is that the single-layer graphene has a direct connection with its neighboring 
+Graphene
+Graphene
+Graphene
+60°
+hBN
+Graphene
+0° G/hBN
+30° G/hBN
+2
+1
+hBN
+30° G/hBN
+1
+0° G/hBN
+2
+30° G/hBN
+1
+0° G/hBN
+2
+hBN
+c
+a
+b
+d
+e
+f
+g
+h
+i
+Graphene
+Graphene
+PCA (x)
+PCA (y)
+PCA (x)
+PCA (y)
+PCA (x)
+PCA (y)
+j
+
+40 rel. 1/cm
+0 rel. 1/cm50
+0°
+G/hBN
+30°
+G/hBN
+40
+0.2
+FWHM (cm-1)
+Twist angle (°)
+30
+20
+10
+0
+5
+10
+15
+20
+Moire samples9 
+ 
+graphite edge (Fig. 2a). In this case, the single-layer graphene must share the same PCA 
+with neighboring graphite edges, which means the PCA of graphite can be used for 
+alignment. Our Raman 2D-band mapping confirms one part belongs to 0 °G/hBN and 
+the other to 30 ° G/hBN, as the FWHM of the 2D peaks is 40 and 20 cm-1, respectively 
+(Fig. 2c). The second case is that the single-layer graphene has no direct connection 
+with the graphite edge, but one edge of graphene is multiples of 30° with the graphite 
+edge. In this case, the graphene and the neighboring graphite also share the same PCA. 
+Therefore, the PCA of graphite can be used for alignment. Raman 2D-band confirms 
+the alignment since they have 0° G/hBN and the second part has 30 °G/hBN (Fig. 2f). 
+The third case is that the single-layer graphene neither has any connection nor is 
+multiples of 30° with the neighboring graphite edge (Fig. 2g). We show that the PCA 
+of graphite can still be used for alignment if they are close to each other since our 
+Raman spectra show that one part is 0° G/hBN and the second part is 30 °G/hBN (Fig. 
+2i).  
+The best advantage of our technique is that the alignment is not limited by the geometry 
+of graphene itself, and any random shape of single-layer graphene flake can be used for 
+fabricating moiré samples, as long as its neighboring graphite can provide the straight 
+edges (Fig. S7). In order to demonstrate the high productivity and accuracy of our 
+strategy, we further fabricate 20 moiré samples, where all the single-layer graphene 
+flakes have a random edge. The histogram shows that the FWHM of all the samples is 
+larger than 40 cm-1, indicating that the accuracy of our alignment is better than ~0.2º. 
+(Fig. 2j, see also Supplementary Fig. S8). In comparison, the alignment accuracy 
+
+10 
+ 
+when using a conventional technique is merely 0.5- 1º (See the summary in Table S1). 
+Compared with single-layer graphene alignment, we find the PCA of neighboring 
+graphite edges can be better for alignment because the thick graphite edge can have a 
+longer and more straight edge, which can be used to determine a better PCA for 
+alignment. Our results show that a straight graphite edge can always ensure the 
+alignment better than 0.2°, while a non-perfect straight edge or short straight edge (5-
+10 𝜇m) will lead to a significant deviation from an ideal alignment (> 0.5°) (Fig. S5). 
+Therefore, using the neighboring graphite edge for alignment, we can not only 
+significantly improve the device yields, but also guarantee the high accuracy of 
+alignment. 
+ 
+Fig. 3 | Control alignment of T-hBN and B-hBN using the neighboring hBN surface. 
+a, Schematics of conventional pick-up and new flip-over technique for double 
+alignment. b,c, Schematics of hBN/graphene/hBN heterostructures with odd (b) and 
+even (c) hBN layers. Lattice models at the high symmetry points of the moiré pattern 
+show the atomic arrangement for each. Purple shading in (c) denotes the overlap of 
+boron (red) and nitrogen (blue) in the T-hBN and B-hBN. d, Optical images of fractured 
+hBN and alignment of T-hBN and B-hBN using the flip-over technique. The black lines 
+profile the T-hBN, and the red lines profile the B-hBN. e, Maps of the FWHM of Raman 
+2D-band of graphene for the single and double alignment using the hBN in (d). f, 
+Optical images of two neighboring hBN and one of hBN have 60° with PCA can also 
+be aligned using the flip-over technique. g, Maps of the FWHM of Raman 2D-band of 
+graphene for the single and double alignment using the hBN in (f). Scale bars, 1 µm (e, 
+a
+L = 3 (odd)
+L = 4 (Even)
+b
+c
+BN1
+BN2
+T-hBN
+B-hBN
+Graphene
+BN1
+BN2
+0° / 0°
+0° / 60°
+Conventional pick-up
+T-hBN
+B-hBN
+Graphene
+BN1
+BN1
+BN2
+Flip-over
+d
+f
+B-hBN
+T-hBN
+B-hBN
+T-hBN
+T-hBN
+B-hBN
+B-hBN
+T-hBN
+e
+g
+0° hBN/G
+30° hBN/G
+Single alignment
+Double alignment
+0° / 0°
+hBN/G/hBN
+30° / 30°
+hBN/G/hBN
+Single alignment
+Double alignment
+30° hBN/G
+30° / 30°
+hBN/G/hBN
+0° hBN/G
+0° / 0°
+hBN/G/hBN
+PCA (x)
+PCA (y)
+PCA (x)
+PCA (y)
+Flip-over
+60 °
+Flip-over
+Rel. 1/cm
+0
+60
+20
+40
+80
+Rel. 1/cm
+0
+60
+20
+40
+80
+C
+B
+N
+B
+N
+N
+N
+C
+N
+B
+N
+N
+B
+B
+N
+
+11 
+ 
+g). 
+Next, we study the control alignment of T-hBN and B-hBN. Because of the uncertain 
+edge chirality of B-hBN, there are again two possible cases when we stack the 0° and 
+30° T-hBN/G on the B-hBN: C1/C3, if T-hBN and B-hBN have the same edges (Fig.S9) 
+and C2/C4, if T-hBN and B-hBN have the different edges (Fig.S10). To secure the 
+crystallographic orientation of T-hBN and B-hBN, we can use the same edge of the 
+same hBN for alignment. However, apart from the edge chirality, we also need to 
+consider the lattice symmetry of each hBN layer. As shown in Fig. 3a, for the 
+conventional pick-up process, we pick up BN1 and stack it on BN2, so the bottom 
+surface of BN1 is in contact with the top surface of BN2. Depending on the surface 
+symmetry determined by the layer number of hBN, the final stack can be 0°/0° or 0°/60° 
+(Fig. 3b, c). The 0°/0° heterostructure has three-fold rotationally symmetric, and the 
+overall structure breaks inversion symmetry, while the 0°/60° heterostructure has a six-
+fold rotationally symmetric and the structure host inversion symmetry. Even though 
+these two structures have the same moiré wavelength, the change of local stack induces 
+a change in the atomic relaxation, consequently leading to totally different bandgaps20,23. 
+ 
+In order to control the alignment of T-hBN and B-hBN, we then develop a “flip-over 
+technique” based on the same crystal edge as well as the same atomic surface. As 
+illustrated in Fig.3a, the key step is that BN2 is flipped over that makes the bottom 
+surface of BN2 exposed to the same bottom surface of BN1. In this case, the same 
+lattice symmetry of T-hBN and B-hBN can be guaranteed. Figure 3d shows the optical 
+
+12 
+ 
+images of two pieces of hBN that can be regarded as T-hBN and B-hBN, respectively. 
+Combining the 30° rotation and flip-over technique, we can first realize the single 
+alignment and then the double alignment, as demonstrated by the increase in FWHM 
+of Raman 2D-band from ~40 cm-1 (single alignment) to ~70 cm-1 (double alignment) at 
+the same area19 (Fig. 3d, see also Fig. S11). In addition, we show that if the two hBN 
+layers are not coming from the same fractured hBN, but their straight edges have the 
+multiples of 30°, which means these two hBN can also share the common PCA (Fig. 
+3f). In this case, we can also realize a perfect double alignment based on the same 
+surface of hBN, as confirmed by the Raman 2D-band shown in the same area (Fig. 3g, 
+see also Supplementary Fig. S11). Therefore, combining the 30° rotation and flip-over 
+technique, we can overcome the uncertainty in the edge chirality and lattice symmetry 
+during rotation alignment, and realize the 100% success rate for the alignment of the 
+double moiré structure. This is in stark contrast to the conventional technique that can 
+only lead to a 12.5% success rate. 
+ 
+1/3
+1/4
+1/5
+1/6
+1/7
+6 10 14
+-14 -10 -6
+-18
+18
+log (Rxx)
+0
+5
+10
+2
+-2
+Vxy
+Vxx
+Vtg
+I
+b
+c
+d
+a
+f
+g
+C1
+ϴt= 0°
+ϴb= 0°
+C3
+ϴt= 30°
+ϴb= 30°
+1/3
+1/4
+1/5
+1/6
+1/7
+6 10 14
+-14 -10 -6
+h
+e
+T-hBN
+B-hBN
+Graphene
+18
+10 14 18
+-10 -6 -2
+6
+-14
+C2
+ϴt= 0°
+ϴb= 30°
+CNP
+CNP
+CNP
+log (Rxx)
+0
+5
+10
+log (Rxx)
+0
+5
+10
+2
+-2
+-18
+2
+-18
+
+9
+0.3
+6.
+E
+0.2
+B
+3
+0.1
+0
+0.0
+-8
+-4
+0
+4
+8
+n/no9
+0.3
+6
+E
+0.2
+B
+3.
+0.1
+0
+0.0
+-8
+-4
+0
+4
+8
+n/no3
+30
+20
+kQ
+Rt
+10
+6
+.4
+-2
+0
+2
+6
+n(1012
+cm230
+9.
+20
+(Uy)
+6
+kQ
+i
+0.
+6
+2
+-10
+2
+6
+n(1012
+2cm-230
+30
+20
+(Uy)
+20
+k21
+10
+6
+n10
+13
+cm29
+E
+6
+B
+3
+0
+-4
+-2
+0
+2
+4
+n(1012
+cm2)13 
+ 
+Fig. 4 | Lattice symmetry and band structure tuned by moiré potential in T-
+hBN/graphene/B-hBN heterostructure. a, Art view of the supermoiré lattice with 
+twist angles (θt and θb) between graphene and T-hBN and B-hBN. b, Schematics of the 
+top-gate device with double moiré. Longitudinal resistance (Left axis) and Hall 
+resistance (right axis) with B = 0.5 T versus carrier density for c, C1 (0°/0°), d, C2 
+(0°/30°), and e, C3 (30°/30°). The insets show the corresponding band structures at the 
+K-point. Landau fan diagram of f, C1, g, C2 and h, C3 plotted in magnetic field (left) 
+and corresponding 𝜙/𝜙0 versus 𝑛/𝑛0. 𝜙/𝜙0 and 𝑛/𝑛0 are the normalized magnetic 
+flux and carrier density, respectively. The top number are the topological index ν to be 
+±2, ±6, ±10, etc. 𝑇 =  2 𝐾. 
+ 
+Finally, to reveal the role of moiré potential in the reconstruction of lattice symmetry 
+and band structure, we study the electronic properties of C1-C3 configurations by top-
+gate devices38,39 (Fig. 4a, b). For double-aligned C1 (0°/0°) device, apart from the 
+neutrality point (CNP), there are also two successive satellite peaks that appear at hole-
+side, locating at -ns and -2ns, where ns=2.3×1012 cm-2 (Fig. 4c). Since ns is the carrier 
+density required to reach the edge of its first Brillouin zone of a moiré pattern λ by 
+𝑛𝑠 =
+8
+√3λ2  8-10, we can calculate the moiré wavelength of C1 of ~14 nm, in good 
+agreement with our Raman and STM measurements. Moreover, our band structure 
+calculation (Inset of Fig. 4c) shows that the double moiré superlattice significantly 
+splits the hole-sided bands at higher energy into minibands with fully developed gaps 
+at electron fillings of − ns, −2 ns and −3 ns, which is consistent with our observation 
+since these satellite peaks in hole-side bands are more developed than in the electron-
+side. Moreover, apart from these band filling peaks, we do not observe any extra peaks 
+which may come from the misalignment18-24. Therefore, we can conclude that the C1 
+sample is a perfect double-aligned sample, where the twist angles between graphene 
+and top/bottom hBN are both close to zero. A similar feature is also observed in another 
+
+14 
+ 
+perfect double-aligned sample (Fig. S12). While for single-aligned C2 (0°/30°) device, 
+we only observe one satellite peak at hole-side, locating at -ns = -2.2×1012 cm-2, 
+corresponding to the moiré wavelength of ~14.5 nm (Fig. 4d). The slightly larger moiré 
+periodicity in C2 can be attributed to the strain effect in the heterostructure19,24. The 
+lack of -2ns peak in C2 suggests that the overlap between the second and third band, as 
+our band structure calculation shows a rather small bandgap (Inset of Fig. 4d). When 
+the moiré potential disappears in C3 (30°/30°) device, there is only one main Dirac 
+point, consistent with our band structure calculation (Inset of Fig. 4e). 
+ 
+When the electrons simultaneously subjected to both a magnetic field and a spatially 
+periodic electrostatic fields, the energy spectrum develops into a Landau fan with a 
+fractal structure known as the Hofstadter butterfly8-10, which can be renormalized into 
+a diagram defined by the Diophantine equation: 
+𝑛
+𝑛0 = 𝑣
+𝜙
+𝜙0 + 𝑠 , where v is the Hall 
+conductivity in units of a conductance quantum e2/h, as indicated by black solid lines 
+in Fig. 4f, g, h, with topological index ν of ±2, ±6, ±10, ±14,..., and s the index of band 
+filling. 𝜙 = 𝐵 ⋅ 𝐴 is flux per moiré unit area at magnetic field B, and 𝜙0 = ℎ/𝑒 is a 
+flux quantum with ℎ  being the Planck’s constant. Figure 4f shows the fracture 
+spectrum of C1 (0°/0°) device. The straight minigaps arises at 
+𝜙
+𝜙0𝑞 (𝑞 = 1, 2,3, … ) , 
+resulting in Brown-Zak (BZ) oscillations with the fundamental period field of Bf = 
+24.49 T (𝜙0/A). Thus, we can calculate the moiré periodicity of D1 of 13.97 nm, which 
+is consistent with the first band filling density at -ns = -2.35 × 1012 cm-2. Similarly, the 
+BZ oscillations for C2 device has the Bf = 23 T (Fig. 4g). We can then calculate the 
+
+15 
+ 
+moiré periodicity of C2 to be 14.417 nm, also consistent with the first band filling -ns 
+= - 2.2 × 1012 cm-2. When the moiré periods disappear in C3 device, the BZ oscillations 
+also disappear, as there is only the Landau levels spectrum (Fig. 4h). These results 
+suggest that the control alignment of moiré patterns, acting as a periodic moiré potential, 
+can tune the lattice symmetry and engineer the band structure. 
+  
+In conclusion, we develop a generic strategy to overcome the edge chirality and lattice 
+symmetry uncertainty in rotation alignment. Moreover, we show that the neighboring 
+graphite edge can be used for better alignment of the stacking structures, significantly 
+improving the device yield and alignment accuracy. Compared with previous 
+conventional techniques, the current technique is easier and reliable to operate with 
+robust and definite control of alignment. Considering the emerging area of 
+“twistronics”, our technique can be beneficial for much effort in this area in many 
+laboratories. For example, our strategy can also be applied to the family of transition 
+metal dichalcogenide (TMD) semiconductors3,4 such as WSe2/WS2 moiré superlattice, 
+where a prior measurement of edge chirality in each crystal is necessary before 
+alignment40-43. To show the universality of our technique, we also extend our technique 
+to other correlated systems, like low-angle twisted bilayer graphene (Supplementary 
+Fig. S13) and ABC-stacked trilayer graphene (Supplementary Fig. S14). We believe 
+our technique can help effort in exploring the physics of strong electronic correlations 
+and non-trivial band topology in these moiré materials. 
+ 
+
+16 
+ 
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+ 
+ 
+
+19 
+ 
+Acknowledgements 
+This work is supported by the Ministry of Education (MOE) Singapore under the 
+Academic Research Fund Tier 2 (Grant No. MOE-T2EP50120-0015), by the Agency 
+for Science, Technology and Research (A*STAR) under its Advanced Manufacturing 
+and Engineering (AME) Individual Research Grant (IRG) (Grant No. A2083c0054), 
+and by the National Research Foundation (NRF) of Singapore under its NRF-ISF joint 
+program (Grant No. NRF2020-NRF-ISF004-3518). K.W. and T.T. acknowledge 
+support from the JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 
+21H05233). 
+Author contributions 
+A.A. conceived and supervised the project. J.X.H. designed and performed the 
+experiments. J.Y.T. developed 30°-rotation alignment techniques and provided support 
+and training on device fabrication process. M.M.A.E and U.C. carried out theoretical 
+calculations under the supervision of S.A. Y.T.Z., J.Y.C., R.T. and J.B.L. helped to 
+prepare sample and make the device. Z.H.W. helped the data analysis and interpretation. 
+T.T. and K.W. provided the bulk hBN crystals. J.G. helped STM measurement under 
+the supervision of A.T.S.W. J.X.H. and A. A. analyzed the experimental data and wrote 
+the manuscript. All authors discussed the results and commented on the manuscript. 
+ 
+Competing interests 
+The authors declare no competing interests. 
+Additional information 
+
+20 
+ 
+Supplementary information is available for this paper at. 
+Reprints and permissions information is available at. 
+Correspondence should be addressed to A. Ariando. 
+Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims 
+in published maps and institutional affiliations. 
+ 
+Data availability. The data that support the plots within this paper and other finding of 
+this study are available from the corresponding authors upon reasonable request. 
+
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+page_content='2†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Jian Gou1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Yuntian Zheng1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Zihao Wang3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Jiayu Chen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Reshmi Thottathil1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Jiangbo  Luo1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Kenji Watanabe4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Takashi Taniguchi5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Andrew Thye Shen Wee1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Shaffique  Adam1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Ariando Ariando1*  1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' National University of Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Singapore 117542  2Centre for Advanced 2D Materials and Graphene Research Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' National  University of Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Singapore 117551  3Department of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' National University of Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Singapore 117575,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Singapore  4Research Center for Functional Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Japan  5International Center for Materials Nanoarchitectonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' National Institute for  Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Email: ariando@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='sg  2  The supermoiré lattice, built by stacking two moiré patterns, provides a platform  for creating flat mini-bands and studying electron correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' An ultimate  challenge in assembling a graphene supermoiré lattice is in the deterministic  control of its rotational alignment, which is made highly probabilistic due to the  random nature of the edge chirality and crystal symmetry of each component layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  In this work, we present an experimental strategy to overcome this challenge and  realize the controlled alignment of double-aligned hBN/graphene/hBN supermoiré  lattice, where graphene is precisely aligned with both top hBN and bottom hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Remarkably, we find that the crystallographic edge of neighboring graphite can  be used to better guide the stacking alignment, as demonstrated by the controlled  production of 20 moiré samples with an accuracy better than ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='2º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Employing  this technique, we are first time to fabricate the perfect double-aligned graphene  supermoiré lattice and to observe the sharp resistivity peaks at band filling of 0, -  4 and -8 electrons per moiré unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Finally, we extend our technique to other  strongly correlated electron systems, such as low-angle twisted bilayer graphene  and ABC-stacked trilayer graphene, providing a strategy for flat-band  engineering in these moiré materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  3  The moiré superlattice, created by stacking van der Waals (vdW) heterostructures with  a controlled twist angle1-6, enables the engineering of electronic band structures and  provides a platform for investigating exotic quantum states, both in the weakly  interacting electron systems7-10, as well as recently in the strongly correlated electron  systems11-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Particularly, when two moiré superlattices contact and interface together,  the overlay of double moiré will create a new structure called supermoiré lattice,  strongly modifying the lattice symmetry and electronic band structure18-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Compared  with the single moiré potential, the double moiré potential will further break the lattice  symmetry and create isolated flat moiré minibands, providing a strategy for the band  flattening effect23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Recently, evidence of possible correlated states was observed in the  double-aligned graphene supermoiré lattice24, which triggered further effort to search  for other correlated phenomena, such as superconductivity and ferromagnetic states, as  observed in twisted graphene systems25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' However, due to the sophisticated stacking and  the lack of control of rotational alignment, searching for these correlated phenomena in  double-aligned supermoiré lattice remains elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  In previous studies of graphene/hexagonal boron nitride (G/hBN) supermoiré lattice,  several techniques have been developed to control the rotation alignment, such as in  situ  rotation  mediated  by  atomic  force  microscope  (AFM)  tips  and  polydimethylsiloxane (PDMS) hemisphere20,21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' However, these techniques are  restricted by either specialized equipment or complicated sample preparation steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Consequently, the optical alignment of straight edges is still the most popular and  4  reliable technique for moiré device fabrication8-10,14-19,22-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Nevertheless, the  conventional optical alignment has two limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' First, the lack of prior  understanding of the crystallographic orientation of graphene or hBN, leading to only  a 50% and 25% success rate for alignment of a single and double moiré structure,  respectively, because the zigzag (armchair) edge of graphene can be unintentionally  aligned to either the zigzag or armchair edge of hBN (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Moreover, the lattice  symmetry of hBN layer also obscures the inversion symmetry of the moiré  heterostructure20,21, further decreasing the success rate to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='5% (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 3b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Second,  the optical alignment highly depends on the crystal edge of graphene itself and usually  it has a short and non-perfect straight edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' This leads to the error in the determination  of principle crystallographic axes (PCA) and decreases the accuracy of alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Consequently, a perfect double-aligned sample, where the twist angles between  graphene and top/bottom hBN are both close to zero, has not been realized so far (See  Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  In this work, we overcome the above two limitations and realize the control alignment  of double-aligned graphene supermoiré lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' First, we use a rotation 30° technique to  control the alignment of top hBN and graphene (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1), while we use a flip-over  technique to control the alignment of top hBN and bottom hBN (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Based on these  two techniques, we can control the lattice symmetry and tune the graphene band  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' In a high-quality perfect double-aligned device with mobility of ~700,000  cm2/Vs at 2 K, we observe sharp resistive peaks at band fillings of 0, -4, -8 electrons  5  per moiré unit cell, consistent with our calculated band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Second, we show that  the neighboring graphite edge can be used to better guide the alignment, as  demonstrated by the controlled production of 20 moiré samples with accuracy better  than ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='2º, significantly improving the productivity and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Finally, we extend  our alignment technique to other correlated systems, such as low-angle twist bilayer  graphene and ABC-stacked trilayer graphene, enabling us to examine moiré potential  effects in these strongly correlated electron systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1 | Control alignment of T-hBN and graphene by rotating 30°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' a, Alignment of  T-hBN, graphene and B-hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The Zigzag (ZG) edge of T-hBN is aligned with the ZG  edge or Armchair (AR) edge of graphene and is then aligned with the ZG or AR edge  of B-hBN, leading to four possible combinations: C1 (0°/0°), C2 (0°/30°), C3 (30°/30°)  and C4 (30°/0°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The middle cartoons are two basic moiré patterns of 0° G/hBN and 30°  G/hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' b,c, Calculated interaction energies for G/hBN heterostructure around 0° and  30°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Total energy (red circles) contributions from intralayer (elastic energy/blue  triangles) and interlayer interactions (adhesive energy/black squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' d-f, Side view  and bottom view of 30° rotational alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' g, Optical image of G/hBN stack on  PDMS stamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The red dash line profiles the outline of 0° G1/hBN and the green dash  line profiles the outline of 30° G2/hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Scale bar, 20 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' h, Spatial map of the FWHM  of 2D-band for the dashed area in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The red color map refers to 0° G1/hBN, while the  green color map refers to 30° G2/hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Scale bar, 5 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' i, The STM topography image  of 0° G1/hBN shows ~14 nm moiré patterns (left, 500mV,15pA), and the topography  image of 30° G2/hBN shows the characteristic of quasicrystal (right, 100mV, 100pA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Scale bar, 10 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  30°  d  e  f  Bottom view  Bottom view  Bottom view  hBN  Graphene  SiO2/Si  Rotate 30°  PDMS  Glass handle  PDMS  Glass handle  PDMS  Glass handle  i  g  b  c  PDMS  T-hBN  B-hBN  Graphene  C1  (0°/ 0°)  C2  (0°/ 30°)  C3  (30°/ 30°)  C4  (30°/ 0°)  a  h  0° G1/hBN 30° G2/hBN  ZG  ZG  AR  ZG AR  ZG AR  0° G/hBN  30° G/hBN  30° G2/hBN  0° G1/hBN  40 rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='1/cm  0 rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='1/cm  0° G/hBN  30° G/hBN  PCA (x)  PCA (y)  hBN  PCA (x)  PCA (y)  hBN  30°  PCA (x)  PCA (y)  hBN  30°  Graphene  Graphene  G1  G2  Graphene  G1 G2  0° G1/hBN  30° G2/hBN  (meV/atom)  (meV/atom)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='01  E  E  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='0  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='5  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='0  (°)  (°)6  First, we study the controlled alignment between top hBN (T-hBN) and graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' In  the optical alignment technique, the G/hBN moiré superlattice is achieved by aligning  the PCA between graphene and hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Based on the crystallographic structures, they can  form two basic moiré patterns: 0° G/hBN and 30° G/hBN (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1a, also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' When  in the assembly of hBN/graphene/hBN sandwich structure, there are four possible  configurations: C1 (ZG/ZG/ZG, 0°/0°), C2 (ZG/ZG/AR, 0°/30°), C3 (ZG/AR/ZG,  30°/30°), C4 (ZG/AR/AR, 30°/0°) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' All possible combinations lead to 50%  single alignment (C2 and C4), 25% double 30° alignment (C3), and 25% double 0°  alignment (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Our theoretical calculations investigate the adhesive energies of two  basic moiré patterns and find that there are two energy minima at 0° and 30° twist angles,  indicating these two moiré patterns are energetically stable states (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' These  calculations also suggest that graphene and hBN tend to self-rotate to 0° or 30° when  stacked together, depending on the initial states31,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Based on these calculations, we develop the so-called “30°-rotation technique” to  simultaneously obtain these two energy-stable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The central concept of this  technique is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1d-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Instead of directly picking up the whole graphene  as in the conventional optical alignment technique, the hBN is aligned partially with a  graphene layer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Thanks to a stronger vdW interaction between graphene and  hBN, the graphene layer can be torn into two pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The graphene area in contact with  hBN can be selectively detached (G1), leaving behind a section of the graphene layer  (G2) on the silicon wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The critical step in our technique is that the left graphene (G2)  7  is rotated manually by a twist-angle of 30° (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1e), and G2 is then stacked at another  location on the same hBN (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' This process results in two G/hBN structures based  on the same hBN layer: G1/hBN and G2/hBN (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Since G1 and G2 are 30° rotated  to each other, one of them must be 0° aligned with hBN, and the other one must be 30°  with hBN (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' To examine our concept, we investigate the resulting G/hBN  structures using optical microscopy, Raman spectroscopy and scanning tunneling  microscopy (STM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Figure 1g shows the optical image of G/hBN stacks, showing two  sections of graphene rotated 30° to each other as indicated by the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Figure  1h shows the respective FWHM mapping of Raman 2D peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The FWHM mapping  of the red area is close to 40 cm-1, indicating the 0° rotation between graphene and  hBN33 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' While the FWHM mapping of the green area is close to 20 cm-1,  indicating the 30° G/hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Moreover, the homogeneous distribution mapping indicates  the spatially uniform twist angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' To further confirm the twist angles, we also use STM  to directly characterize the moiré patterns of G1/hBN and G2/hBN (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The 0°  G1/hBN has a clear moiré wavelength of ~14 nm34,35, while the 30° G2/hBN shows the  character of quasicrystal line, which has 12-fold rotational order but lacks translational  symmetry36,37 (see also Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Moreover, the twist angles are also  confirmed by our transport measurements, as discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Therefore, using this 30°-  rotation technique, we can always obtain the 0° G/hBN without the need to consider  the exact chirality edge of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 2 | Perfect alignment of T-hBN and graphene using the neighboring graphite  edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Optical image of single-layer graphene connecting with a neighboring graphite  edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' b, G/hBN stack after alignment using the graphite edge of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' c, Spatial map of the  FWHM of Raman 2D-band for the dashed area in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' d, single-layer graphene with one  edge has 60° with a neighboring graphite edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' e, G/hBN stack after alignment using  the graphite edge of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' f, Spatial map of the FWHM of Raman 2D-band for the dashed  area in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' g, single-layer graphene without any connection with an adjacent graphite  edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' h, G/hBN stack after alignment using the graphite edge of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' i, Spatial map of the  FWHM of Raman 2D-band for the dashed area in h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Scale bars, 20 µm (a, d, g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 5 µm  (b, e, h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 2 µm (c, f, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' j, Histogram of the FWHM of Raman 2D-band and twist angle  for 20 moiré samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The FWHM of ~20 cm-1 (green color) corresponds to 30° G/hBN,  and the FWHM of ~40 cm-1 (red color) corresponds to 0° G/hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Even though we can overcome the uncertainty in the edge chirality, there are still two  other challenges for optical alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' First, it is incredibly challenging to identify a  single-layer graphene flake with a straight edge, which usually happens in less than 1%  from all exfoliated flakes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Second, the single-layer graphene usually is not  long and straight enough, decreasing the accuracy in the alignment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' To  improve productivity and accuracy, we show that the neighboring graphite edge can be  better for alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The concept is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Figure 2 illustrates three  typical cases of neighboring graphite edges that can be used for perfect alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The  first case is that the single-layer graphene has a direct connection with its neighboring  Graphene  Graphene  Graphene  60°  hBN  Graphene  0° G/hBN  30° G/hBN  2  1  hBN  30° G/hBN  1  0° G/hBN  2  30° G/hBN  1  0° G/hBN  2  hBN  c  a  b  d  e  f  g  h  i  Graphene  Graphene  PCA (x)  PCA (y)  PCA (x)  PCA (y)  PCA (x)  PCA (y)  j  40 rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1/cm  0 rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1/cm50  0°  G/hBN  30°  G/hBN  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='2  FWHM (cm-1)  Twist angle (°)  30  20  10  0  5  10  15  20  Moire samples9  graphite edge (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' In this case, the single-layer graphene must share the same PCA  with neighboring graphite edges, which means the PCA of graphite can be used for  alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Our Raman 2D-band mapping confirms one part belongs to 0 °G/hBN and  the other to 30 ° G/hBN, as the FWHM of the 2D peaks is 40 and 20 cm-1, respectively  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The second case is that the single-layer graphene has no direct connection  with the graphite edge, but one edge of graphene is multiples of 30° with the graphite  edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' In this case, the graphene and the neighboring graphite also share the same PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Therefore, the PCA of graphite can be used for alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Raman 2D-band confirms  the alignment since they have 0° G/hBN and the second part has 30 °G/hBN (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  The third case is that the single-layer graphene neither has any connection nor is  multiples of 30° with the neighboring graphite edge (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 2g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' We show that the PCA  of graphite can still be used for alignment if they are close to each other since our  Raman spectra show that one part is 0° G/hBN and the second part is 30 °G/hBN (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  2i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  The best advantage of our technique is that the alignment is not limited by the geometry  of graphene itself, and any random shape of single-layer graphene flake can be used for  fabricating moiré samples, as long as its neighboring graphite can provide the straight  edges (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' In order to demonstrate the high productivity and accuracy of our  strategy, we further fabricate 20 moiré samples, where all the single-layer graphene  flakes have a random edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The histogram shows that the FWHM of all the samples is  larger than 40 cm-1, indicating that the accuracy of our alignment is better than ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='2º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 2j, see also Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' In comparison, the alignment accuracy  10  when using a conventional technique is merely 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='5- 1º (See the summary in Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Compared with single-layer graphene alignment, we find the PCA of neighboring  graphite edges can be better for alignment because the thick graphite edge can have a  longer and more straight edge, which can be used to determine a better PCA for  alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Our results show that a straight graphite edge can always ensure the  alignment better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='2°, while a non-perfect straight edge or short straight edge (5-  10 𝜇m) will lead to a significant deviation from an ideal alignment (> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='5°) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Therefore, using the neighboring graphite edge for alignment, we can not only  significantly improve the device yields, but also guarantee the high accuracy of  alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 3 | Control alignment of T-hBN and B-hBN using the neighboring hBN surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  a, Schematics of conventional pick-up and new flip-over technique for double  alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' b,c, Schematics of hBN/graphene/hBN heterostructures with odd (b) and  even (c) hBN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Lattice models at the high symmetry points of the moiré pattern  show the atomic arrangement for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Purple shading in (c) denotes the overlap of  boron (red) and nitrogen (blue) in the T-hBN and B-hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' d, Optical images of fractured  hBN and alignment of T-hBN and B-hBN using the flip-over technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The black lines  profile the T-hBN, and the red lines profile the B-hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' e, Maps of the FWHM of Raman  2D-band of graphene for the single and double alignment using the hBN in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' f,  Optical images of two neighboring hBN and one of hBN have 60° with PCA can also  be aligned using the flip-over technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' g, Maps of the FWHM of Raman 2D-band of  graphene for the single and double alignment using the hBN in (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Scale bars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1 µm (e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='L = 3 (odd)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='L = 4 (Even)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='c  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='BN1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='BN2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='T-hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='B-hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Graphene  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='BN1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='BN2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='0° / 0°  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='0° / 60°  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Conventional pick-up  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='T-hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='B-hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Graphene  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='BN1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='BN1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='BN2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Flip-over  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='B-hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='T-hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='B-hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='T-hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='T-hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='B-hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='B-hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='T-hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='g  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='0° hBN/G  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='30° hBN/G  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Single alignment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Double alignment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='0° / 0°  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='hBN/G/hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='30° / 30°  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='hBN/G/hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Single alignment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Double alignment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='30° hBN/G  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='30° / 30°  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='hBN/G/hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='0° hBN/G  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='0° / 0°  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='hBN/G/hBN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='PCA (x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='PCA (y)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='PCA (x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='PCA (y)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Flip-over  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='60 °  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Flip-over  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1/cm  0  60  20  40  80  Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 1/cm  0  60  20  40  80  C  B  N  B  N  N  N  C  N  B  N  N  B  B  N  11  g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Next, we study the control alignment of T-hBN and B-hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Because of the uncertain  edge chirality of B-hBN, there are again two possible cases when we stack the 0° and  30° T-hBN/G on the B-hBN: C1/C3, if T-hBN and B-hBN have the same edges (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='S9)  and C2/C4, if T-hBN and B-hBN have the different edges (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' To secure the  crystallographic orientation of T-hBN and B-hBN, we can use the same edge of the  same hBN for alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' However, apart from the edge chirality, we also need to  consider the lattice symmetry of each hBN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 3a, for the  conventional pick-up process, we pick up BN1 and stack it on BN2, so the bottom  surface of BN1 is in contact with the top surface of BN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Depending on the surface  symmetry determined by the layer number of hBN, the final stack can be 0°/0° or 0°/60°  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 3b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The 0°/0° heterostructure has three-fold rotationally symmetric, and the  overall structure breaks inversion symmetry, while the 0°/60° heterostructure has a six-  fold rotationally symmetric and the structure host inversion symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Even though  these two structures have the same moiré wavelength, the change of local stack induces  a change in the atomic relaxation, consequently leading to totally different bandgaps20,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  In order to control the alignment of T-hBN and B-hBN, we then develop a “flip-over  technique” based on the same crystal edge as well as the same atomic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' As  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='3a, the key step is that BN2 is flipped over that makes the bottom  surface of BN2 exposed to the same bottom surface of BN1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' In this case, the same  lattice symmetry of T-hBN and B-hBN can be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Figure 3d shows the optical  12  images of two pieces of hBN that can be regarded as T-hBN and B-hBN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Combining the 30° rotation and flip-over technique, we can first realize the single  alignment and then the double alignment, as demonstrated by the increase in FWHM  of Raman 2D-band from ~40 cm-1 (single alignment) to ~70 cm-1 (double alignment) at  the same area19 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 3d, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' In addition, we show that if the two hBN  layers are not coming from the same fractured hBN, but their straight edges have the  multiples of 30°, which means these two hBN can also share the common PCA (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  3f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' In this case, we can also realize a perfect double alignment based on the same  surface of hBN, as confirmed by the Raman 2D-band shown in the same area (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 3g,  see also Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Therefore, combining the 30° rotation and flip-over  technique, we can overcome the uncertainty in the edge chirality and lattice symmetry  during rotation alignment, and realize the 100% success rate for the alignment of the  double moiré structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' This is in stark contrast to the conventional technique that can  only lead to a 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='5% success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  1/3  1/4  1/5  1/6  1/7  6 10 14  14 -10 -6  18  18  log (Rxx)  0  5  10  2  2  Vxy  Vxx  Vtg  I  b  c  d  a  f  g  C1  ϴt= 0°  ϴb= 0°  C3  ϴt= 30°  ϴb= 30°  1/3  1/4  1/5  1/6  1/7  6 10 14  14 -10 -6  h  e  T-hBN  B-hBN  Graphene  18  10 14 18  10 -6 -2  6  14  C2  ϴt= 0°  ϴb= 30°  CNP  CNP  CNP  log (Rxx)  0  5  10  log (Rxx)  0  5  10  2  2  18  2  18  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='2  B  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='0  8  4  0  4  8  n/no9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='3  6  E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='2  B  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='0  8  4  0  4  8  n/no3  30  20  kQ  Rt  10  6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='4  2  0  2  6  n(1012  cm230  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  20  (Uy)  6  kQ  i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  6  2  10  2  6  n(1012  2cm-230  30  20  (Uy)  20  k21  10  6  n10  13  cm29  E  6  B  3  0  4  2  0  2  4  n(1012  cm2)13  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 4 | Lattice symmetry and band structure tuned by moiré potential in T-  hBN/graphene/B-hBN heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' a, Art view of the supermoiré lattice with  twist angles (θt and θb) between graphene and T-hBN and B-hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' b, Schematics of the  top-gate device with double moiré.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Longitudinal resistance (Left axis) and Hall  resistance (right axis) with B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='5 T versus carrier density for c, C1 (0°/0°), d, C2  (0°/30°), and e, C3 (30°/30°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The insets show the corresponding band structures at the  K-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Landau fan diagram of f, C1, g, C2 and h, C3 plotted in magnetic field (left)  and corresponding 𝜙/𝜙0 versus 𝑛/𝑛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 𝜙/𝜙0 and 𝑛/𝑛0 are the normalized magnetic  flux and carrier density, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The top number are the topological index ν to be  ±2, ±6, ±10, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 𝑇 =  2 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Finally, to reveal the role of moiré potential in the reconstruction of lattice symmetry  and band structure, we study the electronic properties of C1-C3 configurations by top-  gate devices38,39 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 4a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' For double-aligned C1 (0°/0°) device, apart from the  neutrality point (CNP), there are also two successive satellite peaks that appear at hole-  side, locating at -ns and -2ns, where ns=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='3×1012 cm-2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Since ns is the carrier  density required to reach the edge of its first Brillouin zone of a moiré pattern λ by  𝑛𝑠 =  8  √3λ2  8-10, we can calculate the moiré wavelength of C1 of ~14 nm, in good  agreement with our Raman and STM measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Moreover, our band structure  calculation (Inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 4c) shows that the double moiré superlattice significantly  splits the hole-sided bands at higher energy into minibands with fully developed gaps  at electron fillings of − ns, −2 ns and −3 ns, which is consistent with our observation  since these satellite peaks in hole-side bands are more developed than in the electron-  side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Moreover, apart from these band filling peaks, we do not observe any extra peaks  which may come from the misalignment18-24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Therefore, we can conclude that the C1  sample is a perfect double-aligned sample, where the twist angles between graphene  and top/bottom hBN are both close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' A similar feature is also observed in another  14  perfect double-aligned sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' While for single-aligned C2 (0°/30°) device,  we only observe one satellite peak at hole-side, locating at -ns = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='2×1012 cm-2,  corresponding to the moiré wavelength of ~14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='5 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The slightly larger moiré  periodicity in C2 can be attributed to the strain effect in the heterostructure19,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The  lack of -2ns peak in C2 suggests that the overlap between the second and third band, as  our band structure calculation shows a rather small bandgap (Inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' When  the moiré potential disappears in C3 (30°/30°) device, there is only one main Dirac  point, consistent with our band structure calculation (Inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  When the electrons simultaneously subjected to both a magnetic field and a spatially  periodic electrostatic fields, the energy spectrum develops into a Landau fan with a  fractal structure known as the Hofstadter butterfly8-10, which can be renormalized into  a diagram defined by the Diophantine equation:  𝑛  𝑛0 = 𝑣  𝜙  𝜙0 + 𝑠 , where v is the Hall  conductivity in units of a conductance quantum e2/h, as indicated by black solid lines  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 4f, g, h, with topological index ν of ±2, ±6, ±10, ±14,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=', and s the index of band  filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 𝜙 = 𝐵 ⋅ 𝐴 is flux per moiré unit area at magnetic field B, and 𝜙0 = ℎ/𝑒 is a  flux quantum with ℎ  being the Planck’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Figure 4f shows the fracture  spectrum of C1 (0°/0°) device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The straight minigaps arises at  𝜙  𝜙0𝑞 (𝑞 = 1, 2,3, … ) ,  resulting in Brown-Zak (BZ) oscillations with the fundamental period field of Bf =  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='49 T (𝜙0/A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Thus, we can calculate the moiré periodicity of D1 of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='97 nm, which  is consistent with the first band filling density at -ns = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='35 × 1012 cm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Similarly, the  BZ oscillations for C2 device has the Bf = 23 T (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 4g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' We can then calculate the  15  moiré periodicity of C2 to be 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='417 nm, also consistent with the first band filling -ns  = - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='2 × 1012 cm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' When the moiré periods disappear in C3 device, the BZ oscillations  also disappear, as there is only the Landau levels spectrum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 4h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' These results  suggest that the control alignment of moiré patterns, acting as a periodic moiré potential,  can tune the lattice symmetry and engineer the band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  In conclusion, we develop a generic strategy to overcome the edge chirality and lattice  symmetry uncertainty in rotation alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Moreover, we show that the neighboring  graphite edge can be used for better alignment of the stacking structures, significantly  improving the device yield and alignment accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Compared with previous  conventional techniques, the current technique is easier and reliable to operate with  robust and definite control of alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Considering the emerging area of  “twistronics”, our technique can be beneficial for much effort in this area in many  laboratories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' For example, our strategy can also be applied to the family of transition  metal dichalcogenide (TMD) semiconductors3,4 such as WSe2/WS2 moiré superlattice,  where a prior measurement of edge chirality in each crystal is necessary before  alignment40-43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' To show the universality of our technique, we also extend our technique  to other correlated systems, like low-angle twisted bilayer graphene (Supplementary  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S13) and ABC-stacked trilayer graphene (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' S14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' We believe  our technique can help effort in exploring the physics of strong electronic correlations  and non-trivial band topology in these moiré materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  16  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Andrei, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=', MacDonald, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Graphene bilayers with a twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Andrei, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Mak, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' Semiconductor moiré materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Huang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Excitons in semiconductor moiré superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Xie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Twist engineering of the two-dimensional magnetism in double  bilayer chromium triiodide homostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' 18, 30–36 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Emergent interfacial superconductivity between twisted cuprate  superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Preprint at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Yankowitz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' van der Waals heterostructures combining graphene and  hexagonal boron nitride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Nat Rev Phys 1, 112–125 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' Cloning  of  Dirac  fermions  in  graphene  superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' Hofstadter’s butterfly and the fractal quantum Hall effect in moiré  superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Nature 497, 598–602 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' Massive Dirac fermions and Hofstadter butterfly in a van der Waals  heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' Correlated insulator behavior at half-filling in magic-angle graphene  superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Unconventional superconductivity in magic-angle graphene  superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Nature 556, 43–50 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Emergent ferromagnetism near three-quarters filling in twisted  bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Science 365, 605 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' Serlinc, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Intrinsic quantized anomalous Hall effect in a moiré  heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Signatures of tunable superconductivity in a trilayer graphene moiré  17  superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Nature 572, 215–219 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Evidence of a gate-tunable Mott insulator in a trilayer graphene  moiré superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content='  19  Acknowledgements  This work is supported by the Ministry of Education (MOE) Singapore under the  Academic Research Fund Tier 2 (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' MOE-T2EP50120-0015), by the Agency  for Science, Technology and Research (A*STAR) under its Advanced Manufacturing  and Engineering (AME) Individual Research Grant (IRG) (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' A2083c0054),  and by the National Research Foundation (NRF) of Singapore under its NRF-ISF joint  program (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' NRF2020-NRF-ISF004-3518).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' acknowledge  support from the JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and  21H05233).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Author contributions  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+page_content='  Competing interests  The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Additional information  20  Supplementary information is available for this paper at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Reprints and permissions information is available at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Correspondence should be addressed to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' Ariando.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content="  Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims  in published maps and institutional affiliations." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content='  Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
+page_content=' The data that support the plots within this paper and other finding of  this study are available from the corresponding authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFLT4oBgHgl3EQfmi_p/content/2301.12124v1.pdf'}
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+SACDNet: Towards Early Type 2 Diabetes Prediction
+with Uncertainty for Electronic Health Records
+Tayyab Nasira, Muhammad Kamran Malikb
+aCureMD Research and Development
+bDepartment of Information Technology, Faculty of Computing and Information
+Technology, University of the Punjab, Lahore, PK
+Abstract
+Type 2 diabetes mellitus (T2DM) is one of the most common diseases and a
+leading cause of death. The problem of early diagnosis of T2DM is challenging
+and necessary to prevent serious complications. This study proposes a novel
+neural network architecture for early T2DM prediction using multi-headed self-
+attention and dense layers to extract features from historic diagnoses, patient
+vitals, and demographics. The proposed technique is called the Self-Attention
+for Comorbid Disease Net (SACDNet), achieving an accuracy of 89.3% and
+an F1-Score of 89.1%, having a 1.6% increased accuracy and 1.3% increased
+f1-score compared to the baseline techniques. Monte Carlo (MC) Dropout is
+applied to the SACEDNet to get a bayesian approximation. A T2DM prediction
+framework based on the MC Dropout SACDNet is proposed to quantize the
+uncertainty associated with the predictions.
+A T2DM prediction dataset is
+also built as part of this study which is based on real-world routine Electronic
+Health Record (EHR) data comprising 4,124 diabetic and 181,767 non-diabetic
+examples, collected from 295 different EHR systems running in different parts of
+the United States of America. This dataset is further used to evaluate 7 different
+machine learning and 3 deep learning-based models. Finally, a detailed analysis
+of the fairness of every technique against different patient demographic groups
+is performed to validate the unbiased generalization of the techniques and the
+∗Corresponding author
+Email address: tayyabnasir22@gmail.com (Tayyab Nasir)
+Preprint submitted to arxiv
+January 13, 2023
+arXiv:2301.04844v1  [cs.LG]  12 Jan 2023
+
+diversity of the data.
+Keywords:
+Artificial neural networks, Deep Learning, Corpus generation,
+Healthcare, Uncertainty, Disease Prediction,
+1. Introduction
+Diabetes mellitus or simply diabetes is a set of conditions concerning the
+body’s mechanism of managing blood glucose [1, 2]. There are two major types
+of diabetes mellitus i.e., Type 1 Diabetes mellitus (T1DM) and Type 2 Diabetes
+mellitus (T2DM) [1, 2]. Almost 90% of the diabetics are type 2 diabetics which
+signifies the importance of diagnosing T2DM [1, 2]. T2DM occurs more often
+in middle-aged and older people [1, 2, 3]. However, over the past 20 years, this
+disease is becoming more common in adolescents and even in children [2].
+Symptoms of T2DM include unplanned weight loss, increased thirst (poly-
+dipsia), frequent urination (polyuria), and increased hunger [4, 2, 5, 6]. Other
+symptoms of T2DM involve slow wound healing, fatigue, depression, and high
+blood pressure [7, 2]. Previous studies show higher comorbidity of T2DM with
+various diseases including cardiovascular disease, hypertension, angina, gastric
+ulcer, and hypothyroidism [8, 9, 10, 11, 12]. It can cause serious long-term com-
+plications like affecting the kidneys, eyes, and nerves [13, 14]. Also, it increases
+the risk of heart disease and stroke [15]. Similarly, there are several risk factors
+involved when it comes to T2DM including obesity, ethnicity, older age, high
+blood pressure, family history, and genetics [16, 17, 18].
+According to the International Diabetes Federation (IDF), around 536.6 mil-
+lion people were diabetic in the year 2021 [2]. This number is projected to grow
+to 783.2 million by 2045. As stated almost 90% of total diabetics are diagnosed
+with T2DM. Thus, the total number of type 2 diabetic persons in the year 2021
+was 483 million [2]. Diabetes has a high mortality rate as well. According to
+the IDF, 6.7 million people died due to some complications caused by diabetes
+in 2021 alone [2]. The USA is also impacted hugely by diabetes. A total of 32.2
+million US citizens were diabetic in the year 2021, which amounts to almost
+2
+
+11% of the total US population [2, 19]. The US is at number 4 on the list of
+the countries with the most diabetic patients and spends a fortune every year
+fighting diabetes and its underlying complications [2]. In 2021 a total of 966
+billion US dollars were spent on fighting diabetes which is 316% higher than the
+232 billion US dollars spent in the year 2007 for the same [2].
+Early detection of diabetes is of vital importance for an effective treat-
+ment [20] yet stays a challenge [21]. T2DM is often diagnosed very late when the
+symptoms become severe and complications arise [2, 20]. Diagnosis of diabetes is
+usually delayed even in developed countries. Early diagnosis of T2DM can help
+prevent many serious complications including but not limited to loss of vision,
+kidney failure, amputations of limbs, stroke, and even premature death [22, 23].
+Thus, an early diagnosis of T2DM is essential which can not only save millions
+of dollars every year but also save precious human lives. However, diabetes in
+general and T2DM, in particular, remain undiagnosed in the vast majority [20],
+and according to the National Diabetes Statistics Report, around 8.5 million
+diabetic persons in the US alone are undiagnosed [19].
+Predicting T2DM has been studied previously using supervised and unsu-
+pervised machine learning algorithms [24, 25, 26, 12, 14]. Most of the research
+work has used PIMA Indian diabetes dataset for training and evaluating T2DM
+predictive systems [27, 28, 29, 30, 31]. PIMA dataset has a very limited number
+of examples, which raises the concern of a lack of versatility in the dataset that
+is required for better generalization of the models. Also, the dataset contains
+examples belonging to only the female PIMA Indians of Arizona further lim-
+iting the variation in the dataset. Also, some other datasets like the UCI and
+MIMIC-III have been used but such datasets are collected in a more controlled
+environment and may lack the inherent features and characteristics of real-world
+data.
+This work aims to overcome the shortcomings of the existing research in-
+cluding the lack of volume and diversity of datasets, data being collected in a
+controlled environment, and the nature of features like tricep skin thickness and
+blood insulin that are not part of routine EHR data. To this extent following
+3
+
+key contributions are made as part of this study:
+• A novel deep neural network architecture is proposed for the diagnosis of
+T2DM. The proposed technique called the SACDNet uses multi-headed
+self-attention to extract relational features from historic diagnoses and
+uses fully connected dense layers to extract features from vitals and de-
+mographics. The technique then uses a set of fully connected layers to
+predict T2DM using these features.
+• A novel framework is proposed to predict T2DM with associated confi-
+dence that can help in identifying uncertain predictions. The proposed
+framework is built using the proposed SACDNet with MC-Dropout and
+entropy to get uncertainty associated with the predictions.
+• A novel dataset is built collecting data from different EHR systems running
+in different parts of the US. The dataset encloses a set of diverse data
+points that can be used to train a more generalized T2DM predictive
+model.
+• Different machine and deep learning techniques were used to train and
+evaluate T2DM predictive models using the proposed corpus, building
+T2DM predictive systems that rely solely on routine EHR data features
+• Also, a fairness analysis of all the techniques used in the study is presented,
+evaluating each against different demographic groups of data, verifying
+the claimed diversity of the proposed dataset and the fairness of each
+technique.
+The rest of the paper is divided as follows: Section 2 presents related work.
+Section 3 gives an overview of the corpus generation process and its specifica-
+tions. Section 4 presents the novel deep neural network-based technique used
+in this study, along with the details of the proposed framework for diabetes
+prediction. The details of experimentation are given in section 5. The results
+obtained and the analysis performed on the results are presented in Section 6.
+Section 7 provides the conclusions and directions for future work.
+4
+
+Table 1: Details of existing datasets.
+Dataset
+Positive Examples
+Negative Examples
+Total Examples
+PIMA
+179
+358
+537
+UCI
+51,034
+14,806
+65,840
+MIMIC-III
+1,242
+38,456
+39,698
+2. Related Work
+Different machine and deep learning techniques have been previously used
+for the diagnosis of type 2 diabetes [30, 31, 25, 26, 12, 14]. Many datasets have
+been used for training and evaluating machine learning techniques for T2DM
+prediction.
+Three significant datasets have been used previously for T2DM
+detection which are the PIMA Indian Diabetes dataset [32], UCI [33], and the
+MIMIC-III [34]. Out of these, the PIMA Indian diabetes dataset is the most
+widely used. Table 1 gives the details of each dataset.
+A large number of studies have used the PIMA dataset to build T2DM
+prediction models [14].
+The very first study that used machine learning for
+predicting T2DM using the PIMA dataset dates back to 1988 [32]. Techniques
+like KNN, Logistic Regress, Na¨ıve Bayes, Decision Trees, Support Vector Ma-
+chines (SVM), Random Forests, AdaBoosting, and Gradient Boosted Trees have
+all been used to train and evaluate T2DM prediction models using the PIMA
+dataset [24, 27, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 29, 28].
+Apart from
+the conventional machine learning techniques, deep neural networks have also
+been widely used for T2DM prediction. Previous studies have used multi-layer
+perceptron networks to predict T2DM, where PIMA was used for training and
+evaluation [45, 4, 46, 47, 48]. Many other neural network architectures including
+CNN, RNN (both LSTM and GRU), and a combination of CNN-LSTM-based
+deep neural networks were evaluated using the PIMA dataset [30, 31, 49, 14].
+Additionally, many studies have used voting and stacking-based ensembling to
+combine the aforementioned techniques to improve the overall performance of
+T2DM prediction models [24, 50, 35, 44].
+5
+
+Previous studies have also applied machine and deep learning techniques to
+datasets other than PIMA [51, 13, 16, 8, 42, 5, 52, 12]. For example, NHANES
+dataset [53] was used to train an SVM-based model for predicting diabetes [54].
+Similarly, another study combined NHANES dataset [53] with lab results of
+different patients for training Logistic Regression, SVM, Random Forest, and
+Gradient Boosted Trees for predicting T2DM [50].
+Many studies have also
+collected data consisting of the same features as the PIMA dataset from other
+sources to build machine learning-based T2DM predictive models [13, 37].
+Over the past decade, a large number of hospitals and practices have moved
+to digital health records [34, 55]. Previous work has used data from different
+EHRs to build machine learning models [18, 25, 56, 57] for T2DM prediction.
+For example, studies have used Logistic Regression, SVM, Random Forest, Na¨ıve
+Bayes, and Gradient Boosted Trees on EHR data collected from different hospi-
+tals in different parts of the world [58, 59, 60]. Also, deep neural networks have
+been trained on EHR data for T2DM prediction [22, 14]. Most such studies
+that utilize some EHR data rely on clinical components like lab results, histori-
+cal diagnoses, family medical history, or a combination of these. Although data
+collected from EHRs might be significantly larger yet in most of the studies
+data is collected from a single practice or region and might not be an effective
+sample of the data required for better generalization of the machine and deep
+learning tehcniques [61, 62, 63].
+In addition to the aforementioned techniques that rely solely on tabular
+data features, some of the studies also rely on images and signal data for the
+prediction of T2DM. A couple of studies have utilized heart rate variation data
+collected from ECG and built machine and deep learning models to predict
+diabetes [64, 65]. Similarly, images of the retina have been used by a couple of
+studies to predict T2DM using deep CNN and LSTM-based models [66, 67].
+6
+
+3. Dataset
+For this study, the data was provided by CureMD1 which is one of the leading
+EHR providers in the US. Data from 767 different practices was provided, out
+of which 295 unique practices were selected based on the existence of type 2
+diabetic patients. Hence it can be stated that the dataset that is used in this
+study has examples belonging to 295 different EHR environments, operating
+in various parts of the US. The motivation behind using such a dataset is to
+be able to build real-world T2DM prediction systems that rely on real-world
+features coming from routine EHR records, which cannot only provide a non-
+invasive method of early T2DM detection but can also be plugged into any EHR,
+without requiring any additional data from patients.
+The raw data from CureMD’s EHRs contained 5,092,987 unique clinical
+encounters belonging to 1,034,889 different patients.
+All the data attributes
+that contained personal information related to the patients were already masked
+and are not used in any of the experiments or analyses. For this study only the
+following three clinical components are used:
+• Diagnosis: This clinical component consists of a set of ICD-10-CM codes
+that are assigned to a patient during a clinical encounter.
+Each code
+represents a disease, symptom, or injury2 that was found in the patient at
+the time of the encounter. The raw data from the 295 different practices
+contained 29,714 unique ICD-10-CM codes. This clinical component is
+selected because it provides the key information related to the individual
+disease that can either lead to or is comorbid with T2DM.
+• Vitals: The vitals clinical component represents a set of vitals readings
+that were taken during a visit. The aforementioned EHR data contained
+17 different vitals including weight, height, BMI, lean body weight, ideal
+body weight, neck circumference, waist, oxygen saturation, peak expira-
+1https://www.curemd.com/
+2https://www.cdc.gov/nchs/icd/index.htm
+7
+
+tory flow, blood type, blood rh, finger stick, pulse, respiration, tempera-
+ture, systolic blood pressure, and diastolic blood pressure. This clinical
+component was selected because many previous studies relied on the use of
+vitals for predicting the chances of developing T2DM. Vitals like weight,
+BMI, blood pressure, etc., have all been marked as important markers in
+T2DM prediction [31, 52, 22, 50, 62, 54].
+• Demographics:
+The patient’s demographics component consists of at-
+tributes including the patient’s date of birth (which can be used to calcu-
+late the present age of the patient), gender, sexual orientation, and race
+category. Demographical information has been widely used in various dis-
+ease prediction systems and is considered an important feature [54, 16, 68,
+69].
+The CureMD EHR data in its raw form required a certain set of preprocess-
+ing steps to mold it into a form usable for experiments which are discussed in
+the section below.
+3.1. Data Preprocessing
+Data collected from EHRs is sparse, noisy, heterogeneous, and unstruc-
+tured [56, 70, 55]. Such data also presents other challenges including the problem
+of missing data and class imbalance [71, 72]. The raw EHRs data that was ob-
+tained from CureMD had all of these inherent issues. The details of these data
+challenges and the required preprocessing techniques that were used to tackle
+these challenges are discussed below:
+3.1.1. Handling Records with no History
+The aim of this study is to build an early T2DM prediction system that relies
+solely on patients’ historic clinical diagnoses, recorded vitals, and demographics.
+Thus, it is desired to filter out only those patients that have some historic visits
+before being diagnosed with T2DM. However, in the raw EHR data, a large
+number of records were encountered where a patient was marked with T2DM
+8
+
+(indicated with an E11 ICD-10-CM code) in the very first encounter.
+Such
+examples occur in cases when a patient, already diagnosed with T2DM walks
+into a hospital or practice for the first time without any recorded medical history
+in the EHR of the same. Thus, a strategy is required to eliminate such example
+points having no historic encounters prior to the diagnosis of T2DM. For this, all
+the T2DM-diagnosed patients that do not have at least 4 previous encounters
+before being diagnosed with T2DM were filtered out of the data. A total of
+72,626 patients with T2DM were identified in the raw data, out of which only
+5,770 had at least previous 4 encounters before being diagnosed with T2DM.
+For the non-diabetics, the patients with less than 4 encounters were filtered out.
+After the execution of this step, the data set was reduced from 72,626 to
+5,770 T2DM patients. Also, from a total of 962,263 non-diabetic patients, only
+186,865 patients were left.
+Hence, after executing the first step of our pre-
+processing a dataset with 5,770 diabetic and 186,865 non-diabetic patients was
+created.
+3.1.2. Handling Missing Data
+The problem of missing data is widely encountered in EHRs [71, 72]. Differ-
+ent studies have implied different approaches for handling missing data. Tech-
+niques like dropping complete records with missing values, and filling in missing
+values using mean, median, or from the nearest neighbors, have been used in
+many research studies [27, 37, 8, 41, 62, 29, 26]. Raw EHR records were analyzed
+to identify the attributes that were most affected by the problem of missing val-
+ues. As a result of this analysis, it was found that all the attributes that have
+the problem of missing values belonged to the vitals component. To deal with
+missing values a 2-step strategy was applied. In the first step, a missing value
+ratio was calculated for every attribute using equation 1.
+MissingV aluesRatio = NumberOfRecordsWithMissingV alues
+TotalNumberofRecords
+∗ 100
+(1)
+Table 2 presents the details of the missing value ratio of different attributes
+9
+
+Table 2: Missing value ratio for vitals.
+Attribute
+Missing Value Ratio (%)
+Weight
+12.67
+Height
+21.53
+BMI
+28.34
+Lean Body Weight
+64.71
+Ideal Body Weight
+65.14
+Neck Circumference
+92.22
+Waist
+91.80
+Oxygen Saturation
+89.15
+Peak Expiratory Flow
+99.99
+Blood Type
+99.82
+Blood Rh
+99.94
+Finger Stick
+98.98
+Pulse
+28.91
+Respiration
+53.93
+Temperature
+51.66
+Systolic Blood Pressure
+16.48
+Diastolic Blood Pressure
+16.44
+10
+
+in the vitals component. Based on the results presented in the table 2 the vitals
+attributes for which the missing value ratio was higher than 50% were dropped.
+The reason for dropping these attributes with a higher missing value ratio lies
+in the fact that such variables do not possess a sufficient amount of information
+that can be used to impute the missing values in the next step.
+The second step of the applied strategy consists of imputing the missing
+values for the remaining vitals attributes. The following set of steps was used
+to impute the missing values for vitals:
+• Firstly, all the records were grouped based on the patient they belong to.
+• Secondly, the patient groups having no value for any of the vitals attributes
+were dropped, as the missing data cannot be imputed for such patients.
+• Thirdly, for each patient group, the Exponentially Weighted Moving Av-
+erage (EWMA) of every vitals attribute was calculated and stored.
+• Finally, all the missing values were populated by their respective calculated
+values, calculated using the EWMA.
+EWMA was used for data imputation keeping in mind the fact that EHR data
+is temporal in nature. EWMA can be used to assign larger weights to recent
+observations, hence giving more importance to the recent ones as compared to
+the older data points [73].
+Such behavior is desirable and quite effective in
+capturing the temporal nature associated with a patient’s EHR records where
+more weight is given to recent visits and less weight to older visits. For the
+remaining two components i.e., patient demographics and diagnosis the problem
+of missing value did not exist in the raw data. After applying this 2-step strategy
+the dataset size was further reduced to a total of 4,124 diabetic and 181,767 non-
+diabetic patients.
+3.1.3. Creating Single Example Points
+The data of every patient consists of multiple encounters/visits. This step
+deals with merging these multiple encounters of a single patient into one example
+11
+
+point that can be used for training and evaluating machine and deep learning
+techniques. The process of creating the example points is given as follows:
+• All three components i.e., diagnosis, vitals, and demographics, were merged
+into a single record for every patient encounter.
+• For every encounter the age of the patient at the time of the encounter
+was calculated using the patient’s date of birth.
+• Next for every patient all the encounter records were sorted using the date
+of the encounter in ascending order of the date.
+• For every T2DM patient, all the encounter records before the first T2DM
+diagnosis encounter were selected and the associated diagnoses were ex-
+tracted. The ICD-10-CM diagnosis codes were merged together creating a
+single comma-separated string enclosing all the unique historic diagnoses
+of a patient.
+• Next, the vitals from the first T2DM encounter were merged with the
+diagnoses attribute along with the demographic information of the patient,
+creating a single T2DM positive example.
+• Label 1 was also generated for the created T2DM example points.
+• For every non-T2DM patient, all the diagnoses from all the previous en-
+counters were merged to create a single comma-separated diagnoses fea-
+ture. Also, the vitals from the last encounter and the patient’s demo-
+graphic information were merged with the diagnoses feature to create a
+T2DM negative example.
+• Label 0 was generated for the non-T2DM example points.
+3.1.4. Dealing with Class Imbalance
+Class imbalance is a problem that arises when the data is distributed un-
+evenly among the classification categories [74, 75, 76]. In case of class imbal-
+ance, the performance of traditional classification techniques is affected where
+12
+
+the performance of the techniques becomes biased towards the majority class
+and is reduced for the minority class [74, 77, 78]. Many techniques have been
+presented over the years to deal with class imbalance including oversampling,
+undersampling, SMOTE, MUTE, ADASYN [78, 75, 79, 80, 81]. Class imbalance
+is another inherent property of health care and medicine data [82, 62, 22].
+Techniques like SMOTE, ADASYNC, data oversampling, data undersam-
+pling, etc. have been used for balancing data for T2DM prediction as well [22,
+49, 62]. The EHR data that is used as part of this study has a huge class im-
+balance having a total of 4,124 diabetic and 181,767 non-diabetic records. A
+combination of 5-fold cross-validation combined with undersampling is applied
+to handle this class imbalance. A total of 5 non-overlapping samples of size
+4,124 were selected from the non-T2DM examples which are used to train 5
+different variations of every machine learning and deep learning technique. The
+details are discussed in section 5.
+4. Methodology
+A novel deep neural network architecture is proposed for T2DM prediction
+that combines multi-headed self-attention with fully connected layers. The pro-
+posed technique called the Self-Attention for Comorbid Disease Net (SACDNet)
+uses multiheaded self-attention to extract features from the diagnoses vector
+with the intent to find a relation between every pair of diagnoses and uses dense
+layers to extract features from vitals and demographics data. The resultant fea-
+ture vectors are then concatenated and passed through a single dropout layer.
+Finally, a series of fully connected layers are used for T2DM classification.
+4.1. Self Attention
+The concept of attention was introduced to modify the RNN-based Encoder-
+Decoder models [83] where every decoder timestamp will now receive a weighted
+sum of all encoder hidden states called the attention score. The attention score
+is usually a scaled dot product given by the equation 2.
+13
+
+Attention(Q, K, V ) = Softmax(QKT
+√dm
+)V
+(2)
+Where Q, K, and V represent the query, key, and value matrixes, obtained by
+multiplying the inputs sequence X with three different weight matrixes Wq, Wk,
+and Wv respectively. If the query, key, and value are all generated using the
+same input sequence then the above equation 2 becomes self-attention. The
+concept of self-attention involves finding the similarity between all elements in
+the input sequence, in our case, this will be to find a relation between all historic
+diagnoses.
+4.2. Multi-Headed Self Attention
+This involves applying self-attention to the input sequence multiple times,
+obtaining different weight matrixes for the query, key, and value [84]. Thus, it
+provides a more diverse set of relationships between the elements of the input
+sequence. These multiple self-attention outputs are concatenated. For given
+input sequence X multi-headed self-attention for N heads is given by equation
+3.
+MHSA(Q, K, V ) = Concat(Att1(Q, K, V ), Att2(Q, K, V ), ...., AttN(Q, K, V )))WO
+(3)
+Where Attentioni represents self-attention obtained using different Wq, Wk, and
+Wv weight metrixes. N represents the number of heads.
+4.3. SACDNet
+For the given diagnosis matrix D, and the vitals/demographics matrix V
+SACDNet feature extraction is given using equation 4.
+Feature(D, V ) = Concat(MHSA(Q, K, V ), MLP(V ))
+(4)
+Where MLP represents a set of fully connected layers. This feature vector
+is then passed to another set of fully connected layers, for T2DM classification.
+The details of the network are given in the table 3.
+14
+
+Table 3: Architecture of SACDNet.
+No.
+Network Layers
+1
+Diagnosis Input Vector
+Vitals/Demographics Input Vector
+2
+Multiheaded Self Attention (N=3, Query
+Dimension=3, Key Dimension=3, Value
+Dimension=3)
+Fully
+Connected
+(Units=256,
+Activa-
+tion=SELU)
+3
+Fully
+Connected
+(Units=128,
+Activa-
+tion=SELU)
+4
+Concatenate (To a single merged feature vector)
+5
+Dropout (Rate=0.1)
+6
+Fully Connected (Units=64, Activation=SELU)
+7
+Fully Connected (Units=64, Activation=Sigmoid)
+4.4. Monte Carlo (MC) Dropout
+Deep neural networks are deterministic in nature, which is not always de-
+sirable, especially in the case of high-risk systems such as clinical systems [85,
+86, 87].
+It is desirable to have a system that can provide associated confi-
+dence with its prediction. The probability distribution returned by the Soft-
+max function cannot be considered a model’s confidence as it has been proven
+that neural network-based models can make incorrect predictions with higher
+confidence [88, 89, 90, 87]. Epistemic or Model uncertainty estimation using
+MC dropout [88] has been used widely for deep neural network-based mod-
+els [91, 92, 93, 94]. The idea of MC-Dropout is to get a prediction for given
+input X from T highly similar ensemble models and then obtain an average of
+these T different output probability distributions 5. This is achieved by ap-
+plying the concept of dropout regularization at inference time, which in fact
+yields T different distributions of the model’s parameters. Hence, a large dif-
+ference in the prediction of these T different models can be an indicator of
+larger model uncertainty and such can help in obtaining more robust predictive
+confidence [87, 92].
+p(y = 1) = 1
+T
+T
+�
+i
+P(y = 1|x, Wi)
+(5)
+15
+
+Where for given T forward passes from the network we will have T different
+parameter distributions W1, W2, W3, . . . ., WT as given in equation 5.
+4.5. Measuring Predictive Uncertainty
+For the given output probability distributions from T different passes, differ-
+ent measures have been used for uncertainty quantization [91, 95, 92]. Entropy
+has been widely used as a measure of uncertainty quantization [91, 95, 96], where
+entropy for the averaged MC-Dropout results is calculated using the equation
+6.
+H(y) = −
+C
+�
+c
+1
+T
+T
+�
+t
+P(y = cc|x, Wt) ∗ log(
+T
+�
+t
+P(y = cc|x, Wt))
+(6)
+4.6. Proposed Framework
+The proposed framework combines the novel SACDNet with MC-Dropout
+to build a system that can predict T2DM with associated uncertainty. The idea
+is to get an ensemble of 100 different highly similar models, for which an average
+prediction probability for T2DM is calculated 7.
+MC − SACDNet = 1
+T
+T =100
+�
+t
+SACDNett
+(7)
+Next using the entropy metric discussed before uncertainty value for the
+prediction is calculated and based on a specified threshold theta a prediction is
+marked as uncertain or certain. An overview of the framework is given in fig-
+ure 1. It can be observed in the figure 1 the electronic health record of a patient
+is divided into 2 different feature vectors comprising of the historic diagnoses
+and vitals-demographics respectively. These feature vectors are passed through
+100 different representation of the MC-Dropout SACDNet. The generated set
+of probabilities is averaged to get the final probability which is passed through
+a threshold function to predict T2DM. Also, the same mean probability is used
+to quantize uncertainty using entropy.
+16
+
+Figure 1: Proposed framework for T2DM prediction with uncertainty.
+5. Experiments
+This section presents details of experiments performed on the developed
+T2DM corpus.
+Machine learning algorithms, deep learning algorithms, and
+proposed SACDNet are applied to the dataset to train T2DM classifiers. As
+discussed in section 3, the proposed corpus has a huge class imbalance. This
+class imbalance can affect the performance of the classifiers [74, 77, 78]. To deal
+with this problem 5-fold cross-validation was applied obtaining a balanced set
+by randomly undersampling the majority class. 7 different machine learning
+techniques including AdaBoost, Decision Trees, Logistic Regression, Gaussian
+Na¨ıve Bayes, Random Forest, SVM, and Gradient Boosted Trees (XGB) were
+trained using the proposed corpus. Multiple deep neural network-based tech-
+niques were trained using the proposed corpus [47, 46, 31, 46, 48], however, apart
+from our proposed novel technique SACDNet, only two other techniques were
+able to produce satisfactory results, referred to as the FCN and FCN Dropout.
+The FCN network consists of 4 fully connected hidden layers using ReLU ac-
+17
+
+Feature
+Value
+SACDNet1
+SACDNet2
+T=100
+1
+Threshold
+Z
+SACDNet3
+SACDNett
+-T
+Average
+Probability 
+t
+EHR
+Feature
+Value
+Entropy
+Threshold
+H(y)
+Entropy 0
+SACDNetTtivations whereas the FCN Dropout consists of 3 fully connected hidden layers
+with hidden layer one having a dropout rate of 30% and hidden layer two having
+a dropout rate of 20%, where using the same ReLU activations.
+The details of the experimentation process are given as follows:
+• Firstly, all the T2DM positive and negative class examples were separated.
+• Next 5 random samples, without overlap were selected from the non-
+T2DM examples, where the size of each sample selected was equal to
+the size of our T2DM positive examples set.
+• Next 5 different datasets were created combining the T2DM positive set
+with each of the 5 non-T2DM sets.
+• Next for every dataset an 80-20% train test split was done.
+• Every technique was trained using a 5-fold cross-validation, where one of
+the 5 balanced samples was used for each fold.
+• Performance of each model per fold was calculated using the evaluation
+measures discussed in the results section 6.
+• An average performance of every model was also calculated using the
+results from the 5 folds.
+For a fair comparison, none of the models’ hyperparameters were tuned for both
+machine and deep learning techniques
+6. Results and Analysis
+6.1. Evaluation Measures
+For the evaluation of the models presented in this study, 5 widely used met-
+rics were used including Accuracy, F1-Score, Precision, Recall, and Specificity.
+These have also been used for evaluating disease prediction systems, especially
+for T2DM prediction [24, 36, 5, 38, 39, 13, 7, 41, 42, 29, 26, 25]. For the evalu-
+ation of the uncertainty of the model, we have used entropy as discussed in the
+methodology section 4.
+18
+
+Table 4: Mean of 5 folds cross-validation results.
+No.
+Model
+Accuracy
+(%)
+F1-Score
+(%)
+Precision
+(%)
+Recall
+(%)
+Specificity
+(%)
+1
+Ada Boost
+85.8
+86.0
+84.7
+87.4
+85.8
+2
+Decision Tree
+77.1
+77.2
+76.8
+77.5
+77.1
+3
+Logistic Regression
+87.9
+88.0
+87.5
+88.5
+87.9
+4
+Na¨ıve Bayes
+69.6
+74.8
+63.9
+90.1
+69.6
+5
+Random Forest
+85.4
+86.1
+82.0
+90.7
+85.4
+6
+SVM
+87.2
+87.6
+85.2
+90.1
+87.2
+7
+XGB
+86.1
+86.2
+85.5
+86.8
+86.1
+8
+FCN
+87.7
+87.6
+88.4
+86.9
+87.7
+9
+FCN Dropout
+87.3
+87.5
+86.4
+88.5
+87.3
+10
+SACDNet
+89.3
+89.1
+90.5
+87.9
+89.3
+6.2. Comparison Under Fixed Settings
+This section presents a comparison of the performance of all the techniques
+discussed, using different evaluation measures.
+Firstly, the results obtained
+against the test set for each of the 5 data folds are presented in figure 2. Next,
+the average accuracy, f1-score, precision, recall, and specificity are presented in
+table 4.
+Figure 2: Results of 5 folds cross-validation.
+It can be observed from the table 4 that our proposed technique SACD-
+19
+
+Accuracy
+Accuracy
+Accuracy
+F1-Score
+F1-Score
+F1-Score
+Precision
+Precision
+Precision
+Recall
+Recall
+Recall
+Specificity
+Specificity
+Specificity
+Accuracy
+Accuracy
+F1-Score
+F1-Score
+Precision
+Precision
+Recall
+Recall
+Specificity
+SpecificityNet performs better compared with 7 different machine learning and 2 neural
+network-based techniques, achieving a 1.6% increased accuracy and 1.3% in-
+creased F1-Score compared to the other techniques.
+6.3. Fairness Evaluation
+It is desirable to have a T2DM prediction system that is able to perform
+equally well on a diverse set of examples, without having any bias towards
+a certain gender, age, or racial group.
+The proposed dataset collected from
+different EHR systems running in different parts of the US was built to capture
+examples belonging to a larger population. Such a diverse dataset can be helpful
+in building more generalized machine and deep learning models that can perform
+equally well for examples belonging to different demographic groups of patients.
+In order to evaluate this hypothesis, the next set of analyses was performed to
+validate the fairness of each of the techniques used.
+For fairness evaluation, the proposed dataset was divided into certain groups
+based on the associated demographic information of the patients.
+Example
+points were divided based on age, gender, and race. All the techniques were
+evaluated against the individual groups and the results are presented in the
+figure 3.
+It can be observed from the figure 3 that all the techniques have identical
+performance for all the race and gender groups. However, there is a sudden
+drop in the case of precision, recall, specificity, and f1-score for two age groups
+namely the children and preschoolers. The reason for this is the unavailability
+of significant examples belonging to the two aforementioned groups. For exam-
+ple, for preschoolers, there were only 1569 examples in the dataset all of which
+belonged to the non-diabetic class. Hence, there were no diabetic patients be-
+longing to this class, and the precision, recall, specificity, and f1-score, cannot
+be calculated. Similarly, for Children, only 1857 examples were present all of
+which belonged to the non-diabetic class, and hence the same holds. Hence
+the sudden drop does not represent the biasness of the techniques but only the
+unavailability of data for the calculation of precision, recall, specificity, and f1-
+20
+
+Figure 3: Performance of every technique against certain demographic groups.
+score, which can be further validated by no drop in the accuracy for these two
+groups. Also, for the unspecified gender, only one example of a diabetic patient
+and only 6 examples of non-diabetics were encountered which again is causing
+discrepancies in the performance of certain evaluation measures even though the
+accuracy of the models is close to near perfect. Hence it can be stated that all
+the techniques were able to generalize fairly against all the data groups which
+confirm the diversity of the proposed dataset.
+6.4. Uncertainty Analysis
+As discussed in the methodology section 4, the proposed SACDNet is in-
+tended to be part of a complete T2DM prediction framework that can make
+predictions with associated confidence, marking predictions with lower confi-
+dence as uncertain. In order to draw a comparison of the proposed technique and
+its MC-Dropout-based extension figure 4 was plotted. The horizontal straight
+dashed lines represent the probability thresholding for the simple model while
+the vertical straight line represents the entropy threshold for the MC-Dropout
+21
+
+Accuracy
+F1-Score
+Over Eighty
+Asian Americans
+African Amer
+Over Eighty
+Asian Americans
+African Amer
+Aged
+Latino Americans
+Aged
+Latino Americans
+Middle Aged
+Mixed Race
+Middle Aged
+Mixed Race
+Adult
+ Native American
+Adult
+ Native American
+0.40.6/0.8
+Adolescent
+Other Races
+Adolescen
+Other Races
+Chilc
+white Americans
+Child
+white Americans
+Preschoolel
+Feamles
+Preschooler
+Feamles
+Unspecified
+Males
+Unspecified
+Males
+Precision
+Recall
+Over Eighty
+Asian Americans
+African Americans
+Over Eighty
+Asian Americans
+African Americans
+Aged
+Aged
+Middle Age
+lixed Race
+Middle Aged
+ixed Race
+Adult
+Native American
+Adult
+ Native American
+Adolescel
+Other Races
+Other Races
+Child
+white Americans
+Child
+White Americans
+Preschooler
+Feamles
+Preschooler
+Feamles
+Unspecified
+Males
+Unspecified
+Males
+Specificity
+Over Eighty
+Asian Americans
+African Americans
+ModelName
+Aged
++ FCN
+SACDNet
+Decision Tree
+Middle Age
+Mixed Race
+FCN Doprout
+Logistic Regression
+Naive Bayes
+Adult
+Native American
+Random Forest
+SVM
+XGB
+Adolescen
+OtherRaces
+ Ada Boost
+Child
+Preschooler
+Feamles
+Unspecified
+Malesextension. It can be observed from the figure 4 that the MC-Dropout extension
+used with entropy is able to identify more misclassified examples as uncertain,
+making it more reliable.
+Figure 4: Comparison of confidences of SACDNet and MC-SACDNet for misclassified exam-
+ples.
+In order to further clarify the reliability of the MC-Dropout extension tech-
+nique all the examples from the test that were misclassified by either technique
+were separated and the entropy for the output probabilities was calculated. The
+histogram of the calculated entropy is given in figure 5, where each bin repre-
+sents the number of examples per entropy range.
+It can be observed from the figure 5 that a higher entropy, which counts for
+a higher uncertainty, is assigned to more misclassified examples by the proposed
+MC-Dropout-based extension as compared to the simple model. Hence adding
+MC-Dropout further improves the proposed technique, reducing the number of
+overconfident wrong inferences.
+7. Conclusion and Future Work
+Type 2 diabetes mellitus is a chronic disease that is often undiagnosed and
+can lead to severe complications. It is desirable to have a system that can help in
+the early diagnosis of T2DM, reducing the yearly health expenses spent tackling
+22
+
+LD 
+0.B
+6.
+I Predictions
+0.6
+ClassificationDifference
+Correctly Classified by Both
+Misclassified by Dropout Model
+Misclassified by Both
+Misclassified by Sinqle Model
+0.4
+0.2 
+:
+0.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+10
+MC-Dropout EntropyFigure 5: Histogram of entropies for output probabilities of SACDNet and MC-SACDNet.
+complications of undiagnosed T2DM and also saving millions of lives every year
+that are lost due to late diagnosis of T2DM. This study proposes a novel deep
+learning-based model to predict T2DM using routine EHR features. SACDNet
+was able to perform better than the baseline machine and deep learning models.
+The study also presents a complete real-world solution for building an early
+diabetes detection system, developing a diverse corpus from real-world EHR
+data.
+Also, the proposed framework based on MC-Dropout can help make
+predictions with more reliability and abstain from predicting in cases where the
+model is uncertain.
+The proposed technique and framework can be plugged
+into existing EHR software with little or no changes. Also, the methodology
+followed in this study can be used to build predictive software for other diseases
+like hypertension, ischemic heart disease, etc.
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diff --git a/z9E4T4oBgHgl3EQfAguY/content/tmp_files/load_file.txt b/z9E4T4oBgHgl3EQfAguY/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1143 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf,len=1142
+page_content='SACDNet: Towards Early Type 2 Diabetes Prediction  with Uncertainty for Electronic Health Records  Tayyab Nasira, Muhammad Kamran Malikb  aCureMD Research and Development  bDepartment of Information Technology, Faculty of Computing and Information  Technology, University of the Punjab, Lahore, PK  Abstract  Type 2 diabetes mellitus (T2DM) is one of the most common diseases and a  leading cause of death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The problem of early diagnosis of T2DM is challenging  and necessary to prevent serious complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' This study proposes a novel  neural network architecture for early T2DM prediction using multi-headed self-  attention and dense layers to extract features from historic diagnoses, patient  vitals, and demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The proposed technique is called the Self-Attention  for Comorbid Disease Net (SACDNet), achieving an accuracy of 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='3% and  an F1-Score of 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1%, having a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='6% increased accuracy and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='3% increased  f1-score compared to the baseline techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Monte Carlo (MC) Dropout is  applied to the SACEDNet to get a bayesian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' A T2DM prediction  framework based on the MC Dropout SACDNet is proposed to quantize the  uncertainty associated with the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  A T2DM prediction dataset is  also built as part of this study which is based on real-world routine Electronic  Health Record (EHR) data comprising 4,124 diabetic and 181,767 non-diabetic  examples, collected from 295 different EHR systems running in different parts of  the United States of America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' This dataset is further used to evaluate 7 different  machine learning and 3 deep learning-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Finally, a detailed analysis  of the fairness of every technique against different patient demographic groups  is performed to validate the unbiased generalization of the techniques and the  ∗Corresponding author  Email address: tayyabnasir22@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='com (Tayyab Nasir)  Preprint submitted to arxiv  January 13, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='04844v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='LG]  12 Jan 2023  diversity of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Keywords:  Artificial neural networks, Deep Learning, Corpus generation,  Healthcare, Uncertainty, Disease Prediction,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Introduction  Diabetes mellitus or simply diabetes is a set of conditions concerning the  body’s mechanism of managing blood glucose [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' There are two major types  of diabetes mellitus i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=', Type 1 Diabetes mellitus (T1DM) and Type 2 Diabetes  mellitus (T2DM) [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Almost 90% of the diabetics are type 2 diabetics which  signifies the importance of diagnosing T2DM [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' T2DM occurs more often  in middle-aged and older people [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' However, over the past 20 years, this  disease is becoming more common in adolescents and even in children [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Symptoms of T2DM include unplanned weight loss, increased thirst (poly-  dipsia), frequent urination (polyuria), and increased hunger [4, 2, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Other  symptoms of T2DM involve slow wound healing, fatigue, depression, and high  blood pressure [7, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Previous studies show higher comorbidity of T2DM with  various diseases including cardiovascular disease, hypertension, angina, gastric  ulcer, and hypothyroidism [8, 9, 10, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' It can cause serious long-term com-  plications like affecting the kidneys, eyes, and nerves [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Also, it increases  the risk of heart disease and stroke [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Similarly, there are several risk factors  involved when it comes to T2DM including obesity, ethnicity, older age, high  blood pressure, family history, and genetics [16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  According to the International Diabetes Federation (IDF), around 536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='6 mil-  lion people were diabetic in the year 2021 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' This number is projected to grow  to 783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='2 million by 2045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' As stated almost 90% of total diabetics are diagnosed  with T2DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Thus, the total number of type 2 diabetic persons in the year 2021  was 483 million [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Diabetes has a high mortality rate as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' According to  the IDF, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='7 million people died due to some complications caused by diabetes  in 2021 alone [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The USA is also impacted hugely by diabetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' A total of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='2  million US citizens were diabetic in the year 2021, which amounts to almost  2  11% of the total US population [2, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The US is at number 4 on the list of  the countries with the most diabetic patients and spends a fortune every year  fighting diabetes and its underlying complications [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' In 2021 a total of 966  billion US dollars were spent on fighting diabetes which is 316% higher than the  232 billion US dollars spent in the year 2007 for the same [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Early detection of diabetes is of vital importance for an effective treat-  ment [20] yet stays a challenge [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' T2DM is often diagnosed very late when the  symptoms become severe and complications arise [2, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Diagnosis of diabetes is  usually delayed even in developed countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Early diagnosis of T2DM can help  prevent many serious complications including but not limited to loss of vision,  kidney failure, amputations of limbs, stroke, and even premature death [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Thus, an early diagnosis of T2DM is essential which can not only save millions  of dollars every year but also save precious human lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' However, diabetes in  general and T2DM, in particular, remain undiagnosed in the vast majority [20],  and according to the National Diabetes Statistics Report, around 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='5 million  diabetic persons in the US alone are undiagnosed [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Predicting T2DM has been studied previously using supervised and unsu-  pervised machine learning algorithms [24, 25, 26, 12, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Most of the research  work has used PIMA Indian diabetes dataset for training and evaluating T2DM  predictive systems [27, 28, 29, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' PIMA dataset has a very limited number  of examples, which raises the concern of a lack of versatility in the dataset that  is required for better generalization of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Also, the dataset contains  examples belonging to only the female PIMA Indians of Arizona further lim-  iting the variation in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Also, some other datasets like the UCI and  MIMIC-III have been used but such datasets are collected in a more controlled  environment and may lack the inherent features and characteristics of real-world  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  This work aims to overcome the shortcomings of the existing research in-  cluding the lack of volume and diversity of datasets, data being collected in a  controlled environment, and the nature of features like tricep skin thickness and  blood insulin that are not part of routine EHR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' To this extent following  3  key contributions are made as part of this study:  A novel deep neural network architecture is proposed for the diagnosis of  T2DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The proposed technique called the SACDNet uses multi-headed  self-attention to extract relational features from historic diagnoses and  uses fully connected dense layers to extract features from vitals and de-  mographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The technique then uses a set of fully connected layers to  predict T2DM using these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  A novel framework is proposed to predict T2DM with associated confi-  dence that can help in identifying uncertain predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The proposed  framework is built using the proposed SACDNet with MC-Dropout and  entropy to get uncertainty associated with the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  A novel dataset is built collecting data from different EHR systems running  in different parts of the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The dataset encloses a set of diverse data  points that can be used to train a more generalized T2DM predictive  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Different machine and deep learning techniques were used to train and  evaluate T2DM predictive models using the proposed corpus, building  T2DM predictive systems that rely solely on routine EHR data features  Also, a fairness analysis of all the techniques used in the study is presented,  evaluating each against different demographic groups of data, verifying  the claimed diversity of the proposed dataset and the fairness of each  technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  The rest of the paper is divided as follows: Section 2 presents related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Section 3 gives an overview of the corpus generation process and its specifica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Section 4 presents the novel deep neural network-based technique used  in this study, along with the details of the proposed framework for diabetes  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The details of experimentation are given in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The results  obtained and the analysis performed on the results are presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Section 7 provides the conclusions and directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  4  Table 1: Details of existing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Dataset  Positive Examples  Negative Examples  Total Examples  PIMA  179  358  537  UCI  51,034  14,806  65,840  MIMIC-III  1,242  38,456  39,698  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Related Work  Different machine and deep learning techniques have been previously used  for the diagnosis of type 2 diabetes [30, 31, 25, 26, 12, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Many datasets have  been used for training and evaluating machine learning techniques for T2DM  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Three significant datasets have been used previously for T2DM  detection which are the PIMA Indian Diabetes dataset [32], UCI [33], and the  MIMIC-III [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Out of these, the PIMA Indian diabetes dataset is the most  widely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Table 1 gives the details of each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  A large number of studies have used the PIMA dataset to build T2DM  prediction models [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  The very first study that used machine learning for  predicting T2DM using the PIMA dataset dates back to 1988 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Techniques  like KNN, Logistic Regress, Na¨ıve Bayes, Decision Trees, Support Vector Ma-  chines (SVM), Random Forests, AdaBoosting, and Gradient Boosted Trees have  all been used to train and evaluate T2DM prediction models using the PIMA  dataset [24, 27, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 29, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Apart from  the conventional machine learning techniques, deep neural networks have also  been widely used for T2DM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Previous studies have used multi-layer  perceptron networks to predict T2DM, where PIMA was used for training and  evaluation [45, 4, 46, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Many other neural network architectures including  CNN, RNN (both LSTM and GRU), and a combination of CNN-LSTM-based  deep neural networks were evaluated using the PIMA dataset [30, 31, 49, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Additionally, many studies have used voting and stacking-based ensembling to  combine the aforementioned techniques to improve the overall performance of  T2DM prediction models [24, 50, 35, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  5  Previous studies have also applied machine and deep learning techniques to  datasets other than PIMA [51, 13, 16, 8, 42, 5, 52, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' For example, NHANES  dataset [53] was used to train an SVM-based model for predicting diabetes [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Similarly, another study combined NHANES dataset [53] with lab results of  different patients for training Logistic Regression, SVM, Random Forest, and  Gradient Boosted Trees for predicting T2DM [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Many studies have also  collected data consisting of the same features as the PIMA dataset from other  sources to build machine learning-based T2DM predictive models [13, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Over the past decade, a large number of hospitals and practices have moved  to digital health records [34, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Previous work has used data from different  EHRs to build machine learning models [18, 25, 56, 57] for T2DM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  For example, studies have used Logistic Regression, SVM, Random Forest, Na¨ıve  Bayes, and Gradient Boosted Trees on EHR data collected from different hospi-  tals in different parts of the world [58, 59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Also, deep neural networks have  been trained on EHR data for T2DM prediction [22, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Most such studies  that utilize some EHR data rely on clinical components like lab results, histori-  cal diagnoses, family medical history, or a combination of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Although data  collected from EHRs might be significantly larger yet in most of the studies  data is collected from a single practice or region and might not be an effective  sample of the data required for better generalization of the machine and deep  learning tehcniques [61, 62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  In addition to the aforementioned techniques that rely solely on tabular  data features, some of the studies also rely on images and signal data for the  prediction of T2DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' A couple of studies have utilized heart rate variation data  collected from ECG and built machine and deep learning models to predict  diabetes [64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Similarly, images of the retina have been used by a couple of  studies to predict T2DM using deep CNN and LSTM-based models [66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Dataset  For this study, the data was provided by CureMD1 which is one of the leading  EHR providers in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Data from 767 different practices was provided, out  of which 295 unique practices were selected based on the existence of type 2  diabetic patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Hence it can be stated that the dataset that is used in this  study has examples belonging to 295 different EHR environments, operating  in various parts of the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The motivation behind using such a dataset is to  be able to build real-world T2DM prediction systems that rely on real-world  features coming from routine EHR records, which cannot only provide a non-  invasive method of early T2DM detection but can also be plugged into any EHR,  without requiring any additional data from patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  The raw data from CureMD’s EHRs contained 5,092,987 unique clinical  encounters belonging to 1,034,889 different patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  All the data attributes  that contained personal information related to the patients were already masked  and are not used in any of the experiments or analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' For this study only the  following three clinical components are used:  Diagnosis: This clinical component consists of a set of ICD-10-CM codes  that are assigned to a patient during a clinical encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Each code  represents a disease, symptom, or injury2 that was found in the patient at  the time of the encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The raw data from the 295 different practices  contained 29,714 unique ICD-10-CM codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' This clinical component is  selected because it provides the key information related to the individual  disease that can either lead to or is comorbid with T2DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Vitals: The vitals clinical component represents a set of vitals readings  that were taken during a visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The aforementioned EHR data contained  17 different vitals including weight, height, BMI, lean body weight, ideal  body weight, neck circumference, waist, oxygen saturation, peak expira-  1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='curemd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='com/  2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='cdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='gov/nchs/icd/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='htm  7  tory flow, blood type, blood rh, finger stick, pulse, respiration, tempera-  ture, systolic blood pressure, and diastolic blood pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' This clinical  component was selected because many previous studies relied on the use of  vitals for predicting the chances of developing T2DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Vitals like weight,  BMI, blood pressure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=', have all been marked as important markers in  T2DM prediction [31, 52, 22, 50, 62, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Demographics:  The patient’s demographics component consists of at-  tributes including the patient’s date of birth (which can be used to calcu-  late the present age of the patient), gender, sexual orientation, and race  category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Demographical information has been widely used in various dis-  ease prediction systems and is considered an important feature [54, 16, 68,  69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  The CureMD EHR data in its raw form required a certain set of preprocess-  ing steps to mold it into a form usable for experiments which are discussed in  the section below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Data Preprocessing  Data collected from EHRs is sparse, noisy, heterogeneous, and unstruc-  tured [56, 70, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Such data also presents other challenges including the problem  of missing data and class imbalance [71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The raw EHRs data that was ob-  tained from CureMD had all of these inherent issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The details of these data  challenges and the required preprocessing techniques that were used to tackle  these challenges are discussed below:  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Handling Records with no History  The aim of this study is to build an early T2DM prediction system that relies  solely on patients’ historic clinical diagnoses, recorded vitals, and demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Thus, it is desired to filter out only those patients that have some historic visits  before being diagnosed with T2DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' However, in the raw EHR data, a large  number of records were encountered where a patient was marked with T2DM  8  (indicated with an E11 ICD-10-CM code) in the very first encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Such  examples occur in cases when a patient, already diagnosed with T2DM walks  into a hospital or practice for the first time without any recorded medical history  in the EHR of the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Thus, a strategy is required to eliminate such example  points having no historic encounters prior to the diagnosis of T2DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' For this, all  the T2DM-diagnosed patients that do not have at least 4 previous encounters  before being diagnosed with T2DM were filtered out of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' A total of  72,626 patients with T2DM were identified in the raw data, out of which only  5,770 had at least previous 4 encounters before being diagnosed with T2DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  For the non-diabetics, the patients with less than 4 encounters were filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  After the execution of this step, the data set was reduced from 72,626 to  5,770 T2DM patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Also, from a total of 962,263 non-diabetic patients, only  186,865 patients were left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Hence, after executing the first step of our pre-  processing a dataset with 5,770 diabetic and 186,865 non-diabetic patients was  created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Handling Missing Data  The problem of missing data is widely encountered in EHRs [71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Differ-  ent studies have implied different approaches for handling missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Tech-  niques like dropping complete records with missing values, and filling in missing  values using mean, median, or from the nearest neighbors, have been used in  many research studies [27, 37, 8, 41, 62, 29, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Raw EHR records were analyzed  to identify the attributes that were most affected by the problem of missing val-  ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' As a result of this analysis, it was found that all the attributes that have  the problem of missing values belonged to the vitals component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' To deal with  missing values a 2-step strategy was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' In the first step, a missing value  ratio was calculated for every attribute using equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  MissingV aluesRatio = NumberOfRecordsWithMissingV alues  TotalNumberofRecords  ∗ 100  (1)  Table 2 presents the details of the missing value ratio of different attributes  9  Table 2: Missing value ratio for vitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Attribute  Missing Value Ratio (%)  Weight  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='67  Height  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='53  BMI  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='34  Lean Body Weight  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='71  Ideal Body Weight  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='14  Neck Circumference  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='22  Waist  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='80  Oxygen Saturation  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='15  Peak Expiratory Flow  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='99  Blood Type  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='82  Blood Rh  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='94  Finger Stick  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='98  Pulse  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='91  Respiration  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='93  Temperature  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='66  Systolic Blood Pressure  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='48  Diastolic Blood Pressure  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='44  10  in the vitals component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Based on the results presented in the table 2 the vitals  attributes for which the missing value ratio was higher than 50% were dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  The reason for dropping these attributes with a higher missing value ratio lies  in the fact that such variables do not possess a sufficient amount of information  that can be used to impute the missing values in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  The second step of the applied strategy consists of imputing the missing  values for the remaining vitals attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The following set of steps was used  to impute the missing values for vitals:  Firstly, all the records were grouped based on the patient they belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Secondly, the patient groups having no value for any of the vitals attributes  were dropped, as the missing data cannot be imputed for such patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Thirdly, for each patient group, the Exponentially Weighted Moving Av-  erage (EWMA) of every vitals attribute was calculated and stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Finally, all the missing values were populated by their respective calculated  values, calculated using the EWMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  EWMA was used for data imputation keeping in mind the fact that EHR data  is temporal in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' EWMA can be used to assign larger weights to recent  observations, hence giving more importance to the recent ones as compared to  the older data points [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Such behavior is desirable and quite effective in  capturing the temporal nature associated with a patient’s EHR records where  more weight is given to recent visits and less weight to older visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' For the  remaining two components i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=', patient demographics and diagnosis the problem  of missing value did not exist in the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' After applying this 2-step strategy  the dataset size was further reduced to a total of 4,124 diabetic and 181,767 non-  diabetic patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Creating Single Example Points  The data of every patient consists of multiple encounters/visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' This step  deals with merging these multiple encounters of a single patient into one example  11  point that can be used for training and evaluating machine and deep learning  techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The process of creating the example points is given as follows:  All three components i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=', diagnosis, vitals, and demographics, were merged  into a single record for every patient encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  For every encounter the age of the patient at the time of the encounter  was calculated using the patient’s date of birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Next for every patient all the encounter records were sorted using the date  of the encounter in ascending order of the date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  For every T2DM patient, all the encounter records before the first T2DM  diagnosis encounter were selected and the associated diagnoses were ex-  tracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The ICD-10-CM diagnosis codes were merged together creating a  single comma-separated string enclosing all the unique historic diagnoses  of a patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Next, the vitals from the first T2DM encounter were merged with the  diagnoses attribute along with the demographic information of the patient,  creating a single T2DM positive example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Label 1 was also generated for the created T2DM example points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  For every non-T2DM patient, all the diagnoses from all the previous en-  counters were merged to create a single comma-separated diagnoses fea-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Also, the vitals from the last encounter and the patient’s demo-  graphic information were merged with the diagnoses feature to create a  T2DM negative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Label 0 was generated for the non-T2DM example points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Dealing with Class Imbalance  Class imbalance is a problem that arises when the data is distributed un-  evenly among the classification categories [74, 75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' In case of class imbal-  ance, the performance of traditional classification techniques is affected where  12  the performance of the techniques becomes biased towards the majority class  and is reduced for the minority class [74, 77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Many techniques have been  presented over the years to deal with class imbalance including oversampling,  undersampling, SMOTE, MUTE, ADASYN [78, 75, 79, 80, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Class imbalance  is another inherent property of health care and medicine data [82, 62, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Techniques like SMOTE, ADASYNC, data oversampling, data undersam-  pling, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' have been used for balancing data for T2DM prediction as well [22,  49, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The EHR data that is used as part of this study has a huge class im-  balance having a total of 4,124 diabetic and 181,767 non-diabetic records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' A  combination of 5-fold cross-validation combined with undersampling is applied  to handle this class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' A total of 5 non-overlapping samples of size  4,124 were selected from the non-T2DM examples which are used to train 5  different variations of every machine learning and deep learning technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The  details are discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Methodology  A novel deep neural network architecture is proposed for T2DM prediction  that combines multi-headed self-attention with fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The pro-  posed technique called the Self-Attention for Comorbid Disease Net (SACDNet)  uses multiheaded self-attention to extract features from the diagnoses vector  with the intent to find a relation between every pair of diagnoses and uses dense  layers to extract features from vitals and demographics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The resultant fea-  ture vectors are then concatenated and passed through a single dropout layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Finally, a series of fully connected layers are used for T2DM classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Self Attention  The concept of attention was introduced to modify the RNN-based Encoder-  Decoder models [83] where every decoder timestamp will now receive a weighted  sum of all encoder hidden states called the attention score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The attention score  is usually a scaled dot product given by the equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  13  Attention(Q, K, V ) = Softmax(QKT  √dm  )V  (2)  Where Q, K, and V represent the query, key, and value matrixes, obtained by  multiplying the inputs sequence X with three different weight matrixes Wq, Wk,  and Wv respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' If the query, key, and value are all generated using the  same input sequence then the above equation 2 becomes self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The  concept of self-attention involves finding the similarity between all elements in  the input sequence, in our case, this will be to find a relation between all historic  diagnoses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Multi-Headed Self Attention  This involves applying self-attention to the input sequence multiple times,  obtaining different weight matrixes for the query, key, and value [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Thus, it  provides a more diverse set of relationships between the elements of the input  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' These multiple self-attention outputs are concatenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' For given  input sequence X multi-headed self-attention for N heads is given by equation  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  MHSA(Q, K, V ) = Concat(Att1(Q, K, V ), Att2(Q, K, V ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='., AttN(Q, K, V )))WO  (3)  Where Attentioni represents self-attention obtained using different Wq, Wk, and  Wv weight metrixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' N represents the number of heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' SACDNet  For the given diagnosis matrix D, and the vitals/demographics matrix V  SACDNet feature extraction is given using equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Feature(D, V ) = Concat(MHSA(Q, K, V ), MLP(V ))  (4)  Where MLP represents a set of fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' This feature vector  is then passed to another set of fully connected layers, for T2DM classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  The details of the network are given in the table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  14  Table 3: Architecture of SACDNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Network Layers  1  Diagnosis Input Vector  Vitals/Demographics Input Vector  2  Multiheaded Self Attention (N=3, Query  Dimension=3, Key Dimension=3, Value  Dimension=3)  Fully  Connected  (Units=256,  Activa-  tion=SELU)  3  Fully  Connected  (Units=128,  Activa-  tion=SELU)  4  Concatenate (To a single merged feature vector)  5  Dropout (Rate=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1)  6  Fully Connected (Units=64, Activation=SELU)  7  Fully Connected (Units=64, Activation=Sigmoid)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Monte Carlo (MC) Dropout  Deep neural networks are deterministic in nature, which is not always de-  sirable, especially in the case of high-risk systems such as clinical systems [85,  86, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  It is desirable to have a system that can provide associated confi-  dence with its prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The probability distribution returned by the Soft-  max function cannot be considered a model’s confidence as it has been proven  that neural network-based models can make incorrect predictions with higher  confidence [88, 89, 90, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Epistemic or Model uncertainty estimation using  MC dropout [88] has been used widely for deep neural network-based mod-  els [91, 92, 93, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The idea of MC-Dropout is to get a prediction for given  input X from T highly similar ensemble models and then obtain an average of  these T different output probability distributions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' This is achieved by ap-  plying the concept of dropout regularization at inference time, which in fact  yields T different distributions of the model’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Hence, a large dif-  ference in the prediction of these T different models can be an indicator of  larger model uncertainty and such can help in obtaining more robust predictive  confidence [87, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  p(y = 1) = 1  T  T  �  i  P(y = 1|x, Wi)  (5)  15  Where for given T forward passes from the network we will have T different  parameter distributions W1, W2, W3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=', WT as given in equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Measuring Predictive Uncertainty  For the given output probability distributions from T different passes, differ-  ent measures have been used for uncertainty quantization [91, 95, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Entropy  has been widely used as a measure of uncertainty quantization [91, 95, 96], where  entropy for the averaged MC-Dropout results is calculated using the equation  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  H(y) = −  C  �  c  1  T  T  �  t  P(y = cc|x, Wt) ∗ log(  T  �  t  P(y = cc|x, Wt))  (6)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Proposed Framework  The proposed framework combines the novel SACDNet with MC-Dropout  to build a system that can predict T2DM with associated uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The idea  is to get an ensemble of 100 different highly similar models, for which an average  prediction probability for T2DM is calculated 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  MC − SACDNet = 1  T  T =100  �  t  SACDNett  (7)  Next using the entropy metric discussed before uncertainty value for the  prediction is calculated and based on a specified threshold theta a prediction is  marked as uncertain or certain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' An overview of the framework is given in fig-  ure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' It can be observed in the figure 1 the electronic health record of a patient  is divided into 2 different feature vectors comprising of the historic diagnoses  and vitals-demographics respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' These feature vectors are passed through  100 different representation of the MC-Dropout SACDNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The generated set  of probabilities is averaged to get the final probability which is passed through  a threshold function to predict T2DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Also, the same mean probability is used  to quantize uncertainty using entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  16  Figure 1: Proposed framework for T2DM prediction with uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Experiments  This section presents details of experiments performed on the developed  T2DM corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Machine learning algorithms, deep learning algorithms, and  proposed SACDNet are applied to the dataset to train T2DM classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' As  discussed in section 3, the proposed corpus has a huge class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' This  class imbalance can affect the performance of the classifiers [74, 77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' To deal  with this problem 5-fold cross-validation was applied obtaining a balanced set  by randomly undersampling the majority class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' 7 different machine learning  techniques including AdaBoost, Decision Trees, Logistic Regression, Gaussian  Na¨ıve Bayes, Random Forest, SVM, and Gradient Boosted Trees (XGB) were  trained using the proposed corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Multiple deep neural network-based tech-  niques were trained using the proposed corpus [47, 46, 31, 46, 48], however, apart  from our proposed novel technique SACDNet, only two other techniques were  able to produce satisfactory results, referred to as the FCN and FCN Dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  The FCN network consists of 4 fully connected hidden layers using ReLU ac-  17  Feature  Value  SACDNet1  SACDNet2  T=100  1  Threshold  Z  SACDNet3  SACDNett  T  Average  Probability  t  EHR  Feature  Value  Entropy  Threshold  H(y)  Entropy 0  SACDNetTtivations whereas the FCN Dropout consists of 3 fully connected hidden layers  with hidden layer one having a dropout rate of 30% and hidden layer two having  a dropout rate of 20%, where using the same ReLU activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  The details of the experimentation process are given as follows:  Firstly, all the T2DM positive and negative class examples were separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Next 5 random samples, without overlap were selected from the non-  T2DM examples, where the size of each sample selected was equal to  the size of our T2DM positive examples set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Next 5 different datasets were created combining the T2DM positive set  with each of the 5 non-T2DM sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Next for every dataset an 80-20% train test split was done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Every technique was trained using a 5-fold cross-validation, where one of  the 5 balanced samples was used for each fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Performance of each model per fold was calculated using the evaluation  measures discussed in the results section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  An average performance of every model was also calculated using the  results from the 5 folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  For a fair comparison, none of the models’ hyperparameters were tuned for both  machine and deep learning techniques  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Results and Analysis  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Evaluation Measures  For the evaluation of the models presented in this study, 5 widely used met-  rics were used including Accuracy, F1-Score, Precision, Recall, and Specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  These have also been used for evaluating disease prediction systems, especially  for T2DM prediction [24, 36, 5, 38, 39, 13, 7, 41, 42, 29, 26, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' For the evalu-  ation of the uncertainty of the model, we have used entropy as discussed in the  methodology section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  18  Table 4: Mean of 5 folds cross-validation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Model  Accuracy  (%)  F1-Score  (%)  Precision  (%)  Recall  (%)  Specificity  (%)  1  Ada Boost  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='8  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='0  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='7  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='4  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='8  2  Decision Tree  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='2  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='8  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='5  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1  3  Logistic Regression  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='9  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='0  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='5  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='5  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='9  4  Na¨ıve Bayes  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='6  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='8  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='9  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='6  5  Random Forest  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='4  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='0  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='7  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='4  6  SVM  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='2  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='6  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='2  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='2  7  XGB  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='2  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='5  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='8  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1  8  FCN  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='7  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='6  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='4  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='9  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='7  9  FCN Dropout  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='3  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='5  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='4  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='5  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='3  10  SACDNet  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='3  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='5  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='9  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Comparison Under Fixed Settings  This section presents a comparison of the performance of all the techniques  discussed, using different evaluation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Firstly, the results obtained  against the test set for each of the 5 data folds are presented in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Next,  the average accuracy, f1-score, precision, recall, and specificity are presented in  table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Figure 2: Results of 5 folds cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  It can be observed from the table 4 that our proposed technique SACD-  19  Accuracy  Accuracy  Accuracy  F1-Score  F1-Score  F1-Score  Precision  Precision  Precision  Recall  Recall  Recall  Specificity  Specificity  Specificity  Accuracy  Accuracy  F1-Score  F1-Score  Precision  Precision  Recall  Recall  Specificity  SpecificityNet performs better compared with 7 different machine learning and 2 neural  network-based techniques, achieving a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='6% increased accuracy and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='3% in-  creased F1-Score compared to the other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Fairness Evaluation  It is desirable to have a T2DM prediction system that is able to perform  equally well on a diverse set of examples, without having any bias towards  a certain gender, age, or racial group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  The proposed dataset collected from  different EHR systems running in different parts of the US was built to capture  examples belonging to a larger population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Such a diverse dataset can be helpful  in building more generalized machine and deep learning models that can perform  equally well for examples belonging to different demographic groups of patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  In order to evaluate this hypothesis, the next set of analyses was performed to  validate the fairness of each of the techniques used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  For fairness evaluation, the proposed dataset was divided into certain groups  based on the associated demographic information of the patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Example  points were divided based on age, gender, and race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' All the techniques were  evaluated against the individual groups and the results are presented in the  figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  It can be observed from the figure 3 that all the techniques have identical  performance for all the race and gender groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' However, there is a sudden  drop in the case of precision, recall, specificity, and f1-score for two age groups  namely the children and preschoolers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The reason for this is the unavailability  of significant examples belonging to the two aforementioned groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' For exam-  ple, for preschoolers, there were only 1569 examples in the dataset all of which  belonged to the non-diabetic class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Hence, there were no diabetic patients be-  longing to this class, and the precision, recall, specificity, and f1-score, cannot  be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Similarly, for Children, only 1857 examples were present all of  which belonged to the non-diabetic class, and hence the same holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Hence  the sudden drop does not represent the biasness of the techniques but only the  unavailability of data for the calculation of precision, recall, specificity, and f1-  20  Figure 3: Performance of every technique against certain demographic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  score, which can be further validated by no drop in the accuracy for these two  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Also, for the unspecified gender, only one example of a diabetic patient  and only 6 examples of non-diabetics were encountered which again is causing  discrepancies in the performance of certain evaluation measures even though the  accuracy of the models is close to near perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Hence it can be stated that all  the techniques were able to generalize fairly against all the data groups which  confirm the diversity of the proposed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Uncertainty Analysis  As discussed in the methodology section 4, the proposed SACDNet is in-  tended to be part of a complete T2DM prediction framework that can make  predictions with associated confidence, marking predictions with lower confi-  dence as uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' In order to draw a comparison of the proposed technique and  its MC-Dropout-based extension figure 4 was plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The horizontal straight  dashed lines represent the probability thresholding for the simple model while  the vertical straight line represents the entropy threshold for the MC-Dropout  21  Accuracy  F1-Score  Over Eighty  Asian Americans  African Amer  Over Eighty  Asian Americans  African Amer  Aged  Latino Americans  Aged  Latino Americans  Middle Aged  Mixed Race  Middle Aged  Mixed Race  Adult  Native American  Adult  Native American  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
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+page_content='Adolescen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='OtherRaces  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='Ada Boost  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='Child  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='Preschooler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='Feamles  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='Unspecified  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='Malesextension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' It can be observed from the figure 4 that the MC-Dropout extension  used with entropy is able to identify more misclassified examples as uncertain,  making it more reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Figure 4: Comparison of confidences of SACDNet and MC-SACDNet for misclassified exam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  In order to further clarify the reliability of the MC-Dropout extension tech-  nique all the examples from the test that were misclassified by either technique  were separated and the entropy for the output probabilities was calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' The  histogram of the calculated entropy is given in figure 5, where each bin repre-  sents the number of examples per entropy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  It can be observed from the figure 5 that a higher entropy, which counts for  a higher uncertainty, is assigned to more misclassified examples by the proposed  MC-Dropout-based extension as compared to the simple model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Hence adding  MC-Dropout further improves the proposed technique, reducing the number of  overconfident wrong inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Conclusion and Future Work  Type 2 diabetes mellitus is a chronic disease that is often undiagnosed and  can lead to severe complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' It is desirable to have a system that can help in  the early diagnosis of T2DM, reducing the yearly health expenses spent tackling  22  LD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='B  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  I Predictions  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='6  ClassificationDifference  Correctly Classified by Both  Misclassified by Dropout Model  Misclassified by Both  Misclassified by Sinqle Model  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='2  :  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='6  10  MC-Dropout EntropyFigure 5: Histogram of entropies for output probabilities of SACDNet and MC-SACDNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  complications of undiagnosed T2DM and also saving millions of lives every year  that are lost due to late diagnosis of T2DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' This study proposes a novel deep  learning-based model to predict T2DM using routine EHR features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' SACDNet  was able to perform better than the baseline machine and deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  The study also presents a complete real-world solution for building an early  diabetes detection system, developing a diverse corpus from real-world EHR  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  Also, the proposed framework based on MC-Dropout can help make  predictions with more reliability and abstain from predicting in cases where the  model is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  The proposed technique and framework can be plugged  into existing EHR software with little or no changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Also, the methodology  followed in this study can be used to build predictive software for other diseases  like hypertension, ischemic heart disease, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='  References  [1] Types  of  diabetes  mellitus,  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='webmd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content='com/diabetes/  types-of-diabetes-mellitus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
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+page_content=' Ghahramani, Deep bayesian active learning with image  data, in: International Conference on Machine Learning, PMLR, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
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+page_content='  [96] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Lukovnikov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Daubener, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
+page_content=' Fischer, Detecting compositionally out-of-  distribution examples in semantic parsing, in: Findings of the Association  for Computational Linguistics: EMNLP 2021, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}
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+page_content='  34' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfAguY/content/2301.04844v1.pdf'}