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+arXiv:2301.03958v1  [math.AP]  10 Jan 2023
+Rigidity results for the p-Laplacian Poisson problem with Robin
+boundary conditions
+Alba Lia Masiello*, Gloria Paoli
+Abstract
+Let Ω ⊂ Rn be an open, bounded and Lipschitz set. We consider the Poisson problem for
+the p−Laplace operator associated to Ω with Robin boundary conditions. In this setting, we
+study the equality case in the Talenti-type comparison stated in [6]. We prove that the equality
+is achieved only if Ω is a ball and both the function u and the right hand side f of the Poisson
+equation are radial.
+Keywords: Robin boundary conditions, p-Laplace operator, rigidity result, Talenti compari-
+son.
+MSC 2020: 35J92, 35J25, 46E30.
+E-mail address, A.L. Masiello (corresponding author): albalia.masiello@unina.it
+Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli studi di Napoli
+Federico II, Via Cintia, Complesso Universitario Monte S. Angelo, 80126 Napoli, Italy.
+E-mail address, G. Paoli: gloria.paoli@fau.de
+Department of Data Science (DDS) Chair in Dynamics, Control and Numerics (Alexander von
+Humboldt-Professorship), Cauerstr. 11, 91058 Erlangen, Germany.
+1
+Introduction
+Symmetrization techniques in the context of qualitative properties of solutions to second-order
+elliptic boundary value problems are introduced by Talenti in [25]. In this seminal paper, the
+author considers an open, bounded and Lipschitz set Ω ⊂ Rn, the ball Ω♯ with the same measure
+as Ω and the solutions u and v to the following problems
+®
+−∆uD = f
+in Ω,
+uD = 0
+on ∂Ω,
+®
+−∆vD = f ♯
+in Ω♯,
+vD = 0
+on ∂Ω♯,
+(1)
+where f ∈ L2(Ω) is a positive function and f ♯ is its Schwarz rearrangement of f (see Definition
+2.3). In this setting, Talenti proves the following point-wise estimate:
+u♯
+D(x) ≤ vD(x),
+for all x ∈ Ω♯.
+(2)
+For sake of completeness, we observe that this result is proved more generally for a uniformly
+elliptic linear operator in divergence form.
+1
+
+1
+INTRODUCTION
+2
+A version of this result for nonlinear operators in divergence form is contained in [26], which
+includes as a special instance the case of the p-Laplace operator. Moreover, these results are later
+extended, for instance, to the anisotropic elliptic operators in [2], to the parabolic case in [4], and
+to higher order operators in [9, 28].
+Once a comparison result holds, it is natural to ask whether the equality cases can be character-
+ized and, so, if a rigidity result is in force. In [3], the rigidity result linked to problem (1) is proved.
+Indeed, the authors prove that if equality holds in (2), then Ω is a ball, u is radially symmetric
+and decreasing and f = f ♯. Rigidity results for a generic linear, elliptic second order operator can
+be found in [17] and [18]. To the best of our knowledge, rigidity results for nonlinear operators
+with Dirichlet boundary conditions are not present in the literature. In this paper, we obtain, as a
+corollary of our results, the rigidity for the p−Laplace operator with Dirichlet boundary conditions
+in any dimension (see Corollary 3.3).
+For a long time, it was believed that comparison results could not be proved by means of
+spherical rearrangement argument when dealing with Robin boundary conditions, until the recent
+paper [5]. The authors consider the following problems
+
+
+
+
+
+−∆u = f
+in Ω,
+∂u
+∂ν + βu = 0
+on ∂Ω,
+
+
+
+
+
+−∆v = f ♯
+in Ω♯,
+∂v
+∂ν + βv = 0
+on ∂Ω♯,
+and they prove a comparison result involving Lorentz norms of u and v whenever f is a non negative
+function in L2(Ω) and β is a positive parameter. In particular, in the case f ≡ 1, they prove
+∥u∥Lp(Ω) ≤ ∥v∥Lp(Ω♯),
+p = 1, 2,
+and, if n = 2, the pointwise comparison holds:
+u♯(x) ≤ v(x),
+for all x ∈ Ω♯.
+(3)
+In [21], it is proved that (3) is rigid, i.e. the equality case is possible if and only if Ω is a ball
+and u is a radial and decreasing function.
+Generalizations of the results contained in [5] can be found for the anisotropic case in [24], for
+mixed boundary conditions in [1], in the case of the Hermite operator in [13].
+In the present paper, we focus our study on the rigidity for p−Laplace operator. In this case,
+the comparison results are obtained in [6] and the setting is the following.
+Let Ω be a bounded, open and Lipschitz set in Rn, with n ≥ 2. Let p ∈ (1, +∞) and let
+f ∈ Lp′(Ω) be a positive function, where p′ = p/(p − 1). The Poisson problem for the p−Laplace
+operator with Robin boundary conditions is
+
+
+
+
+
+−∆pu := −div(|∇u|p−2∇u) = f
+in Ω
+|∇u|p−2 ∂u
+∂ν + β|u|p−2u = 0
+on ∂Ω,
+(4)
+where ν is the unit exterior normal to ∂Ω and β > 0. A function u ∈ W 1,p(Ω) is a weak solution
+to (4) if
+ˆ
+Ω
+|∇u|p−2∇u∇ϕ dx + β
+ˆ
+∂Ω
+|u|p−2uϕ dHn−1(x) =
+ˆ
+Ω
+fϕ dx,
+∀ϕ ∈ W 1,p(Ω).
+(5)
+
+1
+INTRODUCTION
+3
+The symmetrized problem associated to (4) is the following
+
+
+
+
+
+−∆pv = f ♯
+in Ω♯
+|∇v|p−2 ∂v
+∂ν + β|v|p−2v = 0
+on ∂Ω♯.
+(6)
+In [6] the authors establish a comparison result between suitable Lorentz norms (see Definition
+2.4) of the solutions u and v to problems (4) and (6) respectively. In particular, they prove
+∥u∥Lpk,p(Ω) ≤ ∥v∥Lpk,p(Ω♯),
+∀ 0 < k ≤
+n(p − 1)
+(n − 2)p + n,
+(7)
+and in the case f ≡ 1, they prove
+u♯(x) ≤ v(x),
+1 ≤ p ≤
+n
+n − 1
+(8)
+and
+∥u∥Lpk,p(Ω) ≤ ∥v∥Lpk,p(Ω♯),
+∀ 0 < k ≤
+n(p − 1)
+(n − 2)p + n,
+∀p > 1.
+(9)
+In the present paper, we want to characterize the equality case in (7) and (9), answering to
+the open problem contained in [21]. For simplicity, we state the main Theorem only in the case
+f ∈ Lp′(Ω) positive, since in the case f ≡ 1 the proof is analogous, as we observe in Remark 4.1.
+Theorem 1.1. Let Ω ⊂ Rn be a bounded, open and Lipschitz set and let Ω♯ be the ball centered at
+the origin with the same measure as Ω. Let u be the solution to (4) and let v be a solution to (6).
+If
+∥u∥Lpk,p(Ω) = ∥v∥Lpk,p(Ω♯),
+for some k ∈
+ò
+0,
+n(p − 1)
+(n − 2)p + n
+ò
+(10)
+then, there exists x0 ∈ Rn such that
+Ω = Ω♯ + x0,
+u(· + x0) = v(·),
+f(· + x0) = f ♯(·).
+The idea of the proof of Theorem 1.1 is the following. First of all, we prove that hypothesis
+(10) implies that the superlevel sets of u are balls. The main difficulty is to prove that these balls
+are concentric.
+Differently from the case of the Laplace operator with Dirichlet boundary conditions studied in
+[4, 16], we can’t apply directly the steepest descent method introduced in [8], because it strongly
+relays on the continuity of both the solution and of its gradient. In the case of the p−Laplace
+equation, the continuity of the solution up to the boundary depends on the regularity of the given
+datum f. To overcome this regularity issue we show that u is a solution to a suitable Dirichlet
+problem and it satisfies the Pólya-Szegő inequality with equality sign. Then, we can conclude that
+u is radially symmetric and decreasing, using the classical result contained in [10]. We make use
+of Lemma 3.2, where the rigidity of the Poisson problem for the p-Laplace operator with Dirichlet
+boundary condition is proved under the assumption f ∈ Lp′(Ω) and positive. Up to our knowledge,
+Corollary 3.3 seems to be new in the literature.
+The paper is organized as follows. In Section 2 we recall some definitions about rearrangement
+of functions and we state some lemmas that we will need in the proof of the main theorem. Section
+3 is dedicated to the proof of the main result and we conclude with a list of open problems.
+
+2
+NOTATION AND PRELIMINARIES
+4
+2
+Notation and Preliminaries
+Throughout this article we will denote by |Ω| the Lebesgue measure of an open and bounded
+Lipschitz set of Rn, with n ≥ 2, and by P(Ω) the perimeter of Ω. Since we are assuming that
+∂Ω is Lipschitz, we have that P(Ω) = Hn−1(∂Ω), where Hn−1 denotes the (n − 1)−dimensional
+Hausdorff measure.
+We recall the classical isoperimetric inequality and and we refer the reader, for example, to
+[22, 11, 12, 27] and to the original paper by De Giorgi [15].
+Theorem 2.1 (Isoperimetric Inequality). Let E ⊂ Rn be a set of finite perimeter. Then,
+nω
+1
+nn |E|
+n−1
+n
+≤ P(E),
+(11)
+where ωn is the measure of the unit ball in Rn. Equality occurs if and only if E is (equivalent to)
+a Ball.
+For the following theorem, we refer to [7].
+Theorem 2.2 (Coarea formula). Let Ω ⊂ Rn be an open set with Lipschitz boundary. Let f ∈
+W 1,1
+loc (Ω) and let u : Ω → R be a measurable function. Then,
+ˆ
+Ω
+u(x)|∇f(x)|dx =
+ˆ
+R
+dt
+ˆ
+Ω∩f−1(t)
+u(y) dHn−1(y).
+(12)
+Let us recall some basic notions about rearrangements. For a general overview, see, for instance,
+[19].
+Definition 2.1. Let u : Ω → R be a measurable function, the distribution function of u is the
+function µ : [0, +∞[ → [0, +∞[ defined as the measure of the superlevel sets of u, i.e.
+µ(t) = |{ x ∈ Ω : |u(x)| > t }|.
+Definition 2.2. Let u : Ω → R be a measurable function, the decreasing rearrangement of u is
+the distribution function of µ. We will denote it by u∗(·).
+Remark 2.1. Let us notice that the function µ(·) is decreasing and right continuous and the
+function u∗(·) is its generalized inverse.
+Definition 2.3. The Schwartz rearrangement of u is the function u♯ whose superlevel sets are
+balls with the same measure as the superlevel sets of u.
+We have the following relation between u♯ and u∗:
+u♯(x) = u∗(ωn|x|n),
+where ωn is the measure of the unit ball in Rn, and one can easily check that the functions u, u∗
+e u♯ are equi-distributed, i.e. they have the same distribution function, and it holds
+∥u∥Lp(Ω) = ∥u∗∥Lp(0,|Ω|) = ∥u♯∥Lp(Ω♯),
+for all p ≥ 1.
+
+2
+NOTATION AND PRELIMINARIES
+5
+We also recall the Hardy-Littlewood inequality, an important propriety of the decreasing rear-
+rangement,
+ˆ
+Ω
+|h(x)g(x)| dx ≤
+ˆ |Ω|
+0
+h∗(s)g∗(s) ds.
+So, by choosing h(·) = χ{|u|>t}, one has
+ˆ
+|u|>t
+|g(x)| dx ≤
+ˆ µ(t)
+0
+g∗(s) ds.
+We now introduce the Lorentz spaces (see [28] for more details on this topic).
+Definition 2.4. Let 0 < p < +∞ and 0 < q ≤ +∞. The Lorentz space Lp,q(Ω) is the space of
+those functions such that the quantity:
+∥u∥Lp,q =
+
+
+
+
+
+
+
+p
+1
+q
+ň ∞
+0
+tqµ(t)
+q
+p dt
+t
+ã 1
+q
+0 < q < ∞
+sup
+t>0
+(tpµ(t))
+q = ∞
+is finite.
+Let us observe that for p = q the Lorentz space coincides with the Lp space, as a consequence
+of the Cavalieri’s Principle
+ˆ
+Ω
+|u|p = p
+ˆ +∞
+0
+tp−1µ(t) dt.
+The solutions u to problem (4) and v to problem (6) are both p-superharmonic and, as a conse-
+quence of the strong maximum principle and the lower semicontinuity (see [29, 20]), they achieve
+their minima on the boundary. If we set
+um = min
+Ω u,
+vm = min
+Ω♯ v
+the positiveness of β and the Robin boundary conditions leads to um ≥ 0 and vm ≥ 0. Hence, u
+and v are strictly positive in the interior of Ω. Moreover, we can observe that
+um = min
+Ω u ≤ min
+Ω♯ v = vm,
+(13)
+indeed,
+vp−1
+m P(Ω♯) =
+ˆ
+∂Ω♯ v(x)p−1 dHn−1(x) = 1
+β
+ˆ
+Ω♯ f ♯ dx = 1
+β
+ˆ
+Ω
+f dx
+=
+ˆ
+∂Ω
+u(x)p−1 dHn−1(x)
+≥ up−1
+m P(Ω) ≥ ump−1P(Ω♯).
+Moreover, it holds
+µ(t) ≤ φ(t) = |Ω|,
+∀t ≤ vm.
+(14)
+
+2
+NOTATION AND PRELIMINARIES
+6
+Now, for t ≥ 0, we introduce the following notations:
+Ut = {x ∈ Ω : u(x) > t}
+∂Uint
+t
+= ∂Ut ∩ Ω,
+∂Uext
+t
+= ∂Ut ∩ ∂Ω,
+µ(t) = |Ut|
+and
+Vt =
+¶
+x ∈ Ω♯ : v(x) > t
+©
+,
+∂V int
+t
+= ∂Vt ∩ Ω,
+∂V ext
+t
+= ∂Vt ∩ ∂Ω,
+φ(t) = |Vt|.
+Because of the invariance of the p−Laplacian and of the Schwarz rearrangement of f by rotation,
+the solution v to (6) is radial, so the set Vt are balls.
+Now, we recall some technical Lemmas, proved in [6], that we need in what follows. We recall
+the proof of Lemma 2.3 for reader’s convenience, while we omit the proof of Lemma 2.4 and Lemma
+2.5.
+Lemma 2.3. Let u be the solution to (4) and let v be the solution to (6). Then, for almost every
+t > 0, we have
+γnµ(t)(1− 1
+n)
+p
+p−1 ≤
+Lj µ(t)
+0
+f ∗(s) ds
+å
+1
+p−1 Ç
+−µ′(t) +
+1
+β
+1
+p−1
+ˆ
+∂Uext
+t
+1
+u dHn−1(x)
+å
+(15)
+and
+γnφ(t)(1− 1
+n)
+p
+p−1 =
+Lj φ(t)
+0
+f ∗(s) ds
+å
+1
+p−1 Ç
+−φ′(t) +
+1
+β
+1
+p−1
+ˆ
+∂V ext
+t
+1
+v dHn−1(x)
+å
+.
+(16)
+where γn =
+Ä
+nω1/n
+n
+ä
+p
+p−1.
+Proof. Let t > 0 and h > 0. In the weak formulation (5), we choose the following test function
+ϕ(x) =
+
+
+
+
+
+0
+if u < t
+u − t
+if t < u < t + h
+h
+if u > t + h,
+(17)
+obtaining
+ˆ
+Ut\Ut+h
+|∇u|p dx + βh
+ˆ
+∂Uext
+t+h
+up−1 dHn−1(x) + β
+ˆ
+∂Uext
+t
+\∂Uext
+t+h
+up−1(u − t) dHn−1(x)
+=
+ˆ
+Ut\Ut+h
+f(u − t) dx + h
+ˆ
+Ut+h
+f dx.
+(18)
+Dividing (18) by h, using coarea formula and letting h go to 0, we have that for a.e. t > 0
+ˆ
+∂Ut
+g(x) dHn−1 =
+ˆ
+Ut
+f dx,
+where
+®
+|∇u|p−1
+if x ∈ ∂Uint
+t
+,
+βup−1
+if x ∈ ∂Uext
+t
+.
+(19)
+
+2
+NOTATION AND PRELIMINARIES
+7
+Using the isoperimetric inequality, for a.e. t ∈ [0, +∞) we have
+nω
+1
+nn µ(t)
+n−1
+n
+≤ P(Ut) =
+ˆ
+∂Ut
+dHn−1
+(20)
+≤
+ň
+∂Ut
+g dHn−1(x)
+ã 1
+p Lj
+∂Ut
+1
+g
+1
+p−1
+dHn−1(x)
+å1− 1
+p
+(21)
+=
+ň
+∂Ut
+g dHn−1(x)
+ã 1
+p Lj
+∂Uint
+t
+1
+|∇u| dHn−1(x) +
+1
+β
+1
+p−1
+ˆ
+∂Uext
+t
+1
+u dHn−1(x)
+å1− 1
+p
+(22)
+≤
+Lj µ(t)
+0
+f ∗(s) ds
+å 1
+p Ç
+−µ′(t) +
+1
+β
+1
+p−1
+ˆ
+∂Uext
+t
+1
+u dHn−1(x)
+å1− 1
+p
+,
+(23)
+and, so, (15) follows. Finally, we notice that, if v is the solution to (6), then all the inequalities
+above are equalities, and, consequently, we have (16).
+Lemma 2.4. For all τ ≥ vm, we have
+ˆ τ
+0
+tp−1
+Lj
+∂Uext
+t
+1
+u(x) dHn−1(x)
+å
+dt ≤ 1
+pβ
+ˆ |Ω|
+0
+f ∗(s) ds.
+(24)
+Moreover,
+ˆ τ
+0
+tp−1
+ň
+∂Vt∩∂Ω♯
+1
+v(x) dHn−1(x)
+ã
+dt = 1
+pβ
+ˆ |Ω|
+0
+f ∗(s) ds,
+(25)
+Lemma 2.5 (Gronwall). Let ξ(τ) be a continuously differentiable function, let q > 1 and let C be
+a non negative constant C such that the following differential inequality holds
+τξ′(τ) ≤ (q − 1)ξ(τ) + C
+∀τ ≥ τ0 > 0.
+Then, we have
+ξ(τ) ≤
+Å
+ξ(τ0) +
+C
+q − 1
+ã Å τ
+τ0
+ãq−1
+−
+C
+q − 1
+∀τ ≥ τ0,
+(26)
+and
+ξ′(τ) ≤
+Å(q − 1)ξ(τ0) + C
+τ0
+ã Å τ
+τ0
+ãq−2
+∀τ ≥ τ0.
+(27)
+The following Lemma is contained in [4].
+Lemma 2.6. Let f, g ∈ L2(Ω) be two positive functions. If
+ˆ
+Ω
+fg dx =
+ˆ
+Ω♯ f ♯g♯ dx,
+(28)
+then, for every τ ≥ 0 there exists t ≥ 0 such that we have, up to zero measure set,
+{g > τ} = {f > t}.
+(29)
+
+3
+PROOF OF THEOREM ??
+8
+We conclude this preliminary session, recalling the classical results contained in [10] (see The-
+orem 1.1 and Lemma 2.3). In particular, the result contained in (iii) of Lemma (2.7) gives the
+rigidity of the Pólya-Szegő inequality (see [23]):
+ˆ
+Rn |∇u♯|p dx ≤
+ˆ
+Rn |∇u|p dx,
+∀u ∈ W 1,p(Rn).
+(30)
+Theorem 2.7. Let w ∈ W 1,p(Rn), let σ(t) be its distribution function and let
+wM :=
+®
+∥w∥∞
+if w ∈ L∞(Ω)
++∞
+otherwise.
+Then, the following are true:
+i. For almost all t ∈ (0, wM),
+∞ > −σ′(t) ≥
+ˆ
+w−1(t)
+1
+|∇w|dHn−1
+(31)
+ii. σ is absolutely continuous if and only if
+���
+¶
+|∇w♯| = 0
+©
+∩
+¶
+0 < w♯ < wM
+©��� = 0.
+(32)
+iii. If
+ˆ
+Rn |∇w|p =
+ˆ
+Rn |∇w♯|p,
+(33)
+and (32) holds, then there exist a translate of w♯ which is almost everywhere equal to w.
+Remark 2.2. We observe that in [14], it is proved that the condition
+|{ |∇w| = 0 } ∩ { 0 < w < wM }| = 0
+(34)
+implies (32). So, by (iii) in Lemma 2.7, if we have (33) and (34), there exists a translated of w♯
+which is almost everywhere equal to w.
+Remark 2.3. We observe that the Pólya -Szegő inequality (30) and the relative rigidity result
+(iii) contained in Lemma 2.7 hold also if we assume w ∈ W 1,p
+0 (Ω). Indeed, it is easily proved that
+for every w ∈ W 1,p
+0 (Ω) one has w♯ ∈ W 1,p
+0 (Ω♯).
+3
+Proof of Theorem 1.1
+In order to prove the main Theorem 1.1, we divide the proof into the following steps. First of all,
+we prove that, under the assumptions of Theorem 1.1, equality holds in (15) and this is the content
+of Proposition 3.1. Then, in Proposition 3.4, we prove that equality in (15) implies the fact that Ω
+is a ball and u and f are radial functions. In order to prove this last step, we need the key Lemma
+3.2.
+
+3
+PROOF OF THEOREM ??
+9
+Proposition 3.1. Let u be the solution to (4) and let v be the solution to (6). If there exists k
+k ∈
+ò
+0,
+n(p − 1)
+(n − 2)p + n
+ò
+such that
+∥u∥Lpk,p(Ω) = ∥v∥Lpk,p(Ω♯),
+then equality holds in (15) for almost every t.
+Proof. Since we are assuming that ∥u∥Lpk,p(Ω) = ∥v∥Lpk,p(Ω♯), we have that
+ˆ +∞
+0
+tp−1µ(t)
+1
+k dt =
+ˆ +∞
+0
+tp−1φ(t)
+1
+k dt.
+(35)
+Let us multiply (15) by tp−1µ(t)α, where α = 1
+k −
+Å
+1 − 1
+n
+ã
+p
+p − 1, and let us integrate from 0 to
++∞:
+γn
+ˆ +∞
+0
+tp−1µ
+1
+k (t) dt
+≤
+ˆ +∞
+0
+Lj µ(t)
+0
+f ∗(s) ds
+å
+1
+p−1 Ç
+−µ′(t) +
+1
+β
+1
+p−1
+ˆ
+∂Uext
+t
+1
+u dHn−1
+å
+tp−1µ(t)α dt
+≤
+ˆ +∞
+0
+tp−1µ(t)α
+Lj µ(t)
+0
+f ∗(s) ds
+å
+1
+p−1
+(−µ′(t)) dt + |Ω|α
+pβ
+p
+p−1
+Lj |Ω|
+0
+f ∗(s) ds
+å
+p
+p−1
+,
+(36)
+where in the last inequality we have used µ(t) ≤ |Ω| and (24) in Lemma 2.4. As far as v is
+concerned, it holds
+γn
+ˆ +∞
+0
+tp−1φ
+1
+k (t) dt
+=
+ˆ +∞
+0
+Lj φ(t)
+0
+f ∗(s) ds
+å
+1
+p−1 Ç
+−φ′(t) +
+1
+β
+1
+p−1
+ˆ
+∂V ext
+t
+1
+u dHn−1
+å
+tp−1φ(t)α dt
+=
+ˆ +∞
+0
+tp−1φ(t)α
+Lj φ(t)
+0
+f ∗(s) ds
+å
+1
+p−1
+(−φ′(t)) dt + |Ω|α
+pβ
+p
+p−1
+Lj |Ω|
+0
+f ∗(s) ds
+å
+p
+p−1
+.
+(37)
+We observe that the left-hand-side of (36) and the left-hand-side of (37) are equal from (35). So,
+it follows
+ˆ +∞
+0
+tp−1φ(t)α
+Lj φ(t)
+0
+f ∗(s) ds
+å
+1
+p−1
+(−φ′(t)) dt ≤
+ˆ +∞
+0
+tp−1µ(t)α
+Lj µ(t)
+0
+f ∗(s) ds
+å
+1
+p−1
+(−µ′(t)) dt.
+(38)
+Setting F(l) =
+ˆ l
+0
+ωδ
+�ˆ ω
+0
+f ∗(s) ds
+�
+1
+p−1
+dω, and integrating (38) by parts, we get
+ˆ ∞
+0
+tp−2F(φ(t)) dt ≤
+ˆ ∞
+0
+tp−2F(µ(t)) dt,
+being µ(t) = φ(t) = 0 for t > vM. In [6] (see the proof of Theorem 1.1), it is proved that
+ˆ ∞
+0
+tp−2F(µ(t)) dt ≤
+ˆ ∞
+0
+tp−2F(φ(t)) dt.
+(39)
+
+3
+PROOF OF THEOREM ??
+10
+and we recall here the proof for the reader’s convenience. In order to do that, we multiply (15)
+by tp−1F(µ(t))µ(t)− (n−1)p
+n(p−1) and we integrate between 0 and τ > vm. First, we observe that, by
+the hypothesis k ≤
+n(p − 1)
+(n − 2)p + n, it follows that the function h(l) = F(l)l− (n−1)p
+n(p−1) is non decreasing.
+Hence, we obtain
+ˆ τ
+0
+γntp−1F(µ(t)) dt ≤
+ˆ τ
+0
+�
+−µ′(t)
+�
+tp−1µ(t)− (n−1)p
+n(p−1) F(µ(t))
+Lj µ(t)
+0
+f ∗(s) ds
+å
+1
+p−1
+dt
++ F(|Ω|)|Ω|− p(n−1)
+n(p−1)
+pβ
+p
+p−1
+Lj |Ω|
+0
+f ∗(s) ds
+å
+p
+p−1
+.
+If we integrate by parts both sides of the last expression and we set
+C = F(|Ω|)|Ω|− p(n−1)
+n(p−1)
+pβ
+p
+p−1
+Lj |Ω|
+0
+f ∗(s) ds
+å
+p
+p−1
+,
+we obtain
+τ
+ˆ τ
+0
+γntp−2F(µ(t)) dt + τHµ(τ) ≤
+ˆ τ
+0
+ˆ t
+0
+rp−2F(µ(r)) drdt +
+ˆ τ
+0
+Hµ(t) dt + C,
+(40)
+where
+Hµ(τ) = −
+ˆ +∞
+τ
+tp−2µ(t)− p(n−1)
+n(p−1) F(µ(t))
+ň µ(t)
+0
+f ∗(s) ds
+ã
+1
+p−1
+dµ(t).
+Setting now
+ξ(τ) =
+ˆ τ
+0
+ˆ t
+0
+γnrp−2F(µ(r)) dr +
+ˆ t
+0
+Hµ(t) dt,
+inequality (40) becomes
+τξ′(τ) ≤ ξ(τ) + C.
+So, Lemma 2.5, with τ0 = vm and q=2, gives
+ˆ τ
+0
+γntp−2F(µ(t)) dt + Hµ(τ) ≤
+܈ vm
+0
+tp−2F(µ(t) dt + Hµ(vm) + C
+vm
+ê
+.
+Of course, the inequality holds as equality if we replace µ(t) with φ(t), so we get:
+ˆ τ
+0
+γntp−2F(µ(t)) dt + Hµ(τ) ≤
+ˆ τ
+0
+γnF(φ(t)) dt + Hφ(τ),
+keeping in mind that µ(t) ≤ φ(t) = |Ω| for t ≤ vm. Now, letting τ → ∞, one has
+ˆ ∞
+0
+tp−2F(µ(t))dt ≤
+ˆ ∞
+0
+tp−2F(φ(t))dt,
+since Hµ(τ), Hφ(τ) → 0.
+
+3
+PROOF OF THEOREM ??
+11
+So, we get equality in (36) and, consequently, in (15) for almost every t, indeed
+γn
+ˆ +∞
+0
+tp−1µ
+1
+k (t) dt
+≤
+ˆ +∞
+0
+Lj µ(t)
+0
+f ∗(s) ds
+å
+1
+p−1 Ç
+−µ′(t) +
+1
+β
+1
+p−1
+ˆ
+∂Uext
+t
+1
+u dHn−1
+å
+tp−1µ(t)α dt
+≤
+ˆ +∞
+0
+tp−2F(µ(t)) dt + |Ω|α
+pβ
+p
+p−1
+Lj |Ω|
+0
+f ∗(s) ds
+å
+p
+p−1
+=
+ˆ +∞
+0
+tp−2F(φ(t)) dt + |Ω|α
+pβ
+p
+p−1
+Lj |Ω|
+0
+f ∗(s) ds
+å
+p
+p−1
+=
+ˆ +∞
+0
+Lj φ(t)
+0
+f ∗(s) ds
+å
+1
+p−1 Ç
+−φ′(t) +
+1
+β
+1
+p−1
+ˆ
+∂V ext
+t
+1
+v dHn−1
+å
+tp−1φ(t)α dt
+= γn
+ˆ +∞
+0
+tp−1φ
+1
+k (t) dt = γn
+ˆ +∞
+0
+tp−1µ
+1
+k (t) dt.
+In the following Lemma we prove that a solution to a Dirichlet problem, such that its distribu-
+tion function satisfies the differential equation (42), is necessarily defined on a ball and it has to
+be radial and decreasing.
+Lemma 3.2. Let Ω ⊂ Rn be an open, bounded and Lipschitz set. Let f ∈ Lp′(Ω) be a positive
+function, let w be a weak solution to
+®
+−∆pw = f
+in Ω
+w = 0
+on ∂Ω,
+(41)
+and let σ be the distribution function of w. If σ satisfies the following condition
+γnσ(t)(1− 1
+n)
+p
+p−1 =
+Lj σ(t)
+0
+f ∗(s) ds
+å
+1
+p−1 �
+−σ′(t)
+�
+,
+for a.e. t ∈ [0, wM]
+(42)
+then, there exists x0 such that
+Ω = Ω♯ + x0,
+w(· + x0) = w♯(·),
+f(· + x0) = f ♯(·).
+Proof. First of all, we recall that w is a weak solution to (41) if and only if
+ˆ
+Ω
+|∇w|p−2∇w∇ϕ dx =
+ˆ
+Ω
+fϕ dx,
+∀ϕ ∈ W 1,p
+0 (Ω).
+(43)
+Arguing as in the proof of (15) in Lemma 2.3, choosing the same test function ϕ, defined in (17),
+ϕ(x) =
+
+
+
+
+
+0
+if w < t
+w − t
+if t < w < t + h
+h
+if w > t + h,
+
+3
+PROOF OF THEOREM ??
+12
+one obtains
+ˆ
+∂Wt
+|∇w|p−1 dHn−1 =
+ˆ
+Wt
+f(x) dx ≤
+ˆ σ(t)
+0
+f ⋆(s) ds,
+(44)
+where Wt = {x ∈ Ω : w(x) > t}.
+If we apply the isoperimetric inequality to the superlevel set Wt, the Hölder inequality and the
+Hardy-Littlewood inequality, we get, for almost every t,
+nω
+1
+nn σ(t)
+n−1
+n
+≤ P(Wt) =
+ˆ
+∂Wt
+dHn−1
+(45)
+≤
+ň
+∂Wt
+|∇w|p−1 dHn−1(x)
+ã 1
+p ň
+∂Wt
+1
+|∇w| dHn−1(x)
+ã1− 1
+p
+(46)
+≤
+Lj σ(t)
+0
+f ∗(s) ds
+å 1
+p �
+−σ′(t)
+�1− 1
+p .
+(47)
+So, hypothesis (42) ensures us that equality holds in the isoperimetric inequality (45), in the Hölder
+inequality (46) and in the Hardy-Littlewood inequality (47).
+We now divide the proof into three steps.
+Step 1. Let us prove that the superlevel set { w > t } is a ball for all t ∈ [0, wM). Equality in (45)
+implies that, for almost every t, Wt is a ball. On the other hand, for all t ∈ [0, wM), there exists a
+sequence { tk } such that
+1. tk → t;
+2. tk > tk+1;
+3. {w > tk} is a ball for all k.
+Since { w > t } = ∪k { w > tk } can be written as an increasing union of balls, {w > t} is a ball for
+all t and, in particular, Ω = {w > 0} is a ball too and we obtain that Ω = x0 + Ω♯.
+From now on, we can assume without loss of generality that x0 = 0.
+Step 2. Let us prove that the superlevel sets are concentric balls.
+Equality in (46) implies also equality in Hölder inequality, i.e.
+ˆ
+∂Wt
+dHn−1 =
+ň
+∂Wt
+|∇w|p−1 dHn−1(x)
+ã 1
+p ň
+∂Wt
+1
+|∇w| dHn−1(x)
+ã1− 1
+p
+.
+This means that, for almost every t, |∇w| is constant Hn−1−almost everywhere on ∂Wt , and we
+denote by Ct the (Hn−1−a.e.) constant value of |∇w| on ∂Wt. We claim that Ct ̸= 0 for almost
+every t. Indeed, (44) and the positivity of f ensure us that
+P(Wt)Cp−1
+t
+=
+ˆ
+∂Wt
+|∇w|p−1 dHn−1 =
+ˆ
+Wt
+f(x) dx > 0.
+Integrating (42), we obtain w♯(x) = z(x), for all x ∈ Ω♯, where z is the solution to
+®
+−∆pz = f ♯
+in Ω♯
+z = 0
+on ∂Ω♯,
+(48)
+
+3
+PROOF OF THEOREM ??
+13
+and it has the following explicit form:
+z(x) =
+ˆ |Ω|
+ωn|x|n
+1
+γn
+ň s
+0
+f ⋆(r) dr
+ã1/(p−1)
+1
+s(1−1/n)(p/(p−1)) ds,
+so it easily follows that
+���
+¶
+|∇w♯| = 0
+©
+∩
+¶
+0 < w♯ < wM
+©��� = 0.
+(49)
+Using (ii) in Lemma 2.7, we have that (49) implies the absolutely continuity of σ.
+Now, we denote by C♯
+t the (Hn−1−a.e.) constant value of
+��∇w♯�� on ∂W ♯
+t . We recall that it
+holds
+−σ′(t) =
+ˆ
+∂W ♯
+t
+1
+|∇w♯| = P(∂W ♯
+t )
+C♯
+t
+.
+and, by the absolutely continuity of σ, we have
+−σ′(t) =
+ˆ
+∂Wt
+1
+|∇w| = P(∂Wt)
+Ct
+.
+Since w and w♯ are equi-distributed, we have,
+P(∂Wt)
+Ct
+= P(∂W ♯
+t )
+C♯
+t
+Moreover, since P(∂Wt) = P(∂W ♯
+t ), we have that Ct = C♯
+t. So, by the coarea formula, we get
+ˆ
+Ω
+|∇w|p dx =
+ˆ +∞
+0
+ˆ
+∂Wt
+|∇w|p−1 dHn−1 =
+ˆ +∞
+0
+Cp−1
+t
+P(Wt) dt dHn−1
+=
+ˆ +∞
+0
+Ä
+C♯
+t
+äp−1 P(Wt) dt dHn−1 =
+ˆ +∞
+0
+ˆ
+∂W ♯
+t
+|∇w♯|p−1 dHn−1 =
+ˆ
+Ω♯|∇w♯|p dx.
+By (iii) in Lemma 2.7, we conclude that u = u♯.
+Step 3. Let us prove that f is radial and decreasing.
+Equality in (47) reads, for almost every t,
+ˆ
+Wt
+f(x) dx =
+ˆ σ(t)
+0
+f ∗(s) ds.
+Moreover, for all τ ∈ [0, wM), there exists a sequence { τk } such that
+1. τk → τ;
+2. τk > τk+1;
+3.
+ˆ
+Wτk
+f(x) dx =
+ˆ σ(τk)
+0
+f ∗(s) ds,
+
+3
+PROOF OF THEOREM ??
+14
+and, by the continuity of σ(·), we have
+ˆ σ(τ)
+0
+f ∗(s) ds = lim
+k
+ˆ σ(τk)
+0
+f ∗(s) = lim
+k
+ˆ
+Wτk
+f(x) dx =
+ˆ
+Wτ
+f(x) dx.
+By Lemma 2.6, we have that for all τ, there exists ατ such that
+{w > τ} = {f > ατ}.
+Consequently, we have that also f is radial and decreasing, so f = f ♯.
+As a direct consequence of Lemma 3.2, we obtain the rigidity for the p−Laplace operator with
+Dirichlet boundary conditions.
+Corollary 3.3. Let Ω ⊂ Rn be an open, bounded and Lipschitz set. Let f ∈ Lp′(Ω) be a positive
+function and let w and z be weak solutions respectively to
+®
+−∆pw = f
+in Ω
+w = 0
+on ∂Ω,
+®
+−∆pz = f ♯
+in Ω♯
+z = 0
+on ∂Ω♯.
+(50)
+If w♯(x) = z(x), for all x ∈ Ω♯, then there exists x0 ∈ Rn such that
+Ω = Ω♯ + x0,
+w(· + x0) = z(·),
+f(· + x0) = f ♯(·).
+Proof. From the proof of Lemma 3.2, it follows that the distribution function of w, denoted by σ,
+satisfies
+nω
+1
+nn σ(t)
+n−1
+n
+≤
+Lj σ(t)
+0
+f ∗(s) ds
+å 1
+p �−σ′(t)�1− 1
+p .
+(51)
+Now, we integrate (51) from 0 to t, obtaining
+u∗(t) =
+ˆ |Ω|
+σ(t)
+1
+γn
+ň s
+0
+f ⋆(r) dr
+ã1/(p−1)
+1
+s(1−1/n)(p/(p−1)) ds = z∗(t).
+So, if w♯ = z, we have w∗ = z∗, and consequently we obtain equality in (51) for almost every
+t ∈ [0, wM]. We can conclude by applying Lemma 3.2.
+Now, using Lemma 3.2, we are in position to conclude the proof of the main Theorem.
+Proposition 3.4. Let Ω ⊂ Rn be an open, bounded and Lipschitz set and let Ω♯ be the ball with the
+same measure as Ω. Let u be the solution to (4) and let µ be its distribution function. If equality
+holds in (15), then there exists x0 ∈ Rn such that
+Ω = Ω♯ + x0,
+u(· + x0) = v(·),
+f(· + x0) = f ♯(·).
+
+3
+PROOF OF THEOREM ??
+15
+Proof. Firstly, we claim that the superlevel sets { u > t } are balls for every t ∈ [0, uM). Equality
+in (15) implies the equality in (20), i.e.
+nω
+1
+nn µ(t)
+n−1
+n
+= P(Ut),
+for a. e. t ∈ [0, uM]
+that means that almost every superlevel set is a ball. Arguing as in Step 1 of Lemma 3.2, we
+can conclude that every superlevel set is a ball, so, Ω = {u > um} is a ball and we obtain that
+Ω = x0 + Ω♯.
+Let us observe that for every t, s ∈ [um, uM] with t < s, as both Ut and Us are balls, we have
+that ∂Ut ∩ ∂Us contains at most one point. In particular, the function w = u − um is a weak
+solution to the Dirichlet problem (41) in Ω.
+We claim that σ(t) = |{ w > t }| satisfies (42).
+Since { w > t } = { u > t + um }, we have
+σ(t) = µ(t + um) for all t ∈ [0, uM − um]. Moreover, we have
+ˆ
+∂Ut
+1
+u dHn−1 = 0,
+∀t > um
+So, using the fact that we have equality in (15) by hypothesis, we get
+γnσ(t)(1− 1
+n)
+p
+p−1 = γnµ(t + um)(1− 1
+n)
+p
+p−1
+=
+Lj µ(t+um)
+0
+f ∗(s) ds
+å
+1
+p−1 Ç
+−µ′(t + um) +
+1
+β
+1
+p−1
+ˆ
+∂Uext
+t+um
+1
+u dHn−1(x)
+å
+=
+Lj σ(t)
+0
+f ∗(s) ds
+å
+1
+p−1 �−σ′(t)� ,
+for all t ∈ (0, uM − um). So, we can conclude by Lemma 3.2.
+We conclude now with the proof of the main Theorem.
+Proof of Theorem 1.1. From Proposition 3.1, we have that the hypothesis of Theorem 1.1
+∥u∥Lpk,p(Ω) = ∥v∥Lpk,p(Ω♯),
+for some k ∈
+ò
+0,
+n(p − 1)
+(n − 2)p + n
+ò
+implies the following equality for almost every t ∈ (0, uM)
+γnµ(t)(1− 1
+n)
+p
+p−1 =
+Lj µ(t)
+0
+f ∗(s) ds
+å
+1
+p−1 Ç
+−µ′(t) +
+1
+β
+1
+p−1
+ˆ
+∂Uext
+t
+1
+u dHn−1(x)
+å
+,
+where µ(t) is the distribution function of u.
+Now, we are in position to apply Proposition 3.4, and, so, there exists x0 ∈ Rn such that
+Ω = Ω♯ + x0,
+u(· + x0) = v(·),
+f(· + x0) = f ♯(·).
+
+4
+REMARKS AND OPEN PROBLEMS
+16
+4
+Remarks and open problems
+Remark 4.1. In [6] the authors also prove that in the case f ≡ 1, it holds
+∥u∥Lpk,p(Ω) ≤ ∥v∥Lpk,p(Ω♯),
+if 0 < k ≤
+n(p − 1)
+n(p − 1) − p.
+(52)
+We stress that the proof of Theorem 1.1 can be adapted to case f ≡ 1, regardless of the fact that
+now the admissible k varies in a wider range.
+Open problem 4.2. Below we present a list of open problems and work in progress.
+• Generalize the rigidity results in the anisotropic setting, starting from the comparison proved
+in [24].
+• Generalize the rigidity results to other problems, such as the ones investigated in [1], [13].
+Acknowledgements
+The authors Alba Lia Masiello and Gloria Paoli are supported by GNAMPA of INdAM. The
+author Gloria Paoli is supported by the Alexander von Humboldt Foundation with an Alexander
+von Humboldt research fellowship.
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+

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+Training Differentially Private Graph Neural
+Networks with Random Walk Sampling
+Morgane Ayle
+Technical University of Munich
+morgane.ayle@tum.de
+Jan Schurchardt
+Technical University of Munich
+j.schuchardt@tum.de
+Lukas Gosch
+Technical University of Munich
+l.gosch@tum.de
+Daniel Zügner
+Technical University of Munich
+zuegnerd@in.tum.de
+Stephan Günnemann
+Technical University of Munich
+s.guennemann@tum.de
+Abstract
+Deep learning models are known to put the privacy of their training data at risk,
+which poses challenges for their safe and ethical release to the public. Differentially
+private stochastic gradient descent is the de facto standard for training neural
+networks without leaking sensitive information about the training data. However,
+applying it to models for graph-structured data poses a novel challenge: unlike
+with i.i.d. data, sensitive information about a node in a graph cannot only leak
+through its gradients, but also through the gradients of all nodes within a larger
+neighborhood. In practice, this limits privacy-preserving deep learning on graphs
+to very shallow graph neural networks. We propose to solve this issue by training
+graph neural networks on disjoint subgraphs of a given training graph. We develop
+three random-walk-based methods for generating such disjoint subgraphs and
+perform a careful analysis of the data-generating distributions to provide strong
+privacy guarantees. Through extensive experiments, we show that our method
+greatly outperforms the state-of-the-art baseline on three large graphs, and matches
+or outperforms it on four smaller ones.
+1
+Introduction
+The introduction of Graph Neural Networks (GNNs) has enabled the training of Deep Learning (DL)
+models on graph-structured data and for various tasks such as node classification, link prediction or
+graph classification. However, similar to DL models trained on image [1] or text data [2, 3], GNNs
+leak information about their training data [4–6], such as the features of a node, or which nodes are
+connected by an edge.
+In this paper, we analyze the privacy of GNNs under the lens of Differential Privacy (DP)
+[7]. In particular, we ensure the privacy of all nodes’ features in a graph. While DP-SGD [8] is the de
+facto standard for training DL models with DP, its transfer to GNNs is not straightforward given the
+non-i.i.d. nature of the data. Indeed, since an L-layer GNN typically uses the L-hop neighborhood
+of a node during the forward pass, the gradient of a node does not depend on that node alone, but
+on all nodes in its neighborhood. While some works [9, 10] have attempted to apply DP to GNNs,
+most of them focus on edge-level DP. Methods that can be applied to feature-level DP suffer from
+2022 Trustworthy and Socially Responsible Machine Learning (TSRML 2022) co-located with NeurIPS 2022.
+arXiv:2301.00738v1  [cs.LG]  2 Jan 2023
+
+loose privacy guarantees [9], or rely on custom GNN architectures [10]. We propose an adaptation
+of DP-SGD to train GNNs with feature-level DP while attenuating the aforementioned problem
+and preserving a high model utility. We experimentally demonstrate that our method can offer
+significantly stronger privacy guarantees than prior work, particularly on large graphs.
+2
+Background
+2.1
+Differential privacy
+(ϵ, δ)-DP
+Differential Privacy (DP) [7] is a notion of privacy that allows data analysts to extract
+useful statistics from a dataset, without leaking too much information about the samples in it. More
+formally, given two neighboring datasets D and D′ – denoted D ∼ D′ – that differ by one sample
+(either by deleting, adding or modifying a sample), a randomized algorithm M with co-domain Y
+is (ϵ, δ)-DP if for all O ⊆ Y , and for all D ∼ D′, Pr[M(D) ∈ O] ≤ exp(ϵ)Pr[M(D′) ∈ O] + δ.
+The parameters ϵ and δ are the privacy budget parameters: the smaller their values, the better the
+privacy guarantees.
+(α, γ)-RDP
+An alternative definition of DP is Rényi Differential Privacy (RDP) [11]. A randomized
+algorithm M is said to be γ-RDP of order α – or (α, γ)-RDP – if for any D ∼ D′ it holds that
+Dα(M(D), M(D′)) ≤ γ, where Dα =
+1
+α−1 log Ex∼Q
+�
+P (x)
+Q(x)
+�α
+is the Rényi divergence of order α
+which measures the similarity of the distributions P and Q. Note that if M is (α, γ)-RDP, then it
+is also (ϵ, δ)-DP for any 0 < δ < 1 where ϵ = fRDP→DP(α, γ, δ) = γ + log( α−1
+α ) − log δ+log α
+α−1
+[12].
+We rely on (α, γ)-RDP during our analysis, but report our results in terms of (ϵ, δ)-DP following
+prior work.
+The Gaussian mechanism
+Given an algorithm A with real-valued output space A : ND → Rd,
+the Gaussian mechanism privatizes the algorithm by adding Gaussian noise to the outputs of A,
+i.e. M = Gσ (A (D)) = A(D) + N(0, σ2). Given that the ℓ2 sensitivity of A is ∆2A(D) =
+maxD∼D′ ∥A(D) − A(D′)∥2, the mechanism satisfies (α, γ(α))-RDP, with γ(α) = α(∆2A)2
+2σ2
+.
+Intuitively, this indicates that the larger the sensitivity of the function, the more noise needs to be
+added to obtain a small privacy budget, and therefore the worse the final performance will be. A
+small sensitivity is therefore desirable.
+Amplification by sub-sampling
+A useful property of DP (and RDP) is that, given a mechanism S
+that samples a sub-set of the dataset D, applying a private mechanism to S(D) leads to better privacy
+guarantees than applying it to the entire dataset D. Intuitively, this is due to the fact that subsampling
+introduces a non-zero chance of an added or modified sample to not be processed by the randomized
+algorithm. Typically, S is assumed to be a Poisson or uniform sampling over the dataset. Poisson
+sampling is typically used when the neighboring datasets differ in size, while uniform sampling is
+used otherwise. In this paper, we rely on uniform sampling.
+2.2
+Differential privacy in deep learning
+Differentially Private Stochastic Gradient Descent (DP-SGD) [13, 14, 8] is the foundation of many
+works [9, 2, 15] that apply DP to deep learning. It privatizes the weights of a model with respect to
+the input dataset at every iteration of training, and then accumulates the privacy budget being spent
+over all iterations. One private training iteration consists of batching a set of samples, computing the
+gradient on each sample independently, clipping the norm of each gradient vector to a maximum norm
+C, calculating the entire gradient by adding calibrated Gaussian noise, and finally performing an
+update step. The clipping step is used to bound the sensitivity of the gradients to changes in the input.
+Then, assuming that two neighboring datasets D and D′ differ in the features of one sample, the
+sensitivity of the total gradient on a batch of i.i.d. samples is bounded by 2C. Through batching (i.e.
+sub-sampling the dataset using a sampling mechanism S), amplification by sub-sampling theorems
+[16, 17] can be exploited to get better privacy guarantees at every iteration. Finally, assuming each
+iteration t is (α, γt)-RDP, the overall training is then (α, �T
+t=0 γt)-RDP [11] where T is the total
+number of iterations.
+2
+
+2.3
+Graph neural networks
+Definition
+In the following, we define a graph as G = {X, A}, where X ∈ RN×d is the feature
+matrix in which each row corresponds to one node’s feature vector, and A ∈ {0, 1}N×N is the
+adjacency matrix in which Aij is 1 if there exists an edge between nodes i and j and 0 otherwise.
+Note that we only consider undirected graphs, therefore A = AT . Graph Neural Networks (GNNs)
+are a class of models that learn a mapping f : G → Z ∈ RN×d′, where Z is an updated feature
+matrix of G that can be used for various downstream tasks. Each layer of a GNN typically consists of
+two steps: 1) in the aggregation step, information about the neighborhood of every node is gathered;
+2) in the update step, the feature vector of every node is updated based on its current feature vector
+and the aggregated neighborhood information.
+The receptive field
+The receptive field of a node in a GNN is defined as the region in the input
+graph that influences the GNN’s predictions for that specific node. For a GNN with L layers, the
+receptive field of a node v is the L-hop neighborhood of v. Thus, for a graph with maximum node
+degree K, the largest possible receptive field size of any node v is RF(v) = �L
+l=0 Kl = KL+1−1
+K−1
+,
+i.e. the receptive field grows exponentially with the number of layers of the GNN.
+2.4
+Differential privacy in graph neural networks
+Given that graphs contain two types of attributes – node features and edges – multiple levels of DP
+[18, 9, 10] can be considered: edge-level DP, where the edges between nodes are private; feature-level
+DP, where the features of nodes are private; and node-level DP, where both the features and edges of
+nodes are private. In this work, we focus on feature-level DP using DP-SGD. Contrary to traditional
+i.i.d. datasets, samples in a graph (i.e. nodes) are not independent: changing the features of one
+node affects the gradients of all nodes within the receptive field of the modified node. In fact, the
+sensitivity of the total gradient on a graph is bounded by 2 KL+1−1
+K−1
+C (see Appendix A), which
+grows exponentially with the number of layers L. Given that the Gaussian mechanism adds noise
+proportional to the sensitivity of the total gradient, this can lead to large amounts of noise being
+added during training, which in turn leads to poor final model utility.
+3
+Related work
+In [19], a node-level differentially private GNN is trained by perturbing features and edges locally
+before sending them to a global server. This setup is called local DP, and differs from our notion of
+DP where a central learner is trusted with the real data. The authors in [15] propose to split the graph
+into disjoint sub-graphs using uniform node sampling, then treat each sub-graph as an independent
+sample. Note that, contrary to our method which considers privacy at the individual node feature
+level, their approach treats the entire graph as a datapoint to privatize, rather than providing privacy
+for the individual nodes in the graph. The method in [10] privatizes GNNs at both the node-level
+and edge-level. However, their approach only applies to the GNN architecture they propose and
+not to arbitrary GNNs, unlike our proposed method. Furthermore, it does not resolve the issue of
+exponentially growing sensitivity in transductive learning scenarios. For a survey on DP on graph
+data, refer to [20]. Finally, the authors of [9] propose to reduce the sensitivity of a GNN’s gradients
+by bounding the maximum degree K of the graph. However, this does not resolve the exponential
+growth with the number of layers. Therefore, they still obtain loose privacy guarantees (ϵ = 20).
+Since this method is the closest to our setup, we compare our approach to theirs in our experiments.
+4
+Methodology
+4.1
+Approach
+We propose to adapt DP-SGD to the graph domain to ensure that the weights of a GNN are private
+with respect to the nodes’ features, while overcoming the problem of requiring exponentially more
+noise with a growing network depth. In the following, we define two graphs G and G′ as neighbors if
+they share the same structure A and number of nodes N but differ in one row of the feature matrix
+X corresponding to the modified node ˜v. We want to train the GNN such that for all G ∼ G′,
+3
+
+Figure 1: Our general sampling method. Starting with a graph, we generate subgraphs by first
+sampling a root node (depicted in red), and then sampling one or more random walks starting from
+the root node. Every node appears in exactly one subgraph. Before every iteration, we batch m many
+subgraphs, where m = 2 in this case. Root nodes are used as training nodes, while remaining nodes
+are used for aggregation in the GNN only.
+Dα(M(G), M(G′)) ≤ γ, where M is a randomized algorithm that returns the weights of the GNN.
+To adapt DP-SGD to the graph domain, we propose to pre-process the graph into sets of
+independent subgraphs that do not affect each others’ gradients, so that the sensitivity of the total
+gradient on any batch depends on the gradient of one subgraph only. We summarize our training
+procedure in Algorithm 1. More precisely, we pre-process the graph into a set of M disjoint
+subgraphs GS = {s1, s2, . . . , sM}, i.e. subgraphs that do not have any nodes in common, using
+sampling method S. Each subgraph si consists of two components: 1) one training node vi, and
+2) a set of neighbors N (vi) that is used for the aggregation step of the GNN. At training time, for
+every iteration t, we create a batch by sampling m subgraphs uniformly at random from the set of
+subgraphs GS. We then compute the gradients ∇wtL(vj, N (vj)) on all training nodes and clip
+the norm of each to a value C. We compute the total gradient by summing individual gradients and
+adding Gaussian noise. Finally, we update the weights.
+Due to the disjointness of subgraphs, changing one node’s features – whether it is a train-
+ing node or a neighbor – will affect at most one subgraph (i.e. sample) in the batch, which reduces
+the upper bound on the sensitivity of the total gradient to 2C. Since we sample subgraphs uniformly
+at random, we can leverage the strong amplification by sub-sampling theorem [17], i.e. account for
+the possibility of the gradient not being affected if the modified node ˜v is not part of the batch.
+We generate these disjoint subgraphs via random walk sampling, which is an effective way
+of training GNNs [21]. We choose random walk sampling, since it ensures that nodes form a
+connected subgraph of a training node’s neighborhood, while limiting the number of nodes being
+sampled from that neighborhood (i.e. from the receptive field). In the following, we propose three
+different random-walk-based sampling methods, which we later compare in our experimental results.
+Furthermore, we derive for each sampling method a tight upper bound on the probability of sampling
+the modified node ˜v in a batch, which is required for applying the amplification by subsampling
+theorem in [17].
+4.2
+Sampling methods
+Our three sampling methods consist of pre-processing the graph into a set of M disjoint subgraphs
+GS = {s1, s2, . . . , sM}, and then generating a batch B ⊆ GS by sampling m subgraphs uniformly
+at random. An overview of our general approach is depicted in Figure 1. Given a graph with M
+generated disjoint subgraphs, the true probability of sampling node ˜v is P[˜v] =
+1
+M , since we know
+that a node is in exactly one of the M subgraphs. However, to ensure differential privacy, we require
+a bound that holds for all possible graphs and any run of the sampling procedure. Thus, we use the
+upper bound P[˜v] =
+1
+M ≤
+1
+Mmin where Mmin is the minimum number of subgraphs that can be
+generated in any graph of N nodes. Then, the probability of sampling ˜v in a batch of m subgraphs
+using sampling mechanism S is at most PS[˜v] ≤
+m
+Mmin .
+4
+
+pre-process
+batchAlgorithm 1 DP-SGD with random walk sampling
+Input: Graph G = {V, E}, sampling method S, loss function L, initial model weights w0, noise
+standard deviation σ, gradient clipping norm C, number of iterations T, frequency at which to
+re-sample subgraphs in DRW-D i
+GS = S(G)
+▷ Generate subgraphs from graph G using sampling method S
+for t in [0, T) do
+if t % i == 0 and S == DRW-D then
+GS = S(G)
+end if
+Sample m subgraphs uniformly at random from GS to form batch B
+for sj in B do
+▷ sj is a subgraph
+Compute ∇wtL(vj, N (vj))
+gt(vj) = clip (∇wtL (vj, N (vj)) , C)
+▷ Compute and clip individual gradients in B
+end for
+gt(B) =
+1
+|B|
+���
+sj∈B gt(vj)
+�
++ N(0, σ2)
+�
+▷ Add noise to the gradients
+wt+1 = update(wt, gt(B))
+▷ Update weights based on optimizer being used
+end for
+Disjoint random walks
+The first sampling method we propose is called Disjoint Random Walks
+(DRW). We pre-process the graph once before training and then generate batches at every iteration
+using the same set of subgraphs. Each subgraph consists of one random walk of length L (refer to
+Appendix B for a pseudo-code). A random walk of length L contains at most L + 1 nodes, and
+generating random walks that all have maximal length would result in the minimum number of
+random walks, since a node can only appear in one random walk. Therefore, we get Mmin = ⌈ N
+L+1⌉
+and P[˜v] ≤
+1
+⌈
+N
+L+1 ⌉. Finally, the upper bound probability of sampling a node ˜v is PDRW[˜v] ≤
+m
+⌈
+N
+L+1 ⌉.
+Disjoint random walks with restarts
+To create better subgraphs that contain more nodes for
+aggregation, we also propose Disjoint Random Walks with Restarts (DRW-R). Similary to DRW,
+this sampling method generates subgraphs once before training by using random walks, but instead
+of sampling one random walk per training node we sample R of them (refer to Appendix B for a
+pseudo-code). Given a random walk length of L and R restarts, the minimum number of subgraphs
+is Mmin = ⌈
+N
+1+R×L⌉ where 1 + R × L is the maximum size of one subgraph when all random
+walks have length L, and the probability of sampling node u in a batch of size m is therefore
+PDRW-R[u] ≤
+m
+⌈
+N
+1+R×L ⌉.
+Disjoint random walks with dynamic re-sampling
+Finally, we propose a third sampling method
+in which we pre-process the graph into disjoint subgraphs every ith iteration instead of once before
+training, where i is a hyper-parameter that is chosen based on the cost of the sampling procedure
+on each dataset. This allows us to increase the diversity of subgraphs used for training, and prevent
+overfitting on the subgraphs generated in one run of the sampling procedure. We call this procedure
+DRW-D, where D stands for Dynamically re-sampling random walks. The probability of sampling
+node ˜v is the same as in DRW, namely PDRW-D[˜v] = PDRW[˜v] ≤
+m
+⌈
+N
+L+1 ⌉. Note that this method
+consists simply of re-running the subgraph generation process DRW at every ith iteration instead of
+once before training, which is reflected in Algorithm 1.
+5
+Experimental results
+Experimental setup
+We report our results on seven datasets, both in the transductive and the
+inductive settings. The dataset sizes in terms of total nodes range from small (Cora [22], Citeseer
+[22]) to medium (PPI [21], Pubmed [22]) to large (Flickr [21], Arxiv [21], Reddit [21]), or in number
+of training nodes from small (Pubmed, Citeseer, Cora) to medium (PPI) to large (Flickr, Arxiv,
+Reddit). We report the exact number of nodes as well as some additional dataset characteristics
+in Appendix C. We focus on the node classification task, and report our results in terms of F1
+Micro score, a metric equivalent to accuracy except on PPI which is a multi-label classification task.
+Following prior work, we report our privacy budget using ϵ and a fixed δ per dataset (see Appendix
+5
+
+Table 1: Comparison between the F1 Micro score (%) achieved by a basic GCN and MLP, the FDP
+baseline, and our proposed method with multiple sampling methods. All DP methods are trained with
+a target budget of ϵ ≤ 8.
+Layers
+Width
+Dataset
+Cora
+CiteSeer
+PPI
+PubMed
+Flickr
+Arxiv
+Reddit
+GCN (non-DP)
+1
+-
+69.8
+59.5
+46.2
+68.7
+45.6
+59.7
+92.5
+2
+256
+77.3
+63.7
+58.9
+72.9
+51.3
+69.1
+94.7
+512
+76.6
+62.2
+60.7
+72.9
+51.3
+69.5
+94.7
+MLP (non-DP)
+1
+-
+43.0
+37.6
+45.2
+61.3
+45.7
+52.3
+67.7
+2
+256
+47.3
+36.1
+52.1
+61.5
+36.2
+52.6
+69.8
+512
+44.8
+39.3
+53.6
+63.3
+38.4
+52.0
+69.7
+FDP (DP)
+1
+-
+17.1
+17.5
+38.4
+39.6
+33.6
+43.8
+56.7
+2
+256
+17.6
+21.5
+40.7
+41.4
+42.5
+31.9
+43.7
+512
+23.2
+22.1
+40.0
+41.2
+42.4
+30.2
+42.3
+Ours
+DRW (DP)
+1
+-
+19.9
+20.6
+40.2
+41.7
+42.1
+59.2
+81.4
+2
+256
+17.2
+20.9
+38.7
+40.3
+48.7
+59.6
+80.2
+512
+24.9
+21.3
+37.9
+41.1
+47.9
+59.2
+81.8
+DRW-D (DP)
+1
+-
+19.8
+20.6
+40.1
+41.7
+42.2
+59.2
+81.4
+2
+256
+17.2
+21.3
+38.6
+40.2
+48.5
+59.7
+80.2
+512
+25.0
+21.7
+37.9
+41.2
+47.8
+59.3
+81.5
+DRW-R (DP)
+1
+-
+18.3
+19.2
+40.0
+40.3
+42.3
+59.1
+82.0
+2
+256
+17.3
+20.7
+38.2
+40.4
+48.3
+59.7
+81.0
+512
+24.5
+21.3
+36.9
+40.4
+48.5
+59.4
+82.2
+C). Given a target ϵ, we keep training while tracking the (α, γt) privacy budget being spent until we
+reach ϵ = fRDP→DP(α, �T ′
+t=0 γt, δ) at iteration T ′.
+We compare our proposed methodology with each sampling method to three baselines: 1) A basic
+GCN trained with random walk sampling; 2) A basic MLP trained with uniform node sampling; and
+3) The method proposed in [9] which we call FDP for Feature-level DP. Note that while they train
+their models up to an ϵ of 20, we only train them until ϵ = 8, since a very large ϵ does not have much
+value in terms of privacy.
+Discussion
+Table 5 summarizes our results. A GCN trained without DP always outperforms the
+ones trained with DP, which is expected since clipping gradients and especially adding Gaussian noise
+decreases the utility of the final model. However, in some cases our method can almost match the
+utility of the basic GCN, whereas the FDP baseline struggles. For example, DRW sampling on Flickr
+can reach up to 48.7% accuracy – which corresponds to 95% of the baseline GCN’s performance
+– whereas FDP reaches only 42.5% accuracy – which corresponds to 83% of the baseline GCN’s
+performance. Similarly, our method achieves 87% of the GCN’s performance on the challenging
+dataset Reddit, while FDP can only reach 60% of the GCN’s performance. This shows that our
+sub-sampling approach is effective at solving the exponential growth of the receptive field while
+approaching the utility of the non-DP GCN baseline, which makes our method attractive for real
+world applications. That being said, our method uses a smaller amount of training nodes than what
+is available at every iteration, even when computational complexity is not an issue (i.e. on small
+graphs). The effect of this reduction in training training samples is exacerbated on small graphs that
+do not require batching in non-DP training, which leads to our method performing on-par with the
+FDP baseline on small datasets.
+Comparison with variable privacy budget
+Finally, in Figure 2 we expand on our previous results
+by reporting the accuracy at various ϵ checkpoints during training. We report the best results that
+our method achieved across all sampling methods and compare to the FDP baseline. On all datasets,
+our method largely outperforms FDP across multiple epsilon values. Moreover, FDP cannot achieve
+an epsilon lower than 2, whereas our method does while sometimes outperforming FDP at higher
+privacy budgets.
+6
+
+(a)
+(b)
+(c)
+Figure 2: F1 Micro Score vs. epsilon achieved by FDP and our best sampling method for a) Flickr, b)
+Arxiv and c) Reddit datasets.
+6
+Conclusion
+We proposed a novel way of training differentially private graph neural networks. Since graphs
+consist of inter-connected nodes that influence each other’s gradients during training, naively adapting
+traditional DP methods to graph neural networks can result in unnecessarily large amounts of noise
+being added to the model during training, which in turn leads to poor utility of the model. We
+proposed an adapted version of DP-SGD that uses random-walk based sub-sampling to overcome
+this problem and introduced three sampling methods that generate disjoint subgraphs. For each
+sampling method, we derived an upper bound on the probability of sampling a modified node in
+a batch to apply the amplification by sub-sampling theorem and obtain tighter privacy guarantees.
+Our method achieves a better privacy-utility trade-off compared to the state-of-the-art baseline FDP
+across multiple datasets, especially for large datasets. A necessary future work direction in this field
+is to attempt to solve the performance issue on small datasets, which is especially exacerbated on
+GNNs. For example, pre-training the models on public datasets [2] or using variable signal-to-noise
+ratios during training are ways of improving the utility in DP. Moreover, different sampling methods
+that do not necessarily focus on random walks can be explored.
+References
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+confidence information and basic countermeasures. In Proceedings of the 22nd ACM SIGSAC
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+and Li Zhang. Deep learning with differential privacy. In Proceedings of the 2016 ACM SIGSAC
+conference on computer and communications security, pages 308–318, 2016.
+7
+
+0.48
+FDP
+Ours
+0.46
+F1 Micro Score
+0.44
+0.42
+0.40
+0.38
+0.36
+0.34
+1
+2
+3
+4
+5
+6
+7
+8
+Epsilon0.60
+0.55
+0.50
+Micro Score
+0.45
+0.40
+0.35
+F1
+0.30
+FDP
+0.25
+Ours
+0.20
+1
+2
+3
+4
+5
+6
+7
+8
+Epsilon0.8
+FDP
+Ours
+0.7
+Fl Micro Score
+0.6
+0.5
+0.4
+0.3
+1
+2
+3
+4
+5
+6
+7
+8
+Epsilon[9] Ameya Daigavane, Gagan Madan, Aditya Sinha, Abhradeep Guha Thakurta, Gaurav Aggarwal,
+and Prateek Jain. Node-level differentially private graph neural networks. In ICLR 2022
+Workshop on PAIR, 2022.
+[10] Sina Sajadmanesh, Ali Shahin Shamsabadi, Aurélien Bellet, and Daniel Gatica-Perez. Gap:
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+arXiv:2203.00949, 2022.
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+symposium (CSF), pages 263–275. IEEE, 2017.
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+ing interpretations and renyi differential privacy. In International Conference on Artificial
+Intelligence and Statistics, pages 2496–2506. PMLR, 2020.
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+and Prateek Jain. Node-level differentially private graph neural networks. arXiv preprint
+arXiv:2111.15521, 2021.
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+ings of the 2021 ACM SIGSAC Conference on Computer and Communications Security, pages
+2130–2145, 2021.
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+Sok: Differential privacy on graph-structured data. arXiv preprint arXiv:2203.09205, 2022.
+[21] Hanqing Zeng, Hongkuan Zhou, Ajitesh Srivastava, Rajgopal Kannan, and Viktor Prasanna.
+Graphsaint: Graph sampling based inductive learning method. arXiv preprint arXiv:1907.04931,
+2019.
+[22] Prithviraj Sen, Galileo Mark Namata, Mustafa Bilgic, Lise Getoor, Brian Gallagher, and Tina
+Eliassi-Rad. Collective classification in network data. AI Magazine, 29(3):93–106, 2008.
+A
+Upper Bound on Gradient Sensitivity
+We show how to derive the upper bound on the sensitivity of the total gradient on a batch, where gt
+is the function that takes a batch B as input and returns the gradients at iteration t, B and B′ are
+neighboring batches that differ by one sample ˜v, Lv is the loss function on a node v, ∇wLv is the
+8
+
+gradient of the loss on v with respect to the weights of the model, and I[˜v ∈ B] is the indicator
+function which is 1 if ˜v is in the batch and 0 otherwise.
+∆2gt = ∥gt(B) − gt(B′)∥2
+= ∥
+�
+v∈B
+∇wLv −
+�
+v∈B′
+∇wLv∥2
+= ∥(∇wL˜v +
+�
+u∈RF (˜v)\{˜v}
+∇wLu)I[˜v ∈ B] − (∇wL˜v′ +
+�
+u∈RF (˜v′)\{˜v′}
+∇wLu)I[˜v′ ∈ B′]∥2
+≤ ∥(∇wL˜v +
+�
+u∈RF (˜v)\{˜v}
+∇wLu)I[˜v ∈ B]∥2 + ∥(∇wL˜v′ +
+�
+u∈RF (˜v′)\{˜v′}
+∇wL˜v′)I[˜v′ ∈ B′]∥2
+≤ ∥∇wL˜v +
+�
+u∈RF (˜v)\{˜v}
+∇wLu∥2 + ∥∇wL˜v′ +
+�
+u∈RF (˜v′)\{˜v′}
+∇wLu∥2
+≤ ∥∇wL˜v∥2 +
+�
+u∈RF (˜v)\{˜v}
+∥∇wLu∥2 + ∥∇wL˜v′∥2 +
+�
+u∈RF (˜v′)\{˜v′}
+∥∇wLu∥2
+≤ 2|RF(˜v)|C
+≤ 2KL+1 − 1
+K − 1
+C
+(1)
+9
+
+B
+Algorithms
+B.1
+DRW Sampler
+The following algorithm shows how to generate disjoint subgraphs using the Disjoint Random Walks
+(DRW) sampling method (see Section 4.2). To generate a subgraph, we first sample a node v from
+the set of remaining nodes, then remove it from this set. We then construct the set of valid neighbors
+of v, which consists of all nodes that have not been already sampled. We sample the next node v in
+the subgraph from the set of valid neighbors, and repeat the process until we get a random walk of
+length L. We iterate this process until all nodes are included in one subgraph.
+Algorithm 2 DRW Sampler
+Input: Graph G = {V, E}, random walk length L.
+Output: Set of all disjoint subgraphs = ()
+remaining_nodes = {v1, v2, . . . , vN}
+while len(remaining_nodes) != 0 do
+subgraph = []
+v = sample(remaining_nodes, 1)
+▷ uniformly sample over non-sampled nodes
+subgraph.append(v)
+remaining_nodes.remove(v)
+l = 0
+while l < L do
+valid_neighbors = Neighbors(v)
+▷ Neighbors returns all neighbors of a node
+for u in valid_neighbors do
+if u not in remaining_nodes then
+valid_neighbors.remove(u)
+end if
+end for
+if len(valid_neighbors) != 0 then
+v = sample(valid_neighbors, 1)
+▷ uniformly sample a neighbor of v
+else
+break
+end if
+random_walk.append(v)
+remaining_nodes.remove(v)
+l = l + 1
+end while
+subgraphs.add(subgraph)
+end while
+B.2
+DRW-R Sampler
+The following algorithm shows how to generate disjoint subgraphs using the Disjoint Random Walks
+with restarts (DRW-R) sampling method (see 4.2). The main difference to the DRW sampler is that,
+instead of stopping the subgraph generation after one random walk, we sample multiple random
+walks rooted at the same node by re-initializing the starting node of the random walk to the same root
+node of the subgraph R times.
+10
+
+Algorithm 3 DRW-R Sampler
+Input: Graph G = {V, E}, random walk length L.
+Output: Set of all disjoint subgraphs = ()
+remaining_nodes = {v1, v2, . . . , vN}
+while len(remaining_nodes) != 0 do
+subgraph = []
+root = sample(remaining_nodes, 1)
+▷ uniformly sample over non-sampled nodes
+subgraph.append(root)
+remaining_nodes.remove(root)
+for r in range(R) do
+v = root
+l = 0
+while l < L do
+valid_neighbors = Neighbors(v)
+▷ Neighbors returns all neighbors of a node
+for u in valid_neighbors do
+if u not in remaining_nodes then
+valid_neighbors.remove(u)
+end if
+end for
+if len(valid_neighbors) != 0 then
+v = sample(valid_neighbors, 1)
+▷ uniformly sample a neighbor of v
+else
+break
+end if
+subgraph.append(v)
+remaining_nodes.remove(v)
+l = l + 1
+end while
+subgraphs.add(subgraph)
+end for
+end while
+11
+
+C
+Training Hyperparameters
+We run all experiments with three different seeds, Adam optimizer and ReLU activation. We
+summarize the number of roots used for different sampling scenarios in Table 4. For the non-DP
+trainings, we fix the learning rate to 0.01. We perform a grid hyper-parameter search for the trainings
+on all datasets. We experiment with the following hyper-parameters for both DP and non-DP trainings:
+• Number of layers in {1, 2}
+• Width of hidden layers in {256, 512}
+• Maximum graph degree in {2 , 4} for the FDP baseline
+We use the follow hyper-parameters for the DP specific trainings:
+• Learning rate in {0.01, 0.1, 0.2}
+• Clip norm percentage C% in {0.001, 0.01, 0.1}.
+• Noise multiplier λ in {1, 2, 4, 8}. The noise multiplier is the ratio of the standard deviation
+σ of the Gaussian noise added to the gradients to the sensitivity ∆2f of the function f.
+Instead of tuning σ, we tune λ, then fix σ = λ × ∆2f.
+• Delta value δ: we summarize the values used in Table 3
+Table 2: Characteristics of the datasets that we use in our experiments. (s) indicates a single-label
+classification problem, and (m) a multi-label one.
+Nodes
+Feature Size
+Classes
+Training Nodes
+Type
+Cora
+2,708
+1,433
+7 (s)
+140
+Transductive
+Citeseer
+3,327
+3,703
+6 (s)
+120
+Transductive
+PPI
+14,755
+50
+121 (m)
+9,716
+Inductive
+Pubmed
+19,717
+500
+3 (s)
+60
+Transductive
+Flickr
+89,250
+500
+7 (s)
+44,625
+Inductive
+Arxiv
+169,343
+128
+40 (s)
+90,941
+Inductive
+Reddit
+232,965
+602
+41 (s)
+153,932
+Inductive
+Table 3: δ value used for each dataset.
+Dataset
+Cora
+Citeseer
+PPI
+Pubmed
+Flickr
+Arxiv
+Reddit
+δ
+1e-5
+1e-5
+1e-5
+1e-6
+1e-6
+1e-7
+1e-7
+Table 4: Batch sizes used for training based on the sampler, depth of the model, and dataset. Note
+that as a general rule, we used around 20% of total number of training nodes for the large datasets,
+and 50% for the small datasets.
+Sampler
+Depth
+Dataset
+Cora
+CiteSeer
+PPI
+PubMed
+Flickr
+Arxiv
+Reddit
+RW, uniform, PreDRW,
+1
+70
+60
+2,000
+30
+10,000
+20,000
+30,000
+PreDRW-D, DynDRW
+2
+46
+40
+2,000
+20
+10,000
+20,000
+30,000
+PreDRW-R
+1
+46
+40
+2,000
+20
+10,000
+20,000
+30,000
+2
+28
+24
+1,800
+12
+8,000
+18,000
+30,000
+12
+

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+HIP-2022-35/TH
+Vector dark matter in supercooled Higgs portal models
+Mads T. Frandsen∗ and Mattias E. Thing†
+CP3-Origins, University of Southern Denmark, Denmark
+Matti Heikinheimo‡ and Kimmo Tuominen§
+Department of Physics, University of Helsinki,
+P.O.Box 64, FI-00014 University of Helsinki, Finland and
+Helsinki Institute of Physics, P.O.Box 64,
+FI-00014 University of Helsinki, Finland
+Martin Rosenlyst¶
+Rudolf Peierls Centre for Theoretical Physics, University of Oxford,
+1 Keble Road, Oxford OX1 3NP, United Kingdom
+We consider extensions of the Standard Model by a hidden sector consisting of a
+gauge field coupled with a scalar field. Assuming the absence of dimensionful param-
+eters in the tree level potential, radiative symmetry breaking will make the hidden
+sector gauge field massive and induce the electroweak scale of the Standard Model.
+We consider separately dark sector gauge groups U(1)D and SU(2)D, and focus on
+probing the models with a combination of direct detection experiments and gravita-
+tional wave observatories. We find that recent dark matter direct detection results
+significantly constrain the parameter space of the models where they can account for
+the observed dark matter relic density via freeze-out. The gravitational wave signals
+originating from strongly first order electroweak phase transition in these models can
+be probed in future gravitational wave observatories such as LISA. We show how
+the projected results compliment direct detection experiments and can help probe
+parameter space near the neutrino floor of direct detection.
+I.
+INTRODUCTION
+Despite the success of the Standard Model (SM) of particle physics, there are many phenomena
+that it does not explain and that appear to require new particles and interactions. One enigmatic
+phenomenon is the problem of missing mass, which emerged in a wide range of astrophysical
+systems including galaxy clusters [1] and galaxies [2]. One possible solution to the missing mass
+problem is cold dark matter (DM), constituted by a new stable and neutral massive particle. This
+hypothesis provides an excellent parametrisation for 26% of the energy density of the universe in
+addition to the components parametrised as baryonic matter and dark energy [3]. On the other
+hand, the non-gravitational nature of dark matter (DM) remains unknown [4–6].
+The cosmological observations on the light element abundance and cosmic microwave back-
+ground radiation spectrum imply that the Standard Model (SM) degrees of freedom must have
+been in thermal equilibrium in the early universe [7–11]. Whether DM was ever part of the same
+heat bath is not known.
+However, assuming that this was the case, allows for the abundance of dark matter to arise as a
+relic from thermal decoupling in the early universe via interactions between the DM and the SM.
+∗ frandsen@cp3.sdu.dk
+† thing@cp3.sdu.dk
+‡ matti.heikinheimo@helsinki.fi
+§ kimmo.i.tuominen@helsinki.fi
+¶ martin.jorgensen@physics.ox.ac.uk
+arXiv:2301.00041v1  [hep-ph]  30 Dec 2022
+
+2
+Moreover, these interactions offer the prospect of detecting DM in direct detection experiments.
+The most studied example of this paradigm is Weakly Interacting Massive Particle (WIMP). How-
+ever, simplest WIMP models are now very strongly constrained by direct detection experiments.
+It is therefore worthwhile to explore the phenomenology of different types of simple benchmark
+hidden sectors instead coupled with the SM via portal interactions.
+In this paper we analyze two simple models of vector DM, that feature scale invariance of the
+tree-level Lagrangian and are coupled to the SM via the Higgs portal, where one scalar mass
+eigenstate is SM-like, with mass 125.46 ± 0.16 GeV [12]. The other eigenstate is massless at tree
+level but obtains its mass via loop corrections as an effect of radiative symmetry breaking [13].
+This framework of classically scale invariant DM models that feature radiative symmetry breaking,
+mediated to the SM via the Higgs portal, has been explored in literature, see e.g. [14–24].
+In this paper we aim to clarify how simple U(1)D and SU(2)D models of this type can be tested
+with a combination of direct detection and gravitational wave observations.
+Direct detection
+experiments have provided very stringent constraints on interactions of weak scale dark matter
+with nuclei. Currently, the most stringent constraints come from the recent PandaX-4T and LZ
+(2022) experiments [25, 26]. It is well known that radiative symmetry breaking in classically scale
+invariant models typically results in a strongly first order electroweak phase transition (EWPT).
+Such a first order EWPT could be relevant for baryogenesis and produces gravitational wave signals
+which could be observable in upcoming gravitational wave experiments such as LISA [27].
+We present a careful examination of the first order phase transition using different numerical
+packages in order to characterise the theoretical uncertainty in the predictions.
+II.
+DEFINITIONS OF THE MODELS
+We consider two models where the SM is extended with a hidden sector gauge group and a
+new scalar field charged under the gauge group. Spontaneous symmetry breaking of the hidden
+sector gauge group via this scalar leads to new massive vector DM candidates. The first model we
+consider is an U(1)D extension defined by the Langrangian [22],
+LU(1)D = L0
+SM − 1
+4VµνV µν + (DµS)∗(DµS) − V (H, S),
+(1)
+where L0
+SM is the SM Lagrangian without the Higgs potential.
+The covariant derivative is
+Dµ = ∂µ + igVµ and the field strength tensor of the U(1)D vector field is Vµν = ∂µVν − ∂νVµ.
+The scalar potential is given by
+V (H, S) = 1
+6λH(H†H)2 + 1
+6λS(S∗S)2 + 2λHS(H†H)(S∗S).
+(2)
+In principle a kinetic mixing term BµνVµν could be present, but we assume this does not arise. For
+example, the mixing term can be explicitly prohibited by a Z2 symmetry under which Vµ → −Vµ
+and all other fields are singlets. In the unitary gauge the scalar fields are written as
+H = 1
+√
+2
+�
+0
+v1 + h1
+�
+,
+S = 1
+√
+2(v2 + h2),
+(3)
+and upon symmetry breaking vi, (i = 1, 2), becomes nonzero. The SM gauge boson masses are
+determined by the vacuum expectation value (VEV) v1 = 246 GeV while the DM mass is related
+to the VEV v2 via M 2
+V = g2v2
+2.
+The second model we consider is the similar SU(2)D extension defined by the Langrangian [24]
+LSU(2)D = L0
+SM − 1
+4V i
+µνV µν
+i
++ (DµS)†(DµS) − V (H, S),
+(4)
+
+3
+where the DM candidate is now the SU(2)D vector triplet V i
+µ. The covariant derivative and the
+field strength tensor take the forms
+Dµ = ∂µ + igV i
+µti,
+V i
+µν = ∂µV i
+ν − ∂νV i
+µ + gϵi
+jkV j
+µ V k
+ν ,
+(5)
+where ti = σi/2 is the SU(2) generator. In this non-Abelian model, the kinetic mixing is forbidden
+by gauge symmetry. The normalization of the scalar potential is here chosen as
+V (H, S) = λH(H†H)2 + λS(S†S)2 + λHS(H†H)(S†S),
+(6)
+where the scalars are now both complex SU(2) doublets, and in the unitary gauge given by
+H = 1
+√
+2
+�
+0
+v1 + h1
+�
+,
+S = 1
+√
+2
+�
+0
+v2 + h2
+�
+.
+(7)
+In both of the above models the two neutral scalar states mix and the resulting mass eigenstates
+are connected to the gauge eigenstates via a mixing matrix of the form
+�
+h
+hS
+�
+=
+�
+cos α − sin α
+sin α
+cos α
+� �
+h1
+h2
+�
+,
+(8)
+where the mixing angle α describes the mixing between the SM and DM sectors. Generally, this
+angle is restricted to small values, sin α ≲ 0.1.
+The parameters of these models can be written in uniform notation as,
+v2 = cV MV
+g
+,
+sin α = v1
+v
+(9)
+λH = 3M 2
+h
+v2
+1
+cos2 α,
+λS = 3M 2
+h
+v2
+2
+sin2 α,
+λHS = − M 2
+h
+2v1v2
+sin α cos α,
+(10)
+where MV is the mass of the DM candidate, Mh is the SM-Higgs mass and cV = 2 for the SU(2)D
+model and cV = 1 for the U(1)D. We have also defined v2 = v2
+1 + v2
+2.
+The tree-level potential has a flat direction along the scalon hS field direction, while the SM-like
+Higgs h is perpendicular to the flat direction. We can thus consider the loop corrections in the flat
+direction as per the Gildener-Weinberg formalism [13]. The first order loop corrections lead to an
+effective potential of the general form,
+V 1
+eff(hS) =
+1
+64π2
+n
+�
+s=1
+gsM 4
+s
+�
+ln
+�M 2
+s
+Λ2
+�
+− Ci
+�
+,
+(11)
+where Ms refers to tree level masses, gs is the degrees of freedom (with positive values for bosons
+and negative for fermions), n is the number of states, and Λ is a renormalization scale. The scalon
+field is massless at tree level, but obtains a mass from the loop corrections, given by
+M 2
+S =
+1
+8π2v2
+�
+gV M 4
+V + 3M 4
+Z + 6M 4
+W + M 4
+h − 12m4
+t
+�
+,
+(12)
+where gV is the degrees of freedom for the vector boson: gV = 9 for the SU(2)D model and gV = 3
+for the U(1)D. Here MS is the scalon mass for each respective model and MV is the DM candidate.
+Notice that Equation (12) relates the scalon and DM masses. In order for the scalon mass to be
+non-negative, this sets a lower bound for the DM masses. The bound is MV > 240 GeV for the
+SU(2)D model and MV > 185 GeV for the U(1)D model.
+
+4
+III.
+FREEZE-OUT RELIC DENSITY
+The dark matter abundance in the model is determined via the freeze-out mechanism. While
+other possibilities, namely super-cool DM and filtered DM have been considered in the context of
+radiative symmetry breaking models such as those under the present study [28–31], we will see
+that the freeze-out mechanism is operational throughout the parameter space considered in this
+work.
+To see how the observed DM abundance Ωh2 = 0.120 ± 0.001 [3] is generated via the freeze-out
+mechanism, we recall the basic formalism below. The present-day dark matter density is obtained
+from the Boltzmann equation
+dnV
+dt + 3HnV = − ⟨σav⟩
+�
+n2
+V − n2
+V,eq
+�
+,
+(13)
+where nV is the number density of the dark matter, which in equilibrium in the broken phase is
+given as
+neq
+V (T) = gV
+�MV T
+2π
+�3/2
+e�� MV
+T .
+(14)
+Here H is the Hubble parameter and ⟨σav⟩ is the thermally averaged annihilation cross section.
+Equation (13) can be rewritten using entropy conservation, the yield YV = nV
+s , and x = MV
+T
+into
+the form
+dYV
+dx =
+1
+3H
+ds
+dx ⟨σav⟩
+�
+Y 2
+V − Y 2
+V,eq
+�
+,
+(15)
+and solving this equation we obtain the present day yield Y 0
+V that links to the abundance as
+Ωh2 = MV s0Y 0
+V h2
+ρc
+0
+≃ 2.755 · 108MV s0Y 0
+V GeV−1,
+(16)
+where
+s0 = 2.8912 · 109 m−3,
+ρc
+0 = 10.537h2 GeVm−3 for H = h100 km/s/Mpc,
+(17)
+and h = 0.678.
+To solve the Boltzmann equation numerically we use the micrOMEGA package [32]. This software
+uses CalcHEP input files with the models Feynman rules to compute the thermally averaged cross
+section, which we generate with the LanHEP package [33, 34]. The numerical results for the relic
+density for both models can be seen in Figure 1. To asses the validity of the numerical results
+we have compared these to the analytical result, obtained in the non-relativistic limit and under
+the approximation of instantaneous freeze-out. Both of these approximations tend to overestimate
+the relic density. Nevertheless, the analytical result only deviates up to around 10% for the U(1)D
+model and slightly more for the SU(2)D model, considering only the leading annihilation processes
+σ (V V → hShS) for the U(1)D model and σ (V iV j → hShS) plus the semi-annihilation process
+σ
+�
+V iV j → V khS
+�
+for the SU(2)D model.
+From Figure 1 it is evident that both models can reproduce the observed relic density. A larger
+coupling g leads to more efficient annihilation of the vector DM candidate V into scalons hS and
+thus the correct abundance is obtained for a correspondingly higher vector mass MV . In the non-
+Abelian model the semi-annihilation process is taken into account in the analytic approximation
+by defining the effective thermally averaged total annihilation cross section as
+⟨σav⟩ = ⟨σannv⟩ + 1
+2 ⟨σsemi−annv⟩ ,
+(18)
+
+5
+(a) The DM relic density as a function of the mass of
+the U(1)D vector DM candidate for different coupling
+constants, including the Planck collaboration result.
+(b) The DM relic density as a function of the mass of
+the SU(2)D vector DM candidate for different coupling
+constants, including the Planck collaboration result.
+FIG. 1. The red line representing the Planck collaboration result of Ωh2 = 0.120 ± 0.001 is shown in red,
+and both models can match it via a freeze-out relic density [3].
+where the first term is the annihilation and the second term is the semi-annihilation cross section.
+The addition of the semi-annihilation generally leads to more efficient annihilation, and thus
+one would expect the relic density to be lower. However, the SU(2)D result in Figure 1(b) is very
+close to the U(1)D result in Figure 1(a), which indicates that there is not much difference in the
+abundance for the two models considered. The origin of this is that that while the additional
+degrees of freedom in the non-Abelian model increase the relic density, this is balanced by the
+reducing effect of the semi-annihilations. Concretely, the semi-annihilations increases the overall
+thermally averaged total annihilation cross section only by roughly 15%.
+Finally, we comment on the possibility of a freeze-in origin for the DM abundance in these
+models. In the freeze-in regime the DM particle V needs to be feebly coupled to the visible sector,
+so that it does not reach equilibrium with the SM thermal bath in the early universe. To achieve
+this, either the gauge coupling g needs to be very small so that the vector remains decoupled while
+the scalar S is in equilibrium, or the portal coupling λHS can be very small, so that both the vector
+and the scalar remain decoupled from the SM.
+In the first scenario, the typical scale for the gauge coupling would be g ∼ O(10−7), as seen
+from the approximate relation [35]
+YV (T) ∼ g2Mpl
+T ,
+(19)
+where Mpl is the reduced Planck mass. Since this process is IR dominated, the dominant production
+would be at the lowest kinematically allowed temperature T ∼ MV . Thus we can approximate the
+abundance by the replacement T = MV in the above to obtain
+Y 0
+V ∼ g2Mpl
+MV
+.
+(20)
+Consider now the relationship between the coupling and DM mass in Equations (10) and (12).
+If the coupling is g ∼ O(10−7) as necessary for the freeze-in mechanism to work, the VEV, v2,
+
+RelicDensity of U(1)pModel
+g= 0.5
+g = 0.7
+g= 0.9
+100
+Planck
+10~1
+10~2
+250500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Mv[GeV]RelicDensity of SU(2)p Model
+101
+g=0.5
+g= 0.7
+g= 0.9
+Planck
+100
+10-1
+10~2
+250500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+My[GeV]6
+becomes very large and the scalon mass, MS, is approximately zero.
+The presence of a very
+light scalar in the spectrum is potentially problematic, e.g. due to Higgs invisible decays, unless
+suppressed by a small portal coupling. On the other hand, the scenario where the portal coupling
+would be very small, would also require a large hidden sector VEV v2 ≫ v1. If the gauge coupling
+is not very small, then this implies that the DM mass MV becomes very large. In this case the
+hidden sector can only be effectively populated in the broken phase, as there is no scalar mixing in
+the unbroken phase. However, in this scenario there will be large supercooling, as discussed below,
+and the DM production should take place after reheating from thermal inflation. Now the scalar
+VEV is mostly in the S-direction v2 ≫ v1, so that the energy stored in the inflaton field mostly
+goes into S-quanta, but since these are feebly coupled to the SM, the reheating will be very slow
+and the reheating temperature suppressed. Thus, the heavy DM can not be efficiently produced
+after reheating, since Tr ≪ MV . While there might be some way to overcome these apparent
+problems with freeze-in, we do not consider this scenario further in this work.
+IV.
+INFLATION, REHEATING AND SUPERCOOLING
+In the previous section, we discussed the DM abundance in the standard freeze-out scenario.
+The situation may however be more complicated [28, 30, 36, 37], due to a possible phase of thermal
+inflation characteristic of classically scale invariant models with radiative symmetry breaking. The
+thermal history in the models can be summarised in terms of the following temperature thresholds:
+• TFO: The freeze-out temperature of the DM candidate defined roughly by neq
+V ⟨σv⟩ = H.
+• Tn: The nucleation temperature when the probability to nucleate an expanding bubble of
+the broken phase vacuum inside a Hubble horizon becomes of O(1), approximately the
+temperature at which the phase transition completes.
+• Tinf: The temperature at the beginning of thermal inflation defined by ρV = ρrad, where
+ρV is the energy density of the false unbroken vacuum (i.e. the difference in the potential
+between the local minimum at V (S) = 0 and the true minimum at V (S = v)), and ρrad is
+the energy density of the radiation dominated universe. When ρV begins to dominate the
+energy density, inflation begins.
+In the case of the two vector DM models discussed in this paper, the finite temperature potential
+includes the thermal integral summing over the bosons and fermions [38],
+V 1
+eff(hS, T) =
+n
+�
+s=1
+gs
+�
+1
+64π2M 4
+s
+�
+ln
+�M 2
+s
+Λ2
+�
+− Ci
+�
++ T 4
+2π2
+� ∞
+0
+x2 ln
+�
+1 ∓ e−√
+x2±M2s /T 2�
+dx
+�
+.
+(21)
+For some models, it might be necessary to consider the additional ring diagrams for the bosons, but
+for this investigation they can be ignored as they are insignificant [39]. This thermal potential is
+not amenable to an analytic solution, but can be approximated using modified Bessel functions of
+the second kind [23]. We compute the freeze-out temperature, TFO, numerically with micrOMEGA,
+and the nucleation temperature, Tn, numerically using CosmoTransitions and Bubbleprofiler
+(for cross-checking) [32, 40, 41].
+Let us now consider the thermal history of the model depending on the order of the above three
+temperature thresholds. If Tn > TFO, the phase transition completes before DM freeze-out, and
+the freeze-out then takes place as usual in the broken phase. This means that we can calculate
+the relic abundance as presented in the previous section.
+In the opposite case, Tn < TFO there are three scenarios to consider. The filtered DM scenario
+takes place for the ordering TFO > Tn > Tinf. In this situation, there is no thermal inflation, as the
+
+7
+phase transition completes before inflation would begin, but the DM annihilations are immediately
+out of equilibrium after the phase transition, and therefore the abundance is set by the amount of
+DM particles that are able to enter the boundary to the broken phase, as described in [30].
+The supercool DM scenario [28], takes place for TFO > Tinf > Tn. In this situation, there is a
+period of thermal inflation, which ends at Tn. After inflation, the latent heat stored in the false
+vacuum is released to reheat the universe back to temperature Tinf, under the assumption of instant
+reheating, or to a lower reheating temperature for delayed reheating. However, since TFO > Tinf,
+no DM is produced in reheating and the abundance is set by the amount that was present before
+inflation, diluted by the expansion of the scale factor and by the filtering effect as in the above
+scenario.
+Finally, there is the case where Tinf > TFO. In this situation, assuming instant reheating, the
+reheating will bring DM back to equilibrium, and the relic abundance is again obtained via the
+usual freeze-out mechanism as presented in the previous section.
+The inflation temperature is obtained by solving for Tinf from
+∆V (Tinf) = V high
+eff
+(hS, Tinf) − V low
+eff (0, Tinf) = g∗π2
+30 T 4
+inf,
+(22)
+where V high
+eff
+(hS, T) is the true vacuum and V low
+eff (0, T) is the false vacuum. We find that throughout
+the parameter space of interest in this work, we are either in the first or the last situation described
+above, and the DM abundance is thus obtained via the usual freeze-out mechanism in both cases.
+See Appendix A for more on the reheating.
+V.
+DIRECT DETECTION
+In this section, we present the direct detection constraints on the two models. We will see
+that the recent results from the LZ experiment significantly affect the SU(2)D model and that the
+U(1)D model is already very constrained.
+To compute the direct detection cross section, we again use the micrOMEGA package [32]. The
+DM coupling to nucleons arises from the scalar mixing and is mediated via exchange of the SM-like
+Higgs and the scalon in the t-channel leading to a spin-independent cross section with negligible
+difference between protons and neutrons. The results of this computation for both models are
+shown in Figure 2. The correct relic abundance is obtained along the red solid line.
+The purple region is excluded by LHC constraints on to Higgs decays into two scalons [44, 45].
+This process becomes kinetically forbidden for larger DM mass, as larger DM mass leads to larger
+scalon mass as shown in Equation (12). In the orange region, the DM-nucleon cross section is below
+the neutrino floor, and the yellow regions indicate the exclusion limit due to the LZ experiment
+[26], providing a significant improvement over the XENON1T experiment shown in green [42].
+Finally, the grey region shows the projected exclusion limit from XENONnT [43].
+Starting with the U(1)D model we see a small gap in the direct detection limits at the resonance
+region, Mh ≃ 2MV , where the DM mass is around 0.9-1 TeV and the coupling 0.65 ≤ g ≤ 0.7. In
+the middle of this range the direct detection cross section falls below the neutrino floor. Outside of
+the resonance region, the model can not produce an O(1)-fraction of DM without being excluded
+by direct detection, unless the DM mass is well above 10 TeV.
+For the SU(2)D model the new constraints from LZ alter the picture compared to the situation
+with the previous XENON1T limits: the relic abundance line above the resonance region is now
+excluded for DM masses below 7.5 TeV, while prior to the LZ result there were no constraints
+beyond 1 TeV. In the resonance region, we find the nucleation temperature for the phase transition
+below the QCD-scale. This alters the computation for the gravitational wave signal, as the phase
+transition will be completed in conjunction with the QCD phase transition, as discussed in [46, 47].
+This picture slightly changes when including additional scalar self-energy corrections for the SU(2)
+
+8
+(a) Constraints for the the U(1)D model.
+(b) Constraints for the the SU(2)D model.
+FIG. 2. The red line shows the correct relic abundance, Ωh2 = 0.12 [3]. The yellow region is excluded by
+the LZ (2022) experiment [26], the green region is the XENON1T experiment [42], the purple region is
+the LHC constraint for exotic Higgs decay, the orange region is the neutrino floor and the gray region is
+the projected 90% CL constraint from the XENONnT experiment [43].
+model [48]. First, the scalon mass is slightly larger than in our leading order analysis, pushing
+the resonance region in figure 2(b) to the right. Additionally, the correction appears to slightly
+increase the nucleation temperature compared to our results. However, we find that overall the
+resulting gravitational wave (GW) signal is not significantly affected, and the GW signal prediction
+remains comparable to our results presented in the next section.
+For DM mass above 7.5 TeV the model is again allowed by direct detection. In Figure 2 we
+have marked three benchmark points allowed by direct detection with the blue, indigo, and purple
+markers. These points will be used as examples for analyzing the GW signals in the next section.
+VI.
+GRAVITATIONAL WAVES
+The strongly first order phase transition possible in classically scale invariant models is inter-
+esting due to the implications for baryogenesis [49], and due to potentially observable gravitational
+wave (GW) signal.
+To explore the gravitational wave signals, we consider the finite temperature potential in Equa-
+tion (21). This potential contains a barrier between the unbroken false vacuum and the broken
+phase minimum, leading to a first order phase transition. At the nucleation temperature Tn, the
+phase transition will complete via the formation of bubbles of the true vacuum. The expanding
+and colliding bubbles deposit energy in the surrounding plasma, generating gravitational waves as
+described in [50–52].
+For the purpose of solving Equation
+(21) and obtaining the parameters that describe the
+gravitational wave signal, we use the Python package CosmoTransitions[40], with custom modifi-
+cations including a method of computing the β value. The relevant parameters are the latent heat
+normalized with respect to the radiation energy, α, the inverse duration of the phase transition,
+
+Constraints ofU(1)p Model
+1.0
+Qh2 = 0.12
+0.9
+0.8
+9 0.7
+LZ(2022)
+0.6
+LHC
+KENONIT
+0.5
+XENONnT
+0.4
+300
+1000
+2000
+3000
+4000
+Mv[GeV]ConstraintsofSU(2)pModel
+2.0
+Qh2 = 0.12
+1.8
+1.6
+1.4
+1.0
+XENONIT
+0.8
+0.6
+0.4
+300
+1000
+2000
+4000
+8000
+My[GeV]9
+Model
+Benchmark point Parameter CosmoTransitions BubbleProfiler
+U(1)D
+g = 0.66
+MV = 911 GeV
+Tc = 303 GeV
+α
+20740
+92180
+β
+23.8
+39.2
+Tn
+7.04 GeV
+4.78 GeV
+U(1)D
+g = 0.7
+MV = 1028 GeV
+Tc = 336 GeV
+α
+1497
+4597
+β
+36.8
+49.5
+Tn
+15.3 GeV
+11.4 GeV
+SU(2)D
+g = 2.0
+MV = 7530 GeV
+Tc = 2345 GeV
+α
+0.16
+0.22
+β
+289
+301
+Tn
+1430 GeV
+1446 GeV
+.
+FIG. 3. Table with benchmark points used for the discussion of gravitaitonal wave signals. The two first
+benchmark points are from the U(1)D model and the last is from the SU(2)D model.
+β, and the nucleation temperature, Tn, defined as [24, 53],
+α ≡ 1
+ρ
+�
+∆V − T
+4
+d∆V
+dT
+� ����
+Tn
+,
+β
+H ≡ T d(S/T)
+dT
+����
+Tn
+,
+(23)
+where,
+∆V = V high
+eff
+(hS, T) − V low
+eff (hS, T),
+ρ = geπ2
+30 T 4
+n,
+(24)
+where the ge ≈ 103 is the number of effective degrees of freedom during the nucleation at the
+temperature Tn. Finally the Euclidean action is defined as,
+S = 4π
+� ∞
+0
+r2
+�
+1
+2
+�dhφ/S
+dr
+�2
++ Veff(hφ/S)
+�
+dr,
+(25)
+where r is the radial distance from the center of the true vacuum bubble.
+In order to assess the reliability of the results, we make use of two different numerical tools
+for computing the nucleation temperature and the β and α parameters. The parameters α and β
+depend heavily on the nucleation temperature, Tn, so that possible errors on Tn will propagate to
+α and β. For the computation we use CosmoTransitions and BubbleProfiler[40, 41]. As shown
+in the appendix, we obtain a smaller numerical error with CosmoTransitions, but the results of
+both numerical computations agree within uncertainty. In general, we find that for sub-TeV DM
+masses the nucleation temperature in the BubbleProfiler implementation tends to be smaller
+than in CosmoTransitions.
+In Figure 2, we identify three benchmark points allowed by all constraints. These benchmark
+points are shown in 3 corresponding to the indigo diamond, blue square and purple hexagon shown
+in Figure 2.
+Notice that the first point is below one TeV, the trend we observed regarding the performance of
+the two simulation tools is noticeable, and the BubbleProfiler nucleation temperature is signifi-
+cantly below the value obtained from CosmoTransitions, affecting also the α and β parameters.
+At this point, the critical temperature is Tc = 303.
+In summary, both CosmoTransitions and BubbleProfiler show similar behavior for both
+models and are in reasonable agreement. For high masses the latter tool yields slightly higher
+nucleation temperatures and therefore α is also a bit lower and β as indicated by Equation 23.
+
+10
+Having computed the relevant parameters for calculating gravitational waves (GW) spectra, we
+can consider the following equation for computing the total signal,
+Ωtoth2 = Ωcolh2 + Ωswh2 + Ωturbh2,
+(26)
+where the first term is the collision term, the second term is the sound wave term, and the last
+term is the turbulence term. The collisions from the bubbles themselves contribute to the GW
+spectra, but they do not give the most significant contribution. The collisions also produce bulk
+motion in the fluid. This causes sound waves that result in the primary contribution to the GW
+spectra. Finally, there is also some turbulence caused by the collisions which contribute to the
+GW spectra [23, 52, 54]. The relevant equations for computing the collision term are,
+Ωcolh2(f) = 1.67 · 10−5
+�
+α
+1 + α
+�2 H2
+β2
+� ge
+100
+�− 1
+3 0.11κ2
+vv3
+b
+0.42 + v2
+b
+Scol
+Scol(f) =
+3.8
+�
+f
+fcol
+�2.8
+2.8
+�
+f
+fcol
+�3.8
++ 1
+fcol = 16.5 µHz
+0.62
+v2
+b − 0.1vb + 1.8
+β
+H
+Tn
+100 GeV
+� ge
+100
+� 1
+6 ,
+(27)
+similarly, the equations for the sound wave term are
+Ωswh2(f) = 2.65 · 10−6
+�
+α
+1 + α
+�2 H
+β
+� ge
+100
+�− 1
+3 κ2
+vvbSsw
+Ssw(f) =
+� f
+fsw
+�3
+�
+�
+�
+7
+3
+�
+f
+fsw
+�2
++ 4
+�
+�
+�
+3.5
+fsw = 19 µHz 1
+vb
+β
+H
+Tn
+100 GeV
+� ge
+100
+� 1
+6 ,
+(28)
+and lastly, the equations for the turbulence term are
+Ωturbh2(f) = 3.35 · 10−4
+�κturbα
+1 + α
+� 3
+2 H
+β
+� ge
+100
+�− 1
+3 vbSturb
+Sturb(f) =
+�
+f
+fturb
+�3
+�
+1 + 8πf
+h∗
+� �
+1 +
+f
+fturb
+� 11
+3
+fturb = 22.7 µHz 1
+vb
+β
+H
+Tn
+100 GeV
+� ge
+100
+� 1
+6 ,
+(29)
+where the inverse Hubble time, h∗, red-shifted to today, at the GW production is given as
+h∗ = 1.65 · 10−5
+Tn
+100 GeV
+� ge
+100
+� 1
+6 ,
+(30)
+
+11
+FIG. 4. The GW spectra for two different sets of transition parameters for the U(1)D model and one
+for the SU(2)D model (g = 2.0, MV = 7530) computed with CosmoTransitions, dashed lines, and
+BubbleProfiler, dotted lines. The sensitivity curves (C1-C4) of the LISA detector are also shown [27].
+According to this result, the signals from this model are strong enough for LISA to detect the GW signal
+from the phase transition.
+and the two modified efficiency factors can be written as,
+κv =
+α
+0.73 + 0.083√α + α,
+κturb = 0.05κv.
+(31)
+The result of this computation can be seen in Figure 4 for the three benchmark points, two
+from the U(1)D model and one from the SU(2)D.
+The marker shape indicates the parameter as shown in Figure 2. The diamond and square
+shapes are from the U(1)D model. For SU(2)D model we have the high mass case marked by
+the hexagon shape. The projected sensitivity curves (for the configurations C1-C4) for the LISA
+detector are also shown [27], and one can see that for the U(1)D model the signal should be
+detectable by three out of four configurations, but for the SU(2)D model the mass becomes too
+high and we need other future experiments to detect such high DM mass models such as the
+proposed TianQin detector [55].
+VII.
+DISCUSSION AND CONCLUSIONS
+We have investigated two vector DM models in light of existing DM direct detection experiments
+and future GW experiments. Both of the models investigated in this work are already strongly
+constrained by direct detection. For the SU(2)D model this is in particular due to recent results
+
+Gravitational wavespectra of selectedpoints
+10-5
+10
+10~9
+yo
+10-11
+10-13
+C1
+g = 0.66 My= 911 [GeV]
+10-15
+C2
+g= 0.70 My = 1028 [GeV]
+C3
+g = 2.0 My = 7530 [GeV]
+C4
+10~17
+10-4
+10-3
+10-2
+10-1
+f[Hz]12
+from the LZ experiment which has ruled out most of parameter space consistent with a full relic
+abundance from freeze-out in the range MV ∈ (1 − 10) TeV and with XENONnT the DM will
+either be detected or the entire parameter space above the neutrino floor will be ruled out as shown
+in Figure 2.
+GW signals in both models have been discussed in earlier literature. In our analysis we find
+that results differ significantly between different numerical implementations. Recently, the SU(2)D
+model was discussed in [48], and we find that their results for the α and Tn parameters agree with
+our findings.
+Regarding the U(1)D model, it was previously suggested that GW signals could be used to probe
+the model in case the direct detection cross section remains below the neutrino floor [23]. We agree
+with this conclusion, but numerically we find differences to [23] in the GW parameters. While we
+can reproduce the critical temperature reported, the nucleation temperature and the α and β
+parameters differ from those reported in [23]. Their results were obtained with the AnyBubble
+package [56], for which we failed to obtain results in agreement with the other two numerical
+implementations used in this work.
+This raises the question of comparability between the phase transition parameters obtained via
+the various numerical implementations available. This issue has been investigated in [41], where
+a fairly good agreement between BubbleProfiler and CosmoTransitions is observed. This is
+compatible with our findings.
+The finite temperature potential in both cases leads to a strong first order electroweak phase
+transition. The U(1)D model can produce significant GW signals, which can be detected by LISA
+[27] and future experiments would be able to test the SU(2)D model also in the high DM mass
+regime.
+ACKNOWLEDGMENTS
+The financial support from Academy of Finland, project #342777, is gratefully acknowledged.
+MTF and MR acknowledge partial funding from The Independent Research Fund Denmark, grant
+numbers DFF 6108-00623 and DFF 1056-00027B, respectively. MET acknowledges funding from
+Augustinus Fonden, application #22−19584, to cover part of the expenses associated with visiting
+the University of Helsinki for half a year.
+Appendix A: Supercooling, inflation and reheating
+The investigation of the GW spectra leads to the discussion of supercooling in the models
+presented. As shown in the GW section there are orders of magnitude in the difference between
+the critical and nucleation temperature at the low mass scale. As discussed in the other papers, this
+can lead to different kinds of phenomena including inflation, filtering and, reheating [28, 30, 37].
+These effects are expected to affect the GW signal for low masses, and it might effects some of
+the results even presented in Figure 4, but it is beyond this paper to look at the details of this.
+As discussed in a recent paper, the universe could escape inflation via bubble nucleation or via
+quantum tunneling, two different scenarios leading to different GW signals [36].
+We would however like to highlight the fact that strong supercooling from hundreds of GeV to
+the QCD scale might not be a big issue for the models. The bigger the supercooling the greater
+the inflation as the scalon Higgs field will be stuck in a false vacuum acting like a cosmological
+constant. The main constraint for any possible is lover than the max number of e-folds,
+Nmax = 23.8 + ln TR
+TeV,
+(A1)
+
+13
+where TR is the reheating temperature after the inflationary epoch and one finds that this limits the
+temperature to TR < 6.6· 1015 GeV [36]. To compute the reheating temperature, we are interested
+in computing the decay of the inflaton-like field which in this case is the scalon Higgs field S for the
+U(1)D model. Due to the mass constraints, only the scalon Higgs is kinetically allowed to decay
+as Γ(hS → h, h), but this requires a DM mas of MV > 1 TeV. From the Lagrangian, we find that
+the Feynman rule for this vertex and this yields the decay,
+Γ(hS → 2h) =
+�
+M 2
+S − 4M 2
+h
+32πM 2
+S
+|M|2,
+(A2)
+where,
+|M|2 =
+�M 2
+h
+4v1
+(5 + 3 cos(4α)) sin(α)
+�2
+.
+(A3)
+We can furthermore include decays into quarks and leptons,
+Γ(hS → f ¯f/ℓ¯ℓ) = NC
+8π
+m2
+b
+v2
+1
+MS
+�
+1 − 4m2
+b
+M 2
+φ
+sin(α)2.
+(A4)
+where NC = 3 for fermions and NC = 1 for leptons. Using the decays one can calculate the
+reheating temperature, TR, using the following equation [57],
+TR ≈ 0.2
+�200
+g∗
+�1/4 �
+ΓMpl,
+(A5)
+where Mpl is the reduced Planck mass and g∗ = 103.
+Considering a rather low mixing value
+0 ≤ α ≤
+π
+64 and a mass range of 250 GeV ≤ MV ≤ 2500 GeV the reheating temperature is
+somewhere around 0.1-1.6 PeV yielding mass of the scalon field around 1 GeV < MS < 200 GeV.
+This is so hot that the universe will reheat back to a temperature much hotter than the scales
+of freeze-out. It also satisfies the constraint from Equation (A1), thus it is not too hot and not
+causing too much inflation. One can repeat this exercise for the SU(2)D, but the result is roughly
+the same with the main differences being a slightly heavier scalon mass, 1 GeV < MS < 350 GeV,
+and higher reheating temperature 0.1-2 PeV. Conclusively, dark matter production can take place
+via freeze-out as the universe subsequently cools down again.
+Appendix B: Model implementation in CosmoTransitions
+For the implementation of the model in CosmoTransitions we feed in the tree level potentials
+as shown in Equation (2) and (6). Then we manually implement the mass matrix with the massive
+SM bosons, plus the new bosons, and the top quark,
+M 2
+W = g2
+W
+4 h2
+1,
+M 2
+Z = g2
+W + g2
+Z
+4
+h2
+1,
+m2
+t = λ2
+t
+2 h2
+1,
+(B1)
+where the Yukawa coupling of the top quark is λt = 1. The DM candidate have their respective
+implementations for each model where,
+M 2
+V = cV g2h2
+2,
+(B2)
+
+14
+with cV = 1 (1/4) for the U(1)D (SU(2)D) model, and then the scalar mass matrices yield,
+M 2
+h,S(U(1)D) = 1
+4
+�
+h2
+1 (λh + 2λφh) + h2
+2(λφ + 2λφh)
+±
+�
+h4
+1 (λh − 2λφh)2 + h4
+2 (λφ − 2λφh)2 + 2h2
+1h2
+2
+�
+2λhλφh + 28λ2
+φh − λhλφ + 2λφλφh
+�
+�
+(B3)
+M 2
+h,S(SU(2)D) = 1
+4
+�
+h2
+1 (6λH + λHS) + h2
+2(6λS + λHS)
+±
+�
+h4
+1 (λHS − 6λH)2 + h4
+2 (λHS − 6λS)2 + 2h2
+1h2
+2 (6λH(λHS − 6λHS) + λHS(7λHS + 6λS))
+�
+, (B4)
+A custom solution is made for computing the beta value. This is done by simply calculating the
+action divided by the temperature around the point of the nucleation temperature, making a fit
+to those plots, and taken the derivative etc. Some tweaks have been done to the source code to
+make this work, and also to improve the precision at low nucleation temperatures.
+Appendix C: Model implementation in BubbleProfiler
+For this package, we give it the full thermal potential in Equation (21), but instead of evaluating
+the thermal integral an approximation is made using Bessel function[23]. Specifically we use the
+modified Bessel functions, K2(kx), as follows
+� ∞
+0
+x2 ln
+�
+1 ∓ e−√
+x2±M2s /T 2�
+dx = −
+3
+�
+k=1
+x2
+k2K2(kx) −
+2
+�
+k=1
+(−1)kx2
+k2
+K2(kx).
+(C1)
+The BubbleProfiler is written in C++, but comes with a command line interface (CLI). Using
+this one can implement simple potentials like polynomials. In order to avoid implementing all
+these functions, we created a python interface where we implement the model in python. Then
+we create a higher order polynomial fit to the full potential.
+This polynomial is then fed to
+BubbleProfiler via the CLI together with other relevant parameters. We compute several points
+around the nucleation temperature and make a fit to that and from there we determine the β and
+Tn value, and the latter is then used to find α.
+Appendix D: Computing parameters of the EWPT
+The essential computation for the phase transition is finding the relationship between the action
+and temperature. The nucleation temperature condition is defined as
+S(T)
+T
+����
+Tn
+≈ 140,
+(D1)
+thus when the action divided by the temperature is equal to 140. We can compute the action and
+by dividing by the temperature a plot of this relationship can be obtained as seen in Figure 5.
+Given some data points, it is possible to make a fit, and from that read off the Tn value.
+Furthermore, the fit is also a function of S(T)/T, which can thus be used to compute β. Now
+
+15
+(a) Using BubbleProfiler for computing parameters.
+(b) Using CosmoTransitions for computing
+parameters.
+FIG. 5. A comparison of the apparent error when computing the β and Tn parameter when computing
+the EWPT parameters in U(1)D model for g = 0.75 and MV = 1184. Note the temperature range is
+different for the implementation, thus the range is different in the plots.
+recall that α is evaluated at the nucleation temperature and α ∝ 1/T 4
+n, thus, the value of α
+is also highly dependent on the nucleation temperature.
+Since our BubbleProfiler result in
+general yields a slightly higher nucleation temperature we get a lower value of α as discussed in
+the GW section. For lower masses the BubbleProfiler result yield significantly lower nucleation
+temperatures suggesting that our implementation might not be as good in this regime.
+Looking at Figure 5, we see that the apparent error of the BubbleProfiler is significantly
+higher than the error from the CosmoTransitions result. This may be attributed to the fact that
+we used an approximated potential via our custom Python interface instead of implementing the
+model using C++. This leads us to consider the CosmoTransitions as the better result in this
+paper even though BubbleProfiler is claimed to be more accurate [41].
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+

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+arXiv:2301.01700v1  [cs.GT]  4 Jan 2023
+Non-Adaptive Matroid Prophet Inequalities
+Kanstantsin Pashkovich, Alice Sayutina
+University of Waterloo
+Department of Combinatorics & Optimization
+200 University Avenue West
+Waterloo, ON, Canada
+N2L 3G1
+Abstract
+We consider the matroid prophet inequality problem. This problem
+has been extensively studied in the case of adaptive mechanisms. In par-
+ticular, there is a tight 2-competitive mechanism for all matroids [KW12].
+However, it is not known what classes of matroids admit non-adaptive
+mechanisms with constant guarantee.
+Recently, in [CGKM20] it was
+shown that there are constant-competitive non-adaptive mechanisms for
+graphic matroids. In this work, we show that various known classes of
+matroids admit constant-competitive non-adaptive mechanisms.
+1
+Introduction
+Let us consider the classical prophet inequality problem [KS77].
+A gambler
+observes a sequence of non-negative independent random variables X1, X2, . . . ,
+Xn, which correspond to a sequence of values for n items. The gambler knows
+the distributions of X1, X2, . . . , Xn. The gambler is allowed to accept at most
+one item; and the gambler is interested in maximizing the value of the accepted
+item. However, the gambler cannot simply select an item of the maximum value,
+because the values of the n items are revealed to the gambler one by one; and
+each time a value of the current item is revealed the gambler has to make an
+irrevocable choice whether to accept the current item or not.
+What stopping rule the gambler should use to maximize the expected value of
+the item they accept? The gambler knows only the distributions of X1, X2, . . . ,
+Xn while a prophet knows the realization of X1, X2, . . . , Xn. Thus, in contrast
+to the gambler the prophet can always obtain the maximum item’s value. The
+seminal result of Krengel and Sucheston [KS77] showed that the gambler can
+obtain at least a half of the expected value obtained by the prophet.
+The classical prophet inequality problem led to a series of works on different
+variants of the problem. A natural variant of the problem is the generalization
+1
+
+of the problem where a gambler can buy more than one item, but the set of
+bought items should satisfy a known feasibility constraint. Formally, let us be
+given a collection S ⊆ 2[n] of item sets. Then both gambler and prophet can
+select any item set S from S. So S defines a feasibility constraint for selecting
+items. In most standard examples of feasibility constraints, S can be defined as
+a collection of all item sets with cardinality at most k for some natural number k.
+More generally S can be defined as a collection of all independent sets in some
+matroid, in this case we speak about the matroid prophet inequality problem.
+The result in [SC84] showed that in the single-item setting a gambler can
+obtain at least half of the prophet value by using the following threshold-rule:
+determine a constant T as a function of known distributions and accept the first
+item exceeding T . This rule results in a 2-competitive mechanism, similar to
+the adaptive approach of [KS77]. Note, that this approximation guarantee is
+known to be tight. There is also another method to set a threshold presented in
+[KW12], which also results in a 2-competitive mechanism. This was extended
+by Chawla et al. in [CHMS10] and [CGKM20] to the setting of several items.
+The results presented in [KW12] further extend to the matroid prophet in-
+equalities, where accepted items need to form an independent set in a known
+matroid. It leads to a 2-competitive mechanism for every matroid, matching
+the single-item setting result. However, unlike the mechanism in the single-item
+setting, the mechanism for matroids is adaptive: the thresholds for items are
+computed based on the previously accepted items. By [KW12], there also exists
+a constant-competitive adaptive mechanism for feasibility constraints defined as
+an intersection of constant number of matroids. The mechanism by Kleinberg
+and Weinberg was further extended to a 2-competitive mechanism for polyma-
+troids by Dütting and Kleinberg in [DK15].
+Gravin and Wang [GW19] studied the bipartite matching version of this
+problem: in their version, the arriving items are the edges of the (known) bi-
+partite graph. Gravin and Wang obtained a 3-competitive non-adaptive mech-
+anism, which assigns thresholds to each vertex in the graph and an edge is
+accepted only if its weight is at least the sum of the thresholds associated with
+its endpoints.
+Feldman, Svensson and Zenklusen [FSZ16] studied online item selection
+mechanisms called “online contention resolution schemes" (OCRS). They showed
+that given special properties, OCRS translate directly into a constant-competitive
+prophet inequality for the same problem against almighty adversary, i.e. an ad-
+versary which knows in advance realizations of all the items and the random
+bits generated by an algorithm. As a result, they develop a constant-competitive
+mechanism for prophet inequalities of the intersection of a constant number of
+matroids, knapsack and matching constraints. Those mechanisms are “almost”
+non-adaptive in a sense that they fix thresholds for all items, however their mech-
+anisms also impose a subconstraint: an item cannot be accepted if together with
+previously accepted items it forms one of the fixed forbidden sets.
+Finally, in a later version of their paper [FSZ21], they prove that pure non-
+adaptive mechanisms cannot achieve a constant-competitive approximation even
+against a “normal” adversary. They construct a family of gammoid matroids
+2
+
+showing a lower bound of Ω(log n/ log log n) for a guarantee of non-adaptive
+mechanisms on gammoids with n elements.
+There have been works studying similar setups with other goals. Chawla
+et al. [CHMS10] studied a Bayesian item selection process in a fixed item ar-
+rival order or against an adversary in control of the order. They studied it
+from a perspective of the revenue maximization for the auctioneer. The per-
+formance is constant-competitive compared to the well-known Myerson mech-
+anism
+[Mye81], which achieves the largest possible expected revenue among
+truthful mechanisms. The mechanism by Chawla et al. [CHMS10] has an ad-
+vantage that it determines static thresholds together with a subconstraint so
+that each agent can be offered take-it-or-leave-it prices in an online fashion.
+Recently, Chawla et al. [CGKM20] developed a 32-competitive non-adaptive
+mechanism for graphic matroids against adversary item ordering.
+1.1
+Our results
+First, we list the known results for non-adaptive mechanism that were mentioned
+in the previous section.
+Theorem 1 (Uniform Rank 1 Matroid [SC84]). There exists a 2-competitive
+non-adaptive mechanism for single-item setting.
+Theorem 2 (Graphic Matroid [CGKM20]). There exists a 32-competitive
+non-adaptive mechanism for graphic matroids.
+Now let us list our results. In case of a simple graph, i.e. a graph with no
+parallel edges or loops, we can slightly improve the above theorem by considering
+essentially the same mechanism as [CGKM20] but considering a different scaling
+of a point from the matroid polytope. We provide this result for the sake of
+completeness.
+Theorem 3. There exists a 16-competitive non-adaptive mechanism for graphic
+matroids in the case of simple graphs.
+Furthermore, the mechanism [CGKM20] can be generalized to the setting of
+k-column sparse matroids. This result we need later to obtain Theorem 8.
+Theorem 4 (k-Column Sparse Matroids). There exists a (2k+2k)-competitive
+non-adaptive mechanism for k-column sparse matroids.
+Note, that Theorem 2 of [CGKM20] follows from Theorem 4, since a graphic
+matroid is also a 2-column sparse matroid over F2.
+Using analogous approach to the one in [Sot13], we also develop a mechanism
+for cographic matroids.
+Theorem 5 (Cographic Matroids). There exists a 6-competitive non-adaptive
+mechanism for cographic matroids.
+The approach in [Sot13] immediately leads to the following result for γ-sparse
+matroids.
+3
+
+Theorem 6 (γ-Sparse Matroids). There exists a γ-competitive non-adaptive
+mechanism for γ-sparse matroids.
+Combining the above results and using classic Seymour’s decomposition re-
+sults we obtain the following theorem.
+Theorem 7 (Regular Matroids). There exists a 256-competitive non-adaptive
+mechanism for regular matroids.
+Subject to the Structural Hypothesis 1 due to Geelen, Gerards and Whittle,
+which is stated later, we can also derive the following result.
+Theorem 8. Subject to the Structural Hypothesis 1, for every prime number p
+there exists a constant-competitive mechanism for every proper minor-closed
+class of matroids representable over Fp.
+We also would like to observe that some of the recent results on “single
+sample prophet inequalities” (SSPI) lead to non-adaptive constant-competitive
+mechanisms. For this, the single sample required by the gambler in SSPI can
+be directly sampled by our gambler from the available distributions. In partic-
+ular, the results in [AKW19] and [CFPP21] on laminar matroids and truncated
+partition matroids inspired by the mechanism in [MTW16] lead to non-adaptive
+mechanisms for prophet inequalities. To obtain these results, it is crucial that
+the mechanism in [MTW16] does not involve subconstraints, i.e. each item is
+accepted as long as the item is not in the “observation phase”, the item passes
+its threshold based only on the “observation phase” and the item forms an in-
+dependent set with previously accepted items. In comparison, it is not clear
+how from the results on regular matroids in [AKW19] based on the mechanism
+in [DK14] one can obtain non-adaptive mechanisms.
+So the following results can be directly obtained from [AKW19] and [CFPP21],
+respectively.
+Theorem 9 (Laminar Matroid). There exists a 9.6-competitive non-adaptive
+mechanism for laminar matroids.
+Theorem 10 (Truncated Partition Matroid). There exists an 8-competitive
+non-adaptive mechanism for truncated partition matroids.
+2
+Comparison to known results
+Our results for cographic matroids and k-column sparse matroids are obtained
+through modifications of the arguments in [Sot13] and [CGKM20], respectively.
+The results on regular matroids and minor-closed families of matroids follow the
+approach outlined in [HN20] for the secretary problem. As necessary building
+blocks we use our results for cographic and 2-column sparse matroids. Note that
+a biggest challenge for us is the compatibility of non-adaptive thresholds with
+contractions. Indeed, standard tools for deriving mechanisms for contraction
+4
+
+minors need subconstraints, while subconstraints are not permitted in non-
+adaptive mechanisms. To obtain our results, we resolve this issue only in the
+context of matroids representable over finite fields, see arguments in Lemma 8.
+It would be interesting to see whether analogous results for contraction minors
+hold with no assumption about representability over finite fields.
+3
+Preliminaries
+In this paper, we consider the matroid prophet inequality problem, where items
+arrive online in adversarial order. Here, the adversary knows the distributions
+of all X1, X2, . . . , Xn and knows the gambler’s mechanism, but the realization
+of X1, X2, . . . , Xn is not known to the adversary.
+Based on the available
+information, the adversary can decide on the order in which items and their
+values are observed by the gambler.
+3.1
+Prophet inequality
+Definition 1. Let M be a matroid on the ground set [n] := {1, . . . , n}, where
+[n] corresponds to n items. Let ⃗X := (X1, . . . , Xn) be non-negative independent
+random variables representing the values of these n items.
+• For every subset of items S ⊆ [n] we define its weight as follows
+w(S) :=
+�
+i∈S
+Xi.
+• Let PROPHM be the random variable corresponding to the value obtained
+by the prophet
+PROPHM :=
+max
+S∈I(M) w(S) ,
+where I(M) is a collection of independent sets for M.
+• Let EPROPHM be the expectation of the value obtained by prophet
+EPROPHM := E[PROPHM] .
+Definition 2. Let us be given a number α > 0.
+• We call a mechanism α-competitive (alternatively, we say that the mech-
+anism guarantees an α-approximation) on the matroid M if the expected
+value obtained by the gambler via this mechanism is at least 1
+αEPROPHM.
+• We call a mechanism α-competitive (alternatively, we say that the mech-
+anism guarantees an α-approximation) on the matroid class M if this
+mechanism is α-competitive for every matroid M ∈ M.
+5
+
+3.2
+Non-adaptive mechanism
+We say that a mechanism is non-adaptive if it has the following structure:
+• Given the distributions of ⃗X = (X1, . . . , Xn), the mechanism determines
+the values of thresholds ⃗T = (T1, . . . , Tn), where each Ti, i ∈ [n] is a real
+number or +∞.
+• If the value of item i ∈ [n] is observed, the gambler accepts the item i if
+and only both conditions hold:
+1. the observed value of Xi is at least Ti
+2. the item i together with all previously selected items forms an inde-
+pendent set with respect to the matroid M.
+Note, that a non-adaptive mechanism does not change thresholds during
+its course.
+So, none of the thresholds depends on the realization of ⃗X =
+(X1, . . . , Xn).
+Another crucial feature of a non-adaptive mechanism is that the mechanism
+works only with the original matroid M. A non-adaptive mechanism does not
+allow us to define a new matroid M ′, such that a set of items is independent in
+M ′ only if it is independent in M, and modify the condition (2) based on M ′.
+In this work, we focus on non-adaptive mechanisms. From here and later we
+use the term mechanism to refer to non-adaptive mechanisms exclusively.
+Remark 1. In this work, non-adaptive mechanisms are allowed to make non-
+deterministic decisions. Hence, we allow a non-adaptive mechanism to construct
+the thresholds ⃗T = (T1, . . . , Tn) non-deterministically.
+To measure the performance of such a mechanism we use the expected total
+value, where the expectation is taken not only with respect to ⃗X = (X1, . . . , Xn)
+but also with respect to ⃗T = (T1, . . . , Tn).
+3.3
+Matroids
+We provide a review of matroids here.
+Experienced readers should consider
+skipping or skimming this section. For further results about matroids, consider
+consulting [Oxl06].
+A matroid M = (E, S) is a pair of a finite ground set E and a collection S ⊆
+2E of independent sets. The collection S ⊆ 2E of subsets of E satisfies the
+following conditions:
+(i) Empty set is an independent set, so ∅ ∈ S.
+(ii) The collection S is closed with respect to taking subsets, so for all A ⊆
+B ⊆ E if B is in S then A is in S.
+(iii) The collection S satisfies so called augmenation property. In other words,
+for all A, B ⊆ E such that A, B ∈ S and |A| > |B|, there exists c ∈ A \ B
+such that B ∪ {c} ∈ S.
+6
+
+A subset of E is called dependent if it is not in S. The inclusion-maximal
+independent sets are called bases and the inclusion-minimal dependent sets are
+called circuits.
+For every two bases, their cardinalities are equal: for every
+bases A and B of M we have |A| = |B|. A rank function for the matroid M is
+a function rM : 2E → N such that for every A ⊆ E the value rM(A) equals the
+cardinality of an inclusion-maximal independent subset of A. In the cases when
+the choice of the matroid is clear from the context, we write r instead of rM.
+Given a matroid M, we can define the dual matroid M ∗ over the same ground
+set E. A set A is independent for matroid M ∗ if and only if E \ A contains a
+basis of M. An element c ∈ E is called a loop in M if rM(c) = 0. An element
+c ∈ E is called a free element in M if rM∗(c) = 0. To put it another way, an
+element c is free, if and only if for every set A, which is independent in M,
+A∪{c} is also independent in M. We say that elements c and d ∈ E are parallel
+in matroid M, denoted by c ∥ d, if rM(c) = rM(d) = rM({c, d}) = 1. One
+can show that “being parallel” defines an equivalence relation on the non-loop
+elements of M. A matroid is called simple if it has no loops and no parallel
+edges.
+Let M = (E, S) be a matroid and A ⊆ E. The contraction of M by A, de-
+noted as M/A, is a matroid over ground set E\A with the following independent
+sets
+{S ⊆ E \ A : S ∪ A′ ∈ S} ,
+where A′ is an inclusion-maximal independent subset of A.
+The restriction of M to A, denoted as M |A or M \ A, is a matroid over
+the ground set A where a set S ⊆ A is independent in M |A if and only if it is
+independent in M.
+A matroid M ′ is called a simple version of M if M ′ is obtained from M by
+deleting all loops and contracting every parallel class of elements into a single
+element.
+For matroids M, N, we say that N is a minor of M = (E, S) if N is
+isomorphic to M/A\B for some disjoint sets A, B ⊆ E. A matroid class M is
+called minor-closed if for any M ∈ M every minor of M is also in M.
+Let us now list some of the classical examples of matroids, which were ex-
+tensively studied in the context of various mathematical fields.
+• A uniform matroid M = (E, S) of rank k is matroid where
+S := {A ⊆ E : |A| ≤ k} .
+When |E| = n, we denote the uniform matroid of rank k as Uk,n.
+• A graphic matroid over graph G = (V, E) is a matroid M = (E, S), where
+S := {A ⊆ E : A is acyclic} .
+The graphic matroid over graph G is denoted as M(G).
+7
+
+• A cographic matroid over graph G = (V, E) is a dual matroid M = (E, S)
+to the graphic matroid over the same graph G. In this case we have
+S := {A ⊆ E : (V, E\A) has the same number of components as (V, E)} .
+• A vector matroid M = (E, S) is a matroid such that there is a vector
+space V and a map φ : E → V satisfying
+S := {A ⊆ E : multiset φ(A) is linearly independent} .
+Given a field F, we say that M is representable over field F if M is iso-
+morphic to the vector matroid where V is a vector space over field F.
+A matroid is called regular if it is representable over every field. A matroid
+is called binary if it is representable over F2.
+• A k-column sparse matroid M = (E, S) is a matroid such that there is a
+field F and dimension m and a map φ : E → Fm such that
+S := {A ⊆ E : multiset φ(A) is linearly independent over F} ;
+and moreover φ(c) ∈ Fm has at most k nonzero coordinates for every
+c ∈ E.
+• A γ-sparse matroid M = (E, S) is a matroid such that the inequality
+|S| ⩽ γrM(S) holds for every S ⊆ E.
+• A laminar matroid M = (E, S) is a matroid such that there exists a
+laminar family F over the ground set E and there are numbers cF ∈ N,
+F ∈ F such that
+S := {A ⊆ E : |A ∩ F| ≤ cF for every F ∈ F} .
+Moreover, if F = {E, E1, . . . , Ek}, where E1, . . . , Ek form a partition of
+the ground set E, then M is called a truncated partition matroid. Recall,
+that a family F is called laminar if for every A, B ∈ F we have A ⊆ B or
+B ⊆ A or A ∩ B = ∅.
+Given a matroid M = (E, S) we can define the corresponding polytope
+PM ⊆ RE as the convex hull of points corresponding to the characteristic vectors
+of independent sets. The polytope PM is known to admit the following outer
+description [Sch03].
+PM = {x ∈ RE : x ≥ 0 and
+x(S) ≤ rM(S) for every S ⊆ E} ,
+where x(S) stands for �
+c∈S xc.
+For a matroid M = (E, S) and a set A ⊆ E we can define the closure of A
+as the following set
+clM(A) := {c ∈ E | rM(A ∪ {c}) = rM(A)} .
+8
+
+For a matroid M = (E, S), we call the following function ⊓M : E × E → Z
+a local connectivity function
+⊓M(X, Y ) = r(X) + r(Y ) − r(X ∪ Y ) .
+The following function λM : E → Z⩾0 is called a connectivity function
+λM(X) := ⊓M(X, E \ X) = r(X) + r(E \ X) − r(E) .
+Informally, connectivity functions measure dependence with respect to the
+matroid between parts of the ground set. To illustrate it, let us consider the
+connectivity function for vector matroids.
+Suppose M = (E, S) is a vector
+matroid defined by a vector space V and a map φ : E → V . Then we have
+λM(S) =r(S) + r(E \ S) − r(E) =
+dim(span φ(S)) + dim(span φ(E \ S)) − dim(φ(E)) =
+dim ((span φ(S)) ∩ (span φ(E \ S))) .
+3.4
+Ex-ante relaxation to the matroid polytope
+The goal of ex-ante relaxation [FSZ16] or [CGKM20] is to reduce the origi-
+nal problem to the problem where item values are distributed as independent
+Bernoulli random variables. Note, that both problems are using the same ma-
+troid.
+In the original problem item values ⃗X = (X1, . . . , Xn) are independent ran-
+dom variables with known distributions. For i ∈ [n] let Fi be the cumulative
+distribution function of Xi. The reduction of the original problem to a new
+problem is done using a point p in the matroid polytope PM. Let us first show
+that there is a point p ∈ PM with properties that prove to be desirable later
+following the argumentation in [CGKM20].
+Lemma 1. Given a matroid M over the ground set [n] and random variables
+⃗X = (X1, . . . , Xn), there exists p ∈ PM such that
+EPROPHM ⩽
+n
+�
+i=1
+piti ,
+where ti := E[Xi | Xi ⩾ F −1
+i
+(1 − pi)] for every i ∈ [n]1.
+Proof. Let Iopt be a random variable indicating an optimal independent set
+in M with respect to ⃗X = (X1, . . . , Xn). In case when for some realization
+of ⃗X = (X1, . . . , Xn) there are several optimal independent sets, Iopt can be
+selected as any of these sets. For i ∈ [n], let pi be the probability that element
+1Here, we assume that for every i ∈ [n] the event Xi = F −1
+i
+(1 − pi) happens with the zero
+probability, which is true for all continuous distributions. In case of discrete distributions one
+needs to introduce appropriate tie-breaking.
+9
+
+i is in Iopt. Note that p = (p1, . . . , pm) is a convex combination of independent
+sets of M, and so lies in PM.
+Due to EPROPHM = E[�
+i∈Iopt Xi], it remains to show that
+E[
+�
+i∈Iopt
+Xi] ⩽
+n
+�
+i=1
+piti .
+We have
+E[
+�
+i∈Iopt
+Xi] =
+n
+�
+i=1
+P[i ∈ Iopt]E[Xi | i ∈ Iopt] =
+n
+�
+i=1
+piE[Xi | i ∈ Iopt] .
+For every i ∈ [n] we have that ti and E[Xi | i ∈ Iopt] are expectations of the
+same random variable Xi but conditioned on the event Xi ⩾ F −1
+i
+(1−pi) and on
+the event i ∈ Iopt, respectively. Note, that the probability of both these events
+equals pi. However, the expectation of Xi conditioned on Xi ⩾ F −1
+i
+(1 − pi) is
+the “largest” conditional expectation of Xi on an event of probability pi. Thus,
+we have piE[Xi | i ∈ Iopt] ⩽ piti for every i ∈ [n] and so we get the desired
+inequality
+n
+�
+i=1
+piE[Xi | i ∈ Iopt] ⩽
+n
+�
+i=1
+piti .
+Let us show how one can use the point p = (p1, . . . , pn) guaranteed by
+Lemma 1 to reduce the original problem. Let us define independent Bernoulli
+random variables ⃗X′ = (X′
+1, . . . , X′
+n) as follows, for each i ∈ [n]
+X′
+i =
+�
+ti
+with probability pi
+0
+with probability 1 − pi ,
+where ti := E[Xi | Xi ⩾ F −1
+i
+(1 − pi)].
+Let us assume that we have a non-adaptive mechanism for the original ma-
+troid M and item values ⃗X′ = (X′
+1, . . . , X′
+n), which sets nonnegative thresholds
+⃗T ′ = (T ′
+1, . . . , T ′
+n). By definition of ⃗X′ = (X′
+1, . . . , X′
+n), for every i ∈ [n] the
+exact value of T ′
+i is not relevant per se, but it is crucial whether ti ≥ T ′
+i or
+ti < T ′
+i. If for some i ∈ [n] we have T ′
+i > ti then this item i is “inactive” and so
+is never selected by the gambler working with M and ⃗X′ = (X′
+1, . . . , X′
+n).
+The key is to construct a non-adaptive mechanism for the original matroid M
+and item values ⃗X′ = (X′
+1, . . . , X′
+n) with positive thresholds ⃗T ′ = (T ′
+1, . . . , T ′
+n)
+such that for each item i ∈ [n] the probability that i is selected by the gambler
+is at least αpi. Now we can use such a non-adaptive mechanism for the original
+matroid M and item values ⃗X′ = (X′
+1, . . . , X′
+n) to construct a non-adaptive
+α-competitive mechanism for the same matroid M and random variables ⃗X =
+10
+
+(X1, . . . , Xn). Let us define the thresholds ⃗T = (T1, . . . , Tn) as follows, for every
+i ∈ [n]
+Ti :=
+�
++∞
+if ti < T ′
+i
+F −1
+i
+(1 − pi)
+otherwise .
+To see that the thresholds ⃗T = (T1, . . . , Tn) lead to an α-competitive mech-
+anism for M and ⃗X = (X1, . . . , Xn), let us couple random variables X′
+i with
+random variables Xi as follows
+X′
+i :=
+�
+ti
+if Xi ≥ F −1
+i
+(1 − pi)
+0
+otherwise.
+Note that ⃗X′ = (X′
+1, . . . , X′
+n) are independent Bernoulli random variables,
+where for each i ∈ [n] the variable X′
+i equals ti with probability pi and equals 0
+with probability 1 − pi. When ⃗X′ are coupled with ⃗X this way, Xi and X′
+i have
+the same expected value when conditioned on X′
+i being ti. The mechanism with
+thresholds ⃗T selects an item i ∈ [n] when run for ⃗X only if the mechanism with
+thresholds ⃗T ′ selects the item i when run for ⃗X′. Moreover, for both of these
+algorithms, conditionally on the event that the item i is selected the expected
+value of i equals ti. Now, α-competitiveness guarantee of the thresholds ⃗T for
+M and ⃗X follows from Lemma 1.
+4
+Graphic and k-column sparse matroids
+First, we construct a 16-competitive non-adaptive mechanism for graphic ma-
+troids without parallel edges. Our construction is done through the ex-ante re-
+laxation to the matroid polytope, following the works in [FSZ16] or [CGKM20].
+Later, we present a constant-competitive non-adaptive mechanism for k-column
+sparse matroids whenever k is constant.
+4.1
+Graphic matroids
+Now we are ready to provide a 16-competitive non-adaptive mechanism for
+graphic matroid. The provided mechanism is essentially the one constructed
+in [CGKM20] but with saving a factor of 2 in the guarantee, which is achieved
+by rescaling the point from the matroid polytope by 2 and not by 4.
+Let us be given a simple graph G = (V, E) and let us consider the corre-
+sponding graphic matroid M over the ground set E. Recall that a subset of E
+is independent with respect to M if and only if it is acyclic in G. Let us also
+assume that the graph G has n edges and so E = {e1, e2, . . . , en}.
+Lemma 2. Let p = (p1, . . . , pn) be a point in the polytope PM. Thus we assume
+that for every i ∈ [n] the coordinate pi of p corresponds to the edge ei. Then
+there exists an orientation of edges E = {e1, e2, . . . , en} in the graph G = (V, E)
+such that for every vertex v ∈ V we have �
+i∈[n]:ei∈δ−(v) pi ≤ 2.
+11
+
+Proof. Observe that the average degree of a vertex in a forest on |V | vertices is
+at most (2|V | − 2)/|V | = 2 − 1/|V | ⩽ 2.
+Let us use this fact to prove the desired statement by induction on the
+number of vertices in the graph G.
+If the graph G has at most two vertices then the orientation is trivial. Other-
+wise, since p is a convex combination of points corresponding to forests in G, we
+have that the average of the value �
+i∈[n]:ei∈δ(v) pi over all vertices v ∈ V is at
+most 2. Thus there exists a vertex v ∈ V such that we have �
+i∈[n]:ei∈δ(v) pi ≤ 2.
+We orient all edges incident to v as edges in δ−(v), so these edges are incoming
+with respect to v. Then we remove the vertex v and all edges incident to it
+and orient the remaining edges according to the orientation guaranteed by the
+inductive hypothesis.
+Now we present an algorithm for graphic matroids of simple graphs.
+Algorithm 1 A non-adaptive 16-competitive mechanisms for graphic matroids
+of a simple graph
+1: Let p be a point in the polytope PM so that the statement of Lemma 1 is
+satisfied.
+2: Let the edges of the original graph G = (V, E) be oriented so that the
+statement of Lemma 2 is satisfied.
+3: For every edge ei ∈ E, i ∈ [n], mark the edge ei as “discarded" independently
+at random with probability 1/2.
+4: Select a cut S ⊆ V uniformly at random, mark all edges not in [S; S] as
+“discarded". Here, [S; S] stands for the set of edges which are oriented such
+that their tail is in S and their head is in S.
+5: Set thresholds ⃗T = (T1, . . . , Tn) as follows, for each i ∈ [n]
+Ti :=
+�
++∞
+if ei is “discarded”
+F −1
+i
+(1 − pi)
+otherwise .
+Lemma 3. For every i ∈ [n], we have
+P[ei is selected | Xi ≥ Ti and ei is not “discarded”] ≥ 1/2 .
+Proof. Let us assume that the vertex v is the head of the oriented edge ei. Let
+us also assume that ei is not marked as “discarded” and Xi ≥ Ti.
+Since the edge ei is not “discarded”, the edge ei is in the selected set [S; S].
+Hence, every not “discarded” edge incident to v has the vertex v as its head.
+Thus, as long as no other edge with the head at the vertex v is selected by
+the gambler, the gambler has to select ei. We claim, that with probability at
+least 1/2 no other edge with the head at v was selected by the gambler.
+Let I be the event indicating that "the gambler selected an edge ej, j ̸= i
+such that v is the head of ej", in other words “there is j ∈ [n], j ̸= i such that
+12
+
+v is the head of ej and Xj ≥ Tj and ej is not “discarded”". Let J indicate the
+event that "ei is not marked as “discarded” after the selection of the cut", in
+other words, "the head of ei is in S and the tail of ei is in S".
+Let us show
+P[I | J] ≤ 1/2 .
+By the union bound, we have
+P[I | J] ⩽
+�
+j∈[n]\{i}:ej∈δ−(v)
+P[Xj ≥ Tj and ej is not “discarded” | J]
+Note that for each edge ej ∈ δ−(v) we have P[Xj ≥ Tj|J] = pj and we also
+have P[ej is not “discarded”|J] = 1/4. Note that any edge is not “discarded”
+in Step 3 of Algorithm 1 with probability 1/2, and not “discarded” in Step 4
+of Algorithm 1 with probability 1/4. However, since the probabilities are with
+respect to the edge ej ∈ δ−(v) and are counted conditioned on the event J,
+the conditioned probability of not being “discarded” in Step 4 of Algorithm 1
+is 1/2. Moreover, even conditioned on J the events "Xj ≥ Tj" and "ej is not
+“discarded”" are independent events. Thus we have
+�
+j∈[n]\{i}:ej∈δ−(v)
+P[Xj ≥ Tj and ej is not “discarded” | J] ≤
+�
+j∈[n]\{i}:ej∈δ−(v)
+pj/4 ⩽ 1/2 ,
+where the last inequality follows from the orientation.
+We are ready to prove Theorem 3 by showing that Algorithm 1 is a 16-
+competitive for graphic matroids without parallel edges.
+Proof of Theorem 3. By Lemma 3 for every i ∈ [n] the probability of edge ei
+being accepted conditional on Xi ≥ Ti and being not “discarded” is at least 1/2.
+Overall, the probability of edge ei being accepted is at least
+1
+16pi. Thus
+mechanism guarantees at least �n
+i=1
+1
+16piti of the expected total value.
+By
+Lemma 1, we have �
+i∈[n]
+1
+16piti ⩾
+1
+16EPROPHM, finishing the proof.
+4.2
+k-column sparse matroids
+There are known constant-competitive mechanisms for k-column sparse ma-
+troids in the context of the secretary problem [Sot13]. However they do not
+immediately lead to a non-adaptive mechanism of constant competitiveness
+guarantee. The reason for that are not the updated thresholds but implicit
+changes to the considered matroid.
+Here, we present a constant competitive mechanism for k-column sparse
+matroid class for each constant k.
+Note, graphic matroids form a subclass
+of 2-column sparse matroids . Because of their significance, 2-column sparse
+matroids are also known in literature as represented frame matroids. Later, we
+use 2-column sparse matroids to prove results in Section 6.4.
+13
+
+Suppose M is a k-column sparse matroid over field F. In this section, we
+prove that there exists a (2k+2k)-competitive mechanism for M.
+Suppose a k-sparse representation of M = (E, S) is defined by a map φ :
+E → Fd. Note, if for some element t ∈ E the vector φ(t) is a zero vector then c
+is a loop and therefore can be removed from consideration.
+Now we consider an undirected hyper-multigraph G with vertex set [d]. Each
+matroid element t ∈ E induces a hyperedge et in this graph between non-zero
+coordinates of φ(t). Formally, the hyperedge et is defined as follows et := {i ∈
+[d] : φ(t)i ̸= 0}. We say that a vertex i ∈ [d] of the hyper-multigraph G is
+incident to every edge e of G such that i ∈ e. For a vertex i ∈ [d] we denote
+the collection of incident hyperedges by δ(i). The degree of a vertex i in the
+hyper-multigraph G equals |δ(i)|.
+Claim 1. Suppose I is an independent set of the matroid M. Then the average
+degree of a vertex is at most k when one considers the hyper-multigraph with
+vertices [d] and hyperedges {et : t ∈ I}.
+Proof. Observe that |I| ⩽ d because having more than d vectors in d-dimensional
+vector space Fd leads to a a linear dependency.
+Since M is k-column sparse, we have that every edge in {et : t ∈ I} is
+incident to at most k vertices in [d]. Hence, the total degree is at most kd and
+thus the average degree of a vertex is at most k.
+Now we consider orientations of the graph G. An orientation of the graph
+G is a function ϕ which maps every edge et into one vertex of G incident to et.
+We call ϕ(et) to be the head of the edge et, and all other vertices, if any, to
+be tails. For every vertex i ∈ [d] we denote the set of incoming edges by δ−(i),
+formally δ−(i) = {et : ϕ(et) = i, t ∈ E}.
+Lemma 4. Let p be a point in the polytope PM. We assume that for every
+t ∈ E, the coordinate pt of p corresponds to the element t. Then there exists
+an orientation ϕ of hyperedges in the hyper-mulrigraph G such that for every
+vertex i ∈ [d] we have �
+t∈E:et∈δ−(i) pt ⩽ k.
+The proof of Lemma 4 is analogous to the proof of Lemma 2. Now let us
+describe an algorithm for k-column sparse matroids.
+Lemma 5. For every t ∈ E we have
+P[t is selected | Xt ≥ Tt and t is not “discarded”] ≥ 1/2 .
+Proof. Note that item t ∈ E is accepted whenever Xt ≥ Tt and no other item
+was selected from non-discarded edges in δ−(ϕ(t)). By the union bound, for
+every event J we can upper bound the probability that
+P[there j ∈ E \ {t} such that j is selected and ej ∈ δ−(ϕ(t)) | J] ⩽
+�
+j∈E\{t}:ej∈δ−(ϕ(t))
+P[ej is not “discarded” and Xj ≥ Tj | J] .
+14
+
+Algorithm 2 A non-adaptive 2k+2k-competitive mechanisms for k-column
+sparse matroids
+1: Let p be a point in the polytope PM so that the statement of Lemma 1 is
+satisfied.
+2: Let the edges of the hyper-multigraph G be oriented so that the statement
+of Lemma 4 is satisfied.
+3: For every edge ei ∈ E, i ∈ [n], mark the edge ei as “discarded" independently
+at random with probability 1 −
+1
+2k.
+4: Select a cut S ⊆ [d] uniformly at random, mark all edges not in [S; S] as
+“discarded”. Here, [S; S] stands for the set of edges which are oriented such
+that all their tails are in S and their head is in S. In particular, for t ∈ E we
+say that et lies in a cut [S; S] with respect to the orientation ϕ if ϕ(et) ∈ S
+and for every i ∈ et \ {ϕ(et)} we have i ∈ S.
+5: Set thresholds {Tt : t ∈ E} as follows, for each t ∈ E
+Tt :=
+�
++∞
+if t is “discarded”
+F −1
+t
+(1 − pt)
+otherwise .
+Let J indicate the event that "et is not marked as “discarded” after the selection
+of the cut". Then for each j ∈ E \{t} we have P[ej is not “discarded” and Xj ≥
+Tj | J] ≤
+1
+2kpj. By Lemma 4, we have �
+j∈E:ej∈δ−(ϕ(t)) pj ⩽ k, leading to the
+desired inequality.
+Note that the proof of Lemma 5 is analogous to the proof of Lemma 3. We are
+ready to prove Theorem by showing that the Algorithm 2 is a 2k+2k-competitive
+for k-column sparse matroids.
+Proof of Theorem 4. For every item t ∈ E we have P[Xt ≥ Tt] = pt and
+P[t is not “discarded”] ≥
+1
+2k+1k. By Lemma 5, we have that with probability
+at least 1/2 the item t is selected when it is not “discarded” and Xt ≥ Tt. Thus
+the expected total value of Algorithm 2 is at least �
+j∈E
+1
+2k+2kpjtj which is at
+least
+1
+2k+2kEPROPHM by Lemma 1.
+5
+Cographic and gamma-sparse matroids
+5.1
+Cographic matroids
+Let us revisit a mechanism of Soto [Sot13] for the cographic matroid secretary
+problem which is based on the following corollary of Edmond’s matroid parti-
+tioning theorem [Edm65]. This mechanism leads to a non-adaptive mechanism
+for cographic matroids.
+15
+
+Proposition 1. Let G = (V, E) be a three edge-connected graph. Then there
+exist spanning trees H1, H2, H3 in G such that the union of their complements
+contains all the edges E, i.e. E = (E \ H1) ∪ (E \ H2) ∪ (E \ H3).
+Algorithm 3 A non-adaptive 3-competitive mechanisms for cographic matroids
+in the case of three edge-connectivity
+1: Let H1, H2 and H3 be the spanning trees as in Proposition 6.
+2: Uniformly at random select a spanning tree H∗ from H1, H2 and H3. Set
+thresholds {Te : e ∈ E} as follows, for each e ∈ E
+Te :=
+�
++∞
+if e is not in H∗
+0
+otherwise .
+Lemma 6. Let G = (V, E) be a three edge-connected graph and let M be the
+cographic matroid over G. Then Algorithm 3 is a 3-competitive non-adaptive
+mechanism for the matroid M.
+Proof. The expected total value of the mechanism provided by Algorithm 3
+equals E[�
+e∈E\H∗ Xe] which can be estimated as follows
+E[
+�
+e∈E\H∗
+Xe] = 1
+3E[
+�
+i∈[3]
+�
+e∈E\Hi
+Xe] ≥ 1
+3E[
+�
+e∈E
+Xe] ≥ 1
+3EPROPHM .
+The next theorem provides a proof for Theorem 5.
+Theorem 11. Let G = (V, E) be a graph and let M be the cographic matroid
+over G. Then Algorithm 4 is a 6-competitive non-adaptive mechanism for the
+matroid M.
+Proof. We can assume that G does not have bridges, because every such bridge
+is a loop in M. Thus these edges can be selected neither by the gambler nor by
+the prophet. So we can assume G = G′ and M = M ′.
+In the case when each connected component of G is three edge-connected,
+then Algorithm 4 runs Algorithm 3 for each component to obtain a 3-competitive
+non-adaptive mechanism.
+Otherwise, there is one or more pairs of edges e,e′ such that {e, e′} corre-
+sponds to a cut in G. In this case, the edges e,e′ correspond to parallel elements
+of the cographic matroid M.
+Algorithm 4 considers the partition of E into classes of parallel elements C1,
+C2, . . . , Ck. Let us construct the matroid M ′′ from M by contracting all but
+one edge in each class C1, C2, . . . , Ck. Note, that the ground set of M ′′ has k
+elements. Abusing the notation we refer to these elements of the ground set as
+16
+
+Algorithm 4 A non-adaptive 6-competitive mechanisms for cographic matroids
+1: Delete all loops of M to obtain a matroid M ′. Remove all bridges from
+G = (V, E) and obtain a graph G′ = (V ′, E′).
+2: Let C1,. . . , Ck be equivalence classes of M ′ with respect to the relation of
+being parallel. Construct the matroid M ′′ from M ′ by contracting all but
+one edge in each class C1, C2, . . . , Ck. Note, that the ground set of M ′′
+has k elements and matroid M ′′ is the cographic matroid over a graph G′′,
+where each connected component of G′′ is three edge-connected. Abusing
+the notation we refer to the elements of the ground set of M ′′ as C1, C2,
+. . . , Ck.
+3: Let H1, H2 and H3 be forests in G′′ such that the restriction of H1, H2
+and H3 to each connected component of G′′ satisfies Proposition 6 for the
+respective connected component.
+4: Uniformly at random select a forest H∗ from H1, H2 and H3.
+5: For each i ∈ [k] select thresholds T e, e ∈ Ci according to Theorem 1 when
+the gambler is allowed to accept only one item of Ci and the distributions
+of Xe, e ∈ Ci are the same as original distributions of values for e ∈ Ci.
+6: Set thresholds {Te : e ∈ E} as follows, for each e ∈ E
+Te :=
+�
+T e
+if e ∈ Ci and Ci ∈ H∗ for some i ∈ [k]
++∞
+otherwise .
+C1, C2, . . . , Ck. The matroid M ′′ is isomorphic to the cographic matroid over
+a graph G′′, where each connected component of G′′ is three edge-connected.
+Following Lemma 6, Algorithm 4 constructs forests H1, H2, H3 for the graph G′′.
+So Algorithm 4 leads us to a 6-competitive mechanism. Indeed, the prophet
+with M and with the original distributions of Xe, e ∈ E performs exactly
+as the prophet with M ′′ and with the corresponding distributions of X′′
+i :=
+maxe∈Ci Xe, i ∈ [k]. By selecting forests in Algorithm 4 the gambler acheives
+in expectation E[�
+i∈[k] X′′
+i ]/3 when all classes C1, C2, . . . , Ck are singletons.
+However, for classes that are not singletons we need to take into account an-
+other 2 approximation factor with respect to the prophet, who can achieve the
+expected value E[X′′
+i ] for each i ∈ [k], while the gambler is guaranteed in ex-
+pectation to achieve only E[X′′
+i ]/2 for each i ∈ [k].
+5.2
+Gamma-sparse matroids
+Let us also revisit a mechanism of Soto [Sot13] for γ-sparse matroids to verify
+that it directly leads to a non-adaptive mechanism.
+Theorem 12. Let M = (E, S) be a γ-sparse matroid.
+There exists a γ-
+competitive non-adaptive mechanism for M.
+Proof. First observe that the point x := 1/γ lies in the matroid polytope PM.
+17
+
+Indeed, it is non-negative and for every set S ⊆ E(M) we have x(S) = |S|/γ ⩽
+rM(S).
+Then x can be expressed as a convex combination of indicator variables
+corresponding to the independent sets of M.
+In other words, we have x =
+�
+S∈S αS1S for some α ⩾ 0, �
+S∈S αS = 1, where 1S refers to the characteristic
+vector of S.
+Now sample an independent set S in matroid M randomly with probabil-
+ity αS. Let the gambler select all items in S and let the gambler leave all the
+items not in S unselected.
+If Xe is the random variable corresponding to the weight of element e ∈
+E(M), then this mechanism results in a total expected value as follows
+�
+S∈S
+αS
+�
+e∈S
+E[Xe] =
+�
+e∈E
+(1/γ)E[Xe] = E[
+�
+e∈E
+Xe]/γ ⩾ EPROPH/γ ,
+finishing the proof.
+Observe that Proposition 1 implies that for a three edge-connected graph G,
+the cographic matroid of G is 3-sparse. Thus Lemma 6 is a corollary of Theo-
+rem 12.
+Similarly, for a planar graph G the graphic matroid is 3-sparse, leading us
+to the following corollary.
+Corollary 1. Let G is a planar graph and let M be the corresponding graphic
+matroid. There is a 3-competitive non-adaptive mechanism for M.
+6
+Representable matroids
+Many results in the theory of matroids make use of minors coming from re-
+strictions and contractions. To get access to the toolbox provided by matroid
+theory, we need to understand how prophet inequality guarantees change when
+we consider minors.
+6.1
+Preliminaries
+Lemma 7. Let M be a matroid and let matroid N be a restriction of the ma-
+troid M. If there exists an α-competitive non-adaptive mechanism on M, then
+there is an α-competitive non-adaptive mechanism for N.
+Proof. To obtain a mechanism for the matroid N, we can impose thresholds +∞
+for the items that were removed from the ground set to obtain the restriction N
+from the matroid M. The remaining items are assigned the same thresholds in
+both mechanisms.
+A similar result for contractions is harder to obtain in the case of non-
+adaptive mechanisms.
+Indeed, a straightforward approach would require us
+to impose the thresholds +∞ for the contracted items, while using the given
+18
+
+mechanism on the remaining items. Unfortunately, this would also require us
+to “change" the underlying matroid, in other words a gambler might be forced
+to reject an item even though its value is over the assigned threshold and its
+addition to the currently selected items keeps the selected set independent with
+respect to M.
+Because of this difficulty, in this work we provide a matching result for
+contractions only for matroids representable over a finite field. This result is
+sufficient for the purpose of this work.
+Lemma 8. Let M = (E, S) be a matroid representable over the field Fp for
+some p. Let T ⊆ E be a subset of the ground set such that λM(T ) ⩽ k for
+some k.
+Then there exists S ⊆ T so that every set that is independent in M |S is also
+independent in M/T and
+EPROPHM|S ⩾
+1
+pk+1 EPROPHM/T .
+Recall that T stands for the complement of T with respect to the ground set E.
+Proof. Consider the representation of the matroid M over Fp. Let φ : E → Fm
+p
+be the map describing the representation of M. Thus, for every S ⊆ E we have
+that the set φ(S) = {φ(e) ∈ Fm
+p : e ∈ S} is independent over the field Fp if and
+only if S is an independent set for the matroid M.
+Since λM(T ) ⩽ k holds, by definition of λM we have
+rM(T ) + rM(T ) − rM(E) ⩽ k .
+We have rM(R) = dim span(φ(R)) for every R ⊆ E. Thus, we have
+dim span φ(E) = dim span φ(T )+dim span φ(T )−dim
+�
+(span φ(T )) ∩ (span φ(T))
+�
+.
+and so
+dim
+�
+(span φ(T )) ∩ (span φ(T ))
+�
+⩽ k .
+Since we are working over the field Fp, the linear space L := (span φ(T )) ∩
+(span φ(T )) has at most pk vectors. Let C be the orthogonal complement of
+the linear space L in the space span φ(T ). Thus, we can represent span φ(T ) as
+L ⊕ C. For every vector v ∈ span φ(T ) we denote v orthogonal projection to L
+and C by v |L and v |C, respectively.
+For each vector a ∈ L, define the set Ta := {t ∈ T : φ(t) |L= a, φ(t) ̸= a}.
+Note that by definition for every a ∈ L we have Ta ∩ L = ∅. Now let us select
+a uniformly at random from L.
+Claim 2. Ea[EPROPHM|Ta ] ��
+1
+pk EPROPHM/T .
+Proof. To prove the desired inequality, we prove the corresponding inequality
+for any realization of item values. From now on we consider the realization of
+item values fixed and thus we prove the following inequality
+Ea[PROPHM|Ta] ≥ 1
+pk PROPHM/T
+19
+
+Let us consider the set Iopt on which the prophet achieves PROPHM/T .
+Note that the set Iopt does not contain any item e such that φ(e) is in L, because
+every such an item e is a loop in M/T . Thus, the set Iopt can be partitioned
+into sets Iopt,a, a ∈ L where Iopt,a is a subset of Ta.
+The set Iopt is independent in M/T and so Iopt is also independent in M.
+Hence the sets Iopt,a, a ∈ L are also independent in M. Thus for every a ∈ L,
+PROPHM|Ta ⩾ w(Iopt,a). Then we have
+Ea[PROPHM|Ta] ⩾
+�
+a∈L w(Iopt,a)
+|L|
+= 1
+|L|w(Iopt) ⩾ 1
+pk PROPHM/T ,
+finishing the proof of the claim.
+Let us now select a∗ ∈ L such that EPROPHM|Ta is maximized. By the
+previous claim, we have
+PROPHM|Ta∗ ⩾ 1
+pk PROPHM/T .
+Now for every c ∈ C define set Hc := {t ∈ Ta∗ : (φ(t) |C) · c = 1}. Now let us
+select c uniformly at random from C.
+Claim 3. Ec[EPROPHM|Hc ] ≥ 1
+pEPROPHM|T a∗ .
+Proof. To prove the desired inequality, we prove the corresponding inequality
+for any realization of item values. From now on we consider the realization of
+item values fixed and thus we prove the following inequality
+Ec[PROPHM|Hc ] ≥ 1
+pPROPHM|T a∗ .
+Let Iopt be the set corresponding to PROPHM|Ta∗ . Thus, we have that for
+every e ∈ Iopt, φ(e) is not in L and hence φ(e) |C is not the zero vector. Due to
+Pc[c · t = 1] = 1/p, for every t ∈ Ta∗, we have
+Ec[w(Iopt∩Hc)] =
+�
+t∈Iopt
+Pc[c·t = 1]w(t) = 1
+p
+�
+t∈Iopt
+w(t) = 1
+pw(Iopt) = PROPHM|Ta∗ .
+Finally, since Iopt is independent in M so is Iopt ∩ Hc. Thus, we have
+Ec[PROPHM|Hc ] ≥ 1
+pPROPHM|T a∗ ,
+finishing the proof of the claim.
+Now let us select c∗ so that EPROPHM|Hc is maximized and let S∗ := Hc∗.
+Then we have EPROPH(M |S∗) ⩾
+1
+pk+1 EPROPH(M/T ).
+Finally, we need to show that every set independent in M |S∗ is an indepen-
+dent set in M/T . Suppose the contrary, i.e. there exists a set that is independent
+20
+
+in M |S∗ but is not an independent set in M/T . Then span S∗ has a non-trivial
+intersection with span T, suppose x ∈ (span φ(S∗)) ∪ (span φ(T )). Let us show
+that x is a zero vector. Since x ∈ span S∗, we have x = �
+s∈S∗ αsφ(s) for some
+αs ∈ Fp, s ∈ S∗.
+Let us consider the projections of x on C and L. Since x ∈ span φ(T ) we have
+that x lies in L and so x |C is the zero vector. Thus x |C= �
+s∈S∗ αs(φ(s) |C)
+is the zero vector.
+Note that by definition, φ(s) |L= a∗ and c∗ · (φ(s) |C) = 1 hold for every s ∈
+S∗ . Thus over the field Fp we have
+�
+s∈S∗
+αs =
+�
+s∈S∗
+αs(c∗ · (φ(s) |C)) = c∗ ·
+� �
+s∈S∗
+αs(φ(s) |C)
+�
+=
+c∗ · (x |C) = 0 .
+Now let us consider x |L. We have
+x |L=
+�
+s∈S∗
+αs(φ(s) |L) =
+� �
+s∈S∗
+αs
+�
+a∗ ,
+where the last expression equals the zero vector since �
+s∈S∗ αs = 0. Thus we
+have a vector x ∈ L ⊕ C such that both projections x |L and x |C are the zero
+vector. Hence, the vector x is the zero vector, finishing the proof.
+6.2
+Tree Decompositions
+Similarly to the approach [HN20] for the matroid secretary problem, we exten-
+sively use the tree decomposition of matroids. A tree decomposition of bounded
+thickness allows us to construct non-adaptive mechanisms with good approxi-
+mation ratios. Before proceeding with these constructions, let us introduce tree
+decompositions.
+A tree decomposition of a matroid M = (E, S) is a pair (T, X) where T is
+a tree and X = {Xv ⊆ E : v ∈ V (T )}, where sets in X form a partition of
+E. Here, we refer to the vertex and edge sets of the tree T as V (T ) and E(T ),
+respectively.
+Given an edge e = {v1, v2} ∈ E(T ) of the tree T , let T1 and T2 be two
+connected components of T − e, in other word the removal of the edge e from
+T leads to two connected components T1 and T2. The thickness of the edge
+e = (v1, v2) is denoted as λ(e) and is defined as follows
+λ(e) := λM(∪v∈V (T1)Xv) .
+The thickness of the tree decomposition is the maximum thickness of the edge e
+in E(T ).
+Let M be a family of matroids, M be a matroid and (T, X) be a tree decom-
+position of M. We say that tree decomposition (T, X) is M-tree decomposition
+21
+
+if M |clM(Xv)∈ M holds for every v ∈ V (T ). Let tk(M) be a set of matroids
+which have M-tree decomposition of thickness at most k.
+Theorem 13. Let Mα,p be the family of matroids which admit α-competitive
+non-adaptive mechanisms and are representable over the finite field Fp. Then
+for every natural number k and every matroid M in tk(Mα,p), the matroid M
+has an (αpk+1)-competitive non-adaptive mechanism.
+Proof. For a natural number m, let tk,m(Mα,p) be the set of matroids which
+have an Mα,p-tree decomposition (T, X) of thickness at most k satisfying |V (T )| =
+m.
+Let us prove the statement of the lemma by induction on m.
+The base
+case follows from the definition of the family Mα,p and the fact that Mα,p =
+tk,1(Mα,p).
+Let us now show how to do the inductive step. Let us assume m ≥ 2 and
+consider a matroid M = (E, S) in tk,m(Mα,p) with its Mα,p-tree decomposition
+(T, X) of thickness at most k and with |V (T )| = m. Let ℓ be a leaf of the tree T
+and let u be the neighbour of the vertex ℓ in the tree T .
+Observe that the tree (V (T ) \ {ℓ}, E(T )\ {ℓu}) together with the subfamily
+{Xw : w ∈ V (T ) \ {ℓ}} defines an Mα,p-tree decomposition of the matroid
+M \ Xℓ. Thus the matroid M \ Xℓ is in M ∈ tk,m−1(Mα,p). Hence, by the
+inductive hypothesis there are thresholds T ′
+e, e ∈ E \ Xℓ guaranteeing αpk+1-
+competitiveness of the gambler in comparison to the prophet on the matroid M \
+Xℓ.
+Claim 4. There are thresholds T ′′
+e , e ∈ Xℓ leading to an (α · pk+1)-competitive
+non-adaptive mechanism for matroid M |Xℓ, such that the gambler always selects
+a set that is independent in M/Xℓ.
+Proof. By Lemma 8 there exists a set S ⊆ Xl such that every set independent
+in M |S is also independent in the matroid M/Xℓ and
+EPROPHM|S ⩾
+1
+pk+1 EPROPHM/Xℓ .
+By definition of Mα,p and the appearance of Xℓ in the tree decomposition, we
+have that M |Xℓ is in the family Mα,p. By Lemma 7, since S is a subset of Xℓ
+the matroid M |S is also in the family Mα,p. Thus, there are thresholds T ′′
+e ,
+e ∈ S that lead to an α-competitive non-adaptive mechanism on M |S. The
+thresholds T ′′
+e , e ∈ Xℓ \ S can be defined as +∞, finishing the proof of the
+claim.
+Now we can define thresholds Te, e ∈ E for all elements of the matroid M
+as follows
+Te :=
+�
+T ′
+e
+if e ̸∈ Xℓ
+T ′′
+e
+otherwise.
+Let us now demonstrate that such thresholds Te, e ∈ E lead to an (αpk+1)-
+competitive non-adaptive mechanism for M.
+22
+
+First, by the above claim the selected items from Xℓ always form an inde-
+pendent set in M/Xℓ when used with the thresholds Te, e ∈ Xℓ on the matroid
+M |Xℓ. Thus the definition of the thresholds guarantees that in expectation the
+value of selected items from Xℓ is at least EPROPHM|Xℓ/(αpk+1); and in ex-
+pectation the value of selected items from E\Xℓ is at least EPROPHM\Xℓ/(αpk+1).
+To finish the proof, note that we have
+PROPHM|Xℓ + PROPHM\Xℓ ≥ PROPHM
+and so
+EPROPHM|Xℓ + EPROPHM\Xℓ ≥ EPROPHM .
+6.3
+Regular matroids
+In this section, we prove Theorem 7. Before we proceed to the proof, let us
+define key notions related to regular matroids.
+A subset of the matroid’s ground set is called a circuit, if it is an inclusion-
+minimal dependent set. A cycle is a subset of the ground set which can be
+partitioned into a disjoint union of circuits.
+Let M1 = (E1, S1), M2 = (E2, S2) be two binary matroids. Then the matroid
+sum M1△M2 has the ground set E1△E2 and the cycles of M1△M2 are all sets
+of the form C1△C2, where C1 is a cycle for M1 and C2 is a cycle for M2.
+Definition 3. Consider two binary matroids M1 = (E1, S1), M2 = (E2, S2)
+and M = M1△M2.
+1. If |E1 ∩ E2| = 0, and E1 ̸= ∅, E2 ̸= ∅, M is called a 1-sum of M1 and
+M2.
+2. If |E1 ∩ E2| = 1, |E1| ≥ 3, |E2| ≥ 3 and E1 ∩ E2 is not a loop of M1 or
+M2 or their dual matroids, M is called a 2-sum of M1 and M2.
+3. If |E1 ∩ E2| = 3, |E1| ≥ 7, |E2| ≥ 7 and E1 ∩ E2 is a circuit in both M1
+and M2, and E1 ∩ E2 does not contain a circuit in their dual matroids,
+then M is called a 3-sum of M1 and M2.
+Proof of Theorem 7. By Seymour’s regular matroid decomposition theorem [Sey80],
+every regular matroid M can be obtained from graphic, cographic or a special
+matroid R10 through a sequence of 1-sums, 2-sums or 3-sums.
+This gives a tree decomposition (T, X) of thickness at most 2 so that each M |Xv,
+v ∈ V (T ) is either a graphic, cographic or a special matroid R10.
+By performing parallel extensions of the elements to be deleted before each
+2-sum and 3-sum, we construct a matroid M ′, so that M is a restriction of M ′
+and M ′ has a tree decomposition (T, X ′) so that each M ′ |clM′ (X′v), v ∈ V (T )
+is either graphic, cographic or a parallel extension of R10.
+23
+
+By Theorem 2, every graphic matroid has a 32-competitive non-adaptive
+mechanism. By Theorem 5, every cographic matroid has a 6-competitive non-
+adaptive mechanism. Since matroid R10 has ground set of size 10, by Theorem 1
+every parallel extension of R10 has a 20-competitive non-adaptive mechanism.
+Note that by definition every regular matroid is representable over finite
+field F2. Thus, by Theorem 13 with p = 2, k = 2 and α = 32 there is a 256-
+competitive non-adaptive mechanism for matroid M ′. Since M is a restriction
+of M ′, by Lemma 7, there is a 256-competitive non-adaptive mechanism for M,
+finishing the proof.
+6.4
+Minor-closed representable matroid families
+In this section we show that every minor-clossed subclass of matroids repre-
+sentable over Fp has a constant-competitive non-adaptive mechanism, where
+the constant is a function only of p. The proof of this fact is analogous to the
+proof in [HN20].
+Theorem 14 (Theorem 4.3 in [Gee11]). Given natural numbers q ⩾ 2 and
+n ⩾ 1, let M = (E, S) be a matroid with no U2,q+2 or M(Kn) minors. Then
+we have |E| ≤ qq3nrM(E).
+Corollary 2. Given natural numbers q ⩾ 2 and n ⩾ 1, let M = (E, S) be a
+matroid with no U2,q+2 or M(Kn) minors. Then there exists a qq3n-competitive
+non-adaptive mechanism for M.
+Proof. If M has no U2,q+2 or M(Kn) minors, then every restriction of M also
+has no U2,q+2 or M(Kn) minors.
+Thus for every X ⊆ E we have |X| ⩽
+qq3nrM(X). So, M is a qq3n-sparse matroid and by Theorem 12 there exists
+a qq3n-competitive non-adaptive mechanism for M.
+6.4.1
+Projections and lifts
+Let M be a matroid and x be an element of the ground set, which is a not a
+loop and not a free element of the matroid M. Then M/x is called a projection
+of M \ x; M \ x is called a lift of M/x. Note that here and later we write M/x
+and M \ x instead of M/{x} and M \ {x}, repsectively.
+Let M and N be two matroids with the same ground set. We say that the
+distance between M and N is t, denoted by dist(M, N) = t if t is the smallest
+integer such that there exists a sequence of matroids P0, P1, . . . , Pt where
+P0 = M and Pt = N and for every i ∈ [t] the matroid Pi is either a lift or a
+projection of Pi−1.
+Lemma 9. Let N be a lift of the matroid M. If there is an α-competitive non-
+adaptive mechanism for M then there exists a (2α+2)-competitive non-adaptive
+mechanism for N.
+24
+
+Proof. Since N is a lift of M, there exists a matroid L = (E, S) and an element
+x of its ground set, such that M = L/x, N = L\x. Here, x is not a loop and
+not a free element of L.
+Let P be the set of elements in L that are parallel to x, in other words P :=
+{x′ ∈ E : x′ ∥ x}. Note that N |P \{x} is a uniform matroid of rank 1. Note also
+that elements in P \{x} are loops in M and so EPROPHM = EPROPHM\P .
+Let T ′
+e, e ∈ E \ {x} be the thresholds imposed by an α-competitive non-
+adaptive mechanism for the matroid M.
+Let T ′′
+e , e ∈ P be the thresholds
+guaranteeing 2-competitive non-adaptive mechanism as in Theorem 1 for the
+uniform matroid of rank 1 on the ground set P \{x}; and let T ′′
+e , e ∈ E\(P ∪{x})
+be +∞. We select one of these two sets of thresholds for the matroid N as
+described below. The constructed mechanism for the matroid N selects one
+of those two sets at random, where first set of thresholds T ′
+e, e ∈ E \ {x} is
+selected with probability γ := α/(α + 1) and the second set T ′′
+e , e ∈ E \ {x}
+with probability 1 − γ = 1/(α + 1).
+Next part is dedicated to the analysis of how thresholds T ′
+e, e ∈ E \ {x}
+perform on the matroid N.
+Note, that these thresholds are coming from a
+mechanism for the matroid M, while they are used for the matroid N with
+probability γ. We show that the total expected value achieved by thresholds
+T ′
+e, e ∈ E \ {x} on N is at least the total expected value achieved by these
+thresholds on M. For this we can assume that for every realization of item
+values, the orders of items in matroid N and M are the same. To see that
+this assumption is valid, we can assume that the order for N is chosen in an
+adversarial way and is used also as the items order for M.
+Claim 5. Let us assume that the items order for M and N is the same for
+a given realization of item values. Let us also assume that for every item e ∈
+E \ {x} the threshold T ′
+e is used. Then the gambler with matroid N selects all
+items that the gambler with matroid M selects.
+Proof. We fix the item values realization and items order. Let e1, e2,. . . , ek be
+the items with their values being at least their threshold and with the corre-
+sponding order.
+Now we need to show that if the gambler with matroid N selects items
+greedily from e1, e2,. . . , ek starting from e1, then the set of selected items is a
+superset of the items greedily selected by the gambler with matroid M. If both
+gamblers end up selecting exactly the same set of items, then proof of the claim
+is complete. Otherwise consider the first index i ∈ [k] such that the item ei is
+selected by exactly one of the two gamblers. Since N = L\x and M = L/x we
+have that it is only possible if ei is selected by the gambler with the matroid N
+and rejected by the gambler with the matroid M.
+Now we claim that every subsequent item, in other words an item in ei+1, . . . ,
+ek, is either selected by both gamblers or rejected by both gamblers. Suppose
+the contrary and consider the first item ej, i + 1 ≤ j ≤ k that is selected by
+one gambler and rejected by another gambler. Let S := {e1, e2, . . . , ej−1} and
+let T be the set of items selected by the gambler with M from the set S. Thus
+25
+
+the gambler with N selected T ∪ {ei} from the set S. So T ∪ {ei} is a basis of
+(L\x) |S and T is a basis of (L/x) |S. Thus, both T ∪{ei} and T ∪{x} are bases
+of L |S. If only one of the two gamblers accepts the item sj then the matroid
+L |S∪{sj} has two bases of different cardinality, attaining a contradiction and
+finishing the proof.
+Thus we have that the thresholds T ′
+e, e ∈ E\{x} guarantee at least EPROPHM
+as the expected total value of the gambler with N. To prove that the constructed
+mechanism is 1/(2α + 2)-competitive it is enough to show the following claim.
+Note that in our construction we used α-competitive non-adaptive mechanism
+for the matroid M and 2-competitive non-adaptive mechanism for the uniform
+matroid of rank 1 on P \ {x}.
+Claim 6. γ 1
+αEPROPHM + (1 − γ) 1
+2EPROPHP \{x} ⩾
+1
+2α+2EPROPHN
+Proof. Let us consider the inclusion-maximal set Iopt on which the prophet
+achieves PROPHN. Let Copt be a random variable corresponding to the unique
+circuit of Iopt ∪ {x} in L. Recall that x is not a free element of L so such a
+circuit exists and is unique and contains x.
+First consider the events when |Copt| ⩾ 3.
+Note that by definition of a
+circuit, for every y ∈ Copt \ {x} the set (Iopt ∪ {x}) \ {y} is independent in
+L. Hence, for every y ∈ Copt \ {x} the set Iopt \ {y} is independent in M. So
+we have that conditioned on |Copt| ⩾ 3 we have PROPHM ≥ w(Iopt \ {y})
+for every y ∈ Copt \ {x}.
+Let yopt be the random variable representing the
+element in Copt \{x} of smallest value. Then conditioned on |Copt| ⩾ 3, we have
+w(Copt \ {yopt, x}) ≥ w(C \ {x})/2. Thus, conditioned on |Copt| ⩾ 3 we have
+PROPHM ⩾ w(Iopt \ {yopt}) = w(Iopt \ Copt) + w(Copt \ {yopt})
+≥ w(Iopt \ Copt) + 1
+2w(Copt \ {x}) ≥ 1
+2w(Iopt) = 1
+2PROPHN .
+Second consider the event that |Copt| < 3. Since x is not a loop of L by
+definition, we have |Copt| = 2 and so Copt = {x, xopt} for some random variable
+element xopt ∈ P \{x}. For the event |Copt| ≥ 3 let us define the random variable
+element xopt to be an arbitrary element in Copt \ {x}. Thus, if |Copt| < 3 we
+have PROPHP \{x} ≥ w(xopt). Now let us define Jopt := Iopt \ {xopt} and note
+that Jopt is independent in the matroid M. Moreover, since Iopt is the set on
+which the prophet achieves PROPHN, we have that conditioned on |Copt| < 3
+the prophet achieves PROPHM on the set Jopt.
+Combining everything together we have
+γ 1
+αEPROPHM + (1 − γ)1
+2EPROPHP \{x} =
+1
+α + 1EPROPHM +
+1
+2α + 2EPROPHP \{x} ≥
+E
+�w(xopt)
+2α + 2 + PROPHM
+α + 1
+���� |Copt| < 3
+�
+P [|Copt| < 3]
+26
+
++ E
+�PROPHM
+α + 1
+���� |Copt| ⩾ 3
+�
+P [|Copt| ⩾ 3] =
+E
+�w(xopt)
+2α + 2 + w(Iopt \ {xopt})
+α + 1
+���� |Copt| < 3
+�
+P [|Copt| < 3]
++ E
+�PROPHM
+α + 1
+���� |Copt| ⩾ 3
+�
+P [|Copt| ⩾ 3] ≥
+E
+�PROPHN
+2α + 2
+���� |Copt| < 3
+�
+P [|Copt| < 3]
++ E
+�PROPHM
+α + 1
+���� |Copt| ⩾ 3
+�
+P [|Copt| ⩾ 3] ≥
+1
+2α + 2EPROPHN .
+Lemma 10. Let N be a matroid obtained from a matroid M by a sequence of
+t projections. Let L be the set of loops in the matroid N. Let there exist an
+α-competitive non-adaptive mechanism for the matroid M. Then there exists
+a non-adaptive mechanism for N\L such that the expected total value of this
+mechanism is at least
+1
+α·3t EPROPHM\L.
+In the context of Lemma 10, every set that is independent for the matroid N\
+L is also independent for the matroid M \ L. Hence, we have EPROPHM\L ≥
+EPROPHN\L. Thus in case t = 1, Lemma 10 leads us to the following corol-
+lary.
+Corollary 3. Let N be a projection of the matroid M.
+If there is an α-
+competitive non-adaptive mechanism for M then there exists a 3α-competitive
+non-adaptive mechanism for N.
+Proof of Lemma 10. Let us prove the statement by induction.
+Of course, in
+case t = 0 we have M = N and the statement is trivially true.
+Let us now assume that t is at least 1. Let N ′ be a matroid such that N ′
+is obtained from the matroid M by a sequence of t − 1 projections and N is a
+projection of N ′. Since N is a projection of N ′ there is a matroid P = (E, S)
+and x ∈ E such that P \ x = N ′ and P/x = N. Let L′ be the set of loops in
+the matroid N ′.
+By induction hypothesis, there exist thresholds T ′
+e, e ∈ E \ (L′ ∪ {x}) such
+that the gambler with the matroid N ′\L′ achieves at least
+1
+α·3t−1 EPROPHM\L′
+as the expected total value.
+Let us assume that to compute thresholds T ′
+e,
+e ∈ E \(L′ ∪{x}) the values of items in L were set to be 0 while the distribution
+of values for other items remain the same. Since L′ ⊆ L, analogously to Lemma 7
+we can define thresholds
+T ′′
+e :=
+�
++∞
+if e ∈ L
+T ′
+e
+otherwise
+27
+
+such that the gambler with the matroid N ′\L achieves at least
+1
+α·3t−1 EPROPHM\L
+as the expected total value. Let T ′′′
+e , e ∈ E \(L∪{x}) be the thresholds guaran-
+teeing 2-competitive non-adaptive mechanism as in Theorem 1 for the uniform
+matroid of rank 1 on the ground set E \ (L ∪ {x}).
+The constructed mechanism for the matroid N\L selects one of two threshohold
+sets at random, where first set of thresholds T ′′
+e , e ∈ E \ (L ∪ {x}) is selected
+with probability 1/3 and the thresholds T ′′′
+e , e ∈ E \ (L ∪ {x}) with probability
+2/3. Note that the thresholds T ′′
+e , e ∈ E \ (L ∪ {x}) were designed for the
+matroid N ′ \ L but are used for the matroid N \ L; hence less items might be
+selected than when it is used for N ′ \ L. Also note, that the thresholds T ′′′
+e ,
+e ∈ E \ (L ∪ {x}) are used for N \ L but were designed for the uniform matroid
+of rank 1.
+For the analysis, let Ialg be the random variable indicating the items set
+selected by the gambler with matroid N ′ \ L when the thresholds T ′′
+e , e ∈
+E \ (L ∪ {x}) are used. Analogously to a claim in the proof of Lemma 9, we can
+assume that when the thresholds T ′′
+e , e ∈ E\(L∪{x}) are used the gambler with
+N \L select all items in Ialg with an exception for possibly one item. Let xopt be
+the random variable indicating the element of maximum value in E \ (L ∪ {x}).
+To finish the proof it is enough to show the following inequality
+1
+3E[w(Ialg) − w(xopt)] + 2
+3
+1
+2E[w(xopt)] ≥
+1
+α · 3t EPROPHM\L .
+To obtain this inequality we can do estimations as follows
+1
+3E[w(Ialg)−w(xopt)]+2
+3
+1
+2E[w(xopt)] = 1
+3E[w(Ialg)] ≥ 1
+3
+1
+α · 3t−1 EPROPHM\L .
+Now let us combine Corollary 3 and Lemma 9.
+Lemma 11. Let M and N be matroids such that dist(M, N) ≤ t. If there exists
+an α-competitive non-adaptive mechanism for the matroid M with α ≥ 2 then
+there exists a 3tα-competitive non-adaptive mechanism for the matroid N.
+Proof. Note that for α ≥ 2 we have 3α ≥ 2α + 2. Since N can be obtained
+from M by a sequence of t projection and lift steps, we can use Corollary 3 or
+Lemma 9 for each of these steps to obtain the desired competitiveness ratio.
+6.4.2
+Minor-closed families theorem
+Lemma 12 (Lemma 6 in [HN20]). Let p and n be integers such that p ⩽ n − 2
+and p is prime. The matroid U2,n is not representable over the field Fp.
+The following Structural Hypothesis is due to Geelen, Gerards and Whittle.
+The proof of this Structural Hypothesis has not appeared in print.
+28
+
+Hypothesis 1. Let p be a prime number and M is a proper minor-closed class
+of matroids representable over Fp.
+Then there exist k, n, t such that every M ∈ M is a restriction of an Fp-
+representable matroid M ′ having a full tree-decomposition (T, X) of thickness at
+most k so that for every v ∈ V (T ) if M ′ |clM′(Xv) has a M(Kn) minor, then
+there exists a 2-column sparse matroid N with dist(M ′ |clM′ (Xv), N) ⩽ t.
+Proof of Theorem 8. Let k, n, t are as stated in the Structural Hypothesis 1 on
+M.
+Let M1 be the set of matroids on distance t or less from some 2-column
+sparse matroid and are representable over Fp.
+By Theorem 4 all 2-column
+sparse matroids have a 32-competitive non-adaptive mechanism. By Lemma 11
+there exists a (3t · 32)-competitive mechanism for matroids in M1.
+Let M2 be the set of matroids without M(Kn) minor and are representable
+over Fp. By Lemma 12 all matroids in M2 do not have U2,p+2 as a minor.
+Then by Corollary 2, we have that there is a pp3n-competitive non-adaptive
+mechanism for every matroid in M2.
+By the Structural Hypothesis 1 we have that every M ∈ M is a restriction
+of some M ′ with a full tree-decomposition (T, X) of thickness at most k so that
+for every v ∈ V (T ) M ′ |clM′ (Xv)∈ M1 ∪ M2.
+Thus by Theorem 13, matroid M ′ has a γ := (max(3t · 32, pp3n) · pk+1)-
+competitive non-adaptive mechanism. By Lemma 7 the matroid M has also a
+γ-competitive non-adaptive mechanism.
+29
+
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+1
+RISs and Sidelink Communications in Smart Cities:
+The Key to Seamless Localization and Sensing
+Hui Chen, Member, IEEE, Hyowon Kim, Member, IEEE, Mustafa Ammous, Student Member, IEEE,
+Gonzalo Seco-Granados, Senior Member, IEEE, George C. Alexandropoulos, Senior Member, IEEE,
+Shahrokh Valaee, Fellow, IEEE, and Henk Wymeersch, Senior Member, IEEE
+Abstract—A smart city involves, among other elements, intelli-
+gent transportation, crowd monitoring, and digital twins, each of
+which requires information exchange via wireless communication
+links and localization of connected devices and passive objects
+(including people). Although localization and sensing (L&S) are
+envisioned as core functions of future communication systems,
+they have inherently different demands in terms of infrastructure
+compared to communications. Wireless communications gener-
+ally requires a connection to only a single access point (AP),
+while L&S demand simultaneous line-of-sight propagation paths
+to several APs, which serve as location and orientation anchors.
+Hence, a smart city deployment optimized for communication
+will be insufficient to meet stringent L&S requirements. In this
+article, we argue that the emerging technologies of reconfigurable
+intelligent surfaces (RISs) and sidelink communications constitute
+the key to providing ubiquitous coverage for L&S in smart cities
+with low-cost and energy-efficient technical solutions. To this end,
+we propose and evaluate AP-coordinated and self-coordinated
+RIS-enabled L&S architectures and detail three groups of appli-
+cation scenarios, relying on low-complexity beacons, cooperative
+localization, and full-duplex transceivers. A list of practical issues
+and consequent open research challenges of the proposed L&S
+systems is also provided.
+Index Terms—Smart cities, reconfigurable intelligent surfaces,
+localization, sensing, sidelink communication.
+I. INTRODUCTION
+The emerging concept of smart cities aims to improve ac-
+cessibility to public services, advance digitization of the urban
+environment, and monitor various human-oriented processes
+as well as assets, by harmonizing diverse digital technologies
+at a city level [1]. This broad concept integrates various
+independent applications to bring improvements both at the
+societal level (such as smart homes, smart transportation, sup-
+ply chains, and environment monitoring), and at the individual
+level (such as indoor navigation and extended reality (XR)). To
+realize the concept of smart cities, reliable, low-latency, and
+high-speed communication systems (to support information
+exchange and management among interconnected devices), as
+well as accurate localization and sensing (L&S) (to support
+communication and provide situation-awareness services), are
+of great importance. In this article, we use the term localization
+to indicate the position (and possibly orientation) estimation of
+a target user equipment (UE), and the term sensing to specify
+the position estimation of passive objects (i.e., objects without
+networking infrastructure or non-cooperating ones).
+By exploiting the large antenna array sizes and wide
+bandwidth of millimeter-wave/THz systems, recent research
+activities in industry and academia on integrated sensing,
+localization, and communication (ISLAC) are growing. ISLAC
+is able to utilize communication infrastructures and signals
+to enable synergies with L&S for diverse applications. To
+this end, several standardization efforts and 3GPP activities
+have been recently studied, such as the definition of new
+radio positioning requirements, evaluation methodologies, and
+techniques (dependent on radio access technology and not), in
+TR 38.855 [2] as well as the development of Wi-Fi sensing
+technology (in both sub-7 GHz and mmWave spectrum), in
+the IEEE 802.11bf standard [3].
+While high angular and delay resolution (due to large
+arrays and wide signal bandwidths) facilitate L&S tasks,
+signal coverage is one of the major challenges, especially
+for high-frequency systems which suffer from high path loss
+and high blockage probability. In contrast to communications,
+which is possible to work with only one point-to-point link,
+L&S functions necessitate access to multiple access points
+(APs). Reflective reconfigurable intelligent surfaces (RISs)
+constitute a promising emerging technology for extending
+coverage and dynamically programming signal propagation
+with almost zero-energy consumption [4], thus, supporting,
+or even enabling in certain cases, wireless communications,
+as well as L&S tasks in various scenarios [5]. However, since
+they are incapable of generating their own signals and only
+modify the analog waveforms impinging on them, separate
+signal generation sources are needed.
+The sources generating signals for RIS-assisted L&S can
+be: i) APs with full communication capabilities, ii) low-
+complexity beacons that broadcast or receive L&S reference
+signals; or iii) the UEs themselves that are involved in the
+L&S tasks. The latter two cases are particularly relevant when
+power-hungry APs, whose deployment is usually costly, needs
+to be avoided. To fulfill the ubiquitous L&S requirements in
+out-of-coverage areas or partially-covered ones (e.g., indoor
+UEs and vehicles in tunnels), sidelink communications via
+the PC5 interface can be particularly useful [6], as mentioned
+in the 3GPP TR 38.845 report [7]. In addition, cooperative
+localization between several UEs can be employed to enhance
+or enable L&S. Furthermore, when a UE is equipped with
+a full-duplex transceiver [8] —a technology that is widely
+discussed in ISLAC for sensing purposes—, it can both
+simultaneously transmit and receive the signal to localize itself
+with a single RIS anchor.
+In this article, we argue that RIS technology in conjunction
+with sidelink communications can provide seamless L&S so-
+lutions, speeding up the intelligent transformation of cities into
+arXiv:2301.03535v1  [eess.SP]  9 Jan 2023
+
+2
+Beacon-Assisted Localization
+Beacon
+Self-L&S with 
+a Full-Duplex 
+Transceiver
+Beacon-UE Channel
+RIS Anchor Channel
+RIS Localization Channel
+Sidelink Channel
+Sensing Channel
+Scenarios
+Cooperative Localization
+RIS
+A2
+A1
+B1
+B2
+C1
+C2
+C3
+RIS-attached 
+Vehicle
+AP
+A3
+B3
+Figure elements by macrovector on Freepik
+Fig. 1. RIS-enabled seamless L&S scenarios in smart cities: a) beacon-assisted localization, b) cooperative localization, and c) self-L&S with a full-duplex
+transceiver.
+smart entities. We particularly elaborate on the representative
+RIS-enabled L&S scenarios illustrated in Figure 1: beacon-
+assisted localization, cooperative localization, and self-L&S
+with a full-duplex transceiver, describing both AP-coordinated
+and AP-free architectures for different coverage scenarios. We
+discuss the open challenges with the proposed green (low-
+cost and energy-efficient) L&S systems in smart cities and list
+potential directions for future research.
+II. L&S ARCHITECTURES AND PROTOCOLS
+In this section, we describe the different entity types of the
+proposed L&S system for smart cities, relying on RISs and
+sidelink communications, as well as its enabling architectures,
+depending on whether an AP is present for L&S coordination.
+A. Entity Types and Overall Architecture
+In the proposed RIS-enabled L&S system, there are several
+types of entities: APs, beacons, RISs, and UEs, as shown in
+Figure 2. The APs (e.g., a gNB macro base station) provide
+cellular services to the devices in coverage. A beacon could be
+a roadside unit (RSU) that is capable of sending and receiving
+L&S reference signals via sidelink communications. In this
+way, the explicit involvement of expensive and power-hungry
+APs with full communication protocols is not needed for L&S
+purposes. RISs serve as reference anchors to assist ISLAC
+tasks, and are controlled by a dedicated entity responsible
+for realizing communication functions. Finally, UEs may have
+different hardware capabilities (e.g., single/multiple antennas,
+half-/full-duplex) that play different roles in L&S (e.g., a target
+UE to be localized, an assistant UE with known or measured
+location to assist L&S, or a server/coordinator in performing
+L&S tasks). Although all the devices involved in the targeted
+tasks will have sidelink communication capabilities, beacons
+usually have fewer power constraints than UEs, while the
+controllers of reflective RISs are expected to work in low-
+power mode without sending or processing L&S reference
+signals [4]. We further consider that RISs coordinate their
+reflective beamforming, either using time division or phase
+profile codes in the time domain.
+For the scenarios considered in this article, we propose two
+different architectures: one based on AP coordination and the
+other on self-coordination. The former architecture works for
+UEs (and other L&S-related devices) inside a coverage area or
+in partial-covered areas (where sidelink is available), requiring
+a specific AP to serve as a L&S coordinator, e.g., interact
+with the location management function (LMF) via the NR
+positioning protocol A (NRPPa) [9]. The self-coordinated ar-
+chitecture relies explicitly on sidelink communications, being
+particularly suitable for UEs in the out-of-coverage of APs, or
+for UEs connected to APs that cannot meet the latency and
+spatial resolution requirements (e.g., legacy 3G/4G APs). In
+both architectures, L&S signals can be generated by a beacon,
+an assistant UE, or a target UE, depending on the network
+topologies and the specific application scenario.
+
+3
+Beacon-UE Channel
+Sidelink Channel
+RIS Channel
+AP-Coordinated Control Link
+Self-Coordinated Control Link
+L&S Coordinator
+Controller
+RIS-1
+In-coverage
+Partial coverage
+Out-of-coverage
+AP
+Beacon
+UE-1
+UE-2
+UE-3
+UE-4
+UE-5
+RIS-2
+RIS-3
+Fig. 2. UEs performing L&S in different coverage areas. The AP-coordinated
+architecture can be used for UEs located in in-coverage or partial-coverage
+(sidelink communications required) areas. When the UEs and all the L&S
+devices cannot access any surrounding APs (i.e., lying in out-of-coverage
+areas), the self-coordinated architecture is the only option for L&S services.
+B. AP-Coordinated Architecture
+This architecture relies on one or several APs to allocate
+the available radio resources and ensure timing among the
+connected devices, which are the UEs, the RIS controllers, and
+dedicated beacon nodes. This network management scheme is
+similar to the mode-1 in sidelink communications [6], and the
+L&S protocols can be summarized into the following 6 steps:
+1) The target UE triggers an L&S request to the AP (which
+is selected as the task coordinator).
+2) The coordinator exchanges related location information
+with the core network (e.g., via the NRPPa), determines
+L&S configurations (e.g., broadcasting, sidelink, and
+RIS phase profiles), as well as selects and notifies nearby
+beacons, UEs, and RISs that are involved in the L&S
+task.
+3) The coordinator allocates time-frequency resources for
+L&S to all involved devices and triggers the RIS con-
+troller(s) to configure their phase profiles for the whole
+duration of the estimation process.
+4) The beacons and/or UEs transmit L&S reference signals,
+which are reflected by the involved RIS(s) and received
+by the target UE.
+5) The collected measurements are used by the target UE
+for the localization, and/or sensing tasks. Alternatively,
+this computation can be offloaded at the localization
+server (e.g., the coordinating AP or another UE with
+high computational power).
+6) The target UE updates the task coordinator with the
+estimated L&S results; this optional step can serve as
+prior information for future use.
+For the partial-coverage scenarios, the out-of-coverage UEs
+need to establish sidelink communications with devices that
+are covered by APs. Then, the L&S tasks can be performed
+similarly to the aforementioned steps.
+C. Self-Coordinated Architecture
+An AP-free architecture is required for cases where the
+devices involved in L&S tasks are located in out-of-coverage
+areas. Similar to the mode-2 sidelink communications [6],
+L&S tasks can be autonomously realized by selecting a
+specific device as the localization coordinator, as follows:
+1) The target UE discovers nearby devices (e.g., beacons,
+RISs, and other UEs) and obtains their location infor-
+mation (if available).
+2) Based on the discovered neighbors, the target UE de-
+termines a L&S task coordinator (could be itself) and
+notifies it of the L&S configurations.
+3) The target UE triggers a L&S request to the coordi-
+nator and performs the same actions as with the AP-
+coordinated architecture (i.e., steps 2)–6)).
+III. RIS-ENABLED L&S SCENARIOS
+In this section, we will present three representative RIS-
+enabled L&S scenarios for smart city applications relying on
+the aforementioned architectures and protocols. We mainly
+focus on localization applications and list potential position-
+ing scenarios in systems with minimum infrastructure and
+resources. For example, if positioning can be done with a
+single antenna, it can also be achieved using a multi-antenna
+array. The same applies to narrowband (NB)/wideband (WB)
+signals, single/multiple anchors, and availability/unavailability
+of a line-of-sight (LOS) path between the UE and the active
+signal source.
+A. Beacon-Assisted Localization
+With low-complexity beacons, high flexibility in the in-
+stallation and deployment of L&S systems is feasible. A
+typical use case could be a train station with multiple low-
+cost beacons broadcasting L&S reference signals to UEs to
+navigate indoors, via the support of RISs. In this category, we
+consider UE localization (with the aid of one or more RISs)
+and RIS localization (with the aid of several beacons).
+A1) Single-RIS-Enabled UE Localization: In a WB system
+where the LOS channel is available, the delays of the LOS
+and RIS paths can be estimated based on the L&S reference
+signals sent from a beacon. Assuming the RIS and beacon
+states are known, the angle-of-departure (AOD) at the RIS
+can also be estimated. The target UE can then be localized
+by the intersection of a hyperbola (i.e., time-difference-of-
+arrival (TDOA) of the LOS and RIS paths) and the line in
+the direction of the AOD at the RIS. When the target UE
+is equipped with an antenna array, 3D orientation can also
+be estimated based on the estimated angle-of-arrivals (AOAs).
+This is the basic RIS-enabled localization scenario that only
+requires a single low-complexity beacon [10].
+A2) Multi-RIS-Enabled UE Localization: If multiple RISs
+are simultaneously available, the requirements for LOS and
+WB are unnecessary. The AODs from different RISs can be
+estimated and used to localize the UE by intersecting the AOD
+lines. In this way, localization tasks can be completed using
+NB signals, which saves bandwidth resources for communi-
+cations. Figure 3 shows the position error bound (PEB) of the
+target UE with different positions inside a 5 × 10 m2 area. As
+shown, with multiple RISs, the UE is localizable even under
+blockage of the LOS path between the beacon and the UE.
+
+4
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+x axis [m]
+0
+1
+2
+3
+4
+5
+y axis [m]
+Beacon
+RIS
+Wall
+0.001
+0.01
+0.1
+1 [m]
+Fig. 3. Scenarios A1 and A2: PEB (in meters) with different UE locations
+in a multi-RIS-aided localization scenario. The target UE can be localized
+with one RIS and a beacon-UE LOS path, or with at least 2 RISs under LOS
+blockage conditions.
+However, the localization tasks cannot be performed when
+only one anchor (beacon or RIS) is visible to the UE (see
+the yellow triangular area around the point [3, 3] m).
+A3) RIS Localization via Multi-Static Sensing: In a
+scenario where passive UEs or objects are coated with RISs,
+the localization (or sensing, depending on scenarios) can be
+performed semi-passively with only a small amount of energy
+needed for localization coordination and RIS phase profile
+control. Such localization tasks can estimate the positions (and
+orientations) of RIS-coated objects by using several beacons
+with known positions. Note that the geometrical constraints
+can largely reduce the difficulties in these scenarios. For
+example, the orientation of an RIS can be assumed as 1D
+(e.g., placed on the top of a car facing up). In addition, the
+adoption of antenna arrays at the beacons can further simplify
+the RIS localization problem.
+B. Cooperative Localization
+Sidelink communications (e.g., device-to-device (D2D)
+communications, or the vehicular version known as vehicle-
+to-everything (V2X) communications) has been introduced in
+the millimeter-wave band for information exchange between
+vehicles, opening the road for numerous use cases, such
+as platooning, collision avoidance and autonomous driving
+[6]. The combination of RISs and sidelink is expected to
+provide low-latency and high-reliability communications [11].
+Interestingly, it can also be exploited to assist L&S. Sidelink
+communications enables UEs to participate in a cooperative
+manner in the sharing of position and surrounding information
+within a local network, and in performing relative location
+estimation using sidelink signals [12]. We will focus on the
+latter case, where the absolute location can be estimated
+using RIS anchors (i.e., with known position and orientation
+information), without any APs or beacons. A typical scenario
+could be cooperative vehicular networks in urban areas with
+severe AP and GPS signal blockages.
+We next discuss single- and multi-RIS-involved localization
+scenarios, where single-antenna UEs cooperate to estimate
+their positions via WB sidelink signals. We also consider a
+more general scenario where RIS-coated objects are involved,
+resulting in cooperative RIS localization.
+B1) Single-RIS-Enabled Cooperative Localization: Con-
+sider a scenario with several UEs and one RIS anchor, where
+the UEs wish to estimate their positions. We assume that each
+UE can send sidelink signals (i.e., being the transmitter (TX))
+to other UEs, which arrive at the receiver (RX) UEs via two
+paths (i.e., the UE-UE and UE-RIS-UE paths). By proper
+control of the RIS elements, those two paths can be separated,
+and thus, two delay measurements can be obtained. Due to the
+unknown states of the UEs, both the AOD and AOA at the
+RIS for every TX-RX pair of UEs are unknown and cannot
+be directly estimated. However, we can estimate the spatial
+frequency information at the RIS for every TX-RX pair. This
+scenario requires at least three UEs to cooperate and render
+their locations feasibly without ambiguities. Figure 4 compares
+the PEBs for three UEs in an RIS-enabled versus beacon-aided
+(equipped with an antenna array) 3D cooperative localization
+scenario as a function of the number of RIS elements.
+B2) Multi-RIS-Enabled One-Way Sidelink Localization:
+In the scenario with at least two RISs, one-way sidelink
+communications is sufficient to localize both the TX and RX
+UEs. Assume that one UE takes the role of the TX and the
+other UE is the RX. With two RISs, three delay measurements
+can be obtained between the TX and RX via the LOS and
+the two RIS paths. However, that would require an optimal
+joint design of the reflection elements at both RISs to be able
+to separate the paths at the RX. In addition, once the RIS
+paths are separated, we can also estimate the spatial frequency
+information at each RIS. Thus, those collected measurements
+can be utilized to estimate the locations of the TX and RX.
+B3) Cooperative RIS Localization: Let us consider a more
+general scenario where one RIS (or several) with a known state
+is used to localize multiple UEs (with sidelink capabilities) and
+objects (coated with an RIS). This scenario is challenging due
+to the high complexity of the network and a large number of
+unknowns. However, with a proper design of all the involved
+RIS profiles and the transmission protocol, this problem can
+be decomposed into a cooperative localization (see B1) and
+an RIS localization (see A3) problems. Similar to B1, at least
+several UEs (depending on scenarios) need to take the role
+of the TX, and transmit sidelink signals to the other UEs via
+the direct and indirect paths. Once the UEs are localized, the
+RIS localization task can be solved similarly to A3, and the
+estimation results can be refined by processing all the available
+information.
+C. Self-L&S with a Full-Duplex Transceiver
+When a UE is equipped with a full-duplex transceiver (like
+radar) [8], the multi-RIS setup and cooperation between UEs
+are unnecessary. Instead, this UE can perform self-positioning
+with a single RIS and use the multipath components to map the
+environment over time; this process is known as monostatic
+simultaneous localization and mapping (SLAM). It is noted
+that SLAM is not limited to full-duplex UEs, and bistatic
+SLAM can also be performed in use cases A1-A3 and B1-
+B3.
+We next present three beacon-free L&S scenarios with a
+full-duplex UE.
+
+5
+0
+20
+40
+60
+80
+100
+120
+140
+160
+0
+0.5
+1
+1.5
+UE-1 (RIS)
+UE-1 (beacon)
+UE-2 (RIS)
+UE-2 (beacon)
+UE-3 (RIS)
+UE-3 (beacon)
+Number of RIS Elements
+PEB [m]
+Fig. 4.
+PEBs of RIS-enabled vs. beacon-aided cooperative localization for
+different RIS sizes and a fixed random phase profile setup. It is shown that,
+with a sufficient number of RIS elements, an active anchor can be replaced
+with a passive one (RIS) without performance degradation.
+C1) RIS-Enabled Self-Localization: Consider a system
+with a single-antenna UE and an RIS, where the UE transmits
+L&S reference signals and receives their back-scattered ver-
+sions, i.e., the UE-RIS-UE (controlled path) and UE-landmark-
+UE (uncontrolled path) signals. One option for the RIS phase
+profiles is to consider directional reflective beams, which can
+be efficiently designed when the UE position uncertainty (even
+under mobility cases) is available [13]. The delay and angle
+information at the RIS of the UE-RIS-UE channel can be
+estimated for this scenario, and then used to localize the UE. In
+Figure 5, beampatterns at the RIS with two different phase pro-
+files are illustrated, focusing on the UE uncertainty region. As
+demonstrated, the optimized phase profiles of [13] can provide
+sufficient beamforming gain, compared with directional phase
+profiles. This gain can offer improved L&S performance.
+C2) RIS-Enabled SLAM: If a UE is equipped with a
+full-duplex multiple-input multiple-output (MIMO) antenna
+array [8], SLAM can be enabled. Similar to scenario C1, the
+signals from different paths can be resolvable with optimized
+RIS phase profiles and precoders/combiners. The following
+channel parameters can be estimated: i) the signal propagation
+delay, the AOD at the RIS, and the AOA at the UE for the
+UE-RIS-UE channel; as well as ii) the delay and AOA at
+the UE for each UE-landmark-UE channel. In addition to the
+position information obtained from the controlled path (as in
+scenario C1), the angular resolution offered by the UE array
+can be leveraged to map/sense the environment. With multiple
+estimations, the localization and radio mapping performance
+can be improved using state-of-the-art filters (e.g., Poisson
+multi-Bernoulli filter).
+C3) RIS Localization with a Full-Duplex Array: Consider
+the more general scenario from C1 including one anchor
+RIS mounted on a wall, a UE equipped with a full-duplex
+MIMO transceiver, and several objects coated with RISs (e.g.,
+mounted on the front and rear side of a vehicle). In addition to
+the signals reflected from the anchor RIS (as also in scenario
+C1), the UE also receives single-bounce reflected signals from
+the RISs mounted on the objects. The time delay, AOA, and
+the amplitude of the channel gain for each signal path can be
+estimated, which can be used for the localization of both itself
+and the RISs-coated objects. When multiple UEs are present
+and cooperate in the estimation process, the orientation of the
+(a) Directional Phase Profiles
+(b) Optimized Phase Profiles
+Fig. 5. The reflective beamforming gain (in dB) with a 50 × 50 RIS using
+(a) a directional phase profile, and (b) an optimized phase profile via [13].
+The red squares represent the UE angular uncertainty region, which needs to
+be fully covered by an effective beampattern design.
+RISs-coated objects can also be obtained. In a scenario without
+any anchors, this RIS localization can also help in estimating
+the relative locations of the active UE and passive UEs.
+IV. OPEN RESEARCH CHALLENGES
+With the assistance of low-complexity beacons, cooperative
+localization, and full-duplex radios, the L&S coverage for
+smart city applications can be significantly extended. However,
+there exist several practical issues that need to be thoroughly
+investigated. In this section, we discuss the most critical
+challenges with the proposed RIS-enabled L&S system and
+list possible directions for future research.
+A. Anchor Deployment Optimization
+The placement of the anchors (e.g., beacons and RISs) is
+critical to meet the L&S key performance indicator (KPI) re-
+quirements within a service area (e.g., error bounds lower than
+a certain threshold, as shown in Figure 3). The deployment
+involves both the position and orientation optimization of the
+anchors, taking into account the blockage in the surrounding
+environment. RIS-aided SLAM can help in creating such an
+environment map, which can be supported by cooperative
+sidelink UEs. Heuristic optimization solutions can then be
+applied for finding optimal anchor sites, extending approaches
+from the literature [14].
+
+e.100100150evation20
+Gro40
+.M06080J60
+azllBeaiot0
+muthele100100150evation20
+G[o]40
+.=060
+B80ncertain60
+azilReaiol
+Vmuth6
+B. Resource Allocation and Coordination
+RIS-aided L&S systems involve APs, beacons, RISs, and
+UEs, making them inherently heterogeneous. Resource al-
+location for L&S tasks, including power allocation, time-
+frequency allocation, beamforming design, and scheduling
+must be carefully designed to ensure a favorable trade-off
+with conventional communication services. Depending on the
+KPI requirements of the applications that send L&S ser-
+vice requests, new objectives that consider integrated L&S
+and communications should be formulated and satisfied. An
+important part of resource allocation is RIS phase profile
+optimization and multiplexing [10]. Broad RIS beams lead to
+coverage reduction, while narrow pencil beams are sensitive to
+misalignment. Hence, highly adaptive RIS profile designs are
+needed, relying, when possible, on prior UE and object state
+information. When RISs are large, the near-field effects need
+to be taken into consideration and the beamforming designs
+should account for the curvature of arrival, resulting in beam-
+focusing designs. RIS multiplexing can be addressed by time
+multiplexing, temporal coding, and making use of high path
+loss for spatial reuse. The afore-described resource allocation
+problems can be tackled by a combination of traditional
+optimization-based methods (e.g., convex optimization) and
+learning-based methods (e.g., reinforcement learning).
+C. Estimation Algorithms
+From an algorithmic perspective, there are challenges re-
+lated to channel parameter estimation, tracking in dynamic
+environments, and calibration. Channel parameter estimation
+in the presence of severe multipath is difficult since almost
+passive reflective RISs have no local signal processing capa-
+bilities. Moreover, in cooperative localization (scenarios B1
+and B2) and RIS localization (e.g., scenarios A3, B3, and
+C3) tasks, the AOAs/AODs at the RISs are coupled, meaning
+that we can no longer estimate them independently. Instead,
+only spatial frequencies (containing coupled AOAs and AODs
+information) can be obtained, requiring novel algorithms for
+further processing. More refined channel parameter estimation
+also requires accurate channel models and the RISs’ impact
+on them, such as the near-field effect, beam squint effect, RIS
+element failures, and anchor calibration errors.
+Due to mobility, difficult conditions such as signal blockage,
+unresolvable signal paths, and severe path loss will affect L&S
+performance. Multiple RISs can be involved to handle such
+blockages, offering coverage extension. As the carrier fre-
+quency increases, the signal resolutions become higher, and the
+effect of the severe path loss can be mitigated by adopting RISs
+with larger sizes or active RISs with reflection amplification
+capabilities [15]. Sensing also suffers from inherent complica-
+tions, such as an unknown number of objects, unknown types
+of objects, unknown detection probabilities for signal paths,
+extended objects, and multi-bounce observations. Dedicated
+filters should be developed to address these complications and
+get integrated into the L&S framework.
+Finally, in terms of calibration, anchor geometry error and
+hardware impairments (HWIs) are two important aspects. We
+note that the geometrical calibration of an RIS is similar to RIS
+localization (as described in scenarios A3, B3, and C3), which
+requires a calibration agent that incorporates other sources of
+localization estimations (i.e., sensor fusion). While for HWIs,
+the channel model could be too complicated when considering
+each specific impairment (e.g., mutual coupling and phase
+noise), learning-based methods can be considered to unveil
+their impact and drive practical algorithmic designs.
+D. Understanding Anchor Hardware Alternatives
+There are also opportunities to improve L&S coverage via
+variations of the hardware deployed at the beacons, RISs, and
+UEs. On the beacon and UE sides, multi-panel arrays (i.e., 3D
+arrays) could be implemented for further coverage extension.
+On the RIS side, new types of RISs are emerging beyond
+almost passive reflective RISs [15]. As previously mentioned,
+an active RIS can be used to boost the signal energy (i.e.,
+change both the amplitude and phase of the incident signal) for
+improved coverage. A receiving RIS (also known as a hybrid
+RIS or a simultaneously reflecting and sensing RIS) can enable
+parameter estimation at the RIS side, offering extra degrees
+of freedom for the design of L&S estimation approaches.
+Omni-directional RISs, intended to realize simultaneous reflec-
+tion and refraction (i.e., 360◦ coverage), enable simultaneous
+indoor and outdoor 3D localization. A non-reciprocal RIS
+that integrates nonreciprocal phase shifters allows full-duplex
+communications, and a delay-adjustable RIS is capable of
+adjusting the delays of signals reflected by different RIS
+elements, which contributes to the alleviation of the beam
+squint effect. All of these alternatives have implications on
+L&S services and merit further study.
+E. Privacy, Security, and Social Acceptance Issues
+Cooperative L&S require extensive information exchange of
+local measurements between devices, which may cause privacy
+issues. For example, the RX in scenario B2 can estimate the
+position of the TX with a one-way pilot signal transmission. In
+addition, different types of cyber attacks can reduce the L&S
+service availability, or even provide an undetected erroneous
+location estimation, which is unacceptable for safety-critical
+applications. Currently, several security management systems
+have been standardized (e.g., IEEE 1609.2), and security
+threats have been identified for sidelink communications.
+However, the discussions on L&S task-related security issues
+are still at the initial stage, and potential threats need to
+be explored and eliminated. A final aspect related to the
+widespread adoption of RISs lies in their social acceptance.
+RISs should be integrated in a way that they blend into
+the environment (ideally, be transparent). To this end, the
+benefits of RISs to improve safety and reduce electromagnetic
+emissions should be demonstrated and disseminated.
+V. CONCLUSION AND OUTLOOK
+The smart city paradigm constitutes the epitome of the
+widespread adoption of digital services for societal needs.
+It is envisioned to profit people and city-level businesses,
+offering efficient, safe, and comfortable living spaces as well
+
+7
+as everyday-life smart-living applications. To achieve this
+overarching goal, seamless wireless communications among
+diverse devices and L&S are of paramount importance, en-
+abling information exchange, device localization, and mapping
+of the environment. In this article, we discussed the key
+to achieving low-cost and energy-efficient seamless L&S,
+namely, reflective RISs in conjunction with sidelink commu-
+nications. We presented AP-coordinated and AP-free system
+architectures and detailed three RIS-enabled L&S scenarios,
+each including several use cases and most relying on sidelink
+communications. As became apparent, instead of using APs
+with full communication capabilities, low-complexity beacons
+and RISs can be widely-deployed to enable green L&S smart
+city applications. In addition, when multiple UEs with sidelink
+communication capabilities can be connected in the same
+network, cooperative localization can relieve the requirement
+for multiple anchors. Furthermore, when devices are equipped
+with full-duplex transceivers, they can localize themself and
+map their surrounding environment with only a single RIS
+anchor. Finally, an extended list of open research challenges
+relevant to the proposed RIS-enabled seamless L&S concept
+was presented, including the necessity for anchor deployment
+optimization and optimized resource allocation schemes, al-
+gorithmic and privacy issues, as well as the role of multi-
+functional RISs.
+ACKNOWLEDGMENTS
+This work was supported, in part, by the European Com-
+mission through the EU H2020 RISE-6G project under grant
+101017011, and by the 6G-Cities project from Chalmers
+Transport Area of Advance.
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+Hui Chen (hui.chen@chalmers.se) is a postdoctoral researcher at Chalmers
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+Hyowon Kim (hyowon@chalmers.se) is a postdoctoral researcher at Chalmers
+University of Technology, Sweden.
+Mustafa Ammous (mustafa.ammous@mail.utoronto.ca) is a Ph.D. student at
+University of Toronto, Canada.
+Gonzalo Seco-Granados (gonzalo.seco@uab.cat) is a professor at Universitat
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+George C. Alexandropoulos (alexandg@di.uoa.gr) is an assistant professor
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+Henk Wymeersch (henkw@chalmers.se) is a professor at Chalmers Univer-
+sity of Technology, Sweden.
+

8NE1T4oBgHgl3EQf7gWJ/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf,len=455
+page_content='1  RISs and Sidelink Communications in Smart Cities:  The Key to Seamless Localization and Sensing  Hui Chen, Member, IEEE, Hyowon Kim, Member, IEEE, Mustafa Ammous, Student Member, IEEE,  Gonzalo Seco-Granados, Senior Member, IEEE, George C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Alexandropoulos, Senior Member, IEEE,  Shahrokh Valaee, Fellow, IEEE, and Henk Wymeersch, Senior Member, IEEE  Abstract—A smart city involves, among other elements, intelli-  gent transportation, crowd monitoring, and digital twins, each of  which requires information exchange via wireless communication  links and localization of connected devices and passive objects  (including people).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Although localization and sensing (L&S) are  envisioned as core functions of future communication systems,  they have inherently different demands in terms of infrastructure  compared to communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Wireless communications gener-  ally requires a connection to only a single access point (AP),  while L&S demand simultaneous line-of-sight propagation paths  to several APs, which serve as location and orientation anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  Hence, a smart city deployment optimized for communication  will be insufficient to meet stringent L&S requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In this  article, we argue that the emerging technologies of reconfigurable  intelligent surfaces (RISs) and sidelink communications constitute  the key to providing ubiquitous coverage for L&S in smart cities  with low-cost and energy-efficient technical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' To this end,  we propose and evaluate AP-coordinated and self-coordinated  RIS-enabled L&S architectures and detail three groups of appli-  cation scenarios, relying on low-complexity beacons, cooperative  localization, and full-duplex transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' A list of practical issues  and consequent open research challenges of the proposed L&S  systems is also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  Index Terms—Smart cities, reconfigurable intelligent surfaces,  localization, sensing, sidelink communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' INTRODUCTION  The emerging concept of smart cities aims to improve ac-  cessibility to public services, advance digitization of the urban  environment, and monitor various human-oriented processes  as well as assets, by harmonizing diverse digital technologies  at a city level [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' This broad concept integrates various  independent applications to bring improvements both at the  societal level (such as smart homes, smart transportation, sup-  ply chains, and environment monitoring), and at the individual  level (such as indoor navigation and extended reality (XR)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' To  realize the concept of smart cities, reliable, low-latency, and  high-speed communication systems (to support information  exchange and management among interconnected devices), as  well as accurate localization and sensing (L&S) (to support  communication and provide situation-awareness services), are  of great importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In this article, we use the term localization  to indicate the position (and possibly orientation) estimation of  a target user equipment (UE), and the term sensing to specify  the position estimation of passive objects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', objects without  networking infrastructure or non-cooperating ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  By exploiting the large antenna array sizes and wide  bandwidth of millimeter-wave/THz systems, recent research  activities in industry and academia on integrated sensing,  localization, and communication (ISLAC) are growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' ISLAC  is able to utilize communication infrastructures and signals  to enable synergies with L&S for diverse applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' To  this end, several standardization efforts and 3GPP activities  have been recently studied, such as the definition of new  radio positioning requirements, evaluation methodologies, and  techniques (dependent on radio access technology and not), in  TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='855 [2] as well as the development of Wi-Fi sensing  technology (in both sub-7 GHz and mmWave spectrum), in  the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='11bf standard [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  While high angular and delay resolution (due to large  arrays and wide signal bandwidths) facilitate L&S tasks,  signal coverage is one of the major challenges, especially  for high-frequency systems which suffer from high path loss  and high blockage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In contrast to communications,  which is possible to work with only one point-to-point link,  L&S functions necessitate access to multiple access points  (APs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Reflective reconfigurable intelligent surfaces (RISs)  constitute a promising emerging technology for extending  coverage and dynamically programming signal propagation  with almost zero-energy consumption [4], thus, supporting,  or even enabling in certain cases, wireless communications,  as well as L&S tasks in various scenarios [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' However, since  they are incapable of generating their own signals and only  modify the analog waveforms impinging on them, separate  signal generation sources are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  The sources generating signals for RIS-assisted L&S can  be: i) APs with full communication capabilities, ii) low-  complexity beacons that broadcast or receive L&S reference  signals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' or iii) the UEs themselves that are involved in the  L&S tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The latter two cases are particularly relevant when  power-hungry APs, whose deployment is usually costly, needs  to be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' To fulfill the ubiquitous L&S requirements in  out-of-coverage areas or partially-covered ones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', indoor  UEs and vehicles in tunnels), sidelink communications via  the PC5 interface can be particularly useful [6], as mentioned  in the 3GPP TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='845 report [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In addition, cooperative  localization between several UEs can be employed to enhance  or enable L&S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Furthermore, when a UE is equipped with  a full-duplex transceiver [8] —a technology that is widely  discussed in ISLAC for sensing purposes—, it can both  simultaneously transmit and receive the signal to localize itself  with a single RIS anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  In this article, we argue that RIS technology in conjunction  with sidelink communications can provide seamless L&S so-  lutions, speeding up the intelligent transformation of cities into  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='03535v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='SP]  9 Jan 2023  2  Beacon-Assisted Localization  Beacon  Self-L&S with  a Full-Duplex  Transceiver  Beacon-UE Channel  RIS Anchor Channel  RIS Localization Channel  Sidelink Channel  Sensing Channel  Scenarios  Cooperative Localization  RIS  A2  A1  B1  B2  C1  C2  C3  RIS-attached  Vehicle  AP  A3  B3  Figure elements by macrovector on Freepik  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' RIS-enabled seamless L&S scenarios in smart cities: a) beacon-assisted localization, b) cooperative localization, and c) self-L&S with a full-duplex  transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  smart entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' We particularly elaborate on the representative  RIS-enabled L&S scenarios illustrated in Figure 1: beacon-  assisted localization, cooperative localization, and self-L&S  with a full-duplex transceiver, describing both AP-coordinated  and AP-free architectures for different coverage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' We  discuss the open challenges with the proposed green (low-  cost and energy-efficient) L&S systems in smart cities and list  potential directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' L&S ARCHITECTURES AND PROTOCOLS  In this section, we describe the different entity types of the  proposed L&S system for smart cities, relying on RISs and  sidelink communications, as well as its enabling architectures,  depending on whether an AP is present for L&S coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Entity Types and Overall Architecture  In the proposed RIS-enabled L&S system, there are several  types of entities: APs, beacons, RISs, and UEs, as shown in  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The APs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', a gNB macro base station) provide  cellular services to the devices in coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' A beacon could be  a roadside unit (RSU) that is capable of sending and receiving  L&S reference signals via sidelink communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In this  way, the explicit involvement of expensive and power-hungry  APs with full communication protocols is not needed for L&S  purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' RISs serve as reference anchors to assist ISLAC  tasks, and are controlled by a dedicated entity responsible  for realizing communication functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Finally, UEs may have  different hardware capabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', single/multiple antennas,  half-/full-duplex) that play different roles in L&S (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', a target  UE to be localized, an assistant UE with known or measured  location to assist L&S, or a server/coordinator in performing  L&S tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Although all the devices involved in the targeted  tasks will have sidelink communication capabilities, beacons  usually have fewer power constraints than UEs, while the  controllers of reflective RISs are expected to work in low-  power mode without sending or processing L&S reference  signals [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' We further consider that RISs coordinate their  reflective beamforming, either using time division or phase  profile codes in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  For the scenarios considered in this article, we propose two  different architectures: one based on AP coordination and the  other on self-coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The former architecture works for  UEs (and other L&S-related devices) inside a coverage area or  in partial-covered areas (where sidelink is available), requiring  a specific AP to serve as a L&S coordinator, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', interact  with the location management function (LMF) via the NR  positioning protocol A (NRPPa) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The self-coordinated ar-  chitecture relies explicitly on sidelink communications, being  particularly suitable for UEs in the out-of-coverage of APs, or  for UEs connected to APs that cannot meet the latency and  spatial resolution requirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', legacy 3G/4G APs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In  both architectures, L&S signals can be generated by a beacon,  an assistant UE, or a target UE, depending on the network  topologies and the specific application scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  3  Beacon-UE Channel  Sidelink Channel  RIS Channel  AP-Coordinated Control Link  Self-Coordinated Control Link  L&S Coordinator  Controller  RIS-1  In-coverage  Partial coverage  Out-of-coverage  AP  Beacon  UE-1  UE-2  UE-3  UE-4  UE-5  RIS-2  RIS-3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' UEs performing L&S in different coverage areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The AP-coordinated  architecture can be used for UEs located in in-coverage or partial-coverage  (sidelink communications required) areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' When the UEs and all the L&S  devices cannot access any surrounding APs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', lying in out-of-coverage  areas), the self-coordinated architecture is the only option for L&S services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' AP-Coordinated Architecture  This architecture relies on one or several APs to allocate  the available radio resources and ensure timing among the  connected devices, which are the UEs, the RIS controllers, and  dedicated beacon nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' This network management scheme is  similar to the mode-1 in sidelink communications [6], and the  L&S protocols can be summarized into the following 6 steps:  1) The target UE triggers an L&S request to the AP (which  is selected as the task coordinator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  2) The coordinator exchanges related location information  with the core network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', via the NRPPa), determines  L&S configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', broadcasting, sidelink, and  RIS phase profiles), as well as selects and notifies nearby  beacons, UEs, and RISs that are involved in the L&S  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  3) The coordinator allocates time-frequency resources for  L&S to all involved devices and triggers the RIS con-  troller(s) to configure their phase profiles for the whole  duration of the estimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  4) The beacons and/or UEs transmit L&S reference signals,  which are reflected by the involved RIS(s) and received  by the target UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  5) The collected measurements are used by the target UE  for the localization, and/or sensing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Alternatively,  this computation can be offloaded at the localization  server (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', the coordinating AP or another UE with  high computational power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  6) The target UE updates the task coordinator with the  estimated L&S results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' this optional step can serve as  prior information for future use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  For the partial-coverage scenarios, the out-of-coverage UEs  need to establish sidelink communications with devices that  are covered by APs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Then, the L&S tasks can be performed  similarly to the aforementioned steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Self-Coordinated Architecture  An AP-free architecture is required for cases where the  devices involved in L&S tasks are located in out-of-coverage  areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Similar to the mode-2 sidelink communications [6],  L&S tasks can be autonomously realized by selecting a  specific device as the localization coordinator, as follows:  1) The target UE discovers nearby devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', beacons,  RISs, and other UEs) and obtains their location infor-  mation (if available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  2) Based on the discovered neighbors, the target UE de-  termines a L&S task coordinator (could be itself) and  notifies it of the L&S configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  3) The target UE triggers a L&S request to the coordi-  nator and performs the same actions as with the AP-  coordinated architecture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', steps 2)–6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' RIS-ENABLED L&S SCENARIOS  In this section, we will present three representative RIS-  enabled L&S scenarios for smart city applications relying on  the aforementioned architectures and protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' We mainly  focus on localization applications and list potential position-  ing scenarios in systems with minimum infrastructure and  resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' For example, if positioning can be done with a  single antenna, it can also be achieved using a multi-antenna  array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The same applies to narrowband (NB)/wideband (WB)  signals, single/multiple anchors, and availability/unavailability  of a line-of-sight (LOS) path between the UE and the active  signal source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Beacon-Assisted Localization  With low-complexity beacons, high flexibility in the in-  stallation and deployment of L&S systems is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' A  typical use case could be a train station with multiple low-  cost beacons broadcasting L&S reference signals to UEs to  navigate indoors, via the support of RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In this category, we  consider UE localization (with the aid of one or more RISs)  and RIS localization (with the aid of several beacons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  A1) Single-RIS-Enabled UE Localization: In a WB system  where the LOS channel is available, the delays of the LOS  and RIS paths can be estimated based on the L&S reference  signals sent from a beacon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Assuming the RIS and beacon  states are known, the angle-of-departure (AOD) at the RIS  can also be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The target UE can then be localized  by the intersection of a hyperbola (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', time-difference-of-  arrival (TDOA) of the LOS and RIS paths) and the line in  the direction of the AOD at the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' When the target UE  is equipped with an antenna array, 3D orientation can also  be estimated based on the estimated angle-of-arrivals (AOAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  This is the basic RIS-enabled localization scenario that only  requires a single low-complexity beacon [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  A2) Multi-RIS-Enabled UE Localization: If multiple RISs  are simultaneously available, the requirements for LOS and  WB are unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The AODs from different RISs can be  estimated and used to localize the UE by intersecting the AOD  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In this way, localization tasks can be completed using  NB signals, which saves bandwidth resources for communi-  cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Figure 3 shows the position error bound (PEB) of the  target UE with different positions inside a 5 × 10 m2 area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' As  shown, with multiple RISs, the UE is localizable even under  blockage of the LOS path between the beacon and the UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  4  5  4  3  2  1  0  1  2  3  4  5  x axis [m]  0  1  2  3  4  5  y axis [m]  Beacon  RIS  Wall  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='1  1 [m]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Scenarios A1 and A2: PEB (in meters) with different UE locations  in a multi-RIS-aided localization scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The target UE can be localized  with one RIS and a beacon-UE LOS path, or with at least 2 RISs under LOS  blockage conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  However, the localization tasks cannot be performed when  only one anchor (beacon or RIS) is visible to the UE (see  the yellow triangular area around the point [3, 3] m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  A3) RIS Localization via Multi-Static Sensing: In a  scenario where passive UEs or objects are coated with RISs,  the localization (or sensing, depending on scenarios) can be  performed semi-passively with only a small amount of energy  needed for localization coordination and RIS phase profile  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Such localization tasks can estimate the positions (and  orientations) of RIS-coated objects by using several beacons  with known positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Note that the geometrical constraints  can largely reduce the difficulties in these scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' For  example, the orientation of an RIS can be assumed as 1D  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', placed on the top of a car facing up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In addition, the  adoption of antenna arrays at the beacons can further simplify  the RIS localization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Cooperative Localization  Sidelink communications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', device-to-device (D2D)  communications, or the vehicular version known as vehicle-  to-everything (V2X) communications) has been introduced in  the millimeter-wave band for information exchange between  vehicles, opening the road for numerous use cases, such  as platooning, collision avoidance and autonomous driving  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The combination of RISs and sidelink is expected to  provide low-latency and high-reliability communications [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  Interestingly, it can also be exploited to assist L&S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Sidelink  communications enables UEs to participate in a cooperative  manner in the sharing of position and surrounding information  within a local network, and in performing relative location  estimation using sidelink signals [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' We will focus on the  latter case, where the absolute location can be estimated  using RIS anchors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', with known position and orientation  information), without any APs or beacons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' A typical scenario  could be cooperative vehicular networks in urban areas with  severe AP and GPS signal blockages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  We next discuss single- and multi-RIS-involved localization  scenarios, where single-antenna UEs cooperate to estimate  their positions via WB sidelink signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' We also consider a  more general scenario where RIS-coated objects are involved,  resulting in cooperative RIS localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  B1) Single-RIS-Enabled Cooperative Localization: Con-  sider a scenario with several UEs and one RIS anchor, where  the UEs wish to estimate their positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' We assume that each  UE can send sidelink signals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', being the transmitter (TX))  to other UEs, which arrive at the receiver (RX) UEs via two  paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', the UE-UE and UE-RIS-UE paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' By proper  control of the RIS elements, those two paths can be separated,  and thus, two delay measurements can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Due to the  unknown states of the UEs, both the AOD and AOA at the  RIS for every TX-RX pair of UEs are unknown and cannot  be directly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' However, we can estimate the spatial  frequency information at the RIS for every TX-RX pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' This  scenario requires at least three UEs to cooperate and render  their locations feasibly without ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Figure 4 compares  the PEBs for three UEs in an RIS-enabled versus beacon-aided  (equipped with an antenna array) 3D cooperative localization  scenario as a function of the number of RIS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  B2) Multi-RIS-Enabled One-Way Sidelink Localization:  In the scenario with at least two RISs, one-way sidelink  communications is sufficient to localize both the TX and RX  UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Assume that one UE takes the role of the TX and the  other UE is the RX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' With two RISs, three delay measurements  can be obtained between the TX and RX via the LOS and  the two RIS paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' However, that would require an optimal  joint design of the reflection elements at both RISs to be able  to separate the paths at the RX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In addition, once the RIS  paths are separated, we can also estimate the spatial frequency  information at each RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Thus, those collected measurements  can be utilized to estimate the locations of the TX and RX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  B3) Cooperative RIS Localization: Let us consider a more  general scenario where one RIS (or several) with a known state  is used to localize multiple UEs (with sidelink capabilities) and  objects (coated with an RIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' This scenario is challenging due  to the high complexity of the network and a large number of  unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' However, with a proper design of all the involved  RIS profiles and the transmission protocol, this problem can  be decomposed into a cooperative localization (see B1) and  an RIS localization (see A3) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Similar to B1, at least  several UEs (depending on scenarios) need to take the role  of the TX, and transmit sidelink signals to the other UEs via  the direct and indirect paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Once the UEs are localized, the  RIS localization task can be solved similarly to A3, and the  estimation results can be refined by processing all the available  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Self-L&S with a Full-Duplex Transceiver  When a UE is equipped with a full-duplex transceiver (like  radar) [8], the multi-RIS setup and cooperation between UEs  are unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Instead, this UE can perform self-positioning  with a single RIS and use the multipath components to map the  environment over time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' this process is known as monostatic  simultaneous localization and mapping (SLAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' It is noted  that SLAM is not limited to full-duplex UEs, and bistatic  SLAM can also be performed in use cases A1-A3 and B1-  B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  We next present three beacon-free L&S scenarios with a  full-duplex UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  5  0  20  40  60  80  100  120  140  160  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='5  UE-1 (RIS)  UE-1 (beacon)  UE-2 (RIS)  UE-2 (beacon)  UE-3 (RIS)  UE-3 (beacon)  Number of RIS Elements  PEB [m]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  PEBs of RIS-enabled vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' beacon-aided cooperative localization for  different RIS sizes and a fixed random phase profile setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' It is shown that,  with a sufficient number of RIS elements, an active anchor can be replaced  with a passive one (RIS) without performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  C1) RIS-Enabled Self-Localization: Consider a system  with a single-antenna UE and an RIS, where the UE transmits  L&S reference signals and receives their back-scattered ver-  sions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', the UE-RIS-UE (controlled path) and UE-landmark-  UE (uncontrolled path) signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' One option for the RIS phase  profiles is to consider directional reflective beams, which can  be efficiently designed when the UE position uncertainty (even  under mobility cases) is available [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The delay and angle  information at the RIS of the UE-RIS-UE channel can be  estimated for this scenario, and then used to localize the UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In  Figure 5, beampatterns at the RIS with two different phase pro-  files are illustrated, focusing on the UE uncertainty region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' As  demonstrated, the optimized phase profiles of [13] can provide  sufficient beamforming gain, compared with directional phase  profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' This gain can offer improved L&S performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  C2) RIS-Enabled SLAM: If a UE is equipped with a  full-duplex multiple-input multiple-output (MIMO) antenna  array [8], SLAM can be enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Similar to scenario C1, the  signals from different paths can be resolvable with optimized  RIS phase profiles and precoders/combiners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The following  channel parameters can be estimated: i) the signal propagation  delay, the AOD at the RIS, and the AOA at the UE for the  UE-RIS-UE channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' as well as ii) the delay and AOA at  the UE for each UE-landmark-UE channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In addition to the  position information obtained from the controlled path (as in  scenario C1), the angular resolution offered by the UE array  can be leveraged to map/sense the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' With multiple  estimations, the localization and radio mapping performance  can be improved using state-of-the-art filters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', Poisson  multi-Bernoulli filter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  C3) RIS Localization with a Full-Duplex Array: Consider  the more general scenario from C1 including one anchor  RIS mounted on a wall, a UE equipped with a full-duplex  MIMO transceiver, and several objects coated with RISs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=',  mounted on the front and rear side of a vehicle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In addition to  the signals reflected from the anchor RIS (as also in scenario  C1), the UE also receives single-bounce reflected signals from  the RISs mounted on the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The time delay, AOA, and  the amplitude of the channel gain for each signal path can be  estimated, which can be used for the localization of both itself  and the RISs-coated objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' When multiple UEs are present  and cooperate in the estimation process, the orientation of the  (a) Directional Phase Profiles  (b) Optimized Phase Profiles  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The reflective beamforming gain (in dB) with a 50 × 50 RIS using  (a) a directional phase profile, and (b) an optimized phase profile via [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  The red squares represent the UE angular uncertainty region, which needs to  be fully covered by an effective beampattern design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  RISs-coated objects can also be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In a scenario without  any anchors, this RIS localization can also help in estimating  the relative locations of the active UE and passive UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' OPEN RESEARCH CHALLENGES  With the assistance of low-complexity beacons, cooperative  localization, and full-duplex radios, the L&S coverage for  smart city applications can be significantly extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' However,  there exist several practical issues that need to be thoroughly  investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In this section, we discuss the most critical  challenges with the proposed RIS-enabled L&S system and  list possible directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Anchor Deployment Optimization  The placement of the anchors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', beacons and RISs) is  critical to meet the L&S key performance indicator (KPI) re-  quirements within a service area (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', error bounds lower than  a certain threshold, as shown in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The deployment  involves both the position and orientation optimization of the  anchors, taking into account the blockage in the surrounding  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' RIS-aided SLAM can help in creating such an  environment map, which can be supported by cooperative  sidelink UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Heuristic optimization solutions can then be  applied for finding optimal anchor sites, extending approaches  from the literature [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='100100150evation20  Gro40  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='M06080J60  azllBeaiot0  muthele100100150evation20  G[o]40  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='=060  B80ncertain60  azilReaiol  Vmuth6  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Resource Allocation and Coordination  RIS-aided L&S systems involve APs, beacons, RISs, and  UEs, making them inherently heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Resource al-  location for L&S tasks, including power allocation, time-  frequency allocation, beamforming design, and scheduling  must be carefully designed to ensure a favorable trade-off  with conventional communication services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Depending on the  KPI requirements of the applications that send L&S ser-  vice requests, new objectives that consider integrated L&S  and communications should be formulated and satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' An  important part of resource allocation is RIS phase profile  optimization and multiplexing [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Broad RIS beams lead to  coverage reduction, while narrow pencil beams are sensitive to  misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Hence, highly adaptive RIS profile designs are  needed, relying, when possible, on prior UE and object state  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' When RISs are large, the near-field effects need  to be taken into consideration and the beamforming designs  should account for the curvature of arrival, resulting in beam-  focusing designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' RIS multiplexing can be addressed by time  multiplexing, temporal coding, and making use of high path  loss for spatial reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' The afore-described resource allocation  problems can be tackled by a combination of traditional  optimization-based methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', convex optimization) and  learning-based methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', reinforcement learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Estimation Algorithms  From an algorithmic perspective, there are challenges re-  lated to channel parameter estimation, tracking in dynamic  environments, and calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Channel parameter estimation  in the presence of severe multipath is difficult since almost  passive reflective RISs have no local signal processing capa-  bilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Moreover, in cooperative localization (scenarios B1  and B2) and RIS localization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', scenarios A3, B3, and  C3) tasks, the AOAs/AODs at the RISs are coupled, meaning  that we can no longer estimate them independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Instead,  only spatial frequencies (containing coupled AOAs and AODs  information) can be obtained, requiring novel algorithms for  further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' More refined channel parameter estimation  also requires accurate channel models and the RISs’ impact  on them, such as the near-field effect, beam squint effect, RIS  element failures, and anchor calibration errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  Due to mobility, difficult conditions such as signal blockage,  unresolvable signal paths, and severe path loss will affect L&S  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Multiple RISs can be involved to handle such  blockages, offering coverage extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' As the carrier fre-  quency increases, the signal resolutions become higher, and the  effect of the severe path loss can be mitigated by adopting RISs  with larger sizes or active RISs with reflection amplification  capabilities [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Sensing also suffers from inherent complica-  tions, such as an unknown number of objects, unknown types  of objects, unknown detection probabilities for signal paths,  extended objects, and multi-bounce observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Dedicated  filters should be developed to address these complications and  get integrated into the L&S framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  Finally, in terms of calibration, anchor geometry error and  hardware impairments (HWIs) are two important aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' We  note that the geometrical calibration of an RIS is similar to RIS  localization (as described in scenarios A3, B3, and C3), which  requires a calibration agent that incorporates other sources of  localization estimations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', sensor fusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' While for HWIs,  the channel model could be too complicated when considering  each specific impairment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', mutual coupling and phase  noise), learning-based methods can be considered to unveil  their impact and drive practical algorithmic designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Understanding Anchor Hardware Alternatives  There are also opportunities to improve L&S coverage via  variations of the hardware deployed at the beacons, RISs, and  UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' On the beacon and UE sides, multi-panel arrays (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', 3D  arrays) could be implemented for further coverage extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  On the RIS side, new types of RISs are emerging beyond  almost passive reflective RISs [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' As previously mentioned,  an active RIS can be used to boost the signal energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=',  change both the amplitude and phase of the incident signal) for  improved coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' A receiving RIS (also known as a hybrid  RIS or a simultaneously reflecting and sensing RIS) can enable  parameter estimation at the RIS side, offering extra degrees  of freedom for the design of L&S estimation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  Omni-directional RISs, intended to realize simultaneous reflec-  tion and refraction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', 360◦ coverage), enable simultaneous  indoor and outdoor 3D localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' A non-reciprocal RIS  that integrates nonreciprocal phase shifters allows full-duplex  communications, and a delay-adjustable RIS is capable of  adjusting the delays of signals reflected by different RIS  elements, which contributes to the alleviation of the beam  squint effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' All of these alternatives have implications on  L&S services and merit further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Privacy, Security, and Social Acceptance Issues  Cooperative L&S require extensive information exchange of  local measurements between devices, which may cause privacy  issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' For example, the RX in scenario B2 can estimate the  position of the TX with a one-way pilot signal transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In  addition, different types of cyber attacks can reduce the L&S  service availability, or even provide an undetected erroneous  location estimation, which is unacceptable for safety-critical  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Currently, several security management systems  have been standardized (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', IEEE 1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='2), and security  threats have been identified for sidelink communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  However, the discussions on L&S task-related security issues  are still at the initial stage, and potential threats need to  be explored and eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' A final aspect related to the  widespread adoption of RISs lies in their social acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  RISs should be integrated in a way that they blend into  the environment (ideally, be transparent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' To this end, the  benefits of RISs to improve safety and reduce electromagnetic  emissions should be demonstrated and disseminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' CONCLUSION AND OUTLOOK  The smart city paradigm constitutes the epitome of the  widespread adoption of digital services for societal needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  It is envisioned to profit people and city-level businesses,  offering ef���cient, safe, and comfortable living spaces as well  7  as everyday-life smart-living applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' To achieve this  overarching goal, seamless wireless communications among  diverse devices and L&S are of paramount importance, en-  abling information exchange, device localization, and mapping  of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In this article, we discussed the key  to achieving low-cost and energy-efficient seamless L&S,  namely, reflective RISs in conjunction with sidelink commu-  nications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' We presented AP-coordinated and AP-free system  architectures and detailed three RIS-enabled L&S scenarios,  each including several use cases and most relying on sidelink  communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' As became apparent, instead of using APs  with full communication capabilities, low-complexity beacons  and RISs can be widely-deployed to enable green L&S smart  city applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' In addition, when multiple UEs with sidelink  communication capabilities can be connected in the same  network, cooperative localization can relieve the requirement  for multiple anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Furthermore, when devices are equipped  with full-duplex transceivers, they can localize themself and  map their surrounding environment with only a single RIS  anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Finally, an extended list of open research challenges  relevant to the proposed RIS-enabled seamless L&S concept  was presented, including the necessity for anchor deployment  optimization and optimized resource allocation schemes, al-  gorithmic and privacy issues, as well as the role of multi-  functional RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was supported, in part, by the European Com-  mission through the EU H2020 RISE-6G project under grant  101017011, and by the 6G-Cities project from Chalmers  Transport Area of Advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
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+page_content=' Converged Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' 1–32, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  Hui Chen (hui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='chen@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='se) is a postdoctoral researcher at Chalmers  University of Technology, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  Hyowon Kim (hyowon@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='se) is a postdoctoral researcher at Chalmers  University of Technology, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  Mustafa Ammous (mustafa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='ammous@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='ca) is a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' student at  University of Toronto, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  Gonzalo Seco-Granados (gonzalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='seco@uab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='cat) is a professor at Universitat  Autonoma of Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  George C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content=' Alexandropoulos (alexandg@di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='uoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='gr) is an assistant professor  at the Department of Informatics and Telecommunications, National and  Kapodistrian University of Athens, Greece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  Shahrokh Valaee (valaee@ece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='ca) is a professor at University of  Toronto, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='  Henk Wymeersch (henkw@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}
+page_content='se) is a professor at Chalmers Univer-  sity of Technology, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQf7gWJ/content/2301.03535v1.pdf'}

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@@ -0,0 +1,1175 @@
+D-Optimal and Nearly D-Optimal Exact Designs for
+Binary Response on the Ball
+Martin Radloff† and Rainer Schwabe‡
+Abstract: In this paper the results of Radloff and Schwabe (2019a) will be
+extended for a special class of symmetrical intensity functions. This includes
+binary response models with logit and probit link. To evaluate the position
+and the weights of the two non-degenerated orbits on the k-dimensional ball
+usually a system of three equations has to be solved. The symmetry allows
+to reduce this system to a single equation. As a further result, the number of
+support points can be reduced to the minimal number. These minimally sup-
+ported designs are highly efficient. The results can be generalized to arbitrary
+ellipsoidal design regions.
+Key words and phrases: Binary response models, D-optimality, k-dimensional
+ball, logit and probit model, multiple regression models, simplex.
+1. Introduction
+Spherical design spaces can occur in engineering or physics problems where the validity of
+a model may be assumed on a spherical region around a target value. So (linear) models
+on spherical design spaces were investigated early in publications like Kiefer (1961) and
+Farrell et al (1967) which discuss polynomial regression on the ball. These ideas were
+followed up by papers in which also only linear problems were focused. So Lau (1988)
+fitted polynomials on the k-dimensional unit ball by using canonical moments. In Dette
+et al (2005, 2007) and Hirao et al (2015) harmonic polynomials and Zernike polynomials
+were used to be fit on the unit disc (2-dimensional unit ball), the 3- and k-dimensional
+unit ball. On the other hand generalized linear models are also well-examined and used
+in practical application. Logit and probit models, for example, in one dimension on an
+interval have already been investigated by Ford et al (1992) and Biedermann et al (2006).
+But there seems to be no available literature which combines both topics.
+In our publication Radloff and Schwabe (2019b) we took the first step to bring non-
+linearity or generalized linear models, respectively, and spherical design regions together.
+These results were extended to a wider class of non-linear models in our follow-up paper
+Radloff and Schwabe (2019a).
+†corresponding author: Martin Radloff, Institute for Mathematical Stochastics, Otto-von-Guericke-
+University, PF 4120, 39016 Magdeburg, Germany, martin.radloff@ovgu.de
+‡Rainer Schwabe, Institute for Mathematical Stochastics, Otto-von-Guericke-University, PF 4120,
+39016 Magdeburg, Germany, rainer.schwabe@ovgu.de
+arXiv:2301.02859v1  [stat.ME]  7 Jan 2023
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+For better comprehensibility, we will start with the model description and give a brief
+overview of the findings so far. Then we will consider a special class of intensity func-
+tions which allows to reduce the the complexity of finding (locally) D-optimal designs.
+Afterwards we will tackle the problem, that the optimal designs are not exact designs in
+general, by establishing highly efficient designs on the ball.
+2. General Model Description
+As in Radloff and Schwabe (2019b) and Radloff and Schwabe (2019a), where we described
+(locally) D-optimal designs for two special classes of linear and non-linear models on a
+k-dimensional unit ball Bk = {x ∈ Rk : x2
+1 + . . . + x2
+k ≤ 1} with k ∈ N, we solely focus
+(non-linear) multiple regression models, which means the linear predictor is
+f(x)⊤β = β0 + β1x1 + . . . + βkxk
+with regression function f : Bk → Rk+1, x �→ (1, x1, . . . , xk)⊤, and parameter vector
+β = (β0, β1, . . . , βk)⊤ ∈ Rk+1. The one-support-point (or elemental) information matrix
+should be representable in the form
+M(x, β) = λ
+�
+f(x)⊤β
+�
+f(x)f(x)⊤
+with an intensity (or efficiency) function λ which only depends on the value of the linear
+predictor f(x)⊤β. These one-support-point (or elemental) information matrices are the
+base for the information matrix of a (generalized) design ξ with independent observations
+M(ξ, β) =
+�
+M(x, β) ξ(dx) =
+�
+λ
+�
+f(x)⊤β
+�
+f(x)f(x)⊤ξ(dx) .
+Here generalized design means an arbitrary probability measure on the design region Bk.
+These information matrices allow to define the (local) D-optimality, which is one of the
+most popular criteria in experimental design theory. A design ξ∗
+β0 with regular infor-
+mation matrix M(ξ∗
+β0, β0) is called (locally) D-optimal (at β0) if det(M(ξ∗
+β0, β0)) ≥
+det(M(ξ, β0)) holds for all suitable probability measures ξ on the design space — here
+Bk. This optimality criterion can be interpreted as the minimization of the volume of the
+(asymptotic) confidence ellipsoid.
+3. Prior Results
+In Radloff and Schwabe (2016) we stated results on equivariance and invariance.
+By
+rotating the design space Bk — the k-dimensional unit ball — and the parameter space
+Rk+1 in an analogous way the linear predictor of the multiple regression problem reduces
+to
+f(x)⊤β = β0 + β1x1
+and
+β1 ≥ 0 .
+(3.1)
+Using the rotation invariance with fixed x1, this means the invariance to all orthogonal
+transformations in O(k) which let the x1-component unchanged, the (locally) D-optimal
+(generalized) design ξ∗ can be decomposed (ξ∗ = ξ∗
+1 ⊗ η) in a marginal probability measure
+ξ∗
+1 on [−1, 1] for x1 and a probability kernel η given x1. For fixed x1 the kernel η(x1, ·) is
+2
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+the uniform distribution on the surface of a (k − 1)-dimensional ball with radius
+�
+1 − x2
+1
+— the orbit at position x1.
+As a consequence the multidimensional problem collapses to a one-dimensional marginal
+problem. Only the positions of the orbits and their weights have to be determined. To get
+an exact design the uniform orbits have to be discretized, for example, by using regular
+simplices.
+In our first paper — Radloff and Schwabe (2019b) — we started with models where the
+intensity function belongs to the class of monotonous functions. Such models have already
+been investigated in one dimension, for example, by Konstantinou et al (2014) and on
+multidimensional cuboids or orthants by Schmidt and Schwabe (2017). These authors
+gave the following four conditions on the intensity function λ:
+(A1) λ is positive on R and twice continuously differentiable.
+(A2) The first derivative λ′ is positive on R.
+(A3) The second derivative u′′ of u = 1
+λ is injective on R.
+(A4) The function λ′
+λ is non-increasing.
+Condition (A2) is the motivation for the name class of monotonous intensity functions.
+The intensity functions of this class have to satisfy always (A1) to (A3). (A4) is an extra
+condition to guarantee uniqueness. For a concise notation
+q(x1) = λ(β0 + β1x1)
+is used and the properties (A1), (A2), (A3) and (A4) transfer to q for β1 > 0, respectively,
+and vice versa. Poisson regression with intensity function qP(x1) = exp(β0 + β1x1) and
+negative binomial regression as well as special proportional hazard models with censoring,
+see Schmidt and Schwabe (2017), satisfy all four conditions.
+If β1 = 0 then the intensity function q is always a constant. This yields to a (locally)
+D-optimal design as it can be found in linear models. In Pukelsheim (1993, section 15.12)
+such a design consists of the equally weighted vertices of a regular simplex inscribed in the
+unit sphere, the boundary of the design space. The orientation of the simplex is arbitrary.
+The main result for β1 > 0 in Radloff and Schwabe (2019b) is recited for the readers’
+convenience.
+Theorem 1. There is a (locally) D-optimal design for the multiple regression prob-
+lem (3.1) with β1 > 0 and intensity function satisfying (A1)-(A3) which has one support
+point equal to (1, 0, . . . , 0)⊤ and the other k support points are the vertices of an arbi-
+trarily rotated (k − 1)-dimensional regular simplex which is maximally inscribed in the
+intersection of the k-dimensional unit ball and a hyperplane with x1 = x∗
+12.
+• For k ≥ 2 the position x∗
+12 ∈ (−1, 1) is solution of
+q′(x∗
+12)
+q(x∗
+12) = 2 (1 + kx∗
+12)
+k (1 − x∗ 2
+12 ) .
+If additionally (A4) is satisfied, the solution x∗
+12 is unique.
+3
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+• For k = 1 the position x∗
+12 ∈ [−1, 1) is either solution of
+q′(x∗
+12)
+q(x∗
+12) =
+2
+1 − x∗
+12
+,
+if such a solution exists in [−1, 1), or otherwise x∗
+12 = −1.
+If additionally (A4) is satisfied, the solution x∗
+12 is unique.
+The design is equally weighted with
+1
+k+1.
+It should be noted, that for fixed β this theorem does not need (A1) to (A4) on the
+entire real line R. It is enough to have it in the ball and so on x1 ∈ [−1, 1] for q and on
+[β0 − β1, β0 + β1] for λ, respectively. But the model has to satisfy the conditions always
+on the whole real line.
+In our second paper — Radloff and Schwabe (2019a) — the conditions (A2) and (A3)
+were replaced by (A2′) and (A3′) and a fifth property (A5) was added.
+(A2′) λ is unimodal with mode c(A2′)
+λ
+∈ R.
+(A3′) There exists a threshold c(A3′)
+λ
+∈ R so that the second derivative u′′ of u = 1
+λ is
+both injective on (−∞, c(A3′)
+λ
+] and injective on [c(A3′)
+λ
+, ∞).
+(A5) u = 1
+λ dominates z2 asymptotically for z → ∞.
+In this context condition (A2′) means that there exists a c(A2′)
+λ
+∈ R so that λ′ is positive
+on (−∞, c(A2′)
+λ
+) and negative on (c(A2′)
+λ
+, ∞).
+Hence, there is only one local maximum
+which is simultaneously the global maximum. So the class of intensity functions, which
+satisfy (A1), (A2′) and (A3′), is called class of unimodal intensity functions.
+Indeed (A2) or (A3) do not imply (A2′) or (A3′), respectively. As mentioned before, we
+only focus on the unit ball and the interval x1 ∈ [−1, 1] for q or [β0 − β1, β0 + β1] for λ.
+So in our special case (A2) and (A3) can be transferred to (A2′) and (A3′) by using an
+arbitrary cλ > β0 + β1, which means that cq lies outside the interval [−1, 1] and only one
+branch of the function is considered.
+Property (A5) means
+lim
+z→∞
+����
+u(z)
+z2
+���� = ∞ .
+This means that u(z) =
+1
+λ(z) goes faster to (±) infinity than z2 for z → ∞.
+As (A1) to (A4) the conditions (A2′), (A3′) and (A5) transfer from the intensity function
+λ to the abbreviated form q for β1 > 0 and vice versa — analogously c(·)
+q
+=
+c(·)
+λ −β0
+β1
+with
+(·) is (A2′), (A3′) or empty.
+The logit model has the intensity function
+qlogit(x1) =
+exp(β0 + β1x1)
+(1 + exp(β0 + β1x1))2
+and probit model has
+qprobit(x1) =
+φ2(β0 + β1x1)
+Φ(β0 + β1x1) · (1 − Φ(β0 + β1x1))
+4
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+with the density function φ and cumulative distribution function Φ of the standard normal
+distribution. Both models satisfy all five conditions (A1), (A2′), (A3′), (A4), (A5) and
+share a common c(A2′)
+λ
+= c(A3′)
+λ
+= 0, say cλ = 0. Analogously cq = − β0
+β1 for q.
+Beside these two models other models like the complementary log-log model, see Ford
+et al (1992), with intensity function λcomp log log(z) =
+exp(2z)
+exp(exp(z))−1 satisfy all five conditions
+with c(A2′)
+λ
+≈ 0.466011 and c(A3′)
+λ
+≈ 0.049084, but here mode c(A2′)
+λ
+and threshold c(A3′)
+λ
+do
+not coincide.
+We showed that if the (concise) intensity function q satisfies (A1), (A2′), (A3′) and (A5)
+the (locally) D-optimal design ξ∗ = ξ∗
+1 ⊗η is concentrated on exactly two orbits, which are
+the support points of the marginal design ξ∗
+1. The idea of the proof is based on Biedermann
+et al (2006) and Konstantinou et al (2014).
+The next theorem is the main result of our second paper — Radloff and Schwabe (2019a)
+— and is reproduced for the readers’ convenience. It characterizes the positions of the
+two support points of the optimal marginal design ξ∗
+1.
+Theorem 2. For k ≥ 2 the simplified problem (3.1) with β1 > 0 and intensity function q
+satisfying (A1), (A2′), (A3′) and (A5) has a (locally) D-optimal marginal design ξ∗
+1 with
+exactly 2 support points x∗
+11 and x∗
+12 with x∗
+11 > x∗
+12 and weights w1 = ξ∗
+1(x∗
+11) and w2 =
+ξ∗
+1(x∗
+12).
+There are 3 cases:
+(a) If c(A2′)
+q
+> 1 and c(A3′)
+q
+/∈ [−1, 1], then x∗
+11 = 1, w1 =
+1
+k+1, w2 =
+k
+k+1 and x∗
+12 ∈ (−1, 1)
+is solution of
+q′(x∗
+12)
+q(x∗
+12) = 2 (1 + kx∗
+12)
+k (1 − x∗ 2
+12 ) .
+(3.2)
+If additionally (A4) is satisfied, the solution x∗
+12 is unique.
+(b) If c(A2′)
+q
+< −1 and c(A3′)
+q
+/∈ [−1, 1], then x∗
+12 = −1, w1 =
+k
+k+1, w2 =
+1
+k+1 and
+x∗
+11 ∈ (−1, 1) is solution of
+q′(x∗
+11)
+q(x∗
+11) = 2 (−1 + kx∗
+11)
+k (1 − x∗ 2
+11 )
+.
+(3.3)
+If additionally (A4) is satisfied, the solution x∗
+11 is unique.
+(c) Otherwise c(A2′)
+q
+∈ [−1, 1] or c(A3′)
+q
+∈ [−1, 1].
+Let x, y ∈ R with x > y and α ∈
+�
+− 1
+2, 1
+2
+�
+be solution of the equation system:
+q′(x)
+q(x) +
+2
+x−y + (k−1) q′(x) (1−x2) ( 1
+2 −α) + q(x) (−2 x) ( 1
+2 −α)
+q(x) (1−x2) ( 1
+2 −α) + q(y) (1−y2) ( 1
+2 +α) = 0
+(3.4)
+q′(y)
+q(y) −
+2
+x−y + (k−1) q′(y) (1−y2) ( 1
+2 +α) + q(y) (−2 y) ( 1
+2 +α)
+q(x) (1−x2) ( 1
+2 −α) + q(y) (1−y2) ( 1
+2 +α) = 0
+(3.5)
+1
+1
+2 −α −
+1
+1
+2 +α + (k−1)
+q(x) (1−x2) − q(y) (1−y2)
+q(x) (1−x2) ( 1
+2 −α) + q(y) (1−y2) ( 1
+2 +α) = 0
+(3.6)
+5
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+Figure 1: Logit model for k = 3 and β1 = 1: Dependence of x∗
+11 and x∗
+12 (solid lines) and
+the corresponding weights w1 and w2 = 1 − w1 (dashed lines) on −β0 = − β0
+β1 =
+cq ∈ [−1.2, 1.2].
+(c0) If x, y ∈ (−1, 1) with x > y and α ∈ (− 1
+2, 1
+2) is a solution of the equation
+system, the orbit positions are x∗
+11 = x, x∗
+12 = y with weights w1 =
+1
+2 − α
+and w2 = 1
+2 + α.
+(c1) If x
+≥
+1 and y
+∈
+(−1, 1), then x∗
+11
+=
+1, w1
+=
+1
+k+1, w2
+=
+k
+k+1
+and x∗
+12 ∈ (−1, 1) is the solution of the equation (3.2).
+(c2) If y ≤ −1 and x ∈ (−1, 1), then x∗
+12 = −1, w1 =
+k
+k+1, w2 =
+1
+k+1
+and x∗
+11 ∈ (−1, 1) is the solution of the equation (3.3).
+Remark 1. Instead of reproducing the whole theorem for k = 1, only the two main
+changes in case (c) should be mentioned. So the weights are always w1 = w2 = 1
+2 and the
+equation system (3.4)–(3.6) is replaced by
+q′(x)
+q(x) +
+2
+x − y = 0
+and
+q′(y)
+q(y) −
+2
+x − y = 0 .
+(3.7)
+To illustrate this complex issue we revisit the logit model in dimension k = 3 with β1 = 1.
+We (numerically) plot the orbit positions x∗
+11 and x∗
+12 and corresponding weights w1 and
+w2 = 1 − w1 depending on −β0 = − β0
+β1 = cq, see Figure 1. The cases (a) and (b) go along
+with Theorem 1 and the results from Radloff and Schwabe (2019b). The cases (c1) and
+(c2) yield marginal extremum solutions which are identical to (a) and (b). So for these
+four cases there is always an exact minimally supported (locally) D-optimal design. As
+described in Theorem 1, it consists of a pole point in x1 = −1 or else x1 = 1 and the k
+vertices of a (regular) simplex which is maximally inscribed in the non-degenerated orbit.
+But the problematic case is (c0) because the (locally) D-optimal (generalized) design
+consists of two non-degenerated orbits and additionally the weights are rarely appropriate
+for a discretization. In Radloff and Schwabe (2019a) we showed two examples for the logit
+model (k = 3, β1 = 1) from which we derived (nearly) exact designs.
+For −β0 = 0 the two orbit positions are symmetrical around 0, that is x∗
+11 = −x∗
+12 ≈ 0.52,
+and the weights are ξ∗
+1(x∗
+11) = ξ∗
+1(x∗
+12) =
+1
+2. These two orbits were discretized by two
+6
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+2-dimensional simplices — overall 6 equally weighted support points; see Figure 2 (left
+image).
+For −β0 = −0.1 it is x∗
+11 ≈ 0.42, x∗
+12 ≈ −0.62 and ξ∗
+1(x∗
+11) ≈ 0.4297, while 0.4297 ≈ 3
+7.
+We took the rounded design ξ≈ with the same support points x∗
+11 and x∗
+12 but with the
+marginal design ξ≈
+1 (x∗
+11) = 3
+7 and ξ≈
+1 (x∗
+12) = 4
+7. So it was possible to substitute one orbit
+by the vertices of a 2-dimensional simplex (3 points — an equilateral triangle) and one
+by the vertices of a 2-dimensional cube or cross polytope (4 points — a square). Because
+of rounding the design ξ≈ is not optimal but exact and has a high D-efficiency, which
+compares the rounded design ξ≈ and the optimal design ξ∗
+β0 with respect to β0 — here
+p = k + 1 = 4 and β0 = (0.1, 1, 0, 0)⊤:
+EffD(ξ≈, β0) =
+�
+det(M(ξ≈, β0))
+det(M(ξ∗
+β0, β0))
+�1
+p
+≈ 0.999757 .
+These designs are not very satisfactory. On the one hand the number of support points
+is not minimal. On the other hand only special cases have appropriate rational weights
+which allow a discretization or otherwise the optimality is lost by rounding. Therefore we
+want to establish minimal supported exact designs for the case (c0) in this paper. Mostly
+these designs wont be optimal but (highly) efficient.
+But we start with the reduction of the system of three equations in Theorem 2 to only one
+single equation for special unimodal intensity functions — symmetrical unimodal intensity
+functions — which can be found, for example, in binary response models with logit and
+probit link.
+4. Optimal Design for Symmetrical Unimodal
+Intensity Functions
+An interesting observation was made in the discussion section in Radloff and Schwabe
+(2019a). For models with unimodal intensity function in which the mode and threshold
+coincide (c(A2′)
+λ
+= c(A3′)
+λ
+= cλ) and which are symmetrical, also the two orbit positions are
+symmetrical in a certain way, which we want to investigate here. For one dimension this
+has been considered and shown in Ford et al (1992, Section 6.5 and 6.6), but this proof
+cannot be extended to higher dimensions directly.
+Definition 1. An unimodal intensity function in which the mode and threshold coincide
+(c(A2′)
+λ
+= c(A3′)
+λ
+= cλ) will be called symmetrical to cλ if
+λ(cλ + z) = λ(cλ − z)
+for all z ∈ R.
+The intensity functions of the logit and probit models are symmetrical with cλ = 0. But
+the unimodal intensity function of the complementary log-log model has c(A2′)
+λ
+̸= c(A3′)
+λ
+and cannot be symmetrical for this reason.
+Lemma 1. Let the intensity function λ be symmetrical to cλ in the situation of Theo-
+rem 2 (c0).
+7
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+• For given β0 ̸= cλ let r solve
+λ′(cλ+r)
+λ(cλ+r) = −
+−2 k r2 (β2
+1 +c2−r2)+(β2
+1 −c2−r2)2−4 c2 r2
++(β2
+1 −c2+r2)
+�
+(β2
+1 −c2−r2)2+4 (k2−1) c2 r2
+(k+1) r (r+c−β1)(r+c+β1)(r−c+β1)(r−c−β1)
+(4.8)
+with c := cλ − β0. Then
+x = c
+β1
++ r
+β1
+,
+(4.9)
+y = c
+β1
+− r
+β1
+,
+(4.10)
+α =
+−(β2
+1 −c2−r2)+
+�
+(β2
+1 −c2−r2)2+4 (k2−1) c2 r2
+4 (k+1) c r
+(4.11)
+is a solution of the equation system (3.4)–(3.6).
+• For given β0 = cλ it is x =
+r
+β1, y = − r
+β1 and α = 0. Here r is the solution of
+λ′(cλ + r)
+λ(cλ + r) = −
+2 (β2
+1 − k r2)
+(k + 1) r (β2
+1 − r2) .
+(4.12)
+Remark 2. For k = 1, see Remark 1, let λ be symmetrical to cλ. Then x = cλ−β0
+β1
++
+r
+β1
+and y = cλ−β0
+β1
+− r
+β1 with r is solution of
+λ′(cλ + r)
+λ(cλ + r) = −1
+r
+(4.13)
+solve the equation system (3.7).
+Lemma 1, whose proof sketch can be found in Appendix B, and Remark 2 in combination
+with Theorem 2 give (locally) D-optimal designs for models with symmetrical unimodal
+intensity functions. As a result we reduced the system of equations (3.4)–(3.6) to only
+one single equation (4.8).
+But now there is the question if condition (A4) can guarantee a unique solution as in
+Theorem 1 or in Theorem 2 (a) and (b) because Theorem 2 (c), especially (c0), tells
+nothing about uniqueness. But we want to add a remark about the values of r before.
+Remark 3. Since the system of equations (3.4)–(3.6) in Theorem 2 (c0) should have a
+solution with two inner support points for the marginal design, x, y ∈ (−1, 1) is required.
+So
+−1 < cλ − β0
+β1
+± r
+β1
+< 1
+must be valid. This leads with β1 > 0 to r ∈ (−(cλ − β0) − β1, −(cλ − β0) + β1) and r ∈
+((cλ − β0) − β1, (cλ − β0) + β1). Consequently, both intervals must overlap. This happens
+for cλ − β0 > 0 at 0 < cλ − β0 < β1 and for cλ − β0 < 0 at −β1 < cλ − β0 < 0.
+Thus cλ − β0 ∈ (−β1, β1) and in particular β2
+1 > (cλ − β0)2 must hold. Then r is in the
+interval (|cλ − β0| − β1, −|cλ − β0| + β1). But Theorem 2 (c) need x > y and consequently
+r > 0. Hence, r ∈ (0, −|cλ − β0| + β1).
+This remains valid in particular for β0 = cλ, i. e. cλ − β0 = 0. So r ∈ (−β1, β1). With
+r > 0 it is r ∈ (0, β1).
+8
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+Lemma 2. In situation of Lemma 1 let the intensity function λ additionally satisfy
+condition (A4), then equation (4.8), whose right hand side is continuously continued
+in −|cλ − β0| + β1, has a unique solution in r ∈ (0, |cλ − β0| + β1).
+This also holds for β0 = cλ and equation (4.12), which has exactly one solution in r ∈
+(0, β1).
+Remark 4. For k = 1, see Remark 2, and for an intensity function satisfying (A4) there
+is only one solution of (4.13).
+The proof sketch of Lemma 2 can be found in Appendix B. Lemma 2 guarantees a unique
+solution in r ∈ (0, |cλ − β0| + β1). But Remark 3 points out that for Theorem 2 (c0)
+we need r ∈ (0, −|cλ − β0| + β1). This means that the unique solution can result in the
+two-orbit case or in the one-orbit one-pole case of Theorem 2 (c).
+5. Minimally Supported Designs
+In the situation of Theorem 1 and Theorem 2 (a), (b), (c1) and (c2) the designs have
+always the minimal number of support points to estimate the parameter vector β. These
+are k + 1 support points.
+In Radloff and Schwabe (2019a) revisited here in the introductory section we indicated
+exemplarily a (locally) D-optimal design for the logit model on the 3-dimensional ball
+with −β0 = 0 and β1 = 1.
+This design consists of six support points which are the
+vertices of two regular 2-dimensional simplices — equilateral triangles; see Figure 2 (left
+image). But this is not the minimum of support points to estimate the four parameters.
+So the question arises whether it is possible to reduce the number of support points as it
+can be found in the concept of fractional factorial designs, see, for example, Pukelsheim
+(1993, section 15.11). Instead of using all vertices of the hypercube [−1, 1]k as in the
+full factorial design the fractional factorial design picks only a special percentage of these
+points. For k = 3
+(−1, −1, 1)⊤, (−1, 1, −1)⊤, (1, −1, −1)⊤, (1, 1, 1)⊤
+represent a 23−1-fractional factorial design.
+In our issue we do not want to pick four of the six points, but we want to use the
+orthogonality of the spaces spanned by the points (without the x1-component) in the
+two orbits (x1 = −1 and x1 = 1) of the given 23−1-fractional factorial design.
+Here
+span{(−1, 1)⊤, (1, −1)⊤} ⊥ span{(−1, −1)⊤, (1, 1)⊤}.
+The idea for our problem is il-
+lustrated in Figure 2 (right image).
+The spanned spaces by points (without the x1-
+component) in the orbits are orthogonal to each other. And all points span a simplex.
+As stated above a (generalized) design ξ which is rotation invariant with fixed x1 —
+invariant with respect to all orthogonal transformations in O(k) which do not change
+the x1-component — and which has all mass on the unit sphere can be decomposed
+into a marginal design ξ1 on [−1, 1] and a probability kernel η (conditional design), i. e.
+ξ = ξ1 ⊗ η. For fixed x1 the kernel η(x1, ·) is the uniform distribution on the surface of a
+(k − 1)-dimensional ball with radius
+�
+1 − x2
+1 — the orbit at position x1. If x1 ∈ {−1, 1},
+the (k − 1)-dimensional ball with the uniform distribution reduces to a single point and
+represents only a one-point-measure.
+Remembering q(x1) = λ(β0 + β1x1) the related
+9
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+information matrix, see Radloff and Schwabe (2019b), is
+M(ξ1 ⊗ η, β0) =
+�
+�
+�
+�
+q dξ1
+�
+q id dξ1
+�
+q id dξ1
+�
+q id2 dξ1
+O2×(k−1)
+O(k−1)×2
+1
+k−1
+�
+q (1 − id2) dξ1 Ik−1
+�
+�
+�
+(5.14)
+with β0 = (β0, β1, 0, . . . , 0)⊤.
+The information matrix for a design on the k-dimensional unit sphere Sk−1, which is
+based on exactly two orbits, can be determined analogously to this result. Additionally
+the uniform distribution does not cover the the full orbits but only sub-spheres.
+Lemma 3. Let ξ1 be the two-point-measure in x11 and x12 with ξ1(x11) =
+1
+2 − α and
+ξ1(x12) =
+1
+2 + α with α ∈
+�
+− 1
+2, 1
+2
+�
+. Further let η(x11, ·) be a uniform distribution on
+Sm−2
+��
+1 − x2
+11
+�
+× {0}k−m and likewise η(x12, ·) be a uniform distribution on {0}m−1 ×
+Sk−m−1
+��
+1 − x2
+12
+�
+. Then the information matrix is
+M(ξ1 ⊗ η, β0) =
+�
+�
+�
+�
+�
+�
+q dξ1
+�
+q id dξ1
+�
+q id dξ1
+�
+q id2 dξ1
+O2×(k−1)
+O(k−1)×2
+c1 Im−1
+O(m−1)×(k−m)
+O(k−m)×(m−1)
+c2 Ik−m
+�
+�
+�
+�
+�
+(5.15)
+with c1 =
+1
+m−1 q(x11) (1−x2
+11) ( 1
+2 −α) and c2 =
+1
+k−m q(x12) (1−x2
+12) ( 1
+2 +α).
+Now the optimality case in Theorem 2 (c0) on two orbits should be used to investigate
+when both information matrices (5.14) und (5.15) are identical. With that both related
+(generalized) designs would be (locally) D-optimal.
+Lemma 4. Both information matrices (5.14) and (5.15) are identical in the situation of
+Theorem 2 (c0) if and only if α = 1
+2 −
+m
+k+1.
+The proof can be found in Appendix B.
+Consequently both orbits need the weights ξ1(x11) =
+m
+k+1 and ξ1(x12) = k−m+1
+k+1
+to coincide
+both information matrices. This allows an experimental design, which has the same value
+for the D-optimality criterion, consisting of two orbits with m and with k −m+1 support
+Figure 2: Logit model for k = 3 and β1 = 1 and −β0 = 0: discretized (locally) D-optimal
+designs with 6 or 4 support points.
+10
+
+1Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+Figure 3: D-efficiency for the logit model with k = 3 and β1 = 1: comparison of designs
+with exactly k+1 = 4 equally weighted support points in −β0 ∈ (−0.403, 0.403)
+(rounded).
+points. This can be done by two regular simplices — one simplex in dimension m − 1
+and one in dimension k − m.
+So the simplices are the discretizations of the uniform
+distributions on Sm−2
+��
+1 − x2
+11
+�
+× {0}k−m and on {0}m−1 × Sk−m−1
+��
+1 − x2
+12
+�
+.
+Let Sm ∈ Rm×(m+1) be a matrix, where the columns represent the m + 1 vertices of an
+m-dimensional regular simplex (in Rm). Then the columns of the matrix
+�
+�
+�
+x111⊤
+m
+x121⊤
+k−m+1
+R1 Sm−1
+O(m−1)×(k−m+1)
+O(k−m)×m
+R2 Sk−m
+�
+�
+�
+with arbitrary orthogonal transformations R1 ∈ O(m − 1) and R2 ∈ O(k − m) represent
+the support points of such a minimal supported design.
+��
+m + 1
+m
+Im + 1 − √m + 1
+m√m
+1m1⊤
+m
+����� −
+1
+√m 1m
+�
+∈ Rm×(m+1)
+is an example for Sm. In this notation Im stands for the standard simplex which needs
+to be scaled and shifted appropriately so that it is in combination with the last vertex
+− 1
+√m 1m (last column) a regular simplex on the unit sphere Sm−1.
+Finally, we want to look at the D-efficiency, here with β0 = (β0, β1, 0, . . . , 0)⊤,
+EffD(ξ, β0) =
+�
+det(M(ξ, β0))
+det(M(ξ∗
+β0, β0))
+�1
+p
+∈ [0, 1]
+for designs ξ with exactly p = k + 1 equally weighted support points in the region where
+two non-degenerated orbits occur.
+As an example, the logit model with β1 = 1 is used to determine the D-efficiency in
+dimensions k = 3 and k = 6. In Figure 3 and Figure 4 only the regions for −β0 with
+11
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+two non-degenerated orbits in the optimal design (case (c0) in Theorem 2), i. e. −β0 ∈
+(−0.403, 0.403) (rounded) for k = 3 and −β0 ∈ (−0.480, 0.480) (rounded) for k = 6, are
+plotted.
+For this purpose, three different types of exact designs are compared with the (locally)
+D-optimal design ξ∗
+β0.
+The optimal design is a generalized design with real weights.
+Therefore it cannot be discretized as an exact design in general.
+First, the two optimal exact designs with one pole and one orbit, which are discretized as
+a regular (k−1)-dimensional simplex, are used for comparison. The orbit position remains
+unchanged and is determined at the boundary values −β0 ≈ ±0.403 or −β0 ≈ ±0.480.
+See the solid lines in both figures.
+Second, the designs with the same orbit position as the associated design which is (locally)
+optimal for −β0 are the next alternative.
+Only the weights were rounded/shifted to
+integral multiples of
+1
+k+1. See the dotted lines.
+Third, the designs with fixed design weights which are integral multiples of
+1
+k+1 represent
+the last model category. So only the positions of the orbits have to be optimized with
+these fixed design weights. This can be done by solving only the equations (3.4) and (3.5)
+with the selected weights in Theorem 2 (c). Equation (3.6) is omitted. See the dashed
+lines in both plots.
+The Figure 3 reveals for dimension k = 3 that there are only three positions in the
+range −β0 ∈ [−0.403, 0.403] (rounded) where (locally) D-optimal designs with the min-
+imal number of support points — four points — exists. For −β0 ≈ −0.403 this is the
+design consisting of the pole x∗
+12 = −1 and one orbit at x∗
+11 with three points as vertices
+of an equilateral triangle. Then for −β0 = 0 there are two orbits with two points each.
+And, at −β0 ≈ 0.403 the design consists of one orbit at x∗
+12 with three equally weighted
+support points and the pole x∗
+11 = 1. In the span between these optimality positions the
+considered discretizations provide a fairly high efficiency. Using the transition directly
+from pole and orbit to orbit and pole, the efficiency is always greater than 0.988 (intersec-
+tion of the solid lines). If the two orbits are also discretized in between, the efficiency is
+greater than 0.993 (intersection of dotted line and solid lines) or even greater than 0.997
+(intersection of dashed line and solid lines).
+For dimension k = 6, see figure 4, an efficiency of more than 0.986 is possible by stepping
+directly from pole and orbit with six support points to orbit with six design points and
+pole. If the intermediate steps — two orbits with 2 and 5 points, 3 and 4 points, 4 and
+3 points as well as 5 and 2 points — are used, then by simple rounding of the weights to
+integral multiples of
+1
+k+1 an efficiency greater than 0.995 (dotted lines) and with additional
+optimization of the orbit positions even greater than 0.999 (dashed lines) can be achieved.
+6. Conclusion
+In summary it can be postulated that very efficient designs are generated based on only
+k + 1 design points which is the minimal number of support points to estimate the pa-
+rameter vector. It seems that higher dimensions enable designs with higher D-efficiency,
+in particular using the third option of discretization. Here we only considered designs
+with exactly two orbits. Thus it cannot be excluded that there are designs with a better
+efficiency or even (locally) optimal designs which are supported by exactly k + 1 points.
+Maybe these designs have support points which lie not on the orbit but are jittered a little
+bit. This as well as a potential lower efficiency bound needs further investigations.
+12
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+Figure 4: D-efficiency for the logit model with k = 6 and β1 = 1: comparison of designs
+with exactly k+1 = 7 equally weighted support points in −β0 ∈ (−0.480, 0.480)
+(rounded).
+On the other side the reduction of the equation system to one single equation for deter-
+mining (locally) D-optimal design for symmetrical unimodal intensity functions is a nice
+feature and can help to decrease computing costs.
+Also the question of optimal designs on the ball with respect to other optimality criteria
+should be considered in future.
+Finally, we want to emphasize that the established designs do not only work for the
+unit ball. By using the concept of equivariance for linear transformations, say scaling,
+reflecting and rotating, the class of design spaces can be extended to k-dimensional balls
+with arbitrary radius or any k-dimensional ellipsoid.
+Appendix A
+Notation
+Bk
+k-dimensional unit ball
+Bk(r)
+k-dimensional ball with radius r
+Sk−1
+unit sphere, which is the surface of Bk
+Sk−1(r)
+sphere with radius r, which is the surface of Bk(r)
+Ok
+k-dimensional zero-vector
+Ok1×k2
+(k1 × k2)-dimensional zero-matrix
+1k
+k-dimensional one-vector
+Ik
+(k × k)-dimensional identity matrix
+id
+identity function
+13
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+Appendix B
+Proofs
+Proof sketch of Lemma 1. By plugging (4.9) and (4.10) into (3.6) and using the sym-
+metry to simplify, we get
+−2 α (4 c r α+(β2
+1 −c2−r2))+4 (k−1) c r
+� 1
+2 −α
+� � 1
+2 +α
+�
+� 1
+2 −α
+� � 1
+2 +α
+�
+(4 c r α+(β2
+1 −c2−r2))
+= 0 .
+In the numerator there is a polynomial of degree two in α with the two roots α∓(r)
+depending on r:
+α∓(r) :=
+− (β2
+1 − c2 − r2) ∓
+�
+(β2
+1 − c2 − r2)2 + 4 (k + 1) (k − 1) c2 r2
+4 (k + 1) c r
+.
+Now we examine the values of α∓(r) depending on r. Only −|c| − β1, |c| − β1, −|c| + β1
+or |c| + β1 can solve the expression α∓(r) = ± 1
+2. But −|c| − β1 and |c| + β1 are not in the
+interesting region for r. We have
+α− (±(|c| − β1)) = ±1
+2 sign(c)
+and
+α+ (±(|c| − β1)) = ∓1
+2 sign(c) k − 1
+k + 1 .
+Because of limr↗0 α− (r) = sign(c)∞ and limr↘0 α− (r) = − sign(c)∞ the root α−(r) has
+in the interval r ∈ [|c| − β1, −|c| + β1] only values outside (− 1
+2, 1
+2). Hence, α−(r) is not a
+relevant root.
+Since limr→0 α+ (r) = 0 the discontinuity of the root α+(r) in r = 0 can be removed.
+So α+(r) has only values in (− 1
+2, 1
+2) on the interval r ∈ [|c| − β1, −|c| + β1] and α+(r),
+which is (4.11), is the only relevant root.
+After inserting (4.9) and (4.10) into (3.4) as well as (4.9) and (4.10) into (3.5) and sub-
+tracting both obtained equations and simplifying by using the symmetry, we get
+(k + 1) λ′(cλ + r)
+λ(cλ + r)
+= −(k − 1)
+−2 r + α · 4 c
+(β2
+1 − c2 − r2) + α · 4 c r − 2
+r .
+Equation (4.8) follows by plugging α+(r) as α into it and by some simplifications.
+For β0 = cλ, i. e. c = cλ − β0 = 0, we get directly α = 0 by inserting x =
+r
+β1 and y = − r
+β1
+in (3.6) and exploiting the symmetry. This is inserted in (3.4) and in (3.5). The difference
+between these two equations results in (4.12).
+Proof sketch of Lemma 2. This proof is a lot of curve sketching. We start with β0 ̸=
+cλ. The denominator of the right hand side of (4.8) has five roots in r. −|cλ −β0|−β1 < 0
+and |cλ − β0| − β1 < 0 are not in the considered interval (0, |cλ − β0| + β1). In r = −|cλ −
+β0| + β1 there is a discontinuity which can be removed. In r = 0 and in r = |cλ − β0| + β1
+there are two poles. Analyzing these poles for the considered interval we see that the
+values start from −∞ (r ↘ 0) and go up to +∞ (r ↗ |cλ − β0| + β1). Sophisticated
+curve sketching shows that the right hand side of (4.8) is strictly monotonically increasing
+on (0, |cλ − β0| + β1). So it is strictly monotonically increasing and covers (−∞, ∞). In
+combination with (A4) for the left hand side of (4.8) (monotonically decreasing) there is
+exactly one solution.
+For β0 = cλ we can mention that the right hand side of (4.12) is also strictly monotonically
+increasing on (0, β1). Hence, there is only one solution.
+An analogue result holds for the situation in Remark 4.
+14
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+Proof of Lemma 4. Rearranging equation (3.6) equivalently in two ways gives
+q(x12) (1−x2
+12) ( 1
+2 +α) = q(x11) (1−x2
+11) ( 1
+2 −α) k ( 1
+2 +α)−( 1
+2 −α)
+k ( 1
+2 −α)−( 1
+2 +α)
+and
+q(x11) (1−x2
+11) ( 1
+2 −α) = q(x12) (1−x2
+12) ( 1
+2 +α) k ( 1
+2 −α)−( 1
+2 +α)
+k ( 1
+2 +α)−( 1
+2 −α) .
+The two denominators are zero if and only if α = 1
+2 −
+1
+k+1 and α = 1
+2 −
+k
+k+1, respectively.
+But this cannot happen to non-degenerated orbits because 1
+2 −
+k
+k+1 < α < 1
+2 −
+1
+k+1.
+Putting both equations into the diagonal entry of the information matrix (5.14) yield
+1
+k − 1
+�
+q (1 − id2) dξ1
+= q(x11) (1−x2
+11) ( 1
+2 −α)
+�
+1
+k − 1 +
+1
+k − 1 · k ( 1
+2 +α)−( 1
+2 −α)
+k ( 1
+2 −α)−( 1
+2 +α)
+�
+and
+1
+k − 1
+�
+q (1 − id2) dξ1
+= q(x12) (1−x2
+12) ( 1
+2 −α)
+�
+1
+k − 1 · k ( 1
+2 −α)−( 1
+2 +α)
+k ( 1
+2 +α)−( 1
+2 −α) +
+1
+k − 1
+�
+They are identical to the diagonal entries of the information matrix (5.15) in Lemma 3 if
+and only if
+1
+k−1 +
+1
+k−1 · k ( 1
+2 +α)−( 1
+2 −α)
+k ( 1
+2 −α)−( 1
+2 +α) =
+1
+m−1 and
+1
+k−1 · k ( 1
+2 −α)−( 1
+2 +α)
+k ( 1
+2 +α)−( 1
+2 −α) +
+1
+k−1 =
+1
+k−m
+which are both equivalent to α = 1
+2 −
+m
+k+1.
+References
+Biedermann S, Dette H, Zhu W (2006) Optimal designs for dose-response models with
+restricted design spaces. Journal of the American Statistical Association 101:747–759
+Dette H, Melas VB, Pepelyshev A, et al (2005) Optimal designs for three-dimensional
+shape analysis with spherical harmonic descriptors. The Annals of Statistics 33:2758–
+2788
+Dette H, Melas VB, Pepelyshev A (2007) Optimal designs for statistical analysis with
+zernike polynomials. Statistics 41:453–470
+Farrell RH, Kiefer J, Walbran A (1967) Optimum multivariate designs. In: Proceedings
+of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume
+1: Statistics. University of California Press, Berkeley, Calif., pp 113–138
+Ford I, Torsney B, Wu C (1992) The use of a canonical form in the construction of locally
+optimal designs for non-linear problems. Journal of the Royal Statistical Society: Series
+B (Statistical Methodology) 54:569–583
+15
+
+Martin Radloff, Rainer Schwabe
+Exact Designs on the Ball
+Hirao M, Sawa M, Jimbo M (2015) Constructions of φp-optimal rotatable designs on the
+ball. Sankhya A : The Indian Journal of Statistics 77:211–236
+Kiefer JC (1961) Optimum experimental designs v, with applications to systematic and
+rotatable designs. In: Proceedings of the Fourth Berkeley Symposium on Mathematical
+Statistics and Probability, Univ of California Press, pp 381–405
+Konstantinou M, Biedermann S, Kimber A (2014) Optimal designs for two-parameter
+nonlinear models with application to survival models. Statistica Sinica 24:415–428
+Lau TS (1988) d-optimal designs on the unit q-ball. Journal of statistical planning and
+inference 19:299–315
+Pukelsheim F (1993) Optimal Design of Experiments. Wiley Series in Probability and
+Statistics
+Radloff M, Schwabe R (2016) Invariance and equivariance in experimental design for
+nonlinear models. In: Kunert J, Müller CH, Atkinson AC (eds) mODa 11-Advances in
+Model-Oriented Design and Analysis. Springer, p 217–224
+Radloff M, Schwabe R (2019a) Locally d-optimal designs for a wider class of non-linear
+models on the k-dimensional ball with applications to logit and probit models. Statis-
+tical Papers 60:165–177
+Radloff M, Schwabe R (2019b) Locally d-optimal designs for non-linear models on the
+k-dimensional ball. Journal of Statistical Planning and Inference 203:106–116
+Schmidt D, Schwabe R (2017) Optimal design for multiple regression with information
+driven by the linear predictor. Statistica Sinica 27:1371–1384
+16
+

8tE1T4oBgHgl3EQfCAKJ/content/tmp_files/load_file.txt

@@ -0,0 +1,427 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf,len=426
+page_content='D-Optimal and Nearly D-Optimal Exact Designs for  Binary Response on the Ball  Martin Radloff† and Rainer Schwabe‡  Abstract: In this paper the results of Radloff and Schwabe (2019a) will be  extended for a special class of symmetrical intensity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' This includes  binary response models with logit and probit link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' To evaluate the position  and the weights of the two non-degenerated orbits on the k-dimensional ball  usually a system of three equations has to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' The symmetry allows  to reduce this system to a single equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' As a further result, the number of  support points can be reduced to the minimal number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' These minimally sup-  ported designs are highly efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' The results can be generalized to arbitrary  ellipsoidal design regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Key words and phrases: Binary response models, D-optimality, k-dimensional  ball, logit and probit model, multiple regression models, simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Introduction  Spherical design spaces can occur in engineering or physics problems where the validity of  a model may be assumed on a spherical region around a target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' So (linear) models  on spherical design spaces were investigated early in publications like Kiefer (1961) and  Farrell et al (1967) which discuss polynomial regression on the ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' These ideas were  followed up by papers in which also only linear problems were focused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' So Lau (1988)  fitted polynomials on the k-dimensional unit ball by using canonical moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In Dette  et al (2005, 2007) and Hirao et al (2015) harmonic polynomials and Zernike polynomials  were used to be fit on the unit disc (2-dimensional unit ball), the 3- and k-dimensional  unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' On the other hand generalized linear models are also well-examined and used  in practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Logit and probit models, for example, in one dimension on an  interval have already been investigated by Ford et al (1992) and Biedermann et al (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  But there seems to be no available literature which combines both topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  In our publication Radloff and Schwabe (2019b) we took the first step to bring non-  linearity or generalized linear models, respectively, and spherical design regions together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  These results were extended to a wider class of non-linear models in our follow-up paper  Radloff and Schwabe (2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  †corresponding author: Martin Radloff, Institute for Mathematical Stochastics, Otto-von-Guericke-  University, PF 4120, 39016 Magdeburg, Germany, martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='radloff@ovgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='de  ‡Rainer Schwabe, Institute for Mathematical Stochastics, Otto-von-Guericke-University, PF 4120,  39016 Magdeburg, Germany, rainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='schwabe@ovgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='de  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='02859v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='ME]  7 Jan 2023  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  For better comprehensibility, we will start with the model description and give a brief  overview of the findings so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Then we will consider a special class of intensity func-  tions which allows to reduce the the complexity of finding (locally) D-optimal designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Afterwards we will tackle the problem, that the optimal designs are not exact designs in  general, by establishing highly efficient designs on the ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' General Model Description  As in Radloff and Schwabe (2019b) and Radloff and Schwabe (2019a), where we described  (locally) D-optimal designs for two special classes of linear and non-linear models on a  k-dimensional unit ball Bk = {x ∈ Rk : x2  1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' + x2  k ≤ 1} with k ∈ N, we solely focus  (non-linear) multiple regression models, which means the linear predictor is  f(x)⊤β = β0 + β1x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' + βkxk  with regression function f : Bk → Rk+1, x �→ (1, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' , xk)⊤, and parameter vector  β = (β0, β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' , βk)⊤ ∈ Rk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' The one-support-point (or elemental) information matrix  should be representable in the form  M(x, β) = λ  �  f(x)⊤β  �  f(x)f(x)⊤  with an intensity (or efficiency) function λ which only depends on the value of the linear  predictor f(x)⊤β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' These one-support-point (or elemental) information matrices are the  base for the information matrix of a (generalized) design ξ with independent observations  M(ξ, β) =  �  M(x, β) ξ(dx) =  �  λ  �  f(x)⊤β  �  f(x)f(x)⊤ξ(dx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Here generalized design means an arbitrary probability measure on the design region Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  These information matrices allow to define the (local) D-optimality, which is one of the  most popular criteria in experimental design theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' A design ξ∗  β0 with regular infor-  mation matrix M(ξ∗  β0, β0) is called (locally) D-optimal (at β0) if det(M(ξ∗  β0, β0)) ≥  det(M(ξ, β0)) holds for all suitable probability measures ξ on the design space — here  Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' This optimality criterion can be interpreted as the minimization of the volume of the  (asymptotic) confidence ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Prior Results  In Radloff and Schwabe (2016) we stated results on equivariance and invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  By  rotating the design space Bk — the k-dimensional unit ball — and the parameter space  Rk+1 in an analogous way the linear predictor of the multiple regression problem reduces  to  f(x)⊤β = β0 + β1x1  and  β1 ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='1)  Using the rotation invariance with fixed x1, this means the invariance to all orthogonal  transformations in O(k) which let the x1-component unchanged, the (locally) D-optimal  (generalized) design ξ∗ can be decomposed (ξ∗ = ξ∗  1 ⊗ η) in a marginal probability measure  ξ∗  1 on [−1, 1] for x1 and a probability kernel η given x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' For fixed x1 the kernel η(x1, ·) is  2  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  the uniform distribution on the surface of a (k − 1)-dimensional ball with radius  �  1 − x2  1  — the orbit at position x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  As a consequence the multidimensional problem collapses to a one-dimensional marginal  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Only the positions of the orbits and their weights have to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' To get  an exact design the uniform orbits have to be discretized, for example, by using regular  simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  In our first paper — Radloff and Schwabe (2019b) — we started with models where the  intensity function belongs to the class of monotonous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Such models have already  been investigated in one dimension, for example, by Konstantinou et al (2014) and on  multidimensional cuboids or orthants by Schmidt and Schwabe (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' These authors  gave the following four conditions on the intensity function λ:  (A1) λ is positive on R and twice continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (A2) The first derivative λ′ is positive on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (A3) The second derivative u′′ of u = 1  λ is injective on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (A4) The function λ′  λ is non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Condition (A2) is the motivation for the name class of monotonous intensity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The intensity functions of this class have to satisfy always (A1) to (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' (A4) is an extra  condition to guarantee uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' For a concise notation  q(x1) = λ(β0 + β1x1)  is used and the properties (A1), (A2), (A3) and (A4) transfer to q for β1 > 0, respectively,  and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Poisson regression with intensity function qP(x1) = exp(β0 + β1x1) and  negative binomial regression as well as special proportional hazard models with censoring,  see Schmidt and Schwabe (2017), satisfy all four conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  If β1 = 0 then the intensity function q is always a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' This yields to a (locally)  D-optimal design as it can be found in linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In Pukelsheim (1993, section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='12)  such a design consists of the equally weighted vertices of a regular simplex inscribed in the  unit sphere, the boundary of the design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' The orientation of the simplex is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The main result for β1 > 0 in Radloff and Schwabe (2019b) is recited for the readers’  convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' There is a (locally) D-optimal design for the multiple regression prob-  lem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='1) with β1 > 0 and intensity function satisfying (A1)-(A3) which has one support  point equal to (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' , 0)⊤ and the other k support points are the vertices of an arbi-  trarily rotated (k − 1)-dimensional regular simplex which is maximally inscribed in the  intersection of the k-dimensional unit ball and a hyperplane with x1 = x∗  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  For k ≥ 2 the position x∗  12 ∈ (−1, 1) is solution of  q′(x∗  12)  q(x∗  12) = 2 (1 + kx∗  12)  k (1 − x∗ 2  12 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  If additionally (A4) is satisfied, the solution x∗  12 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  3  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  For k = 1 the position x∗  12 ∈ [−1, 1) is either solution of  q′(x∗  12)  q(x∗  12) =  2  1 − x∗  12  ,  if such a solution exists in [−1, 1), or otherwise x∗  12 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  If additionally (A4) is satisfied, the solution x∗  12 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The design is equally weighted with  1  k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  It should be noted, that for fixed β this theorem does not need (A1) to (A4) on the  entire real line R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' It is enough to have it in the ball and so on x1 ∈ [−1, 1] for q and on  [β0 − β1, β0 + β1] for λ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' But the model has to satisfy the conditions always  on the whole real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  In our second paper — Radloff and Schwabe (2019a) — the conditions (A2) and (A3)  were replaced by (A2′) and (A3′) and a fifth property (A5) was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (A2′) λ is unimodal with mode c(A2′)  λ  ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (A3′) There exists a threshold c(A3′)  λ  ∈ R so that the second derivative u′′ of u = 1  λ is  both injective on (−∞, c(A3′)  λ  ] and injective on [c(A3′)  λ  , ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (A5) u = 1  λ dominates z2 asymptotically for z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  In this context condition (A2′) means that there exists a c(A2′)  λ  ∈ R so that λ′ is positive  on (−∞, c(A2′)  λ  ) and negative on (c(A2′)  λ  , ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Hence, there is only one local maximum  which is simultaneously the global maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' So the class of intensity functions, which  satisfy (A1), (A2′) and (A3′), is called class of unimodal intensity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Indeed (A2) or (A3) do not imply (A2′) or (A3′), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' As mentioned before, we  only focus on the unit ball and the interval x1 ∈ [−1, 1] for q or [β0 − β1, β0 + β1] for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  So in our special case (A2) and (A3) can be transferred to (A2′) and (A3′) by using an  arbitrary cλ > β0 + β1, which means that cq lies outside the interval [−1, 1] and only one  branch of the function is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Property (A5) means  lim  z→∞  ����  u(z)  z2  ���� = ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  This means that u(z) =  1  λ(z) goes faster to (±) infinity than z2 for z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  As (A1) to (A4) the conditions (A2′), (A3′) and (A5) transfer from the intensity function  λ to the abbreviated form q for β1 > 0 and vice versa — analogously c(·)  q  =  c(·)  λ −β0  β1  with  (·) is (A2′), (A3′) or empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The logit model has the intensity function  qlogit(x1) =  exp(β0 + β1x1)  (1 + exp(β0 + β1x1))2  and probit model has  qprobit(x1) =  φ2(β0 + β1x1)  Φ(β0 + β1x1) · (1 − Φ(β0 + β1x1))  4  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  with the density function φ and cumulative distribution function Φ of the standard normal  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Both models satisfy all five conditions (A1), (A2′), (A3′), (A4), (A5) and  share a common c(A2′)  λ  = c(A3′)  λ  = 0, say cλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Analogously cq = − β0  β1 for q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Beside these two models other models like the complementary log-log model, see Ford  et al (1992), with intensity function λcomp log log(z) =  exp(2z)  exp(exp(z))−1 satisfy all five conditions  with c(A2′)  λ  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='466011 and c(A3′)  λ  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='049084, but here mode c(A2′)  λ  and threshold c(A3′)  λ  do  not coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  We showed that if the (concise) intensity function q satisfies (A1), (A2′), (A3′) and (A5)  the (locally) D-optimal design ξ∗ = ξ∗  1 ⊗η is concentrated on exactly two orbits, which are  the support points of the marginal design ξ∗  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' The idea of the proof is based on Biedermann  et al (2006) and Konstantinou et al (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The next theorem is the main result of our second paper — Radloff and Schwabe (2019a)  — and is reproduced for the readers’ convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' It characterizes the positions of the  two support points of the optimal marginal design ξ∗  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' For k ≥ 2 the simplified problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='1) with β1 > 0 and intensity function q  satisfying (A1), (A2′), (A3′) and (A5) has a (locally) D-optimal marginal design ξ∗  1 with  exactly 2 support points x∗  11 and x∗  12 with x∗  11 > x∗  12 and weights w1 = ξ∗  1(x∗  11) and w2 =  ξ∗  1(x∗  12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  There are 3 cases:  (a) If c(A2′)  q  > 1 and c(A3′)  q  /∈ [−1, 1], then x∗  11 = 1, w1 =  1  k+1, w2 =  k  k+1 and x∗  12 ∈ (−1, 1)  is solution of  q′(x∗  12)  q(x∗  12) = 2 (1 + kx∗  12)  k (1 − x∗ 2  12 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='2)  If additionally (A4) is satisfied, the solution x∗  12 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (b) If c(A2′)  q  < −1 and c(A3′)  q  /∈ [−1, 1], then x∗  12 = −1, w1 =  k  k+1, w2 =  1  k+1 and  x∗  11 ∈ (−1, 1) is solution of  q′(x∗  11)  q(x∗  11) = 2 (−1 + kx∗  11)  k (1 − x∗ 2  11 )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='3)  If additionally (A4) is satisfied, the solution x∗  11 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (c) Otherwise c(A2′)  q  ∈ [−1, 1] or c(A3′)  q  ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Let x, y ∈ R with x > y and α ∈  �  − 1  2, 1  2  �  be solution of the equation system:  q′(x)  q(x) +  2  x−y + (k−1) q′(x) (1−x2) ( 1  2 −α) + q(x) (−2 x) ( 1  2 −α)  q(x) (1−x2) ( 1  2 −α) + q(y) (1−y2) ( 1  2 +α) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='4)  q′(y)  q(y) −  2  x−y + (k−1) q′(y) (1−y2) ( 1  2 +α) + q(y) (−2 y) ( 1  2 +α)  q(x) (1−x2) ( 1  2 −α) + q(y) (1−y2) ( 1  2 +α) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='5)  1  1  2 −α −  1  1  2 +α + (k−1)  q(x) (1−x2) − q(y) (1−y2)  q(x) (1−x2) ( 1  2 −α) + q(y) (1−y2) ( 1  2 +α) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='6)  5  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  Figure 1: Logit model for k = 3 and ��1 = 1: Dependence of x∗  11 and x∗  12 (solid lines) and  the corresponding weights w1 and w2 = 1 − w1 (dashed lines) on −β0 = − β0  β1 =  cq ∈ [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (c0) If x, y ∈ (−1, 1) with x > y and α ∈ (− 1  2, 1  2) is a solution of the equation  system, the orbit positions are x∗  11 = x, x∗  12 = y with weights w1 =  1  2 − α  and w2 = 1  2 + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (c1) If x  ≥  1 and y  ∈  (−1, 1), then x∗  11  =  1, w1  =  1  k+1, w2  =  k  k+1  and x∗  12 ∈ (−1, 1) is the solution of the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (c2) If y ≤ −1 and x ∈ (−1, 1), then x∗  12 = −1, w1 =  k  k+1, w2 =  1  k+1  and x∗  11 ∈ (−1, 1) is the solution of the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Instead of reproducing the whole theorem for k = 1, only the two main  changes in case (c) should be mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' So the weights are always w1 = w2 = 1  2 and the  equation system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='4)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='6) is replaced by  q′(x)  q(x) +  2  x − y = 0  and  q′(y)  q(y) −  2  x − y = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='7)  To illustrate this complex issue we revisit the logit model in dimension k = 3 with β1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  We (numerically) plot the orbit positions x∗  11 and x∗  12 and corresponding weights w1 and  w2 = 1 − w1 depending on −β0 = − β0  β1 = cq, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' The cases (a) and (b) go along  with Theorem 1 and the results from Radloff and Schwabe (2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' The cases (c1) and  (c2) yield marginal extremum solutions which are identical to (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' So for these  four cases there is always an exact minimally supported (locally) D-optimal design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' As  described in Theorem 1, it consists of a pole point in x1 = −1 or else x1 = 1 and the k  vertices of a (regular) simplex which is maximally inscribed in the non-degenerated orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  But the problematic case is (c0) because the (locally) D-optimal (generalized) design  consists of two non-degenerated orbits and additionally the weights are rarely appropriate  for a discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In Radloff and Schwabe (2019a) we showed two examples for the logit  model (k = 3, β1 = 1) from which we derived (nearly) exact designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  For −β0 = 0 the two orbit positions are symmetrical around 0, that is x∗  11 = −x∗  12 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='52,  and the weights are ξ∗  1(x∗  11) = ξ∗  1(x∗  12) =  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' These two orbits were discretized by two  6  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  2-dimensional simplices — overall 6 equally weighted support points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' see Figure 2 (left  image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  For −β0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='1 it is x∗  11 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='42, x∗  12 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='62 and ξ∗  1(x∗  11) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='4297, while 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='4297 ≈ 3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  We took the rounded design ξ≈ with the same support points x∗  11 and x∗  12 but with the  marginal design ξ≈  1 (x∗  11) = 3  7 and ξ≈  1 (x∗  12) = 4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' So it was possible to substitute one orbit  by the vertices of a 2-dimensional simplex (3 points — an equilateral triangle) and one  by the vertices of a 2-dimensional cube or cross polytope (4 points — a square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Because  of rounding the design ξ≈ is not optimal but exact and has a high D-efficiency, which  compares the rounded design ξ≈ and the optimal design ξ∗  β0 with respect to β0 — here  p = k + 1 = 4 and β0 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='1, 1, 0, 0)⊤:  EffD(ξ≈, β0) =  �  det(M(ξ≈, β0))  det(M(ξ∗  β0, β0))  �1  p  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='999757 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  These designs are not very satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' On the one hand the number of support points  is not minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' On the other hand only special cases have appropriate rational weights  which allow a discretization or otherwise the optimality is lost by rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Therefore we  want to establish minimal supported exact designs for the case (c0) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Mostly  these designs wont be optimal but (highly) efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  But we start with the reduction of the system of three equations in Theorem 2 to only one  single equation for special unimodal intensity functions — symmetrical unimodal intensity  functions — which can be found, for example, in binary response models with logit and  probit link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Optimal Design for Symmetrical Unimodal  Intensity Functions  An interesting observation was made in the discussion section in Radloff and Schwabe  (2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' For models with unimodal intensity function in which the mode and threshold  coincide (c(A2′)  λ  = c(A3′)  λ  = cλ) and which are symmetrical, also the two orbit positions are  symmetrical in a certain way, which we want to investigate here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' For one dimension this  has been considered and shown in Ford et al (1992, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='6), but this proof  cannot be extended to higher dimensions directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' An unimodal intensity function in which the mode and threshold coincide  (c(A2′)  λ  = c(A3′)  λ  = cλ) will be called symmetrical to cλ if  λ(cλ + z) = λ(cλ − z)  for all z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The intensity functions of the logit and probit models are symmetrical with cλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' But  the unimodal intensity function of the complementary log-log model has c(A2′)  λ  ̸= c(A3′)  λ  and cannot be symmetrical for this reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Let the intensity function λ be symmetrical to cλ in the situation of Theo-  rem 2 (c0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  7  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  For given β0 ̸= cλ let r solve  λ′(cλ+r)  λ(cλ+r) = −  −2 k r2 (β2  1 +c2−r2)+(β2  1 −c2−r2)2−4 c2 r2  +(β2  1 −c2+r2)  �  (β2  1 −c2−r2)2+4 (k2−1) c2 r2  (k+1) r (r+c−β1)(r+c+β1)(r−c+β1)(r−c−β1)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='8)  with c := cλ − β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Then  x = c  β1  + r  β1  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='9)  y = c  β1  − r  β1  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='10)  α =  −(β2  1 −c2−r2)+  �  (β2  1 −c2−r2)2+4 (k2−1) c2 r2  4 (k+1) c r  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='11)  is a solution of the equation system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='4)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  For given β0 = cλ it is x =  r  β1, y = − r  β1 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Here r is the solution of  λ′(cλ + r)  λ(cλ + r) = −  2 (β2  1 − k r2)  (k + 1) r (β2  1 − r2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='12)  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' For k = 1, see Remark 1, let λ be symmetrical to cλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Then x = cλ−β0  β1  +  r  β1  and y = cλ−β0  β1  − r  β1 with r is solution of  λ′(cλ + r)  λ(cλ + r) = −1  r  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='13)  solve the equation system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Lemma 1, whose proof sketch can be found in Appendix B, and Remark 2 in combination  with Theorem 2 give (locally) D-optimal designs for models with symmetrical unimodal  intensity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' As a result we reduced the system of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='4)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='6) to only  one single equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  But now there is the question if condition (A4) can guarantee a unique solution as in  Theorem 1 or in Theorem 2 (a) and (b) because Theorem 2 (c), especially (c0), tells  nothing about uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' But we want to add a remark about the values of r before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Since the system of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='4)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='6) in Theorem 2 (c0) should have a  solution with two inner support points for the marginal design, x, y ∈ (−1, 1) is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  So  −1 < cλ − β0  β1  ± r  β1  < 1  must be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' This leads with β1 > 0 to r ∈ (−(cλ − β0) − β1, −(cλ − β0) + β1) and r ∈  ((cλ − β0) − β1, (cλ − β0) + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Consequently, both intervals must overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' This happens  for cλ − β0 > 0 at 0 < cλ − β0 < β1 and for cλ − β0 < 0 at −β1 < cλ − β0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Thus cλ − β0 ∈ (−β1, β1) and in particular β2  1 > (cλ − β0)2 must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Then r is in the  interval (|cλ − β0| − β1, −|cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' But Theorem 2 (c) need x > y and consequently  r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Hence, r ∈ (0, −|cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  This remains valid in particular for β0 = cλ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' cλ − β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' So r ∈ (−β1, β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' With  r > 0 it is r ∈ (0, β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  8  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In situation of Lemma 1 let the intensity function λ additionally satisfy  condition (A4), then equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='8), whose right hand side is continuously continued  in −|cλ − β0| + β1, has a unique solution in r ∈ (0, |cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  This also holds for β0 = cλ and equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='12), which has exactly one solution in r ∈  (0, β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' For k = 1, see Remark 2, and for an intensity function satisfying (A4) there  is only one solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The proof sketch of Lemma 2 can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Lemma 2 guarantees a unique  solution in r ∈ (0, |cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' But Remark 3 points out that for Theorem 2 (c0)  we need r ∈ (0, −|cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' This means that the unique solution can result in the  two-orbit case or in the one-orbit one-pole case of Theorem 2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Minimally Supported Designs  In the situation of Theorem 1 and Theorem 2 (a), (b), (c1) and (c2) the designs have  always the minimal number of support points to estimate the parameter vector β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' These  are k + 1 support points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  In Radloff and Schwabe (2019a) revisited here in the introductory section we indicated  exemplarily a (locally) D-optimal design for the logit model on the 3-dimensional ball  with −β0 = 0 and β1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  This design consists of six support points which are the  vertices of two regular 2-dimensional simplices — equilateral triangles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' see Figure 2 (left  image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' But this is not the minimum of support points to estimate the four parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  So the question arises whether it is possible to reduce the number of support points as it  can be found in the concept of fractional factorial designs, see, for example, Pukelsheim  (1993, section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Instead of using all vertices of the hypercube [−1, 1]k as in the  full factorial design the fractional factorial design picks only a special percentage of these  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' For k = 3  (−1, −1, 1)⊤, (−1, 1, −1)⊤, (1, −1, −1)⊤, (1, 1, 1)⊤  represent a 23−1-fractional factorial design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  In our issue we do not want to pick four of the six points, but we want to use the  orthogonality of the spaces spanned by the points (without the x1-component) in the  two orbits (x1 = −1 and x1 = 1) of the given 23−1-fractional factorial design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Here  span{(−1, 1)⊤, (1, −1)⊤} ⊥ span{(−1, −1)⊤, (1, 1)⊤}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The idea for our problem is il-  lustrated in Figure 2 (right image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The spanned spaces by points (without the x1-  component) in the orbits are orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' And all points span a simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  As stated above a (generalized) design ξ which is rotation invariant with fixed x1 —  invariant with respect to all orthogonal transformations in O(k) which do not change  the x1-component — and which has all mass on the unit sphere can be decomposed  into a marginal design ξ1 on [−1, 1] and a probability kernel η (conditional design), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  ξ = ξ1 ⊗ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' For fixed x1 the kernel η(x1, ·) is the uniform distribution on the surface of a  (k − 1)-dimensional ball with radius  �  1 − x2  1 — the orbit at position x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' If x1 ∈ {−1, 1},  the (k − 1)-dimensional ball with the uniform distribution reduces to a single point and  represents only a one-point-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Remembering q(x1) = λ(β0 + β1x1) the related  9  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  information matrix, see Radloff and Schwabe (2019b), is  M(ξ1 ⊗ η, β0) =  �  �  �  �  q dξ1  �  q id dξ1  �  q id dξ1  �  q id2 dξ1  O2×(k−1)  O(k−1)×2  1  k−1  �  q (1 − id2) dξ1 Ik−1  �  �  �  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='14)  with β0 = (β0, β1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' , 0)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The information matrix for a design on the k-dimensional unit sphere Sk−1, which is  based on exactly two orbits, can be determined analogously to this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Additionally  the uniform distribution does not cover the the full orbits but only sub-spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Let ξ1 be the two-point-measure in x11 and x12 with ξ1(x11) =  1  2 − α and  ξ1(x12) =  1  2 + α with α ∈  �  − 1  2, 1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Further let η(x11, ·) be a uniform distribution on  Sm−2  ��  1 − x2  11  �  × {0}k−m and likewise η(x12, ·) be a uniform distribution on {0}m−1 ×  Sk−m−1  ��  1 − x2  12  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Then the information matrix is  M(ξ1 ⊗ η, β0) =  �  �  �  �  �  �  q dξ1  �  q id dξ1  �  q id dξ1  �  q id2 dξ1  O2×(k−1)  O(k−1)×2  c1 Im−1  O(m−1)×(k−m)  O(k−m)×(m−1)  c2 Ik−m  �  �  �  �  �  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='15)  with c1 =  1  m−1 q(x11) (1−x2  11) ( 1  2 −α) and c2 =  1  k−m q(x12) (1−x2  12) ( 1  2 +α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Now the optimality case in Theorem 2 (c0) on two orbits should be used to investigate  when both information matrices (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='14) und (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='15) are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' With that both related  (generalized) designs would be (locally) D-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Both information matrices (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='14) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='15) are identical in the situation of  Theorem 2 (c0) if and only if α = 1  2 −  m  k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The proof can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Consequently both orbits need the weights ξ1(x11) =  m  k+1 and ξ1(x12) = k−m+1  k+1  to coincide  both information matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' This allows an experimental design, which has the same value  for the D-optimality criterion, consisting of two orbits with m and with k −m+1 support  Figure 2: Logit model for k = 3 and β1 = 1 and −β0 = 0: discretized (locally) D-optimal  designs with 6 or 4 support points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  10  1Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  Figure 3: D-efficiency for the logit model with k = 3 and β1 = 1: comparison of designs  with exactly k+1 = 4 equally weighted support points in −β0 ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='403, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='403)  (rounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' This can be done by two regular simplices — one simplex in dimension m − 1  and one in dimension k − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  So the simplices are the discretizations of the uniform  distributions on Sm−2  ��  1 − x2  11  �  × {0}k−m and on {0}m−1 × Sk−m−1  ��  1 − x2  12  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Let Sm ∈ Rm×(m+1) be a matrix, where the columns represent the m + 1 vertices of an  m-dimensional regular simplex (in Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Then the columns of the matrix  �  �  �  x111⊤  m  x121⊤  k−m+1  R1 Sm−1  O(m−1)×(k−m+1)  O(k−m)×m  R2 Sk−m  �  �  �  with arbitrary orthogonal transformations R1 ∈ O(m − 1) and R2 ∈ O(k − m) represent  the support points of such a minimal supported design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  ��  m + 1  m  Im + 1 − √m + 1  m√m  1m1⊤  m  ����� −  1  √m 1m  �  ∈ Rm×(m+1)  is an example for Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In this notation Im stands for the standard simplex which needs  to be scaled and shifted appropriately so that it is in combination with the last vertex  − 1  √m 1m (last column) a regular simplex on the unit sphere Sm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Finally, we want to look at the D-efficiency, here with β0 = (β0, β1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' , 0)⊤,  EffD(ξ, β0) =  �  det(M(ξ, β0))  det(M(ξ∗  β0, β0))  �1  p  ∈ [0, 1]  for designs ξ with exactly p = k + 1 equally weighted support points in the region where  two non-degenerated orbits occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  As an example, the logit model with β1 = 1 is used to determine the D-efficiency in  dimensions k = 3 and k = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In Figure 3 and Figure 4 only the regions for −β0 with  11  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  two non-degenerated orbits in the optimal design (case (c0) in Theorem 2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' −β0 ∈  (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='403, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='403) (rounded) for k = 3 and −β0 ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='480, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='480) (rounded) for k = 6, are  plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  For this purpose, three different types of exact designs are compared with the (locally)  D-optimal design ξ∗  β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The optimal design is a generalized design with real weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Therefore it cannot be discretized as an exact design in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  First, the two optimal exact designs with one pole and one orbit, which are discretized as  a regular (k−1)-dimensional simplex, are used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' The orbit position remains  unchanged and is determined at the boundary values −β0 ≈ ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='403 or −β0 ≈ ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  See the solid lines in both figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Second, the designs with the same orbit position as the associated design which is (locally)  optimal for −β0 are the next alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Only the weights were rounded/shifted to  integral multiples of  1  k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' See the dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Third, the designs with fixed design weights which are integral multiples of  1  k+1 represent  the last model category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' So only the positions of the orbits have to be optimized with  these fixed design weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' This can be done by solving only the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='5)  with the selected weights in Theorem 2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='6) is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' See the dashed  lines in both plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The Figure 3 reveals for dimension k = 3 that there are only three positions in the  range −β0 ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='403, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='403] (rounded) where (locally) D-optimal designs with the min-  imal number of support points — four points — exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' For −β0 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='403 this is the  design consisting of the pole x∗  12 = −1 and one orbit at x∗  11 with three points as vertices  of an equilateral triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Then for −β0 = 0 there are two orbits with two points each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  And, at −β0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='403 the design consists of one orbit at x∗  12 with three equally weighted  support points and the pole x∗  11 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In the span between these optimality positions the  considered discretizations provide a fairly high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Using the transition directly  from pole and orbit to orbit and pole, the efficiency is always greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='988 (intersec-  tion of the solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' If the two orbits are also discretized in between, the efficiency is  greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='993 (intersection of dotted line and solid lines) or even greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='997  (intersection of dashed line and solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  For dimension k = 6, see figure 4, an efficiency of more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='986 is possible by stepping  directly from pole and orbit with six support points to orbit with six design points and  pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' If the intermediate steps — two orbits with 2 and 5 points, 3 and 4 points, 4 and  3 points as well as 5 and 2 points — are used, then by simple rounding of the weights to  integral multiples of  1  k+1 an efficiency greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='995 (dotted lines) and with additional  optimization of the orbit positions even greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='999 (dashed lines) can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Conclusion  In summary it can be postulated that very efficient designs are generated based on only  k + 1 design points which is the minimal number of support points to estimate the pa-  rameter vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' It seems that higher dimensions enable designs with higher D-efficiency,  in particular using the third option of discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Here we only considered designs  with exactly two orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Thus it cannot be excluded that there are designs with a better  efficiency or even (locally) optimal designs which are supported by exactly k + 1 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Maybe these designs have support points which lie not on the orbit but are jittered a little  bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' This as well as a potential lower efficiency bound needs further investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  12  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  Figure 4: D-efficiency for the logit model with k = 6 and β1 = 1: comparison of designs  with exactly k+1 = 7 equally weighted support points in −β0 ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='480, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='480)  (rounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  On the other side the reduction of the equation system to one single equation for deter-  mining (locally) D-optimal design for symmetrical unimodal intensity functions is a nice  feature and can help to decrease computing costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Also the question of optimal designs on the ball with respect to other optimality criteria  should be considered in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Finally, we want to emphasize that the established designs do not only work for the  unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' By using the concept of equivariance for linear transformations, say scaling,  reflecting and rotating, the class of design spaces can be extended to k-dimensional balls  with arbitrary radius or any k-dimensional ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Appendix A  Notation  Bk  k-dimensional unit ball  Bk(r)  k-dimensional ball with radius r  Sk−1  unit sphere, which is the surface of Bk  Sk−1(r)  sphere with radius r, which is the surface of Bk(r)  Ok  k-dimensional zero-vector  Ok1×k2  (k1 × k2)-dimensional zero-matrix  1k  k-dimensional one-vector  Ik  (k × k)-dimensional identity matrix  id  identity function  13  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  Appendix B  Proofs  Proof sketch of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' By plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='10) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='6) and using the sym-  metry to simplify, we get  −2 α (4 c r α+(β2  1 −c2−r2))+4 (k−1) c r  � 1  2 −α  � � 1  2 +α  �  � 1  2 −α  � � 1  2 +α  �  (4 c r α+(β2  1 −c2−r2))  = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  In the numerator there is a polynomial of degree two in α with the two roots α∓(r)  depending on r:  α∓(r) :=  − (β2  1 − c2 − r2) ∓  �  (β2  1 − c2 − r2)2 + 4 (k + 1) (k − 1) c2 r2  4 (k + 1) c r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Now we examine the values of α∓(r) depending on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Only −|c| − β1, |c| − β1, −|c| + β1  or |c| + β1 can solve the expression α∓(r) = ± 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' But −|c| − β1 and |c| + β1 are not in the  interesting region for r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' We have  α− (±(|c| − β1)) = ±1  2 sign(c)  and  α+ (±(|c| − β1)) = ∓1  2 sign(c) k − 1  k + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Because of limr↗0 α− (r) = sign(c)∞ and limr↘0 α− (r) = − sign(c)∞ the root α−(r) has  in the interval r ∈ [|c| − β1, −|c| + β1] only values outside (− 1  2, 1  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Hence, α−(r) is not a  relevant root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Since limr→0 α+ (r) = 0 the discontinuity of the root α+(r) in r = 0 can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  So α+(r) has only values in (− 1  2, 1  2) on the interval r ∈ [|c| − β1, −|c| + β1] and α+(r),  which is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='11), is the only relevant root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  After inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='10) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='4) as well as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='10) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='5) and sub-  tracting both obtained equations and simplifying by using the symmetry, we get  (k + 1) λ′(cλ + r)  λ(cλ + r)  = −(k − 1)  −2 r + α · 4 c  (β2  1 − c2 − r2) + α · 4 c r − 2  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='8) follows by plugging α+(r) as α into it and by some simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  For β0 = cλ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' c = cλ − β0 = 0, we get directly α = 0 by inserting x =  r  β1 and y = − r  β1  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='6) and exploiting the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' This is inserted in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='4) and in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' The difference  between these two equations results in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Proof sketch of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' This proof is a lot of curve sketching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' We start with β0 ̸=  cλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' The denominator of the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='8) has five roots in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' −|cλ −β0|−β1 < 0  and |cλ − β0| − β1 < 0 are not in the considered interval (0, |cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In r = −|cλ −  β0| + β1 there is a discontinuity which can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In r = 0 and in r = |cλ − β0| + β1  there are two poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Analyzing these poles for the considered interval we see that the  values start from −∞ (r ↘ 0) and go up to +∞ (r ↗ |cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Sophisticated  curve sketching shows that the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='8) is strictly monotonically increasing  on (0, |cλ − β0| + β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' So it is strictly monotonically increasing and covers (−∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In  combination with (A4) for the left hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='8) (monotonically decreasing) there is  exactly one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  For β0 = cλ we can mention that the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='12) is also strictly monotonically  increasing on (0, β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Hence, there is only one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  An analogue result holds for the situation in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  14  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Rearranging equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='6) equivalently in two ways gives  q(x12) (1−x2  12) ( 1  2 +α) = q(x11) (1−x2  11) ( 1  2 −α) k ( 1  2 +α)−( 1  2 −α)  k ( 1  2 −α)−( 1  2 +α)  and  q(x11) (1−x2  11) ( 1  2 −α) = q(x12) (1−x2  12) ( 1  2 +α) k ( 1  2 −α)−( 1  2 +α)  k ( 1  2 +α)−( 1  2 −α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  The two denominators are zero if and only if α = 1  2 −  1  k+1 and α = 1  2 −  k  k+1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  But this cannot happen to non-degenerated orbits because 1  2 −  k  k+1 < α < 1  2 −  1  k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  Putting both equations into the diagonal entry of the information matrix (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='14) yield  1  k − 1  �  q (1 − id2) dξ1  = q(x11) (1−x2  11) ( 1  2 −α)  �  1  k − 1 +  1  k − 1 · k ( 1  2 +α)−( 1  2 −α)  k ( 1  2 −α)−( 1  2 +α)  �  and  1  k − 1  �  q (1 − id2) dξ1  = q(x12) (1−x2  12) ( 1  2 −α)  �  1  k − 1 · k ( 1  2 −α)−( 1  2 +α)  k ( 1  2 +α)−( 1  2 −α) +  1  k − 1  �  They are identical to the diagonal entries of the information matrix (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='15) in Lemma 3 if  and only if  1  k−1 +  1  k−1 · k ( 1  2 +α)−( 1  2 −α)  k ( 1  2 −α)−( 1  2 +α) =  1  m−1 and  1  k−1 · k ( 1  2 −α)−( 1  2 +α)  k ( 1  2 +α)−( 1  2 −α) +  1  k−1 =  1  k−m  which are both equivalent to α = 1  2 −  m  k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content='  References  Biedermann S, Dette H, Zhu W (2006) Optimal designs for dose-response models with  restricted design spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Journal of the American Statistical Association 101:747–759  Dette H, Melas VB, Pepelyshev A, et al (2005) Optimal designs for three-dimensional  shape analysis with spherical harmonic descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' The Annals of Statistics 33:2758–  2788  Dette H, Melas VB, Pepelyshev A (2007) Optimal designs for statistical analysis with  zernike polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Statistics 41:453–470  Farrell RH, Kiefer J, Walbran A (1967) Optimum multivariate designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In: Proceedings  of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume  1: Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' University of California Press, Berkeley, Calif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=', pp 113–138  Ford I, Torsney B, Wu C (1992) The use of a canonical form in the construction of locally  optimal designs for non-linear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Journal of the Royal Statistical Society: Series  B (Statistical Methodology) 54:569–583  15  Martin Radloff, Rainer Schwabe  Exact Designs on the Ball  Hirao M, Sawa M, Jimbo M (2015) Constructions of φp-optimal rotatable designs on the  ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Sankhya A : The Indian Journal of Statistics 77:211–236  Kiefer JC (1961) Optimum experimental designs v, with applications to systematic and  rotatable designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In: Proceedings of the Fourth Berkeley Symposium on Mathematical  Statistics and Probability, Univ of California Press, pp 381–405  Konstantinou M, Biedermann S, Kimber A (2014) Optimal designs for two-parameter  nonlinear models with application to survival models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Statistica Sinica 24:415–428  Lau TS (1988) d-optimal designs on the unit q-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Journal of statistical planning and  inference 19:299–315  Pukelsheim F (1993) Optimal Design of Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Wiley Series in Probability and  Statistics  Radloff M, Schwabe R (2016) Invariance and equivariance in experimental design for  nonlinear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' In: Kunert J, Müller CH, Atkinson AC (eds) mODa 11-Advances in  Model-Oriented Design and Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Springer, p 217–224  Radloff M, Schwabe R (2019a) Locally d-optimal designs for a wider class of non-linear  models on the k-dimensional ball with applications to logit and probit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Statis-  tical Papers 60:165–177  Radloff M, Schwabe R (2019b) Locally d-optimal designs for non-linear models on the  k-dimensional ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Journal of Statistical Planning and Inference 203:106–116  Schmidt D, Schwabe R (2017) Optimal design for multiple regression with information  driven by the linear predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}
+page_content=' Statistica Sinica 27:1371–1384  16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfCAKJ/content/2301.02859v1.pdf'}

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@@ -0,0 +1,2499 @@
+arXiv:2301.01712v1  [math.PR]  4 Jan 2023
+1
+Mesoscopic eigenvalue statistics for Wigner-type matrices
+Volodymyr Riabov∗
+Institute of Science and Technology Austria
+volodymyr.riabov@ist.ac.at
+Abstract. We prove a universal mesoscopic central limit theorem for linear eigenvalue statistics of a Wigner-
+type matrix inside the bulk of the spectrum with compactly supported twice continuously differentiable test
+functions. The main novel ingredient is an optimal local law for the two-point function T(z, ζ) and a general
+class of related quantities involving two resolvents at nearby spectral parameters.
+Date: January 5, 2023
+Keywords and phrases: Wigner-type matrix, mesoscopic eigenvalue statistics, central limit theorem
+2010 Mathematics Subject Classification: 60B20, 15B52
+1
+Introduction
+In the study of the eigenvalue distribution of large random matrices, the most celebrated analog of
+the Law of Large Numbers is the Wigner semicircle law [25]. It states that the empirical density of
+eigenvalues converges to a deterministic limit known as the semicircle distribution ρsc. More explicitly,
+if H is an N ×N Wigner matrix and f is a sufficiently smooth test function, then the linear eigenvalue
+statistics N −1 Tr f(H) converge in probability to
+�
+R f(x)ρsc(x)dx in the large N limit.
+The corresponding Central Limit Theorem (CLT) asserts that the asymptotic fluctuations of the
+linear eigenvalue statistics Tr f(H)−E [Tr f(H)] are Gaussian. The absence of the N −1/2 normalization
+factor, appearing in the classical CLT, can be viewed as a manifestation of the strongly-correlated
+nature of the eigenvalues. For the special case of f(x) = (x − z)−1 with Im z ̸= 0, this result was
+obtained by Khorunzhy, Khoruzhenko and Pastur [16]. Johansson obtained the CLT for invariant
+ensembles with arbitrary polynomial potentials in [15].
+In [4], Bai and Yao used martingale CLT
+to establish the result for Wigner matrices with analytic test functions. The proof for bounded test
+functions f with bounded derivatives appeared in the work of Lytova and Pastur [22]. In subsequent
+works, different moment conditions on the matrix and regularity conditions on the test function were
+studied extensively by many authors, e.g., [6, 18, 23, 24].
+While fixed test functions represent macroscopic averaging in the spectrum, one can introduce N-
+dependent scaling and consider scaled test functions of the form f(x) = g(η−1
+0 (x − E0)), where E0
+is a fixed reference energy in the bulk, η0 ≡ η0(N) ≪ 1 is a scaling parameter, and g is compactly
+supported. Then Tr f(H) involves only about Nη0 eigenvalues of H. In particular, on mesoscopic
+scales, corresponding to N −1 ≪ η0 ≪ 1, the limiting variance is given by the square of the
+˙H1/2
+norm of g.
+Mesoscopic test functions were first studied by Boutet de Monvel and Khorunzhy in
+[7] for the Gaussian Orthogonal Ensemble, with subsequent extension to real Wigner matrices in [8]
+with N −1/8 ≪ η0 ≪ 1. In [13], He and Knowles proved the CLT for Wigner matrices with general
+mesoscopic test functions for all scaling parameters N −1 ≪ η0 ≪ 1.
+∗Supported by the ERC Advanced Grant ”RMTBeyond” No. 101020331
+
+The result was extended to ensembles of greater generality in the more recent works, see, e.g., [5]
+and [20]. In particular, Li and Xu obtained mesoscopic CLT for generalized Wigner matrices 1 in the
+bulk and at the spectral edge with C2
+c test functions in the full range of scales [21].
+Finally, Landon, Lopatto, and Sosoe proved the bulk CLT for the much more general ensemble
+of Wigner-type matrices in [17] for two classes of C∞ test functions. For a special class of globally
+supported regularized bump functions , the proof is performed via resolvent techniques for large scales
+and extended to the entire mesoscopic range using Dyson Brownian motion (DBM) dynamics. For
+the more conventional compactly supported scaled test functions, the bulk CLT is established on all
+mesoscopic scales N −1 ≪ η0 ≪ 1 using a combination of DBM and Green’s function comparison.
+Wigner-type matrices were first introduced in [2]; they have centered entries Hjk independent up
+to the symmetry constraint H = H∗. The matrix of variances S, defined by Sjk := E
+�
+|Hjk|2�
+, is
+assumed to be flat, i.e., Sjk ∼ N −1 and satisfy a piece-wise H¨older regularity condition (see (B)).
+As the main step towards CLT in the present paper, we prove the optimal averaged and entry-wise
+local laws (Corollary 3.3) for the two-point function T , defined by
+Txy(z, ζ) :=
+�
+a̸=y
+SxaGay(z)Gya(ζ),
+x, y ∈ {1, . . ., N},
+(1.1)
+where G(z) is the resolvent of H. The corresponding result in the simpler setting of generalized Wigner
+matrices was obtained in [21]. Using the optimal local law for T (z, ζ), we prove the bulk mesoscopic
+CLT for Wigner-type matrices in the full range of scales N −1 ≪ η0 ≪ 1 for compactly supported C2
+scaled test functions (Theorem 2.2). Our proof relies entirely on resolvent methods, circumventing the
+DBM dynamics used in [17].
+Understanding T (z, ζ) is the crucial ingredient for the CLT as it was realized in [17]. In fact, a
+suboptimal entry-wise local law for Txy(z, ζ) was proved in Proposition 5.1 of [17]. If one relies solely
+on resolvent methods, this local law provides sufficient control for mesoscopic CLT only on scales
+η0 ≫ N −1/5. The main reason for this limitation is that the error term in [17] contains the norm of
+the inverted stability operator (defined in (4.5)). In the present paper, we show that this factor can be
+removed by separating the destabilizing eigendirection corresponding to the smallest eigenvalue of the
+stability operator. Using this method, we prove a local law for a general class of quantities involving
+two resolvents (Theorem 3.2) and deduce the optimal averaged and entry-wise local laws for T (z, ζ).
+In particular, this allows us to obtain the CLT on all mesoscopic scales without relying on DBM.
+The main difficulty lies in the fact that the deterministic approximation of the resolvent for Wigner-
+type matrices is not a multiple of the identity matrix, contrary to the generalized Wigner case [21].
+Consequently, the destabilizing direction is no longer parallel to the vector of ones, and generally, no
+closed-form expression is known for the corresponding eigenprojector. It is important to note that for
+the deformed Wigner matrices studied in [20], the deterministic approximation is also not a multiple
+of the identity, but Sjk = N −1. Therefore, the two-point function can be expressed as the square of
+the resolvent and can be studied using the local law, similarly to the standard Wigner case.
+Instead of approximating the destabilizing direction to circumvent this difficulty, we use a contour
+integral representation for the eigenprojector. It allows us to extend the decomposition approach of
+[21] to the Wigner-type ensembles. This method benefits from yielding an integral representation for
+the variance on all mesoscopic scales, under weaker regularity conditions on the test function than in
+[17], and relying only on resolvent methods.
+The paper is organized in the following way. Section 2 contains the precise definition of the model
+and the statement of our main mesoscopic CLT result, Theorem 2.2. In Section 3, we present our main
+technical result, the optimal local law for two-point functions in Theorem 3.2. In Section 4, we collect
+notations and preliminary results to which we refer throughout the paper. In Section 5, we deduce
+Theorem 2.2 from Propositions 5.1 and 5.2, and prove Proposition 5.1 using a local law for T (z, ζ)
+(Corollary 3.3) as an input. The proofs of Theorem 3.2 and Corollary 3.3 are presented in Section 6.
+In Section 7, we prove Proposition 5.2, which relates the variance of the linear eigenvalue statistics to
+the ˙H1/2-norm.
+Acknowledgments.
+I would like to express my gratitude to L´aszl´o Erd˝os for suggesting the project and supervising my
+work. I am also thankful to Yuanyuan Xu and Oleksii Kolupaiev for many helpful discussions.
+1Generalized Wigner matrices are characterized by a flat doubly-stochastic matrix of variances S. Unlike the Wigner
+case, the entries Sjk are not assumed to be equal. The limiting eigenvalue distribution remains semicircular.
+2
+
+2
+Model and Main Result
+We begin with the definition of Wigner-type matrices originally introduced in Section 1.1 of [2].
+Definition 2.1 (Wigner-type matrices). Let H = (Hjk)N
+j,k=1 be an N × N matrix with independent
+entries up to the Hermitian symmetry condition H = H∗ satisfying
+E [Hjk] = 0.
+(2.1)
+We consider both real and complex Wigner-type matrices. In case the matrix H is complex we assume
+additionally that Re Hjk and Im Hjk are independent and E[H2
+jk] = 0 for k ̸= j.
+Denote by S the matrix of variances Sjk := E[|Hjk|2], and assume it satisfies
+cinf
+N
+≤ Sjk ≤ Csup
+N ,
+(A)
+for all j, k ∈ {1, . . ., N} and some strictly positive constants Csup, cinf.
+We assume a uniform bound on all other moments of
+√
+NHjk, that is, for any p ∈ N there exists
+a positive constant Cp such that
+E
+�
+|
+√
+NHjk|p�
+≤ Cp
+(2.2)
+holds for all j, k ∈ {1, . . ., N}.
+Additionally, we assume that S satisfies a H¨older regularity condition1, that is,
+|Sjk − Sj′k′| ≤ L
+N
+�|j − j′| + |k − k′|
+N
+�1/2
+,
+(B)
+for all j, j′, k, k′ ∈ {1, . . ., N} and some positive constant L. The constants cinf, Csup, Cp and L are
+independent of N.
+2.1
+Central Limit Theorem for Mesoscopic Linear Eigenvalue Statistics
+Theorem 2.2. (c.f. Theorem 2.5 in [17]) Let g be a C2
+c (R) test function. Let ε0 be a small fixed
+constant and let N −1+ε0 ≤ η0 ≤ N −ε0, and let E0 be a fixed reference energy in the bulk of the
+spectrum, that is, ρ(E0) ≥ ε0 (here ρ is the density of states to be defined in (3.3) below ). Define the
+scaled test function f to be
+f(x) := g
+�x − E0
+η0
+�
+,
+(2.3)
+then
+Tr f(H) − E [Tr f(H)]
+d−→ N
+�
+0,
+1
+2βπ2 ∥g∥2
+˙H1/2
+��
+,
+(2.4)
+where β = 1 and β = 2 corresponds to real symmetric and complex Hermitian H, respectively.
+Remark 2.3. We remark that the universal limiting variance in (2.4) coincides with the corre-
+sponding formulas for standard Wigner matrices [13], where Sjk = N −1, mj(z) = msc(z) for all
+j, k ∈ {1, . . . , N}, and msc(z) is the Stieltjes transform of the semicircle law.
+1As stated in [2], assumption (B) can be weakened to piece-wise 1/2-H¨older regularity condition for some positive
+constant L on finitely many intervals, in the sense that
+max
+a,b
+max
+j,j′∈(NIb)
+max
+k,k′∈(NIa) N3/2
+|Sjk − Sj′k′|
+|j − j′|1/2 + |k − k′|1/2 ≤ L,
+where {Ia}n
+a=1 is a fixed finite partition of [0, 1] into smaller intervals, and (NIa) denotes the set of positive integers j
+such that j/N lies in Ia.
+3
+
+3
+Local Laws for the Two-point Functions
+In this section, we introduce our main technical result, local laws for quantities that involve two
+resolvents of a Wigner-type matrix. Our prime motivation is to study the function T (z, ζ) defined in
+(1.1), but our methods allow us to estimate a more general class of quantities, namely
+�
+a̸=y
+waGαa(z)Gaβ(ζ),
+�
+b
+�
+a̸=b
+WabGba(z)Gab(ζ),
+(3.1)
+for fixed indices α, β, y, and deterministic weights wa, Wab satisfying |wa|, |Wab| ≤ cN −1 for some
+constant c > 0.
+Here G(z) := (H − z)−1 denotes the resolvent of H.
+Objects of this type were
+first studied in [11] in the setting of random band matrices. We obtain the estimates in the sense of
+stochastic domination.
+Definition 3.1. (Definition 2.1 in [12]) Let X = X (N)(u) and Y = Y(N)(u) be two families of random
+variables possibly depending on a parameter u ∈ U (N). We say that Y stochastically dominates X
+uniformly in u if for any ε > 0 and D > 0 there exists N0(ε, D) such that for any N ≥ N0(ε, D),
+sup
+u∈U(N) P
+�
+X (N)(u) > N εY(N)(u)
+�
+< N −D.
+We denote this relation by X ≺ Y or X = O≺(Y).
+We consider spectral parameters z lying in the domain D, defined by
+D := {z ∈ C : N −1+τ ≤ | Im z| ≤ τ −1, | Re z| ≤ τ −1},
+(3.2)
+for a fixed τ > 0. As in Theorem 2.2, our analysis is limited to the bulk of the spectrum, which we
+define via the self-consistent density of states ρ(E) ≡ ρN(E). The density ρ(E) is recovered by the
+Stieltjes inversion formula,
+ρ(E) := π−1 lim
+η→+0 Im m(E + iη),
+(3.3)
+where m(z) := N −1 �N
+j=1 mj(z), and m(z) = (mj(z))N
+j=1 is the unique (Theorem 4.1 in [2]) solution
+to the vector Dyson equation
+−1
+m(z) = z + Sm(z),
+Im m(z) Im z > 0.
+(3.4)
+Let I be the set on which ρ(E) is positive. Theorem 4.1 of [2] guarantees that I consists of a finite
+union of open intervals (a(j), b(j)). Then for κ > 0, we define the bulk domain by
+Dκ := {z ∈ D : Re z ∈ Iκ},
+Iκ :=
+�
+j
+[a(j) + κ, b(j) − κ].
+(3.5)
+In particular, for all z ∈ Dκ, ρ(z) ≥ C(κ) for some constant C(κ) > 0. Given E0 as in Theorem 2.2,
+we choose κ so that E0 ∈ I2κ.
+Theorem 3.2. There exists a positive constant ǫ = ǫκ which is independent of N, such that for all
+z, ζ in Dκ with | Re ζ − Re z| ≤ ǫ, and deterministic vectors w ∈ CN satisfying ∥w∥∞ ≤ cN −1, the
+following estimate holds,
+�
+a̸=y
+waGαa(z)Gaβ(ζ) = δαβ
+�
+m(z)m(ζ)
+�
+1 − Sm(z)m(ζ)
+�−1w
+�
+α − δαβδαy[m(z)m(ζ)w]α
++ O≺
+�
+(Ψ(z) + Ψ(ζ))(Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0} min{Θ(z), Θ(ζ)})
+�
+,
+(3.6)
+where the vector m is identified with the diagonal operator diag (m).
+Under the same conditions on z, ζ, for any deterministic N ×N matrix W satisfying |Wab| ≤ cN −1
+for all a, b, the following estimate holds,
+�
+b
+�
+a̸=b
+WabGba(z)Gab(ζ) = Tr
+�
+m(z)m(ζ)Sm(z)m(ζ)
+�
+1 − Sm(z)m(ζ)
+�−1W
+�
++ NO≺
+�
+(Ψ(z) + Ψ(ζ))Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0}Θ(z)Θ(ζ)
+�
+.
+(3.7)
+4
+
+Here Ψ(z) and Θ(z) denote control parameters defined as
+Ψ(z) :=
+�
+| Im m(z)|
+N|η|
++
+1
+N|η|,
+Θ(z) :=
+1
+N|η|,
+z = E + iη ∈ C\R.
+(3.8)
+Theorem 3.2 implies the following averaged and entry-wise local laws for T (z, ζ) from (1.1) .
+Corollary 3.3. Let z, ζ satisfy the assumptions of Theorem 3.2.
+The entries Txy(z, ζ) admit the
+estimate
+Txy(z, ζ) =
+�
+(Sm(z)m(ζ))2 �
+1 − Sm(z)m(ζ)
+�−1�
+xy
++ O≺
+�
+(Ψ(z) + Ψ(ζ))
+�
+Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0} min{Θ(z), Θ(ζ)}
+��
+.
+(3.9)
+Furthermore, for all deterministic N × N matrices A, the following equality holds
+Tr[A T (z, ζ)] = Tr[A
+�
+1 − Sm(z)m(ζ)
+�−1�
+Sm(z)m(ζ)
+�2]
++ N ∥A∥ℓ∞→ℓ∞ O≺
+�
+(Ψ(z) + Ψ(ζ))Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0}Θ(z)Θ(ζ)
+�
+.
+(3.10)
+Remark 3.4. The error estimates in the entry-wise local law (3.6), and hence in (3.9) are optimal.
+Indeed, for Sjk := N −1, which corresponds to the standard Wigner matrices, and ζ = ¯z, a simple
+calculation using the Ward identity shows that
+Txy(z, ¯z) = N −1| Im z|−1 Im msc(z) − N −1|msc(z)|2 + O≺
+�
+Θ(z)Ψ(z)
+�
+.
+(3.11)
+The error estimate in (3.7) is not optimal; it can be improved to
+O≺
+�
+N(Ψ(z) + Ψ(ζ))2�
+Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0}NΘ(z)Θ(ζ)
+��
+(3.12)
+However, (3.7) is sufficient for establishing the CLT, so for the sake of brevity, we do not present the
+proof of (3.12) in full detail. We only indicate the necessary ingredients in Remark 6.8 below.
+4
+Notations and Preliminaries
+4.1
+Notations
+For a vector x = (xj)N
+j=1 ∈ CN we use the standard definitions of ℓ2 and ℓ∞ norms, namely,
+∥x∥2 =
+� N
+�
+j=1
+|xj|2
+�1/2
+,
+∥x∥∞ = max
+j
+|xj|.
+For a linear operator T : CN → CN, we denote its matrix norms induced by ℓ2 and ℓ∞ norms,
+respectively, by
+∥T ∥ℓ2→ℓ2 =
+sup
+∥x∥2=1
+∥T x∥2 ,
+∥T ∥ℓ∞→ℓ∞ =
+sup
+∥x∥∞=1
+∥T x∥∞ .
+For two vectors x, y ∈ CN we use angle brackets to denote the ℓ2 scalar product, while for a single
+vector x ∈ CN angle brackets denote the average of its coordinates
+⟨x, y⟩ =
+N
+�
+j=1
+¯xjyj,
+⟨x⟩ = 1
+N
+N
+�
+j=1
+xj.
+We use xy to denote a coordinate-wise product of vectors x and y,
+(xy)j = xjyj,
+j ∈ {1, . . . , N}.
+Similarly, for a given vector x with non-zero entries, 1x denotes a coordinate-wise multiplicative inverse
+� 1
+x
+�
+j
+= 1
+xj
+,
+j ∈ {1, . . ., N}.
+5
+
+We use 1 to denote the vector of ones (1, . . . , 1)t in CN.
+For a measurable function f : R → R we use the standard definition of the Lp norms for p ≥ 1,
+and the following definition of the ˙H1/2 norm
+∥f∥ ˙H1/2 =
+
+
+��
+R2
+|f(x) − f(y)|2
+|x − y|2
+dxdy
+
+
+1/2
+.
+For two deterministic quantities X, Y ∈ R depending on N, we write X ≪ Y if there exists ε, N0 > 0
+such that |X| ≤ N −ε|Y | for all N ≥ N0. Similarly, we write X ≲ Y if there exists a constant C, N0 > 0
+such that |X| ≤ C|Y | for all N ≥ N0, and X ∼ Y if both X ≲ Y and Y ≲ X hold.
+We use C and c to denote constants, the precise value of which is irrelevant and may change from
+line to line.
+4.2
+Local Law for the Resolvent
+In this subsection, we summarize the facts on Wigner-type matrices that we use throughout our proofs.
+Majority of these results were obtained in [1] (see also [3]), but we refer to their concise versions from
+[2] adapted for the Wigner-type setting.
+Lemma 4.1. (Theorem 4.1 in [2]) The solution m(z) of (3.4) satisfies the following properties:
+(1) For every j ∈ {1, . . ., N} there exists a generating probability measure νj(dx) such that
+mj(z) =
+�
+R
+νj(dx)
+x − z .
+(4.1)
+(2) If the matrix of variances S satisfies conditions (A) and (B), then for all z ∈ C\R, the solution
+admits the following bounds
+∥m(z)∥∞ ≤
+c
+1 + |z|,
+����
+1
+m(z)
+����
+∞
+≤ C(1 + |z|).
+(4.2)
+We now state the optimal averaged and isotropic local laws for Wigner-type matrices.
+Theorem 4.2. (Corollary 1.8 in [2]) Let w, x, y be deterministic vectors in CN satisfying ∥w∥∞ = 1
+and ∥x∥2 = ∥y∥2 = 1. Then the following estimates hold uniformly in z ∈ D:
+N −1��Tr
+�
+w(G(z) − m(z))
+��� ≺ Θ(z),
+��⟨x, (G(z) − m(z))y⟩
+�� ≺ Ψ(z),
+(4.3)
+where vectors m and w are associated with corresponding diagonal matrices.
+In particular, it follows from the isotropic local law (4.3) that for any j, k ∈ {1, . . ., N},
+|Gjk(z) − δjkmj(z)| ≺ Ψ(z).
+(4.4)
+4.3
+Preliminary Bounds on the Stability Operator
+A significant part of our proof revolves around the stability operator, originally introduced in [1], that
+emerges when studying the two-point function T (z, ζ) defined in (1.1). In this subsection, we collect
+the known bounds on the stability and related operators.
+The stability operator (1 − Sm(z)m(ζ)) is defined by the matrix with entries
+(1 − Sm(z)m(ζ))jk := δjk − Sjkmk(z)mk(ζ),
+j, k ∈ {1, . . ., N},
+z, ζ ∈ C\R.
+(4.5)
+Throughout this paper we use m (and various functions of m, such as Im m, |m|, m−1, m′) to
+denote both a vector (mj)N
+j=1 and the corresponding multiplication operator, i.e., diag
+�
+(mj)N
+j=1
+�
+. Note
+that this notation agrees with the point-wise multiplication of two vectors if the first multiplicand is
+interpreted as an operator. We stress which interpretation is used whenever ambiguity may arise.
+6
+
+The analysis of the stability operator relies on the corresponding saturated self-energy operator F,
+studied in [17], that depends on two spectral parameters z, ζ, and is defined as
+Fjk(z, ζ) := |mj(z)mj(ζ)|1/2Sjk|mk(z)mk(ζ)|1/2,
+j, k ∈ {1, . . ., N},
+z, ζ ∈ C\R.
+(4.6)
+The following statements encompass the main properties of F and preliminary bounds on the stability
+operator.
+Proposition 4.3. (Proposition 4.3 in [17], c.f. Proposition 7.2.9 and Lemma 7.4.4 in [9]) For any
+z, ζ ∈ C, the principal eigenvalue of F defined in (4.6) is positive and simple, the corresponding ℓ2-
+normalized eigenvector v(z, ζ) has strictly positive entries. The norm of F admits the following upper
+bound
+∥F(z, ζ)∥ℓ2→ℓ2 ��� 1 − 1
+2
+�
+| Im z| ⟨v(z, z), |m(z)|⟩
+⟨v(z, z), | Im m(z)|
+|m(z)| ⟩
++ | Im ζ| ⟨v(ζ, ζ), |m(ζ)|⟩
+⟨v(ζ, ζ), | Im m(ζ)|
+|m(ζ)| ⟩
+�
+.
+(4.7)
+If |z|, |ζ| ≲ 1, then the entries of v(z, ζ) are comparable in size, that is
+cκ ≤
+√
+Nvj(z, ζ) ≤ Cκ,
+j ∈ {1, . . ., N},
+(4.8)
+and moreover, let Gap (F) denote the difference between the two largest eigenvalues of |F| =
+√
+FF ∗,
+then Gap (F) admits the bound
+Gap (F) ≥ �δ,
+(4.9)
+where �δ is a constant that depends only on the constants in conditions (A), (B) and κ.
+Furthermore, for a fixed κ > 0 and z, ζ ∈ Dκ there exists a positive constant �cκ such that
+∥F(z, ζ)∥ℓ2→ℓ2 ≤ 1 − �cκ (| Im z| + | Im ζ|) ,
+(4.10)
+Proposition 4.4. (Proposition 4.6 and Lemma 4.7 in [17]) Let z, ζ ∈ C, such that |z|, |ζ| ≲ 1 and
+Re z, Re ζ ∈ Iκ, then
+��(1 − Sm(z)m(ζ))−1��
+ℓ2→ℓ2 +
+��(1 − Sm(z)m(ζ))−1��
+ℓ∞→ℓ∞ ≲
+1
+| Im z| + | Im ζ|.
+(4.11)
+If additionally Im z Im ζ > 0, the estimate is improved to
+��(1 − Sm �m)−1��
+ℓ∞→ℓ∞ ≤ Cκ,
+(4.12)
+where Cκ > 0 is a positive constants dependent on κ.
+Finally, we state the bounds on the stability operator in the special case of ζ = z, which is related
+to the derivative of m via the (vector) identity m′(z) = (1 − m2(z)S)−1m2(z), obtained by taking the
+derivative of (3.4).
+Lemma 4.5. (Lemma 5.9 in [1], Lemma 7.3.2 in [9]) Let C > 0 be a positive constant, then for
+z ∈ C\R with |z| ≤ C we have
+��(1 − m2(z)S)−1��
+ℓ2→ℓ2 +
+��(1 − m2(z)S)−1��
+ℓ∞→ℓ∞ ≲ |ρ(z)|−2,
+(4.13)
+where ρ(z) = π−1⟨Im m(z)⟩ is the harmonic extension of ρ(E) defined in (3.3).
+Therefore for all z ∈ C\R with Re z ∈ Iκ we have
+∥m′(z)∥∞ ≲ 1.
+(4.14)
+4.4
+Cumulant Expansion Formula
+Lemma 4.6. (Section II in [7], Lemma 3.1 in [13]) Let h be a real-valued random variable with finite
+moments, let f be a C∞(R) function. Then for any ℓ ∈ N the following expansion holds,
+E [h · f(h)] =
+ℓ
+�
+j=0
+1
+j!c(j+1)(h) E
+� dj
+dhj f(h)
+�
++ Rℓ+1,
+(4.15)
+7
+
+where c(j) is the j-th cumulant of h defined by
+c(j)(h) = (−i)j dj
+dtj
+�
+log E
+�
+eith������
+t=0
+,
+and the remainder term Rℓ+1 satisfies
+|Rℓ+1| ≤ Cl E
+�
+|h|ℓ+2�
+sup
+|x|≤M
+|f (ℓ+1)(x)| + Cl E
+�
+|h|ℓ+2 · 1|h|>M
+� ���f (ℓ+1)(x)
+���
+∞ ,
+(4.16)
+for any M > 0.
+We apply formula (4.15) with h equal to the matrix element Hjk. Correspondingly, in the real
+case (β = 1), C(p) denotes the matrix of p-th cumulants of H, C(p)
+jk := C(p)(Hjk). In the complex case
+(β = 2), C(p) is used as a notational shortcut and denotes the sum of matrices of p-th cumulants of
+real and imaginary parts of H, that is C(p)
+jk := C(p)(Re Hjk) + C(p)(Im Hjk).
+5
+Proof of the Main Result
+Proof of Theorem 2.2. We divide the proof into two parts contained in the following propositions. We
+indicate their analogs in the settings of [21] and [17] in parenthesis.
+Proposition 5.1. (c.f. Theorem 2.2 in [21] and (5.76) in [17]) Let η0, ε0 > 0 and E0 satisfy the
+assumptions of Theorem 2.2, let f be a scaled test function defined in (2.3), and let φ(λ) be the
+characteristic function of Tr f(H) − E [Tr f(H)],
+φ(λ) := E [exp{iλ (Tr f(H) − E [Tr f(H)])}] ,
+λ ∈ R.
+(5.1)
+Then its derivative φ′(λ) satisfies the following equation,
+φ′(λ) = −λφ(λ)V (f) + O≺
+�
+N −1/2η−1/2
+0
+(1 + |λ|4) + (1 + |λ|)N −ε0/2�
+,
+λ ∈ R,
+(5.2)
+provided c ≤ V (f) ≤ C for some positive N-independent constants c and C.
+Here the variance V (f) for a scaled test function f is defined by
+V (f) := 1
+π2
+�
+Ω0
+�
+Ω′
+0
+∂ �f(ζ)
+∂¯ζ
+∂ �f(z)
+∂¯z
+K(z, ζ)d¯ζdζd¯zdz,
+(5.3)
+where for z, ζ ∈ C/R the kernel K(z, ζ) is defined by
+K(z, ζ) := 2
+β
+∂
+∂ζ Tr
+�m′(z)
+m(z)
+�
+1 − Sm(z)m(ζ)
+�−1
+�
++
+�
+1 − 2
+β
+�
+Tr [Sm′(z)m′(ζ)] + 1
+2
+∂2
+∂z∂ζ
+�
+m(z)m(ζ), C(4)m(z)m(ζ)
+�
+,
+(5.4)
+with C(4) denoting the matrix of fourth cumulants C(4)
+jk . The integration domains Ω0, Ω′
+0 in (5.3) are
+defined as
+Ω0 := {z ∈ C : | Im z| > N −ε0/2η0},
+Ω′
+0 := {z ∈ C : | Im z| > 2N −ε0/2η0},
+(5.5)
+and �f is the quasi-analytic extension of f, defined by
+�f(x + iη) = χ(η) (f(x) + iηf ′(x)) ,
+(5.6)
+where χ : R → [0, 1] is an even C∞
+c (R) function supported on [−1, 1], satisfying χ(η) = 1 for |η| < 1/2.
+Proposition 5.2. (c.f. Lemma 6.7 in [17]) Let E0, η0 satisfy the conditions of Theorem 2.2. Let f be
+the scaled test function with g ∈ C2
+c (R) given in (2.3), and let V (f) be the variance defined in (5.3),
+then
+V (f) =
+1
+2βπ2 ∥g∥2
+˙H1/2 + O
+�
+η0 log N + N −ε0�
+.
+(5.7)
+8
+
+Proposition 5.2 implies that V (f) satisfies the condition of Proposition 5.1, hence
+φ′(λ) = −λφ(λ)V (f) + o (1) ,
+(5.8)
+as N → ∞, for any fixed λ ∈ R. It then follows by L´evy’s continuity theorem that Tr f(H)−E [Tr f(H)]
+converges in distribution to a centered Gaussian with variance (2βπ2)−1 ∥g∥2
+˙H1/2. Therefore, to estab-
+lish Theorem 2.2, it suffices to show that Propositions 5.1 and 5.2 hold, which is done in Sections 5.1
+and 7, respectively.
+Remark 5.3. We restrict the proof to the real symmetric (β = 1) matrices for the sake of presentation.
+The complex Hermitian (β = 2) case differs solely in replacing the cumulant expansion formula (Lemma
+4.6) with its complex analog. The obvious modifications are left to the reader.
+5.1
+Characteristic Function of Linear Eigenvalue Statistics
+Proof of Proposition 5.1. Using standard techniques of the characteristic function method imported
+from, e.g., Section 5.2 of [17] (see also Section 4.2 of [19] and references therein), we can obtain the
+following series of estimates on the characteristic function of the linear eigenvalue statistics φ(λ) and
+its derivative φ′(λ). The proof is a relatively straightforward modification of similar arguments in [17],
+so we defer it to Appendix A.
+Lemma 5.4. Let φ(λ) be the characteristic function defined in (5.1), then, under the conditions of
+Theorem 2.2, the following estimates hold
+φ(λ) = E [�e(λ)] + O≺
+�
+N −ε0/2�
+,
+φ′(λ) = i
+π
+�
+Ω0
+∂ �f
+∂¯z E [�e(λ) {1 − E} [Tr G(z)]] d¯zdz + O≺
+�
+|λ|N −ε0/2�
+,
+(5.9)
+where
+�e(λ) := exp
+�iλ
+π
+�
+Ω′
+0
+∂ �f
+∂¯z {1 − E} [Tr G(z)] d¯zdz
+�
+.
+(5.10)
+Furthermore, for all z ∈ Dκ, we have
+E [�e(λ) {1 − E} [Tr G(z)]] = E [�e(λ) {1 − E} T (z, z)] + 2iλ
+π E
+�
+�e(λ)
+�
+Ω′
+0
+∂ �f
+∂¯ζ
+∂
+∂ζ T (z, ζ)d¯ζdζ
+�
++ iλ
+π E [�e(λ)]
+�
+Ω′
+0
+∂ �f
+∂¯ζ Tr [Sm′(z)m′(ζ)] d¯ζdζ
++ iλ
+2π E [�e(λ)]
+�
+Ω′
+0
+∂ �f
+∂¯ζ
+∂2
+∂z∂ζ
+�
+m(z)m(ζ), C(4)m(z)m(ζ)
+�
+d¯ζdζ
++ O≺
+�
+(1 + |λ|4)(NΨ(z)Θ(z) + Ψ(z)η−1/2
+0
+)
+�
+,
+(5.11)
+where the random function T (z, ζ) is defined as
+T (z, ζ) := Tr
+�m′(z)
+m(z) T (z, ζ)
+�
+.
+(5.12)
+We now proceed to estimate the first two terms on the right-hand side of (5.11) in such a way
+that E [�e(λ)] factors out. By definition of the scaled test function (2.3), the support of �f is contained
+inside a vertical strip centered at E0 of width ∼ η0, hence we limit the further analysis to the regime
+| Re ζ − Re z| ≲ η0 ≪ ǫ, where ǫ is defined in the statement of Theorem 3.2. We estimate the function
+T (z, ζ) using Corollary 3.3 with weight matrix A := m′(z)
+m(z) . It follows from the bounds (4.2) and (4.14)
+that ∥A∥ℓ∞→ℓ∞ ≲ 1, hence for all z, ζ ∈ Dκ with Re z, Re ζ ∈ supp(f),
+T (z, ζ) = Tr
+�m′(z)
+m(z)
+�
+1 − Sm(z)m(ζ)
+�−1�
+Sm(z)m(ζ)
+�2
+�
++ E(z, ζ),
+(5.13)
+9
+
+where the error term E(z, ζ) is analytic in both variables and admits the bound
+E(z, ζ) ≺ NΨ2(z)Ψ(ζ) + NΨ(z)Ψ2(ζ) + 1{Im z Im ζ<0}NΘ(z)Θ(ζ).
+(5.14)
+It follows from (5.13) and (5.14) for ζ = z that
+E [�e(λ){1 − E} [T (z, z)]] ≺ NΨ(z)3,
+(5.15)
+yielding the desired bound on the first term on the right-hand side of (5.11).
+We now estimate the second term in (5.11). Fix z ∈ Dκ, and consider ζ that lie in Ω′
+0 defined in
+(5.5). Differentiating (5.13) with respect to ζ yields
+∂
+∂ζ T (z, ζ) = ∂
+∂ζ Tr
+�m′(z)
+m(z)
+�
+1 − Sm(z)m(ζ)
+�−1�
+Sm(z)m(ζ)
+�2
+�
++ ∂
+∂ζ E(z, ζ).
+(5.16)
+To bound the derivative of the error term E(z, ζ), we use the following technical lemma.
+Lemma 5.5. (Lemma 5.5 in [17]) Let K(z) be a holomorphic function on C\R, then for all z ∈ C\R
+and any p ∈ N,
+����
+∂pK
+∂zp (z)
+���� ≤ Cp| Im z|−p
+sup
+|ζ−z|≤| Im z|/2
+|K(ζ)|,
+(5.17)
+where Cp > 0 is a constant depending only on p.
+Lemma 5.5 applied to the estimate (5.14) implies that the error term ∂ζE(z, ζ) admits the bound
+∂
+∂ζ E(z, ζ) ≺ N| Im ζ|−1�
+Ψ(z)2Ψ(ζ) + Ψ(z)Ψ(ζ)2 + Θ(z)Θ(ζ)
+�
+.
+(5.18)
+To proceed we require another technical lemma.
+Lemma 5.6. (c.f. Lemma 4.4 in [19]) Let f be the scaled test function defined in (2.3). Let Ω be a
+domain of the form
+Ω := {z ∈ C : cN −τ ′η0 < | Im z| < 1, a < Re z < b},
+(5.19)
+such that supp(f) ⊂ (a, b) and τ ′, c are positive constants. Let K(z) be a holomorphic function on Ω
+satisfying
+|K(z)| ≤ C| Im z|−s,
+z ∈ Ω,
+(5.20)
+for some 0 ≤ s ≤ 2. Then there exists a constant C′ > 0 depending only on g in (2.3), χ in (5.6), and
+s, such that
+����
+�
+Ω
+∂ �f
+∂¯z (x + iy)K(x + iy)dxdy
+���� ≤ CC′η1−s
+0
+log N.
+(5.21)
+Proof of Lemma 5.6. It follows from (2.3) that ∥f∥1 ∼ η0, ∥f ′∥1 ∼ 1, ∥f ′′∥1 ∼ η−1
+0 . In case 1 ≤ s ≤ 2
+the inequality (5.21) follows from Lemma 4.4 in [19]. For 0 ≤ s < 1, the proof is conducted along the
+same lines, except the integration by parts is performed twice in the regime η0 ≤ | Im z| ≤ 1.
+Lemma 5.6 and the matrix identity (1−X)−1X2 = (1−X)−1−X −1 yield the following expression.
+E
+�
+�e(λ)
+�
+Ω′
+0
+∂ �f
+∂¯ζ
+∂T
+∂ζ d¯ζdζ
+�
+= E [�e(λ)]
+�
+Ω′
+0
+∂ �f
+∂¯ζ
+∂
+∂ζ Tr
+�m′(z)
+m(z)
+�
+1 − Sm(z)m(ζ)
+�−1
+�
+d¯ζdζ
+− E [�e(λ)]
+�
+Ω′
+0
+∂ �f
+∂¯ζ Tr
+�
+Sm′(z)m′(ζ)
+�
+d¯ζdζ
++O≺
+�
+N 1/2Ψ(z)2η−1/2
+0
++ Ψ(z)η−1
+0
++ Θ(z)η−1
+0
+�
+,
+(5.22)
+Finally, from (5.11) and (5.22), combined with (5.9) we conclude that
+φ′(λ) = −λV (f) E [�e(λ)] + �E(λ),
+(5.23)
+10
+
+where V (f) is defined in (5.3), and �E(λ) is the total error term collected from previous derivations
+and integrated over d¯zdz. Lemma 5.6 together with error estimates in (5.9), (5.11), (5.15) and (5.18)
+provides the following bound on the error term
+�E = O≺
+�
+N −1/2η−1/2
+0
+(1 + |λ|4) + |λ|N −ε0/2�
+.
+(5.24)
+Under the conditions of Proposition 5.1 V (f) is bounded, hence we conclude from the first estimate
+in (5.9) and (5.23) that (5.2) holds. This concludes the proof of Proposition 5.1.
+6
+Proof of the Local Laws for Two-point Functions
+In this section, we derive all the tools necessary to prove Theorem 3.2 and its specification for the two-
+point function T (z, ζ), Corollary 3.3. To make the notation more concise we introduce the convention
+G ≡ G(z), �G ≡ G(ζ), m ≡ m(z), �m ≡ m(ζ), �Ψ ≡ Ψ(ζ), Ψ ≡ Ψ(z), Θ ≡ Θ(z), �Θ ≡ Θ(ζ).
+For a deterministic matrix W with entries |Wab| ≲ N −1, the quantity �
+a̸=y WaxGαa �Gaβ can be
+readily estimated in two special cases. First, if each column of W is proportional to the vector of ones,
+i.e., Wab = wb depends only on b, then the summation over a yields wx([G �G]αβ − Gαy �Gyβ), and the
+estimate follows from the resolvent identity and the local laws in Theorem 4.2. Second, if the entries
+of X := (1 − Sm �m)−1W are bounded by CN −1, then one can obtain the estimate from Lemma 6.1
+below. We show that these two special cases are exhaustive in the sense that any W can be represented
+as their linear combination with controlled coefficients.
+To this end, we prove that in the relevant regime, the operator (1 − Sm �m) has a very small
+destabilizing eigenvalue and an order one spectral gap above it. Moreover, if Π is the eigenprojector
+corresponding to the principal eigenvalue of (1 − Sm �m), then the ℓ∞ → ℓ∞-norm of the restriction
+of (1 − Sm �m)−1 to the kernel of Π is also an order one quantity. Finally, we show that the vector of
+ones 1 is sufficiently separated from the kernel of Π.
+6.1
+Stable Direction Local Law
+For any N × N deterministic matrix W, and any indices x, y, α, β, we define the quantities
+Fxy
+αβ(W) :=
+�
+a̸=y
+WaxGαa �Gaβ,
+f xy
+α (W) := mα �mα([(1 − Sm �m)−1W]αx − δαyWαx).
+(6.1)
+We prove the following estimate.
+Lemma 6.1. For any z, ζ ∈ Dκ and any deterministic N × N matrix X,
+Fxy
+αβ((1 − Sm �m)X) = δαβf xy
+α ((1 − Sm �m)X) + O≺
+�
+N ∥X∥max Ψ�Ψ(Ψ + �Ψ)
+�
+.
+(6.2)
+provided ∥X∥max := max
+j,k |Xjk| ≲ 1.
+We use the following self-improving mechanism for stochastic domination bounds, borrowed, e.g.,
+from [14].
+Lemma 6.2. (Lemma 6.3 in [14]) Let X be a random variable such that 0 ≤ X ≺ N C for some C > 0,
+and let Ξ ≥ 0 be a deterministic quantity. Suppose there exists a constant q ∈ [0, 1), such that for any
+Φ satisfying Ξ ≤ Φ ≤ N C, and any d ∈ N, we have the implication
+X ≺ Φ
+=⇒
+E
+�
+|X|2d�
+≺
+2d
+�
+k=1
+�
+ΦqΞ1−q)k E
+�
+|X|2d−k�
+,
+(6.3)
+then X ≺ Ξ.
+Proof of Lemma 6.1. Let Y := (1 − Sm �m)X, then the quantity we need to estimate is [GY ]yx =
+Fxy
+yy (Y ). It follows from the local law in the form (4.4) that
+Fxy
+αβ(Y ) ≺ N ∥X∥max Ψ�Ψ =: Λ.
+(6.4)
+11
+
+Let Φ be a deterministic control parameter admitting the bounds (Ψ + �Ψ)Λ ≤ Φ ≤ Λ, such that
+Fxy
+αβ(Y ) − δαβf xy
+α (Y ) ≺ Φ.
+(6.5)
+It follows trivially from (6.4) and (6.5) that
+Fxy
+αβ(Y ) ≺ Φ + δαβΛ.
+(6.6)
+Let ∂jk denote the partial derivative with respect to the matrix element Hjk, then the partial derivatives
+of Fxy
+αβ are given by
+∂abFxy
+αβ(Y ) = −(1 + δab)−1(GαaFxy
+bβ (Y ) + GαbFxy
+aβ(Y ) + Fxy
+αb (Y ) �Gaβ + Fxy
+αa(Y ) �Gbβ).
+(6.7)
+We combine the vector Dyson equation (3.4) and the resolvent identity zG = HG − 1 to obtain
+�Gaβ = − �ma
+�
+b
+�
+Hab �Gbβ + Sab �mb �Gaβ
+�
++ �maδaβ.
+(6.8)
+Let d ∈ N, define P ≡ P(d − 1, d) := (Fxy
+αβ(Y ) − δαβf xy
+α (Y ))d−1(Fxy
+αβ(Y ) − δαβf xy
+α (Y ))d. For any
+p ∈ N, define Mp := E
+�
+|Fxy
+αβ(Y ) − δαβf xy
+α (Y )|p�
+. Plugging (6.8) into the definition (6.1) and applying
+the cumulant expansion formula of Lemma 4.6, we obtain
+E
+�
+Fxy
+αβ(X)P
+�
+=
+�
+a̸=y
+ma �maXax E
+�
+Fay
+αβ(S)P
+�
++ δαβf xy
+α (Y ) E[P] + δαβδβySyym2
+y �m2
+yXyx E[P]
+(6.9a)
++ E
+��
+a̸=y
+�
+b
+�maXaxSab
+�
+Gαa( �Gbb − �mb) �Gaβ + Gαb(Gaa − ma) �Gbβ
+�P
+�
+(6.9b)
++ E
+��
+a̸=y
+�
+b̸=a
+�maXaxSabGαa
+�
+Gba + �Gba
+� �GbβP
+�
++ R2
+(6.9c)
++
+�
+a̸=y
+Xaxma �maSay E
+��
+Gαy �Gyβ − δαyδyβmy �my
+�P
+�
+(6.9d)
++ δβ̸=y �mβXβx E
+�
+(Gαβ − δαβmβ)P
+�
+− E
+��
+a̸=y
+�maXaxGαa
+�
+b
+Sab �Gbβ∂abP
+�
+,
+(6.9e)
+where R2 is the total error coming from the higher order cumulants, and all unrestricted summations
+are from 1 to N. We successively bound the terms (6.9b)-(6.9e) appearing on the right-hand side of
+(6.9). By condition (A), local law (4.4), upper bound (4.2), and (6.5), it follows that the terms (6.9b)
+and the first term in (6.9c) are bounded by O≺((Ψ + �Ψ)ΛM2d−1). Similarly, the term (6.9d) and the
+first term in (6.9e) are bounded by O≺(∥X∥max (Ψ + �Ψ)M2d−1).
+We bound the second term in (6.9e). It follows by (A), (4.4), bounds (4.2), (6.6), and (6.7) that
+�
+b
+Sab �Gbβ∂abP ≺ (Ψ + �Ψ + δαa + δaβ)�ΨΦM2d−2.
+(6.10)
+Hence, the second term in (6.9e) is bounded by O≺
+�
+(Ψ + �Ψ)ΛΦM2d−2
+�
+. Finally, it is easy to check
+using estimates (4.16), (6.6) and identity (6.7), together with condition (A) and (4.2), that the error
+term R2 ≺ (Ψ + �Ψ)ΛM2d−1 + (Ψ + �Ψ)ΛΦM2d−2 + (Ψ + �Ψ)ΛΦ2M2d−3.
+Observe that the first term on the right-hand side of (6.9a) can be expressed as
+�
+a̸=y
+ma �maXax E
+�
+Fay
+αβ(S)P
+�
+= E
+�
+Fay
+αβ(X)P
+�
+− E
+�
+Fay
+αβ(Y )P
+�
+− my �myXyx E
+�
+Fyy
+αβ(S)P
+�
+,
+(6.11)
+where the last term is bounded by O≺(N −1ΛM2d−1). Combining (6.9) and (6.11) yields
+E
+�
+|Fxy
+αβ(Y ) − δαβf xy
+α (Y )|2d�
+≺
+�
+Ψ + �Ψ
+�
+ΛΦ2M2d−3,
+(6.12)
+for any control parameter Φαβ,y satisfying (6.5). Hence, by Lemma 6.2,
+Fxy
+αβ(Y ) = δαβf xy
+α (Y ) + O≺
+�
+Λ(Ψ + �Ψ)
+�
+,
+(6.13)
+which concludes the proof of Lemma 6.1.
+12
+
+Remark 6.3. If z and ζ are in the same (upper or lower) half-plane, Lemma 6.1 implies Theorem 3.2.
+Indeed, the bound (4.12) in Proposition 4.4 shows that provided η�η > 0, X := (1−Sm �m)−1W satisfies
+|Xjk| ≲ N −1. Applying Lemma 6.1 to X = (1 − Sm �m)−1W then yields (3.6), and (3.7) follows by
+summing (3.6). We turn to the case of z and ζ lying in different (upper and lower) half-planes.
+6.2
+Stability Operator Analysis
+In this subsection we obtain all the properties of the stability operator (1 − Sm(z)m(ζ)) that we use
+in combination with Lemma 6.1 to finish the proof of Theorem 3.2 for z, ζ lying in opposite half-planes,
+as outlined in the beginning of Section 6.
+For two spectral parameters z, ζ, let η := Im z, and �η := Im ζ. Without loss of generality, we assume
+in the following that Re z ∈ Iκ, η > 0 and Re ζ ∈ Iκ, �η < 0. For the remainder of this subsection, we
+use the following notation
+F ≡ F(z) := |m(z)|S|m(z)|,
+B ≡ B(z, ζ) := 1 − Sm(z)m(ζ),
+B0 ≡ B0(z) := 1 − S|m(z)|2 = |m(z)|−1(1 − F)|m(z)|.
+(6.14)
+We view the operator B as a perturbation of B0 = B(z, ¯z), since |ζ − ¯z| is small. We deduce the
+desired properties of B from those of B0, which, in turn, follow from the lower bound on the spectral
+gap of F found in (4.9).
+Let {ψj}N
+j=1 denote the eigenvalues of F (with multiplicity) in descending order. Then, by Per-
+ron–Frobenius theorem, the principal eigenvalue ψ1 is real, and it coincides with the spectral radius
+∥F∥ℓ2→ℓ2. Furthermore, by taking the imaginary part of the vector Dyson equation (3.4) and multi-
+plying both sides by |m| coordinate-wise, we obtain
+�
+1 − F
+�Im m
+|m| = η|m|.
+(6.15)
+Furthermore, by condition (A), for every j we have (S Im m)j ∼ ⟨Im m(z)⟩ ∼ ρ(z), where ρ(z) is
+the harmonic extension of the self-consistent density of states ρ(x) defined in (3.3) into C. Hence by
+taking the imaginary part of (3.4), we get
+Im mj
+|mj|
+∼ |mj|(ρ(z) + η), ,
+j ∈ {1, . . . , N}.
+(6.16)
+Therefore, by (6.15) and (6.16), 1 − ψ1 ≲ η. Together with an upper bound (4.10) on ∥F∥ℓ2→ℓ2, this
+implies that 1 − ψ1 ∼ η. It follows from (4.9) that the principal eigenvalue of F is separated from the
+rest of the spectrum by an annulus, i.e., there exist r > 0 and δ > 0 independent of z and N such that
+|1 − ψ1| < r − δ,
+and
+|1 − ψj| > r + δ,
+j ∈ {2, . . . , N}.
+(6.17)
+In the remainder of this subsection, we show that for all ζ sufficiently close to ¯z, the eigenvalue of
+B with the smallest modulus is also separated from the rest of the spectrum by an annulus of order
+one width.
+Using the argument principle and Jacobi’s formula, one can express the number of eigenvalues
+(with multiplicity) of a matrix X inside a domain Ω by a contour integral
+NX(Ω) =
+1
+2πi
+�
+∂Ω
+Tr(w − X)−1dw.
+(6.18)
+To show the eigenvalue separation for B, we begin by estimating the norm of the resolvent of B inside
+the annulus
+Ar,δ := {w ∈ C : r − 3δ/4 ≤ |w| ≤ r + 3δ/4},
+(6.19)
+with r and δ as in (6.17).
+13
+
+Claim 6.4. There exists ε1 > 0 and �C > 0 independent of N and z such that
+���(w − B(z, ζ))−1��� ≤ �C
+(6.20)
+holds for all w ∈ Ar,δ and all ζ such that Re ζ ∈ Iκ, Im ζ < 0 and |ζ − ¯z| ≤ ε1. (The norm ∥·∥ is
+induced by either ℓ2 or ℓ∞.)
+Proof. Observe that
+��(w − B)−1�� ≤
+���
+�
+1 − (w − B0)−1(B − B0)
+�−1���
+��(w − B0)−1��.
+Since (w − B0)−1 = −|m|−1(1 − w − F)−1|m| and |m| ∼ 1, (6.17) implies that
+��(w − B0)−1�� ≤
+C
+min
+j
+|ψj − w| ≤ 4C
+δ ,
+w ∈ Ar,δ.
+(6.21)
+From the uniform bounds (4.2), (4.14) on |m| and |m′| we have ∥B − B0∥ ≲ |ζ − ¯z|, which implies
+that there exists ε1 > 0 such that
+∀ζ : |ζ − ¯z| ≤ ε1, ∥B − B0∥ ≤
+δ
+8C ,
+(6.22)
+where C is the constant in (6.21).
+It follows immediately that
+���
+�
+1 − (w − B0)−1(B − B0)
+�−1��� ≤ 2 and hence
+��(w − B)−1�� ≤ 8C
+δ .
+(6.23)
+Claim 6.4 implies that for any sufficiently large fixed N the integrand in (6.18) with X := B is
+uniformly bounded in Ω := Ar,δ for all ζ such that |ζ − ¯z| ≤ ε1, hence by analyticity
+NB(z,ζ)(Ar,δ) = 0,
+|ζ − ¯z| ≤ ε1.
+(6.24)
+Since the eigenvalues of B(z, ζ) are continuous in ζ, (6.24) implies that no eigenvalue can move between
+the two connected components of C\Ar,δ, which together with (6.17) yields the following claim.
+Claim 6.5. For any sufficiently large N, the equalities
+NB({|w| < r − 3δ/4}) = NB0({|w| < r − 3δ/4}) = 1,
+NB({|w| > r + 3δ/4}) = NB0({|w| > r + 3δ/4}) = N − 1,
+(6.25)
+hold for any ζ such that Re ζ ∈ Iκ, Im ζ < 0 and |ζ − ¯z| ≤ ε1.
+Claim 6.5 now allows us to define the principal eigenprojector Π of B as a contour integral
+Π ≡ Π(z, ζ) :=
+1
+2πi
+�
+|ξ|=r
+(ξ − B(z, ζ))−1dξ.
+(6.26)
+Claim 6.5 asserts that the contour {|ξ| = r} encircles exactly one eigenvalue of B with multiplicity,
+hence Π is a rank one eigenprojector.
+We now prove that the restriction of B−1 to the range of (1 − Π) is bounded by a constant.
+Claim 6.6. For all z, ζ such that Re z, Re ζ ∈ Iκ, Im z Im ζ < 0 and |ζ − ¯z| ≤ ε1,
+��B−1(1 − Π)
+��
+ℓ∞→ℓ∞ ≤ �c,
+(6.27)
+where �c depends only on the constants in conditions (A), (B) and κ.
+Proof. By expression (6.26) for Π we have
+B−1(1 − Π) = − 1
+2πi
+�
+|ξ|=r
+1
+ξ (ξ − B)−1dξ
+(6.28)
+14
+
+Hence the norm of B−1(1 − Π) is bounded by
+��B−1(1 − Π)
+��
+ℓ∞→ℓ∞ ≤ 1
+2π
+2π
+�
+0
+���
+�
+reiθ − B
+�−1���
+ℓ∞→ℓ∞ dθ ≤ 8C
+δ ,
+(6.29)
+using the bound in Claim 6.4 on the circle {|ξ| = r} which lies inside Ar,δ.
+Finally, we show that the vector of ones is sufficiently separated from the kernel of Π. This ensures
+a stable decomposition of the space into the direct sum of the range of (1 − Π) and the span of 1, so
+we can apply the local laws to each of the components separately.
+Claim 6.7. There exists ε > 0 independent of N and z such that for all ζ with Re ζ ∈ Iκ, Im ζ < 0
+and |ζ − ¯z| ≤ ε,
+∥Π1∥∞
+∥Π∥ℓ∞→ℓ∞ ≥ c,
+(6.30)
+where c > 0 is a constant independent of N and z.
+Proof. Define the projector Π0 corresponding to B0 via (6.26). Then Π0 = |m|−1�Π0|m|, where �Π0 is
+the orthoprojector corresponding to the principal eigenvalue of the Hermitian operator F.
+Since |m| ∼ 1 we have ∥Π0∥ℓ∞→ℓ∞ ≤ C0. Moreover, by Proposition 4.3, the ℓ2-normalized eigenvector
+v corresponding to the principal eigenvalue of F has entries vj ≥ 0 with vj ∼ N −1/2, hence
+∥Π01∥∞ =
+���|m|−1�Π0|m|1
+���
+∞ =
+��|m|−1v
+��
+∞ ⟨v, |m|⟩ ≥ c0,
+(6.31)
+where c0 > 0 is a constant independent of N and z.
+Similarly to the proof of (6.22), for any γ ∈ (0, 1] there exists εγ > 0, such that the bound
+∥B − B0∥ℓ∞→ℓ∞ ≤ γ δ
+8C
+(6.32)
+holds for all ζ ∈ D−
+κ with |ζ − ¯z| ≤ εγ. Here δ is defined in (6.17) and C > 0 is the constant in (6.21).
+We choose εγ to be smaller than ε1 of Claim 6.4, then for all ζ with Re ζ ∈ Iκ, Im ζ < 0 such that
+|ζ − ¯z| ≤ εγ we have
+∥Π − Π0∥ℓ∞→ℓ∞ ≤ r
+2π
+2π
+�
+0
+��(reiθ − B)−1 − (reiθ − B0)−1��
+ℓ∞→ℓ∞ dθ
+≤ r
+2π
+2π
+�
+0
+��(reiθ − B)−1(B − B0)(reiθ − B0)−1��
+ℓ∞→ℓ∞ dθ
+≤ r · 8C
+δ · γ δ
+8C · 4C
+δ
+= γ 4Cr
+δ
+.
+(6.33)
+Here we used inequalities (6.21) and (6.23) in the second to last step. We set the value of γ to be
+γ0 := min
+�
+1, c0δ
+8Cr
+�
+, which guarantees that
+∥Π1∥∞ ≥
+��∥Π01∥∞ − ��Π − Π0∥ℓ∞→ℓ∞ ∥1∥∞
+�� ≥ c0 − γ0
+4Cr
+δ
+≥ c0
+2 .
+(6.34)
+Finally, observe that
+∥Π∥ℓ∞→ℓ∞ ≤ ∥Π0∥ℓ∞→ℓ∞ + ∥Π − Π0∥ℓ∞→ℓ∞ ≤ C0 + c0/2.
+(6.35)
+This proves the claim with c := c0/(2C0 + c0).
+15
+
+6.3
+Finishing the Proof of Theorem 3.2
+Proof of Theorem 3.2. Recall that the objective is to estimate the quantities defined in (3.1).
+In-
+stead of estimating �
+a̸=y waGαa �Gaβ directly, it is more convenient to work with objects of the type
+�
+a̸=y WaxGαa �Gaβ, since they generalize quantities appearing in both (3.6) and (3.7). The redundant
+index x can be eliminated by setting Wax := wa.
+In the case Im z Im ζ > 0, (3.6) and (3.7) follow immediately from (4.12) and Lemma 6.1 (see
+Remark 6.3). Therefore, we focus on the case Im z Im ζ < 0.
+Since Π has rank one and Claim 6.7 asserts that Π1 ̸= 0, the kernel of Π together with 1 span
+CN. Therefore we can decompose each column of the matrix W into a linear combination of 1 and an
+element of ker Π, that is, there exists an N × N matrix Y and a vector s ∈ CN such that
+W = Y + 1s∗,
+ΠY = 0.
+(6.36)
+We multiply the first equality in (6.36) by Π from the left, apply both sides to the a-th standard basis
+vector ea of CN and take the ℓ∞-norm to deduce
+∥ΠWea∥∞ = |sa| ∥Π1∥∞ ,
+a ∈ {1, . . ., N}.
+(6.37)
+By assumption, ∥W∥max ≲ N −1, hence ∥Wea∥∞ ≲ N −1. Using Claim 6.7 we get
+|sa| ≲ N −1 ∥Π∥ℓ∞→ℓ∞
+∥Π1∥∞
+≲ N −1,
+a ∈ {1, . . . , N}.
+(6.38)
+We combine (6.36) and the resolvent identity in the form (z − ζ)G �G = G − �G to obtain
+�
+a̸=y
+WaxGαa �Gaβ =
+�
+a̸=y
+YaxGαa �Gaβ + gy
+αβ¯sx,
+gy
+αβ := Gαβ − �Gαβ
+z − ζ
+− Gαy �Gyβ.
+(6.39)
+Define the N × N matrix X := (1 − Sm �m)−1 Y . It follows from (6.36) that Y = (1 − Π)Y , hence
+X = (1 − Sm �m)−1(1 − Π)Y . Furthermore, estimates ∥W∥max ≲ N −1, (6.36), and (6.38) imply that
+|Yab| ≲ N −1 for all a and b. Since by Claim 6.6
+��(1 − Sm �m)−1(1 − Π)
+��
+ℓ∞→ℓ∞ ≲ 1, we conclude that
+∥X∥max = max
+a,b |Xab| ≲ N −1.
+(6.40)
+First, using (6.40), we can apply Lemma 6.1 to the first term in (6.39) to obtain
+�
+a̸=y
+YaxGαa �Gaβ = δαβmα �mα([(1 − Sm �m)−1Y ]αx − δαyYαx) + O≺
+�
+Ψ2 �Ψ + Ψ�Ψ2�
+.
+(6.41)
+Using (6.36), we proceed by computing
+mα �mα[(1 − Sm �m)−1Y ]αx =
+�
+m �m
+�
+1 − Sm �m
+�−1 (W − 1s∗)
+�
+αx
+=
+�
+m �m
+�
+1 − Sm �m
+�−1W
+�
+αx −
+�
+m �m
+�
+1 − Sm �m
+�−11
+�
+α¯sx.
+(6.42)
+Finally, it follows from subtracting the vector Dyson equations (3.4) for z and ζ that
+m �m
+�
+1 − Sm �m
+�−11 = m − �m
+z − ζ .
+(6.43)
+Next, we estimate the second term in (6.39). Applying the local law in the form (4.4), we obtain
+gy
+αβ = δαβ
+mα − �mα
+z − ζ
+− δαβδαymα �mα + O≺
+�
+(|η| + |�η|)−1(Ψ + �Ψ)
+�
+,
+(6.44)
+where we used that |z − ζ| ≥ |η| + |�η|, since η�η < 0. Combining (6.38), (6.39), and (6.41)-(6.44) yields
+�
+a̸=y
+WaxGαa �Gaβ = δαβ
+�
+m �m
+�
+1 − Sm �m
+�−1W
+�
+αx − δαβδαy[m �mW]αx
++ O≺
+�
+(Ψ + �Ψ)(Ψ�Ψ + min{Θ, �Θ})
+�
+,
+(6.45)
+16
+
+which proves (3.6) by setting Wax := wa.
+To prove (3.7), we observe that by setting x = y = α = β = b in (6.39) and summing over b yields
+�
+b
+�
+a̸=b
+WabGba �Gab =
+�
+b
+�
+a̸=b
+YabGaa �Gab+⟨s, g⟩,
+gb := Gbb − �Gbb
+z − ζ
+−Gbb �Gbb, b ∈ {1, . . ., N}. (6.46)
+To estimate ⟨s, g⟩, we use (6.38) and the averaged local law (4.3) to obtain
+�
+s, g
+�
+=
+�
+s, m − �m
+z − ζ
+− m �m
+�
++ O≺
+�
+(|η| + |�η|)−1(Θ + �Θ)
+�
+,
+(6.47)
+where we used that |z − ζ| ≥ |η| + |�η|, since η�η < 0.
+Setting x = y = α = β = b in (6.41), summing over b, using the identities (6.42) and (6.43), and
+combining the result with (6.47), we deduce that
+�
+b
+�
+a̸=b
+WabGba �Gab = Tr
+�
+m �mSm �m
+�
+1 − Sm �m
+�−1W
+�
++ NO≺
+�
+Ψ�Ψ(Ψ + �Ψ) + Θ�Θ
+�
+,
+(6.48)
+where we used that (|η| + |�η|)−1(Θ + �Θ) = NΘ�Θ. This establishes (3.7) and concludes the proof of
+Theorem 3.2.
+Remark 6.8. We outline the steps needed to achieve the optimal error estimate (3.12). First, one
+needs to adapt the proof of Theorem 3.2. More specifically, replace the decomposition (6.36) with
+W = Y + 1s∗ + q1∗, such that Π(z, ζ)Y = Y Πt(ζ, z) = 0,
+(6.49)
+where Π(z, ζ) is the destabilizing eigenprojector defined in (6.26). The terms involving s and q are
+handled using the averaged local law (4.3), similarly to (6.47).
+For the remaining term, R := �
+y Fyy
+yy , we adapt the mechanism of Lemma 6.1 by using the
+following iterative scheme.
+In the first step, we apply an expansion similar to (6.9) to the partial
+derivative ∂jkR. This improves the error in the estimate on R by a factor of (Ψ + �Ψ)1/2. If we expand
+∂lp∂jkR in a similar manner, we gain another (Ψ + �Ψ)1/4. Iterating this approach we can estimate R
+with an error stochastically dominated by NΨ�Ψ(Ψ + �Ψ)2−2−d for any given integer d (where d is the
+maximal order of expanded partial derivatives). By Definition 3.1, this is sufficient to establish (3.12).
+Similar arguments in the context of random band matrices can be found in [10].
+Proof of Corollary 3.3. Estimate (3.9) on Txy(ζ, z) follows from (3.6) by setting α = β = y and
+wa := Sxa. Estimate (3.10) on Tr[AT (z, ζ)] follows from (3.7) by setting W := SAt, which satisfies
+|Wab| ≲ N −1 ∥A∥ℓ∞→ℓ∞. This concludes the proof of Corollary 3.3.
+Remark 6.9. Note that estimates (3.6) and (3.7) (also with the improved error term (3.12)) hold
+without omission of indices in the a summation. Indeed, it follows from Theorems 3.2 and 4.2 that
+�
+a
+waGαa �Gaβ = δαβ
+�
+m �m
+�
+1 − Sm �m
+�−1w
+�
+α + O≺
+�
+(Ψ + �Ψ)(Ψ�Ψ + 1{η�η<0} min{Θ, �Θ})
+�
+,
+�
+a,b
+WabGba �Gab = Tr
+�
+m �m
+�
+1 − Sm �m
+�−1W
+�
++ O≺
+�
+N(Ψ + �Ψ)Ψ�Ψ + 1{η�η<0}NΘ�Θ
+�
+.
+(6.50)
+7
+Proof of Proposition 5.2
+In this section, we compute the variance V (f) defined in (5.3) for mesoscopic C2
+c test functions f. In
+[17], the limiting variance was computed for several types of C∞ test functions, including compactly
+supported ones; however, V (f) is computed with an O(1) error (see, e.g., Lemma 6.7 in [17]), which
+is not negligible in the setting of the present paper. To obtain effective error bounds, we augment the
+proof laid out in [17] by performing further integration by parts in the integral representation of V (f),
+thus eliminating the f ′ terms, improving the error by a factor of O(η0).
+Throughout this section, we adhere to the notation m ≡ m(z), �m ≡ m(ζ), η := Im z, �η := Im ζ.
+17
+
+The stability operator (1 − Sm �m) can be expressed in terms of the self-saturated energy operator
+F, defined in (4.6), via the following identity
+1 − Sm �m = |m �m|−1/2 (U∗ − F(z, ζ)) |m �m|1/2U,
+U := m �m
+|m �m|.
+(7.1)
+Furthermore, by (4.9), the operator F can be decomposed such that
+F(z, ζ) = ψ1(z, ζ) v(z, ζ)
+�
+v(z, ζ)
+�∗ + A(z, ζ),
+A(z, ζ)v(z, ζ) = 0,
+∥A(z, ζ)∥ℓ2→ℓ2 ≤ 1 − �δ,
+(7.2)
+where ψ1, v is the principal eigenvalue-eigenvector pair of F, and �δ is the constant in (4.9).
+Let R ≡ R(z, ζ) denote (U∗(z, ζ) − A(z, ζ))−1. In the sequel, we drop the arguments and write
+A ≡ A(z, ζ). Lower bound (4.8) and the inequality in (7.2) imply that
+∥R∥ℓ2→ℓ2 + ∥R∥ℓ∞→ℓ∞ ≲ 1.
+(7.3)
+In the following lemma, we collect the perturbative estimates on the saturated self-energy operator F
+and related quantities established in [17].
+Lemma 7.1. (Proposition 6.5, (6.52), (6.60), (6.71), and (6.67) in [17]) Let w, ζ1, ζ2 be spectral
+parameters in Iκ + i[−1, 1], and let F be the operator defined in (4.6), then the principal eigenvalue-
+eigenvector pair ψ1, v of F satisfies
+∥v(w, ζ1) − v(w, ζ2)∥ℓ2→ℓ2 + |ψ1(w, ζ1) − ψ1(w, ζ2)| ≲ |ζ1 − ζ2|.
+(7.4)
+Furthermore, for operator A defined in (7.2), we have the estimate
+∥F(w, ζ1) − F(w, ζ2)∥ℓ2→ℓ2 + ∥A(w, ζ1) − A(w, ζ2)∥ℓ2→ℓ2 ≲ |ζ1 − ζ2|.
+(7.5)
+Let z := x + iη, ζ := y − iη, with x, y ∈ Iκ, 0 ≤ η ≤ 1, then
+ψ1
+�
+v, Rm′
+m U∗Rv
+�
+= ψ1(z, z)
+�
+v(z, z)m′
+m v(z, z)
+�
++ O(|x − y|)
+(7.6)
+Let ω ≡ ω(z, ζ) := 1 − ψ1⟨v, Rv⟩, then
+ω(z, ζ) = 1 − ψ1(z, z) + ψ1(z, z)(x − y)
+�
+v(z, z)m′
+m v(z, z)
+�
++ O(|x − y|2),
+(7.7)
+Moreover, there exists ε > 0 independent of N, such that for all x, y ∈ Iκ satisfying |x − y| ≤ ε,
+|ω(z, ζ)| ≳ η + |x − y|.
+(7.8)
+Finally, for z := x + iη with x ∈ Iκ, the following identity holds
+lim
+η→+0
+�
+v(z, z)m′
+m v(z, z)
+�
+= iπ
+2 ρ(x)
+����
+Im m(x + i0)
+|m(x)|
+����
+−2
+2
+(7.9)
+By our choice of κ, E0 is in the interior of the bulk interval Iκ, defined in (3.5) , hence if we define
+ˆε := min{ε/4, dist(E0, R\Iκ)}, then ˆε ∼ 1. Furthermore, since the function g is compactly supported,
+we assume that supp(f) ⊂ [E0 − ˆε, E0 + ˆε] for large N.
+Lemma 7.2. Let η∗ ≡ η∗(N) satisfy 0 < η∗ ≤ N −100, then V (f), defined in (5.3), admits the estimate
+V (f) =
+1
+4π2
+��
+[E0−ˆε,E0+ˆε]2
+(f(y) − f(x))2 �K(x + iη∗, y − iη∗)dxdy + O
+�
+η0 + N −ε0�
+,
+(7.10)
+where
+�K(z, ζ) := −2 Re Tr
+�m′
+m (1 − Sm �m)−1Sm �m′(1 − Sm �m)−1
+�
+.
+(7.11)
+18
+
+In preparation for the proof of Lemma 7.2 we define an auxiliary function L(z, ζ)
+L(z, ζ) := Llog(z, ζ) + L1(z, ζ),
+Llog(z, ζ) := −2 log det {1 − Sm �m} ,
+L1(z, ζ) := − Tr [Sm �m] + 1
+2
+�
+m �m, C(4)m ��m
+�
+,
+(7.12)
+where log is the principal branch of the complex logarithm, and C(4) is the matrix of the fourth cumu-
+lants of H. By Jacobi’s formula for the derivative of the determinant, it follows from the definitions
+of L and K, that for all z, ζ ∈ C\R
+∂2
+∂ζ∂z L(z, ζ) = K(z, ζ).
+(7.13)
+Furthermore, by condition (A) and the upper bound (4.2), it follows that
+|Llog(z, ζ)| ≤π + log |det {1 − Sm �m}| ≲ 1 + Tr
+�
+(1 − Sm �m)∗ (1 − Sm �m) − I
+�
+≲ 1,
+(7.14)
+where in the last line we used
+�
+(1 − Sm �m)∗ (1 − Sm �m) − I
+�
+jj ≲ N −1.
+The partial derivatives of L1 contribute only sub-leading terms to L. Indeed, we have the estimates
+L1(z, ζ) ≲ 1,
+∂
+∂zL1(z, ζ) ≲ 1,
+∂2
+∂ζ∂z L1(z, ζ) ≲ 1,
+(7.15)
+where we used the moment condition (2.2) to bound Sjk and C(4)
+jk , (4.2) to get the upper bound
+m, �m ≲ 1, and (4.14) to obtain m′, �m′ ≲ 1, since [E0 + ˆε, E0 − ˆε] ⊂ Iκ.
+The following claim collects the bounds on K and ∂zL that together with (7.14) enable integration
+by parts in the definition (5.3) of the variance V (f), which is the essence of Lemma 7.2.
+Claim 7.3. (Proposition 6.2 and Proposition 6.6 in [17]) Let K(z, ζ) and L(z, ζ) be as defined in (5.4)
+(with β = 1) and (7.12) respectively, then for all z, ζ ∈ C\R with Re z, Re ζ ∈ [E0 − ˆε, E0 + ˆε] and
+| Im z|, | Im ζ| ≤ 1 we have
+K(z, ζ) ≲ 1 + 1{η�η<0}(|η| + |�η|)−2,
+∂
+∂z L(z, ζ) ≲ 1 + (| Re z − Re ζ| + |η| + |�η|)−1,
+(7.16)
+where η := Im z, �η := Im ζ.
+Proof of Lemma 7.2. Define Ω∗ := {z ∈ C : 1 > | Im z| > η∗}. Recall the definition of V (f) from (5.3).
+First, we prove that
+V (f) = 1
+π2
+�
+Ω∗
+�
+Ω∗
+∂ �f(ζ)
+∂¯ζ
+∂ �f(z)
+∂¯z
+K(z, ζ)d¯ζdζd¯zdz + O
+�
+N −ε0�
+.
+(7.17)
+It follows from (5.6) that
+∂ �f
+∂¯z = 1
+2
+�
+−ηχ′(η)f ′(x) + i
+�
+ηχ(η)f ′′(x) + χ′(η)f(x)
+��
+.
+(7.18)
+Moreover, for all z with | Im z| < 1/2, (7.18) and the properties of χ in (5.6) imply
+∂ �f
+∂¯z = i Im z
+2
+f ′′(Re z).
+(7.19)
+Let V∗(f) denote the integral on right hand side of (7.17), and define η1 := N −ε0/2η0. It follows
+from the first inequality in (7.16), and (7.19) that
+|V (f) − V∗(f)| ≲
+��
+R2
+|f ′′(x)f ′′(y)| dxdy
+η1
+�
+η∗
+2η1
+�
+η∗
+η�η
+(η + �η)2 d�ηdη.
+(7.20)
+19
+
+Note that η�η ≤ (η + �η)2/4, hence the integral over d�ηdη is bounded by η2
+1/2, and since ∥f ′′∥1 ∼ η−1
+0 ,
+(7.17) is established.
+We write z := x + iη, ζ := y + i�η and plug (7.13) into the expression (7.17) for V (f). Using the
+fact that ∂zu = −i∂ηu for any holomorphic function u(z), and integrating by parts in η, we obtain
+V (f) = i
+π2
+��
+R2
+dxdy
+�
+|�η|>η∗
+∂ �f(ζ)
+∂¯ζ
+�
+|η|>η∗
+∂2 �f(z)
+∂η∂¯z
+∂
+∂ζ L(z, ζ)d�ηdη
+− i
+π2
+��
+R2
+dxdy
+�
+|�η|>η∗
+∂ �f(ζ)
+∂¯ζ
+�
+η=±η∗
+∂ �f
+∂¯z (x + iη) ∂
+∂ζ L(z, ζ)d�η + O
+�
+N −ε0�
+.
+(7.21)
+The second estimate in (7.16), expression (7.18) and the estimates ∥f ′′∥1 ∼ η−1
+0 , ∥f ′∥1 ∼ 1, ∥f∥1 ∼ η0
+imply that the boundary term in (7.21) is dominated by O≺(η∗η−2
+0 ), which is smaller than O (N −ε0).
+Similarly, integrating the first term on the right hand side of (7.21) by parts in �η we get
+V (f) = − 1
+π2
+�
+Ω∗
+�
+Ω∗
+∂2 �f(z)
+∂¯z∂η
+∂2 �f(ζ)
+∂¯ζ∂�η L(z, ζ)d¯ζdζd¯zdz
++ 1
+π2
+��
+R2
+dxdy
+�
+|η|>η∗
+∂2 �f(z)
+∂η∂¯z
+�
+�η=±η∗
+∂ �f
+∂¯ζ (y + i�η)L(z, y + i�η)dη + O
+�
+N −ε0�
+.
+(7.22)
+It follows from (7.14) and the expression (7.18) that the boundary term (the second line of (7.22)) is
+again dominated by O≺(N −ε0).
+We apply Stokes’ theorem to (7.22) twice: once in z and once in ζ. Considering that ∂η �f(z) vanishes
+on the boundary of Ω∗ except for the lines {Im z = ±η∗}, this results in
+V (f) = 1
+4π2
+��
+R2
+�
+η,�η=±η∗
+sign (η�η) ∂ �f(x + iη)
+∂η
+∂ �f(y + i�η)
+∂�η
+L(x + iη, y + i�η)dxdy + O
+�
+N −ε0�
+= −
+1
+2π2
+��
+R2
+f ′(x)f ′(y) �L(x, y)dxdy + O
+�
+N −ε0�
+,
+(7.23)
+where
+�L(x, y) := Re [L(x + iη∗, y + iη∗) − L(x + iη∗, y − iη∗)]
+(7.24)
+We restrict the integrations in (7.23) to [E0 − ˆε, E0 + ˆε], since this interval contains the support of f.
+Furthermore, for all y ∈ supp(f), y − E0 ≲ η0, hence |y − E0 ± ˆε| ∼ 1. By symmetry of L(z, ζ), and
+the second estimate in (7.16) it follows that
+∂
+∂y
+�L(E0 ± ˆε, y) ≲ 1,
+y ∈ supp(f).
+(7.25)
+We write f ′(y) = ∂y (f(y) − f(x)), perform integration by parts in y and integrate the boundary term
+by parts in x to obtain
+V (f) = 1
+2π2
+E0+ˆε
+�
+E0−ˆε
+E0+ˆε
+�
+E0−ˆε
+f ′(x) (f(y) �� f(x)) ∂
+∂y
+�L(x, y)dxdy
++
+1
+4π2
+E0+ˆε
+�
+E0−ˆε
+(f(x))2 ∂
+∂x
+�
+�L(x, E0 + ˆε) − �L(x, E0 − ˆε)
+�
+dx + O
+�
+N −ε0�
+.
+(7.26)
+Since ∥f∥2
+2 ≲ η0, it follows from (7.25) that the second integral in (7.26) is O (η0).
+Similarly,
+integrating (7.26) by parts in x and using (7.26) to substitute one of the emerging itegrals for
+−V (f) + O (N −ε0 + η0), we get
+2V (f) = 1
+2π2
+E0+ˆε
+�
+E0−ˆε
+E0+ˆε
+�
+E0−ˆε
+(f(y) − f(x))2
+∂2
+∂x∂y
+�L(x, y)dxdy + O
+�
+η0 + N −ε0�
+,
+(7.27)
+20
+
+where we again used (7.25) to estimate the boundary term. For any holomorphic function u(z) of
+z = x + iη, we have ∂xu = Re[∂zu], hence ∂x∂y �L(x, y) = Re [K(x + iη∗, y + iη∗) − K(x + iη∗, y − iη∗)].
+Finally, in view of in view of the first estimate in (7.16), ∂z∂ζLlog(x + iη∗, y + iη∗) ≲ 1, so its
+contribution is also bounded by O≺(η0 ∥g∥2
+2 + η2
+0 ∥g∥2
+1). Moreover, it follows from the last estimate in
+(7.15) that we can replace K(x+iη∗, y −iη∗) by ∂z∂ζLlog(x+iη∗, y −iη∗), since the contribution of the
+remaining terms is bounded by O≺(η0 ∥g∥2
+2 + η2
+0 ∥g∥2
+1). This concludes the proof of Lemma 7.2.
+Once Lemma 7.2 is established, we can follow the method of Lemma 6.7 in [17] to finish the proof
+of Proposition 5.2.
+Fix x, y ∈ [E0 − ˆε, E0 + ˆε] and write z := x + iη∗, ζ := y − iη∗, as in (7.10). It follows from (7.1)
+and (7.2) that the kernel �K(z, ζ) can be written as
+�K(z, ζ) = −2 Re Tr
+�m′
+m U∗�
+R + ψ1
+ω Rvv∗R
+�
+F �m′
+�m
+�
+R + ψ1
+ω Rvv∗R
+��
+,
+(7.28)
+where ω is defined in (7.7). Expanding the brackets in (7.28), collecting like terms according to the
+powers of ω−1, and using the cyclic property of trace yields
+�K(z, ζ) = −2 Re
+�ψ2
+1
+ω2
+�
+v, Rm′
+m U∗Rv
+��
+v, RF �m′
+�m Rv
+��
++ O
+�
+1 + ω−1�
+,
+(7.29)
+since Tr
+� m′
+m U∗RF �
+m′
+�
+m R
+�
+, Tr
+� m′
+m U∗RF �
+m′
+�
+m Rvv∗R
+�
+, and Tr
+� m′
+m U∗Rvv∗RF �
+m′
+�
+m R
+�
+are all O(1). The first
+scalar product in (7.29) can be estimated using (7.6).
+We compute the second scalar product in (7.29). It follows from uniform bounds (4.2) and (4.14)
+that ∥m(z)− m(¯ζ)∥∞ ≲ |x− y|, and hence ∥U(z, ζ) − 1∥ℓ2→ℓ2 ≲ |x− y|. Together with estimates (7.5)
+and (7.4), this yields
+ψ1
+�
+v, RF �m′
+�m Rv
+�
+= ⟨v(ζ, ζ), F(ζ, ζ) �
+m′
+�m v(ζ, ζ)
+�
++ O(|x − y|),
+(7.30)
+where we used the identity R(¯ζ, ζ)v(ζ, ζ) = (1 − A(ζ, ζ))−1v(ζ, ζ) = v(ζ, ζ).
+It follows from the estimate on v in (7.4) that ∥v(ζ, ζ) − v(y, y)∥2 ≲ η∗. Vector v(y, y) is the ℓ2-
+normalization of |m(y)|−1 Im m(y + i0), hence it satisfies F(y, y)v(y, y) = v(y, y) by (3.4). Therefore
+using (4.14) and the lower bound in (7.5), we obtain
+∥F(ζ, ζ)v(ζ, ζ) − v(ζ, ζ)∥2 ≲ η∗.
+(7.31)
+Substituting (7.31) into (7.30) yields
+ψ1
+�
+v, RF �m′
+�m Rv
+�
+= ⟨v(ζ, ζ), �m′
+�m v(ζ, ζ)
+�
++ O(|x − y| + η∗),
+(7.32)
+Combining (7.28) with estimates (7.4), (7.6), (7.8) and (7.32) yield
+�K(z, ζ) = −2 Re
+�ψ1(z, z)ψ1(ζ, ζ)
+ω2
+�
+v(z, z)m′
+m v(z, z)
+�
+⟨v(ζ, ζ), �m′
+�m v(ζ, ζ)
+��
++ O(1 + ω−1).
+(7.33)
+It follows by (7.9) and (7.7) that
+lim
+η∗→+0
+�K(x + iη∗, y − iη∗) = 2|x − y|−2 + O(|x − y|−1).
+(7.34)
+Since f ∈ C2
+c (R), (7.33) implies that the integrand in (7.10) is uniformly bounded in η∗ ∈ [0, N −100].
+Therefore, we can take the limit η∗ → 0 in (7.10), and apply the boundary estimate (7.34) to obtain.
+V (f) =
+1
+2π2
+��
+[E0−ˆε,E0+ˆε]2
+(f(x) − f(y))2
+(x − y)2
+dxdy + O
+�
+η0 log N + N −ε0�
+,
+(7.35)
+because the contribution of O(|x − y|−1) to the integral (7.10) is bounded by O(η0 log N).
+21
+
+Finally, the contribution of the regime (x, y) /∈ [E0 − ˆε, E0 + ˆε]2 to the integral
+��
+R2
+(f(x) − f(y))2
+(x − y)2
+dxdy = ∥f∥2
+˙H1/2 = ∥g∥2
+˙H1/2 ,
+(7.36)
+is bounded by O≺(η0), therefore
+V (f) =
+1
+2π2 ∥g∥2
+˙H1/2 + O
+�
+η0 log N + N −ε0�
+.
+(7.37)
+This concludes the proof of Proposition 5.2.
+Appendix A
+Proof of Lemma 5.4
+We use the Helffer–Sj¨ostrand representation to express the linear eigenvalue statistics in terms of the
+resolvent of H (see Section 4.2 in [19] for references),
+{1 − E} [Tr f(H)] = 1
+2π
+�
+C
+∂ �f
+∂¯z {1 − E} [Tr G(z)] d¯zdz.
+(A.1)
+The characteristic function φ then admits the form
+φ(λ) = E [e(λ)] ,
+e(λ) := exp
+�
+iλ 1
+2π
+�
+C
+∂ �f
+∂¯z {1 − E} [Tr G(z)] d¯zdz
+�
+,
+λ ∈ R,
+(A.2)
+and its derivative φ′ is given by
+φ′(λ) = E
+�
+e(λ) i
+2π
+�
+C
+∂ �f
+∂¯z {1 − E} [Tr G(z)] d¯zdz
+�
+,
+λ ∈ R.
+(A.3)
+As observed in [19], the regime | Im z| ≤ N −ε0/2η0, referred to as the ultra-local scales, does not
+contribute to the integrals in (A.2) and (A.3). This yields the estimates (5.9) (see equations (4.21)
+and (4.22) in [19] for further detail).
+It remains to show that (5.11) holds.
+Applying the cumulant expansion formula (4.15) to the
+quantity E [�e(λ) {1 − E} [Gjj(z)]] yields the following lemma.
+Lemma A.1. (Lemma 5.7 in [17]) For all z ∈ D defined in (3.2) and j ∈ {1, . . ., N} we have
+−1
+mj(z) E [�e(λ) {1 − E} [Gjj(z)]] = − mj(z)
+N
+�
+k=1
+Sjk E [�e(λ) {1 − E} [Gkk(z)]]
+− E [�e(λ) {1 − E} [Tjj(z, z)]]
++ E
+� N
+�
+k=1
+SjkGkj(z)∂�e(λ)
+∂Hjk
+�
+− 1
+2
+N
+�
+k=1
+C(4)
+jk mj(z)mk(z) E
+�∂2�e(λ)
+∂H2
+jk
+�
++ O≺
+�
+(1 + |λ|4)
+�
+Ψ(z)Θ(z) + N −1Ψ(z)η−1/2
+0
+��
+,
+(A.4)
+where η0 is from (2.3), and for a, b ∈ {1, . . ., N}, z, ζ ∈ C\R, Txy(z, ζ) is defined in (1.1).
+Let gj := E [�e(λ) {1 − E} [Gjj(z)]] and let rj denote the right-hand side of (A.4) without the first
+term, then (A.4) reads
+��
+1 − Sm2(z)
+�g
+�
+j = −mj(z)rj. The operator
+�
+1 − Sm2(z)
+�
+can be inverted to
+22
+
+deduce that gj = −
+��
+1 − Sm2(z)
+�−1 m(z)r
+�
+j, where m(z) is interpreted as a multiplication operator
+acting on the vector r. Summing over j, we obtain
+E [�e(λ) {1 − E} [Tr G(z)]] =
+N
+�
+j=1
+gj = −
+N
+�
+j,k=1
+��
+1 − Sm2(z)
+�−1�
+jk mk(z)r k = −
+N
+�
+j=1
+m′
+j(z)
+mj(z)rj,
+(A.5)
+where in the last step we applied the identity m′(z)/m2(z) = (1 − Sm2(z))−11. The second term on
+the right-hand side of (A.4) contributes the first term to the right hand side of (5.11), which, as we
+show in Section 6, is negligible. Therefore, it suffices to estimate the contribution of the third and
+fourth terms on the right-hand side. The necessary estimates on the partial derivatives of �e(λ) are
+collected in the following lemma.
+Lemma A.2. (Lemma 5.6 in [17]) For all j, k ∈ {1, . . . , N} we have
+∂�e(λ)
+∂Hjk
+= −iλ
+π
+2
+1 + δjk
+�e(λ)
+�
+Ω′
+0
+∂ �f
+∂¯ζ
+∂Gkj(ζ)
+∂ζ
+d¯ζdζ.
+(A.6)
+Moreover, for all p ∈ N, the following bound holds
+����
+∂p�e(λ)
+∂Hp
+jk
+���� = O≺
+�
+(1 + |λ|)p�
+,
+(A.7)
+and for k ̸= j
+����
+∂�e(λ)
+∂Hjk
+���� = O≺
+�
+N −1/2(1 + |λ|)η−1/2
+0
+�
+.
+(A.8)
+Second derivatives with k ̸= j are given by
+∂2�e(λ)
+∂H2
+jk
+= 2iλ
+π �e(λ)
+�
+Ω′
+0
+∂ �f
+∂¯ζ
+∂ {mj(ζ)mk(ζ)}
+∂ζ
+d¯ζdζ + O≺
+�
+N −1/2(1 + |λ|)2η−1/2
+0
+�
+.
+(A.9)
+The form in which we write the error terms in Lemmas A.1 and A.2 slightly differs from their
+original form in [17] because we have already applied the estimate ∥f ′′∥1 ∼ η−1
+0 . The leading term in
+(A.9) results in the third line of (5.11).
+Using Lemmas A.2 and 5.6 we proceed to estimate the third term on the right hand side of (A.4).
+Lemma A.3. (c.f. Equation (5.65) of Lemma 5.8 in [17]) For all z ∈ D defined in (3.2) and all
+j ∈ {1, . . . , N} we have
+E
+� N
+�
+k=1
+SjkGkj(z)∂�e(λ)
+∂Hjk
+�
+= − 2iλ
+π E
+�
+�e(λ)
+�
+Ω′
+0
+∂ �f
+∂¯ζ
+∂Tjj(z, ζ)
+∂ζ
+d¯ζdζ
+�
+− iλ
+π Sjj E [�e(λ)]
+�
+Ω′
+0
+∂ �f
+∂¯ζ m′
+j(ζ)mj(z)d¯ζdζ + O≺
+�Ψ(z)(1 + |λ|)
+Nη1/2
+0
+�
+.
+(A.10)
+Proof of Lemma A.3. In view of (1.1), multiplying (A.6) by SjkGkj(z), summing over k ̸= j and
+taking expectations gives the first term on the right hand side of (A.4). For the remaining k = j term,
+observe that the function K(ζ) := Gjj(ζ) − mj(ζ) is analytic in C\R and is stochastically dominated
+by Ψ(ζ) in D. Applying Lemma 5.5 with p = 1 to K(ζ), we obtain
+∂Gjj(ζ)
+∂ζ
+= m′
+j(ζ) + O≺
+�
+| Im ζ|−1Ψ(ζ)
+�
+.
+(A.11)
+Plugging (A.11) into (A.6) with k = j and applying Lemma 5.6 with K(ζ) := ∂ζGjj(ζ) − m′
+j(ζ) with
+s = 3/2, we get
+∂�e(λ)
+∂Hjj
+= −iλ
+π �e(λ)
+�
+Ω′
+0
+∂ �f
+∂¯ζ m′
+j(ζ)d¯ζdζ + O≺
+�
+1 + |λ|)N −1/2η−1/2
+0
+�
+.
+(A.12)
+23
+
+where we used the the fact that |e(λ)| = 1 and the first line of (5.9) to bound |�e(λ)| by O≺(1).
+Multiplying (A.12) by SjjGjj(z) and using the local law (4.4) to estimate Gjj(z) gives the second
+term on the right hand side of (A.4). Application of the local law (4.4) is justified by (A.7) with p = 1.
+This concludes the proof of Lemma A.3.
+Summing up the leading terms in (A.10) results in the second and third terms on the right-hand
+side of (5.11). Collecting all the error terms, the estimate in (5.11) now follows from (4.13), (A.5),
+(A.7) (A.9) and Lemma A.3. This concludes the proof of Lemma 5.4.
+References
+[1]
+Oskari Ajanki, L´aszl´o Erd˝os, and Torben Kr¨uger. Quadratic Vector Equations On Complex
+Upper Half-Plane. Memoirs of the American Mathematical Society 261.1261 (2019).
+[2]
+Oskari Ajanki, L´aszl´o Erd˝os, and Torben Kr¨uger. Universality for general Wigner-type matrices.
+Probability Theory and Related Fields 169 (2015), pp. 667–727.
+[3]
+Oskari Ajanki, Torben Kr¨uger, and L´aszl´o Erd˝os. Singularities of Solutions to Quadratic Vector
+Equations on the Complex Upper Half-Plane. Communications on Pure and Applied Mathematics
+70 (2017), pp. 1672–1705.
+[4]
+Zhidong Bai and Jian-Feng Yao. On the convergence of the spectral empirical process of Wigner
+matrices. Bernoulli 11 (2005), pp. 1059–1092.
+[5]
+Zhigang Bao, Kevin Schnelli, and Yuanyuan Xu. Central Limit Theorem for Mesoscopic Eigen-
+value Statistics of the Free Sum of Matrices. International Mathematics Research Notices 2022.7
+(2020), pp. 5320–5382.
+[6]
+Zhigang Bao and Junshan Xie. CLT for Linear Spectral Statistics of Hermitian Wigner Matrices
+with General Moment Conditions. Theory of Probability & Its Applications 60.2 (2016), pp. 187–
+206.
+[7]
+Anne Marie Boutet de Monvel and Alexei Khorunzhy. Asymptotic distribution of smoothed
+eigenvalue density. I. Gaussian random matrices. Random Oper. Stochastic Equations 7 (1999),
+pp. 1–22.
+[8]
+Anne Marie Boutet de Monvel and Alexei Khorunzhy. Asymptotic distribution of smoothed
+eigenvalue density. II. Wigner random matrices. Random Oper. Stochastic Equations 7 (1999),
+pp. 149–168.
+[9]
+L´aszl´o Erd˝os. The matrix Dyson equation and its applications for random matrices. Random
+matrices. Vol. 26. IAS/Park City Math. Ser. 2019, pp. 75–158.
+[10]
+L´aszl´o Erd˝os, Antti Knowles, and Horng-Tzer Yau. Averaging Fluctuations in Resolvents of
+Random Band Matrices. Annales Henri Poincar´e 14 (2013), pp. 1837–1926.
+[11]
+L´aszl´o Erd˝os, Antti Knowles, Horng-Tzer Yau, and Jun Yin. Delocalization and Diffusion Profile
+for Random Band Matrices. Communications in Mathematical Physics 323 (2013), 367––416.
+[12]
+L´aszl´o Erd˝os, Antti Knowles, Horng-Tzer Yau, and Jun Yin. The local semicircle law for a general
+class of random matrices. Electronic Journal of Probability 18.none (2013), pp. 1–58.
+[13]
+Yukun He and Antti Knowles. Mesoscopic eigenvalue statistics of Wigner matrices. The Annals
+of Applied Probability 27 (2017), pp. 1510–1550.
+[14]
+Yukun He and Matteo Marcozzi. Diffusion profile for random band matrices: a short proof. J.
+Stat. Phys. 177 (2019), pp. 666–716.
+[15]
+Kurt Johansson. On fluctuations of eigenvalues of random Hermitian matrices. Duke Mathemat-
+ical Journal 91 (1998), pp. 151–204.
+[16]
+Alexei Khorunzhy, Boris Khoruzhenko, and Leonid Pastur. Asymptotic properties of large ran-
+dom matrices with independent entries. Journal of Mathematical Physics 37 (1996), pp. 5033–
+5060.
+[17]
+Benjamin Landon, Patrick Lopatto, and Philippe Sosoe. Single eigenvalue fluctuations of general
+Wigner-type matrices. 2021. arXiv:2105.01178.
+24
+
+[18]
+Benjamin Landon and Philippe Sosoe. Almost-optimal bulk regularity conditions in the CLT for
+Wigner matrices. 2022. arXiv:2204.03419.
+[19]
+Benjamin Landon and Philippe Sosoe. Applications of mesoscopic CLTs in random matrix theory.
+The Annals of Applied Probability 30 (2020), pp. 2769–2795.
+[20]
+Yiting Li, Kevin Schnelli, and Yuanyuan Xu. Central limit theorem for mesoscopic eigenvalue
+statistics of deformed Wigner matrices and sample covariance matrices. Annales de l’Institut
+Henri Poincar´e, Probabilit´es et Statistiques 57 (2021), pp. 506–546.
+[21]
+Yiting Li and Yuanyuan Xu. On fluctuations of global and mesoscopic linear statistics of gener-
+alized Wigner matrices. Bernoulli 27 (2021), pp. 1057–1076.
+[22]
+Anna Lytova and Leonid Pastur. Central limit theorem for linear eigenvalue statistics of random
+matrices with independent entries. The Annals of Probability 37 (2009), pp. 1778–1840.
+[23]
+Mariya Shcherbina. Central Limit Theorem for linear eigenvalue statistics of the Wigner and
+sample covariance random matrices. Zh. Mat. Fiz. Anal. Geom. 7 (2011), pp. 176–192.
+[24]
+Philippe Sosoe and Percy Wong. Regularity conditions in the CLT for linear eigenvalue statistics
+of Wigner matrices. Advances in Mathematics 249 (2013), pp. 37–87.
+[25]
+Eugene Wigner. Characteristics Vectors of Bordered Matrices with Infinite Dimensions II. Annals
+of Mathematics 65.2 (1957), pp. 203–207.
+25
+

BtAyT4oBgHgl3EQfR_cQ/content/tmp_files/2301.00075v1.pdf.txt

@@ -0,0 +1,597 @@
+ s
+er
+ne
+ngi
+E
+al 
+ci
+han
+c
+e
+M
+of Iranian Society of 
+ 
+ce
+en
+er
+f
+Con
+ 
+al
+on
+i
+nat
+nter
+I 
+ual
+n
+An
+ 
+th
+30
+The 
+ 
+10 to 12 May, 2022, Tehran, Iran.
+ 
+ 
+ISME2022-IC1332
+ 
+ 
+10 to 12 May, 2022 
+ 
+ 
+Optimal Motion Generation of the Bipedal Under-Actuated Planar Robot for Stair Climbing 
+ 
+Aref Amiri 1,  Hassan Salarieh 2 
+ 
+1Graduate Student, Sharif University of Technology, Tehran; aref.amiri@mech.sharif.edu  
+2Professor, Sharif University of Technology, Tehran; salarieh@sharif.edu  
+ 
+Abstract 
+The importance of humanoid robots in today's world is 
+undeniable, one of the most important features of 
+humanoid robots is the ability to maneuver in 
+environments such as stairs that other robots can not 
+easily cross. A suitable algorithm to generate the path 
+for the bipedal robot to climb is very important. In this 
+paper, an optimization-based method to generate an 
+optimal stairway for under-actuated bipedal robots 
+without an ankle actuator is presented. The generated 
+paths are based on zero and non-zero dynamics of the 
+problem, and according to the satisfaction of the zero 
+dynamics constraint in the problem, tracking the path is 
+possible, in other words, the problem can be 
+dynamically feasible. The optimization method used in 
+the problem is a gradient-based method that has a 
+suitable 
+number 
+of 
+function 
+evaluations 
+for 
+computational processing. This method can also be 
+utilized to go down the stairs. 
+ 
+Keywords: Bipedal robot, under-actuated, optimization, 
+motion planning 
+ 
+Introduction 
+Inspired by human body physics, bipedal robots have 
+many degrees of freedom and can perform various 
+actions with their joint movements. Bipedal robots can 
+adapt to different environments that other wheeled 
+robots are unable to move. The study of path (trajectory) 
+generation methods as a reference for the output of the 
+control problem of bipedal robots in this regard is 
+essential. For the bipedal robot to climb the stairs, it is 
+necessary to analyze the movement of them ascending 
+the stairs and to examine the method of planning the 
+bipedal robot to move and to determine the position of 
+feet for walking on the stairs [1]. 
+     So far, researches have been done on how to go up 
+and downstairs and find a suitable or optimal path for 
+bipedal robots. Various papers using optimization 
+algorithms and considering the robot angles as 
+polynomial functions tried to design an optimal path for 
+a 6-degree bipedal robot [2]. Some articles have even 
+paths planned for multi-legged robots to cross the stairs 
+[3]. Some articles also used stability criteria such as 
+ZMP in designing their paths [4-7]. But this method is 
+only appliable for robots that have feet (soles) with ankle 
+joint actuators, which often have much lower speed in 
+maneuvering than under-actuated robots without feet, 
+and of course, due to the relatively large feet have more 
+wasted energy. Some articles also derive their initial path 
+using data based on motion capturing and then try to 
+optimize their results by combining optimization 
+methods [8]. However, according to the existing 
+literature, few articles have attempted to design a 
+holonomic path for under-actuated bipedal robots 
+without feet. Due to the importance of optimal motion 
+planning, a lot of work has been done in recent years in 
+this area. 
+     In this paper, the problem of motion planning is 
+investigated to find the optimal paths for under-actuated 
+bipedal robots to step on the stairs, the results obtained 
+as a control output will cause the robot to move properly 
+and optimally. This article consists of three sections. In 
+the first part, the dynamic model of the bipedal robot is 
+derived. In the second part, the constraints of the 
+optimization problem are examined, in the third part, the 
+cost function and method of optimal problem solving 
+and finding a suitable movement gate are examined. In 
+the fourth section, the results are presented and 
+discussed, and at the end, the research of this article is 
+summarized as the conclusion. 
+ 
+Dynamics equation  
+The dynamic model of the robot is shown in Figure 
+1. The robot has 7 degrees of freedom and 5 links, 
+each leg has two joints (one in the knee and the 
+other in the hip) and 3 degrees of freedom. We 
+assume that the contact of the tip of the leg is the 
+point.  
+ 
+Figure 1. Planar bipedal robot 
+ 
+
+0.2 
+10 to 12 May, 2022 
+     The robot's motion is planar and the robot has 4 
+actuators, two actuators at the knees and two actuators 
+at the junction of the hip and the trunk so that there is 
+one actuator between each leg and trunk. It is assumed 
+that by hitting the tip of the swing leg on the ground, the 
+other leg rises from the ground, in other words, the 
+robot has no double support phase. So, when moving on 
+the stairs, no time is wasted for placing both feet on the 
+ground. Therefore, the hybrid dynamic equations of a 
+robot are a combination of a single support phase and 
+collision phase. The equations of the hybrid model are 
+as follows: 
+( )
+( )
+:
+(
+)
+x
+f x
+g x u
+x
+x
+x
+x
+
+
+
+
+ 
+
+
+
+ 
+ 
+
+
+                                 (1) 
+     The vector 
+: (
+,
+)
+T
+T T
+x
+q q
+
+ consists of the vector of 
+generalized coordinates and their derivatives.  is a map 
+to find the states of the system exactly after the collision, 
+and the positive and negative symbols indicate the states 
+of the system before and after the collision. The switch 
+condition is as follows: 
+
+
+2
+2
+( , )
+|
+( )
+0,
+( )
+0
+v
+h
+q q
+x P q
+P
+q
+ 
+
+
+
+                          (2) 
+     In equation (2), 
+2
+h
+P  represents the horizontal position 
+of the swing leg and 
+2
+v
+P  represents its vertical position. 
+     The dynamic equations of the robot before and after 
+the collision and in the single support phase can be 
+written as follows:  
+ 
+( )
+( , )
+( )
+q
+q q
+q
+q
+M
+q
+C
+q
+G
+B u
+
+
+
+                                      (3) 
+     Matrix B is also a pre-multiplication matrix in the 
+torque vector and is not a square matrix due to the 
+under-actuation of the system. 
+     In Equation 1, there is an expression called zero 
+dynamics, and it is easy to separate this term if the 
+generalized coordinates of the system are written in 
+relative terms (as has been done in this paper). The 
+satisfaction of this constraint is important in two ways. 
+First, if this constraint is not satisfied, the problem of 
+optimizing the input torques is practically ambiguous, 
+because these torques are not really applicable to the 
+problem. Although it may lead to a feasible kinematic 
+equation (kinematically possible), it is not feasible in 
+terms of control (open-loop), i.e. it is not dynamically 
+possible. 
+ 
+Optimization problem 
+The most important constraint of the problem, called 
+zero dynamics, was introduced in the previous section. 
+Other constraints in this issue are important to plan the 
+robot movement in the best way; the constraints of the 
+optimization problem are generally classified into two 
+general modes of constraints based on dynamics and 
+constraints based on kinematics. 
+ 
+1. Dynamic constraints: 
+     Torque limit: because the torque generators have a 
+certain limit (inequality constraint). 
+     Zero dynamic: the importance of which was 
+mentioned earlier (equality constraint). 
+     Coefficient of friction limit: for the robot to move on 
+real environments, the ratio of horizontal force to 
+vertical force should not be more or less than a certain 
+limit. In other words, the coefficient of friction required 
+for stepping should not exceed a certain limit that can 
+not be implemented in real environments. (inequality 
+constraint). 
+2. Kinematic constraints: 
+     Configuration: As an initial and final condition, the 
+robot needs to move from an initial configuration to a 
+final configuration. The best option is for the initial and 
+final state to be the same so that the robot has 
+periodicity in its movement and the best footprint is in 
+the middle of each stair Figure 2 (equality constraint). 
+ 
+height
+width
+clearance
+best
+footprint
+r1
+r2
+ 
+Figure 2. Stair properties 
+ 
+     Angular velocity limit: Because motors have limited 
+angular velocity production. (inequality constraint) 
+     Contact in single support phase: The robot is in 
+contact with the ground during the single support phase 
+and the acceleration of the contact point in the 
+horizontal and vertical direction during this period is 
+zero. (equality constraint) 
+     Swing leg collision: The robot swing leg during the 
+single-phase phase, except at the beginning and end of 
+the phase, should not collide with the ground, on the 
+other hand, should have a suitable distance to the 
+obstacles. 
+     Knees movement limitation: To create maximum 
+similarity to human movement, the robot knee should 
+not be opened and closed too much. 
+Failure to satisfy any of the above constraints will cause 
+problems 
+in 
+creating 
+optimal 
+and 
+appropriate 
+movement. 
+ 
+Optimization method 
+This optimization is a nonlinear, constrained, and single-
+objective problem. 
+     Cost function: To find the optimal path, various cost 
+functions are considered, for example, the norm of 
+torque input, system input energy, and cost of transport 
+are common options. In this paper, we consider the 
+norm of torque inputs as the cost function. By this 
+choice, the torques are rational in size and will have 
+proper distribution (If the optimization problem is 
+solved properly). 
+4
+2
+0
+0
+(
+( ))
+T
+i
+i
+J
+u
+d
+
+
+
+
+
+
+                                             (4) 
+     In the above equation, T is the length of the time 
+period. 
+     Selection of optimization variables: Optimization 
+variables can have different types, one of the best 
+choices 
+is 
+the 
+paths 
+followed 
+by 
+generalized 
+coordinates. Here our choice is a time-varying path as a 
+function of polynomials. The polynomial functions are 
+
+ 
+10 to 12 May, 2022 
+uniform and smooth, and they are also simple for 
+deriving. 
+4
+,
+0
+( )
+n
+i
+k
+k i
+i
+q t
+t
+
+
+
+
+                                                (5) 
+     The degree of this polynomial must be chosen in 
+such a way that the number of optimization parameters, 
+which are the same as the number of polynomial 
+coefficients, are appropriate (minimum value to have a 
+smooth motion satisfied the mentioned constraints). In 
+this article, we choose the function of order 4 to have 
+freedom of action in terms of the optimization problem 
+and also not to make the number of optimization 
+parameters of the problem irrational and complicated. 
+     Method of solving the optimization problem: This 
+optimization problem is solved by Variable Metric 
+methods for constrained optimization. This method is a 
+gradient-based method, which provides a desirable and 
+fast solution. Another advantage of this method is to not 
+get out easily from the feasible area [9]. 
+ 
+Results and Discussion 
+Following the model and algorithm presented above, a 
+bipedal robot has been simulated to climb the stairs. The 
+height of the stairs is considered 20cm and the width of 
+the stairs is 40cm. The robot model specifications are in 
+accordance with Table 1. The initial and final angles of 
+the bipedal robot as a configuration are given in Table 
+2. Here the initial and final configurations are intuitively 
+obtained from the human configuration. The speed of 
+crossing each step is .5 seconds. The torque limit 
+applied to the system is 150 N.m and the maximum 
+angular velocity of the motors 10 rad/sec can be.     
+ 
+ 
+Table 1. Rabbit robot properties [10] 
+Symbol 
+Value 
+m1, m5 
+3.2 kg 
+m2, m4 
+6.8 kg 
+m3 
+20 kg 
+I1, I5 
+0.93 kg-m2 
+I2, I4 
+1.08 kg-m2 
+I3 
+2.22 kg-m2 
+l1, l5 
+0.4 m 
+l2, l4 
+0.4 m   
+l3 
+0.625 m 
+d1, d5 
+0.128 m 
+d2, d4 
+0.163 m 
+d3 
+0.2 m 
+ 
+ 
+ 
+Table 2. The initial and final configuration 
+Parameters 
+Initial value(rad) 
+Final value(rad) 
+q1 
+0.2618 
+0.1964 
+q2 
+1.3140 
+0 
+q3 
+-1.2267 
+0.0219 
+                       q4 
+-0.0219 
+1.2267 
+q5 
+0 
+1.3140 
+ 
+ 
+Figure 3. Input torques 
+ 
+     According to Figure 3, the torques have a good 
+margin from the saturation and compared to other 
+articles and research reviewed in the introduction, more 
+optimal results have been obtained, also zero dynamics 
+(
+v ) in a very good way is satisfied. 
+ 
+Figure 4. Friction coefficient 
+ 
+     According to Figure 4, it is clear that the generated 
+path needs the maximum coefficient of friction .69 to 
+slip, so on all surfaces that have a coefficient of friction 
+higher than .69 there is the ability to move.  
+ 
+Figure 5. Angles vs. angular velocities 
+ 
+     According to Figure 5, the generated paths, due to 
+the nature of the polynomial functions, have a smooth 
+
+ 
+10 to 12 May, 2022 
+and non-breaking behavior, and the angular velocities 
+are far from their saturation limit. 
+ 
+ 
+Figure 6. Stick diagram of the climbing a stair up 
+ 
+     As can be seen in Figure 6, the robot's movement is 
+quite normal and very similar to human movement. The 
+trunk is kept in a good position and also the tip of the 
+feet and other links do not touch the surfaces except at 
+the beginning and at the end of the movement. 
+According to the sum of the presented results, the 
+generated path is an optimal path for the proper gait of 
+the under-actuated bipedal robot. 
+                                                                         
+Conclusions 
+In this article, we present a method to generate optimal 
+motion for a bipedal robot, we used this method to find 
+the paths that the 'rabbit' robot by tracking them can 
+optimally climb stairs. This process consists of 3 parts: 
+robot dynamic extraction (because optimization is based 
+on the model), design of constraints based on dynamics 
+and kinematics, and optimization. As a result of the 
+problem, a series of virtual holonomic paths were 
+extracted in which the zero hybrid dynamics of the 
+problem is also satisfied, so tracking the paths are 
+possible for under-actuated robots. 
+In the future, we plan to use a new method called impact 
+invariance to design the above path, which guarantees 
+the periodicity of the proposed paths. 
+ 
+ 
+References 
+ 
+[1] Goldfarb, Nathaniel, Charles Bales, and Gregory S. 
+Fischer. "Toward Generalization of Bipedal Gait 
+Cycle During Stair Climbing Using Learning From 
+Demonstration." IEEE Transactions on Medical 
+Robotics and Bionics 3.2 (2021): 446-454. 
+[2] Kweon Soo Jeon, Ohung Kwon, Jong Hyeon Park. 
+Optimal trajectory generation for a biped robot 
+walking 
+a 
+staircase 
+based 
+on 
+genetic 
+algorithms[C]//Proceedings 
+of 
+2004 
+IEEE/RSJ 
+International Conference on Intelligent Robots and 
+Systems, Sendai, Japan: IEEE, 2004: 2837- 2842. 
+[3] Cebe, Oguzhan, et al. "Online dynamic trajectory 
+optimization and control for a quadruped robot." 
+2021 IEEE International Conference on Robotics 
+and Automation (ICRA). IEEE, 2021. 
+[4] Kim, Eun-Su, Jo-Hwan Kim, and Jong-Wook Kim. 
+"Generation of optimal trajectories for ascending 
+and descending a stair of a humanoid based on 
+uDEAS." 2009 IEEE International Conference on 
+Fuzzy Systems. IEEE, 2009. 
+[5] Sugahara, Yusuke, et al. "Walking up and down 
+stairs carrying a human by a biped locomotor with 
+parallel mechanism." 2005 IEEE/RSJ International 
+Conference on Intelligent Robots and Systems. 
+IEEE, 2005. 
+[6] Zhang, Qin, et al. "Action generation of a biped 
+robot climbing stairs." 2013 IEEE International 
+Conference on Mechatronics and Automation. 
+IEEE, 2013. 
+[7] Kim, E., T. Kim, and J-W. Kim. "Three-dimensional 
+modelling of a humanoid in three planes and a 
+motion scheme of biped turning in standing." IET 
+control theory & applications 3.9 (2009): 1155-
+1166. 
+[8] Powell, Matthew J., Huihua Zhao, and Aaron D. 
+Ames. "Motion primitives for human-inspired 
+bipedal robotic locomotion: walking and stair 
+climbing." 2012 IEEE International Conference on 
+Robotics and Automation. IEEE, 2012.Misc, A., 
+2003. Miscellaneous Title. On the WWW, May. 
+URL http://www.abc.edu. 
+[9] Powell, Michael JD. "A fast algorithm for 
+nonlinearly constrained optimization calculations." 
+Numerical analysis. Springer, Berlin, Heidelberg, 
+1978. 144-157.. 
+[10] Chevallereau, Christine, et al. "Rabbit: A testbed for 
+advanced control theory." IEEE Control Systems 
+Magazine 23.5 (2003): 57-79. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+2
+1.5
+0.5
+-0.5
+-0.5
+0
+0.5
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BtAyT4oBgHgl3EQfR_cQ/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf,len=186
+page_content='s  er  ne  ngi  E  al  ci  han  c  e  M  of Iranian Society of  ce  en  er  f  Con  al  on  i  nat  nter  I  ual  n  An  th  30  The  10 to 12 May, 2022, Tehran, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  ISME2022-IC1332  10 to 12 May, 2022  Optimal Motion Generation of the Bipedal Under-Actuated Planar Robot for Stair Climbing  Aref Amiri 1,  Hassan Salarieh 2  1Graduate Student, Sharif University of Technology, Tehran;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' aref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='amiri@mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='edu  2Professor, Sharif University of Technology, Tehran;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' salarieh@sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content="edu  Abstract  The importance of humanoid robots in today's world is  undeniable, one of the most important features of  humanoid robots is the ability to maneuver in  environments such as stairs that other robots can not  easily cross." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' A suitable algorithm to generate the path  for the bipedal robot to climb is very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' In this  paper, an optimization-based method to generate an  optimal stairway for under-actuated bipedal robots  without an ankle actuator is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The generated  paths are based on zero and non-zero dynamics of the  problem, and according to the satisfaction of the zero  dynamics constraint in the problem, tracking the path is  possible, in other words, the problem can be  dynamically feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The optimization method used in  the problem is a gradient-based method that has a  suitable  number  of  function  evaluations  for  computational processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' This method can also be  utilized to go down the stairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Keywords: Bipedal robot, under-actuated, optimization,  motion planning  Introduction  Inspired by human body physics, bipedal robots have  many degrees of freedom and can perform various  actions with their joint movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Bipedal robots can  adapt to different environments that other wheeled  robots are unable to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The study of path (trajectory)  generation methods as a reference for the output of the  control problem of bipedal robots in this regard is  essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' For the bipedal robot to climb the stairs, it is  necessary to analyze the movement of them ascending  the stairs and to examine the method of planning the  bipedal robot to move and to determine the position of  feet for walking on the stairs [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  So far, researches have been done on how to go up  and downstairs and find a suitable or optimal path for  bipedal robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Various papers using optimization  algorithms and considering the robot angles as  polynomial functions tried to design an optimal path for  a 6-degree bipedal robot [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Some articles have even  paths planned for multi-legged robots to cross the stairs  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Some articles also used stability criteria such as  ZMP in designing their paths [4-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' But this method is  only appliable for robots that have feet (soles) with ankle  joint actuators, which often have much lower speed in  maneuvering than under-actuated robots without feet,  and of course, due to the relatively large feet have more  wasted energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Some articles also derive their initial path  using data based on motion capturing and then try to  optimize their results by combining optimization  methods [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' However, according to the existing  literature, few articles have attempted to design a  holonomic path for under-actuated bipedal robots  without feet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Due to the importance of optimal motion  planning, a lot of work has been done in recent years in  this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  In this paper, the problem of motion planning is  investigated to find the optimal paths for under-actuated  bipedal robots to step on the stairs, the results obtained  as a control output will cause the robot to move properly  and optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' This article consists of three sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' In  the first part, the dynamic model of the bipedal robot is  derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' In the second part, the constraints of the  optimization problem are examined, in the third part, the  cost function and method of optimal problem solving  and finding a suitable movement gate are examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' In  the fourth section, the results are presented and  discussed, and at the end, the research of this article is  summarized as the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Dynamics equation  The dynamic model of the robot is shown in Figure  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The robot has 7 degrees of freedom and 5 links,  each leg has two joints (one in the knee and the  other in the hip) and 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' We  assume that the contact of the tip of the leg is the  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Planar bipedal robot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content="2  10 to 12 May, 2022  The robot's motion is planar and the robot has 4  actuators, two actuators at the knees and two actuators  at the junction of the hip and the trunk so that there is  one actuator between each leg and trunk." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' It is assumed  that by hitting the tip of the swing leg on the ground, the  other leg rises from the ground, in other words, the  robot has no double support phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' So, when moving on  the stairs, no time is wasted for placing both feet on the  ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Therefore, the hybrid dynamic equations of a  robot are a combination of a single support phase and  collision phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The equations of the hybrid model are  as follows:  ( )  ( )  :  (  )  x  f x  g x u  x  x  x  x  \uf02d  \uf02b  \uf02d  \uf02d  \uf0ec \uf03d  \uf02b  \uf0cf\uf047  \uf0ef  \uf053 \uf0ed  \uf03d \uf044  \uf0ce\uf047  \uf0ef\uf0ee  (1)  The vector  : (  ,  )  T  T T  x  q q  \uf03d  consists of the vector of  generalized coordinates and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' \uf044 is a map  to find the states of the system exactly after the collision,  and the positive and negative symbols indicate the states  of the system before and after the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The switch  condition is as follows:  \uf07b  \uf07d  2  2  ( , )  |  ( )  0,  ( )  0  v  h  q q  x P q  P  q  \uf047 \uf03d  \uf0ce  \uf03d  \uf03e  (2)  In equation (2),  2  h  P  represents the horizontal position  of the swing leg and  2  v  P  represents its vertical position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  The dynamic equations of the robot before and after  the collision and in the single support phase can be  written as follows:  \uf028 \uf029  ( )  ( , )  ( )  q  q q  q  q  M  q  C  q  G  B u  \uf02b  \uf02b  \uf03d  (3)  Matrix B is also a pre-multiplication matrix in the  torque vector and is not a square matrix due to the  under-actuation of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  In Equation 1, there is an expression called zero  dynamics, and it is easy to separate this term if the  generalized coordinates of the system are written in  relative terms (as has been done in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The  satisfaction of this constraint is important in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  First, if this constraint is not satisfied, the problem of  optimizing the input torques is practically ambiguous,  because these torques are not really applicable to the  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Although it may lead to a feasible kinematic  equation (kinematically possible), it is not feasible in  terms of control (open-loop), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' it is not dynamically  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Optimization problem  The most important constraint of the problem, called  zero dynamics, was introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Other constraints in this issue are important to plan the  robot movement in the best way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' the constraints of the  optimization problem are generally classified into two  general modes of constraints based on dynamics and  constraints based on kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Dynamic constraints:  Torque limit: because the torque generators have a  certain limit (inequality constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Zero dynamic: the importance of which was  mentioned earlier (equality constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Coefficient of friction limit: for the robot to move on  real environments, the ratio of horizontal force to  vertical force should not be more or less than a certain  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' In other words, the coefficient of friction required  for stepping should not exceed a certain limit that can  not be implemented in real environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' (inequality  constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Kinematic constraints:  Configuration: As an initial and final condition, the  robot needs to move from an initial configuration to a  final configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The best option is for the initial and  final state to be the same so that the robot has  periodicity in its movement and the best footprint is in  the middle of each stair Figure 2 (equality constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  height  width  clearance  best  footprint  r1  r2  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Stair properties  Angular velocity limit: Because motors have limited  angular velocity production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' (inequality constraint)  Contact in single support phase: The robot is in  contact with the ground during the single support phase  and the acceleration of the contact point in the  horizontal and vertical direction during this period is  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' (equality constraint)  Swing leg collision: The robot swing leg during the  single-phase phase, except at the beginning and end of  the phase, should not collide with the ground, on the  other hand, should have a suitable distance to the  obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Knees movement limitation: To create maximum  similarity to human movement, the robot knee should  not be opened and closed too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Failure to satisfy any of the above constraints will cause  problems  in  creating  optimal  and  appropriate  movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Optimization method  This optimization is a nonlinear, constrained, and single-  objective problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Cost function: To find the optimal path, various cost  functions are considered, for example, the norm of  torque input, system input energy, and cost of transport  are common options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' In this paper, we consider the  norm of torque inputs as the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' By this  choice, the torques are rational in size and will have  proper distribution (If the optimization problem is  solved properly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  4  2  0  0  (  ( ))  T  i  i  J  u  d  \uf074  \uf074  \uf03d  \uf03d  \uf0e5  \uf0f2  (4)  In the above equation, T is the length of the time  period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Selection of optimization variables: Optimization  variables can have different types, one of the best  choices  is  the  paths  followed  by  generalized  coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Here our choice is a time-varying path as a  function of polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The polynomial functions are  10 to 12 May, 2022  uniform and smooth, and they are also simple for  deriving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  4  ,  0  ( )  n  i  k  k i  i  q t  t  \uf061  \uf03d  \uf03d  \uf03d\uf0e5  (5)  The degree of this polynomial must be chosen in  such a way that the number of optimization parameters,  which are the same as the number of polynomial  coefficients, are appropriate (minimum value to have a  smooth motion satisfied the mentioned constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' In  this article, we choose the function of order 4 to have  freedom of action in terms of the optimization problem  and also not to make the number of optimization  parameters of the problem irrational and complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Method of solving the optimization problem: This  optimization problem is solved by Variable Metric  methods for constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' This method is a  gradient-based method, which provides a desirable and  fast solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Another advantage of this method is to not  get out easily from the feasible area [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Results and Discussion  Following the model and algorithm presented above, a  bipedal robot has been simulated to climb the stairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The  height of the stairs is considered 20cm and the width of  the stairs is 40cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The robot model specifications are in  accordance with Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The initial and final angles of  the bipedal robot as a configuration are given in Table  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Here the initial and final configurations are intuitively  obtained from the human configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The speed of  crossing each step is .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The torque limit  applied to the system is 150 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='m and the maximum  angular velocity of the motors 10 rad/sec can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Rabbit robot properties [10]  Symbol  Value  m1, m5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='2 kg  m2, m4  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='8 kg  m3  20 kg  I1, I5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='93 kg-m2  I2, I4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='08 kg-m2  I3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='22 kg-m2  l1, l5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='4 m  l2, l4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='4 m  l3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='625 m  d1, d5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='128 m  d2, d4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='163 m  d3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='2 m  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The initial and final configuration  Parameters  Initial value(rad)  Final value(rad)  q1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='2618  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='1964  q2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='3140  0  q3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='2267  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='0219  q4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='0219  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='2267  q5  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='3140  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Input torques  According to Figure 3, the torques have a good  margin from the saturation and compared to other  articles and research reviewed in the introduction, more  optimal results have been obtained, also zero dynamics  (  v\uf074 ) in a very good way is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Friction coefficient  According to Figure 4, it is clear that the generated  path needs the maximum coefficient of friction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='69 to  slip, so on all surfaces that have a coefficient of friction  higher than .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='69 there is the ability to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Angles vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' angular velocities  According to Figure 5, the generated paths, due to  the nature of the polynomial functions, have a smooth  10 to 12 May, 2022  and non-breaking behavior, and the angular velocities  are far from their saturation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=" Stick diagram of the climbing a stair up  As can be seen in Figure 6, the robot's movement is  quite normal and very similar to human movement." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' The  trunk is kept in a good position and also the tip of the  feet and other links do not touch the surfaces except at  the beginning and at the end of the movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  According to the sum of the presented results, the  generated path is an optimal path for the proper gait of  the under-actuated bipedal robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content="  Conclusions  In this article, we present a method to generate optimal  motion for a bipedal robot, we used this method to find  the paths that the 'rabbit' robot by tracking them can  optimally climb stairs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' This process consists of 3 parts:  robot dynamic extraction (because optimization is based  on the model), design of constraints based on dynamics  and kinematics, and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' As a result of the  problem, a series of virtual holonomic paths were  extracted in which the zero hybrid dynamics of the  problem is also satisfied, so tracking the paths are  possible for under-actuated robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  In the future, we plan to use a new method called impact  invariance to design the above path, which guarantees  the periodicity of the proposed paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  References  [1] Goldfarb, Nathaniel, Charles Bales, and Gregory S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Fischer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' "Toward Generalization of Bipedal Gait  Cycle During Stair Climbing Using Learning From  Demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='" IEEE Transactions on Medical  Robotics and Bionics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='2 (2021): 446-454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  [2] Kweon Soo Jeon, Ohung Kwon, Jong Hyeon Park.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Optimal trajectory generation for a biped robot  walking  a  staircase  based  on  genetic  algorithms[C]//Proceedings  of  2004  IEEE/RSJ  International Conference on Intelligent Robots and  Systems, Sendai, Japan: IEEE, 2004: 2837- 2842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  [3] Cebe, Oguzhan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' "Online dynamic trajectory  optimization and control for a quadruped robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='"  2021 IEEE International Conference on Robotics  and Automation (ICRA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  [4] Kim, Eun-Su, Jo-Hwan Kim, and Jong-Wook Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  "Generation of optimal trajectories for ascending  and descending a stair of a humanoid based on  uDEAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='" 2009 IEEE International Conference on  Fuzzy Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' IEEE, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  [5] Sugahara, Yusuke, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' "Walking up and down  stairs carrying a human by a biped locomotor with  parallel mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='" 2005 IEEE/RSJ International  Conference on Intelligent Robots and Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  IEEE, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  [6] Zhang, Qin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' "Action generation of a biped  robot climbing stairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='" 2013 IEEE International  Conference on Mechatronics and Automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  IEEE, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  [7] Kim, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Kim, and J-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' "Three-dimensional  modelling of a humanoid in three planes and a  motion scheme of biped turning in standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='" IET  control theory & applications 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='9 (2009): 1155-  1166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  [8] Powell, Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=', Huihua Zhao, and Aaron D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  Ames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' "Motion primitives for human-inspired  bipedal robotic locomotion: walking and stair  climbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='" 2012 IEEE International Conference on  Robotics and Automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' IEEE, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='Misc, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=',  2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Miscellaneous Title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' On the WWW, May.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  URL http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  [9] Powell, Michael JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' "A fast algorithm for  nonlinearly constrained optimization calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='"  Numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' Springer, Berlin, Heidelberg,  1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' 144-157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='.  [10] Chevallereau, Christine, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content=' "Rabbit: A testbed for  advanced control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='" IEEE Control Systems  Magazine 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='5 (2003): 57-79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
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+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfR_cQ/content/2301.00075v1.pdf'}
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GNE0T4oBgHgl3EQfRQAC/content/tmp_files/2301.02203v1.pdf.txt

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+arXiv:2301.02203v1  [math.CO]  5 Jan 2023
+DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC
+GROUP BY PRIME POWERS
+SARAH PELUSE AND KANNAN SOUNDARARAJAN
+In memory of Chandra Sekhar Raju
+Abstract. Let k be a positive integer. We show that, as n goes to infinity, almost every
+entry of the character table of Sn is divisible by k. This proves a conjecture of Miller.
+1. Introduction
+It is a standard fact that the irreducible characters of Sn take only integer values for every
+natural number n. In 2017, Miller [11] computed the character tables of Sn for all n ≤ 38
+and looked at various statistical properties of these integers as n grew. His computations
+suggested that
+(1) the density of even entries seemed to tend to 1,
+(2) the density of entries divisible by 3, the density of entries divisible by 5, and the
+density of entries divisible by 7 seemed to increase as n grew,
+(3) about half of the nonzero entries were positive,
+(4) and the density of zeros in the character table seemed to decrease as n grew, but not
+very quickly.
+Based on this first observation, Miller [11, 13] conjectured that as n goes to infinity, almost
+every entry of the character table of the symmetric group Sn is even.
+Following partial
+progress due to McKay [10], Gluck [5], and Morotti [14], the first author proved this conjec-
+ture in [15]. Based on the second observation, Miller [11, 13] also conjectured, more generally,
+that for any fixed prime p, almost every entry of the character table of Sn is a multiple of
+p as n goes to infinity. We proved this conjecture in [16], with a uniform upper bound for
+the number of entries not divisible by a fixed prime. Recently, Miller [12] conjectured, even
+more generally, that for any fixed prime power q, almost every entry of the character table
+of Sn is a multiple of q as n goes to infinity. In this paper, we prove this most general of
+Miller’s conjectures.
+Theorem 1.1. Let n be large and q ≤ 10−3 log n/(log log n)2 be a prime power. The number
+of entries in the character table of Sn that are not divisible by q is at most
+O
+�
+p(n)2 exp(−(log log n)2)
+�
+.
+It follows immediately from Theorem 1.1 and the union bound that almost every entry of
+the character table of Sn is divisible by any fixed integer as n goes to infinity.
+Corollary 1.2. Let k be any positive integer. Then, as n goes to infinity, the proportion of
+entries in the character table of Sn that are not divisible by k tends to 0.
+Our methods do not seem to shed any light on Miller’s third and fourth observations.
+Most interesting to us is the question of what proportion of character table entries are zero,
+1
+
+2
+SARAH PELUSE AND KANNAN SOUNDARARAJAN
+and it is not clear from Miller’s data whether the proportion is decreasing to zero or some
+positive constant. Combining the Murnaghan–Nakayama rule and an old result of Erd˝os
+and Lehner [2] on the distribution of the largest part of a uniformly random partition of n
+produces a proportion of
+1
+log n zeros in the character table of Sn, and it appears that no lower
+bound of a larger order of magnitude is known. In the related setting of finite simple groups
+of Lie type, Larsen and Miller [7] have shown that almost every character table entry is zero
+as the rank goes to infinity.
+Acknowledgments.
+The first author is partially supported by the NSF Mathematical
+Sciences Postdoctoral Research Fellowship Program under Grant No. DMS-1903038 and by
+the Oswald Veblen Fund. The second author is partially supported by a grant from the
+National Science Foundation, and a Simons Investigator Grant from the Simons Foundation.
+We thank David Speyer for drawing our attention to Lemma 2.1.
+2. Proof outline
+For any partitions λ and µ of n, let χλ
+µ denote the value of the irreducible character of
+Sn corresponding to λ on the conjugacy class of elements with cycle type corresponding to
+µ. In [16], our argument proceeded by combining two key facts: (i) if µ contains a part
+substantially larger than the typical largest part of a random partition, then χλ
+µ = 0 for
+almost every λ, and (ii) if ν is another partition of n that is obtained from µ by combining
+p parts of the same size m into one part of size pm, then χλ
+µ ≡ χλ
+ν (mod p) for every λ.
+We showed that, for almost every µ, repeatedly combining p parts of the same size in this
+manner produces a partition �µ containing a very large part, large enough so that χλ
+�µ must
+be zero for almost every λ. Our main result on the divisibility of character values by primes
+then followed from the fact that χλ
+µ ≡ χλ
+�µ (mod p) for every λ.
+The second key fact generalizes to a congruence of character value modulo prime powers
+in a straightforward manner.
+Lemma 2.1. Let pr be a power of the prime p. Suppose that µ is a partition of n, and that
+ν is another partition of n obtained from µ by replacing pr parts of the same size m by pr−1
+parts of size pm. Then for all partitions λ of n, we have
+χλ
+µ ≡ χλ
+ν
+(mod pr).
+However, when r > 1, it is no longer the case that starting from a typical partition µ of n
+and repeatedly combining pr parts of the same size m into pr−1 parts of size pm produces a
+partition �µ containing a part substantially larger than the largest part of a typical partition
+of n. The argument from [16] that worked for primes thus breaks down for all other prime
+powers.
+The key idea used to overcome this barrier is a new condition for character values of the
+symmetric group to be divisible by a fixed prime power, which we prove by exploiting certain
+symmetries that appear after applying the Murnaghan–Nakayama rule multiple times.
+Theorem 2.2. Let n, m1, . . . , mr be distinct positive integers. Let µ be a partition of n
+containing parts of size m1, . . . , mr, each appearing at least pr−1 times. If λ is a (�r
+i=1 kimi)-
+core partition of n for all r-tuples (k1, . . . , kr) of integers 0 ≤ k1, . . . , kr ≤ pr−1 for which
+some ki = pr−1, then
+pr | χλ
+µ.
+
+DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP
+3
+Starting with a partition µ of n, repeatedly combine pr parts of the same size m into pr−1
+parts of size pm, until the process terminates in a partition �µ where no part appears more
+than pr − 1 times. As a preliminary to applying Theorem 2.2 we show that for a typical
+partition µ, the resulting partition �µ will have r parts that are suitably large, and with each
+appearing at least pr−1 times.
+Proposition 2.3. Starting with a partition µ of n, repeatedly replace every occurrence of pr
+parts of the same size m by pr−1 parts of size pm until we arrive at a partition ˜µ where no
+part appears more than pr − 1 times. Then, except for
+O
+�
+p(n) exp
+�
+−n1/20pr��
+initial partitions µ, the partition �µ contains at least r distinct parts m1, . . . , mr, each appear-
+ing at least pr−1 times and satisfying
+pr−1mj >
+�
+1 + 1
+6pr
+�√
+6
+2π
+√n log n.
+This holds uniformly for pr ≤ 10−3 log n/(log log n)2.
+The significance of the lower bound on pr−1mj in Proposition 2.3 is that it lies beyond the
+threshold of values t such that almost every partition of n is a t-core.
+Lemma 2.4. Let 1 ≤ L ≤ log n/ log log n be a real number. Then, for any given integer t
+with
+t ≥
+�
+1 + 1
+L
+�√
+6
+2π
+√n log n,
+all but
+O
+�
+p(n) log n
+n1/2L
+�
+partitions of n are t-cores.
+We can swiftly deduce our main result, Theorem 1.1, from the results stated above.
+Deducing Theorem 1.1. Let µ be a partition of n, and suppose that �µ is as in Proposition 2.3.
+Then, for all but at most
+O
+�
+p(n) exp
+�
+−n1/20pr��
+choices of µ, the partition �µ contains at least r distinct parts m1, . . . , mr, each appearing at
+least pr−1 times and satisfying
+(2.1)
+pr−1mj >
+�
+1 + 1
+6pr
+�√
+6
+2π
+√n log n.
+Consider any r-tuple (k1, . . . , kr) with 0 ≤ k1, . . . , kr ≤ pr−1 and ki = pr−1 for some i.
+Then k1m1 + . . . + krmr also exceeds the bound in (2.1), so that by Lemma 2.4 all but
+O(p(n)(log n)/n
+1
+2L) partitions λ of n are (k1m1 + . . . + krmr)-cores. Since there are at most
+r(pr−1 + 1)r−1 such r-tuples (k1, . . . , kr), by the union bound we see that all but at most
+O
+�
+p(n) log n
+n1/12pr r
+�
+pr−1 + 1
+�r−1
+�
+partitions λ of n are (k1m1 + . . . + krmr)-cores for all choices of the r-tuple (k1, . . . , kr).
+
+4
+SARAH PELUSE AND KANNAN SOUNDARARAJAN
+Theorem 2.2 now shows that pr divides χλ
+�µ, and since χλ
+µ ≡ χλ
+�µ (mod pr) by Lemma 2.1, it
+also follows that pr divides χλ
+µ. Putting everything together, we conclude that the number
+of partitions λ and µ with pr ∤ χλ
+µ is at most
+O
+�
+p(n)2�
+exp(−n1/(20pr)) +
+1
+n1/13pr r
+�
+pr−1 + 1
+�r−1 ��
+= O
+�
+p(n)2 exp(−(log log n)2)
+�
+,
+in the range pr ≤ 10−3 log n/(log log n)2.
+□
+The rest of the paper is organized as follows.
+We will prove Lemmas 2.1 and 2.4 in
+Section 3, Theorem 2.2 in Sections 4, 5, 6, and 7, and Proposition 2.3 in Sections 8 and 9.
+3. Proofs of Lemmas 2.1 and 2.4
+We begin by proving the two lemmas stated in the previous section.
+Proof of Lemma 2.1. We claim that if Q ∈ Z[x1, . . . , xk] is a polynomial with integer coeffi-
+cients, then
+Q(x1, . . . , xk)pr ≡ Q(xp
+1, . . . , xp
+k)pr−1
+(mod pr).
+As is well known, we may write
+(3.1)
+Q(x1, . . . , xk)p = Q(xp
+1, . . . , xp
+k) + p · R(x1, . . . , xk)
+for some R ∈ Z[x1, . . . , xk], which establishes the claim when r = 1. For r > 1, raise both
+sides of (3.1) to the power pr−1, and expand using the binomial theorem:
+Q(x1, . . . , xk)pr = (Q(xp
+1, . . . , xp
+k) + p · R(x1, . . . , xk))pr−1
+= Q(xp
+1, . . . , xp
+k)pr−1 +
+pr−1
+�
+ℓ=1
+�pr−1
+ℓ
+�
+Q(xp
+1, . . . , xp
+k)pr−1−ℓ(pR(x1, . . . , xk))ℓ.
+Note that for 1 ≤ ℓ ≤ pr−1
+pℓ
+�pr−1
+ℓ
+�
+= pℓpr−1
+ℓ
+�pr−1 − 1
+ℓ − 1
+�
+≡ 0
+(mod pr),
+since the power of p dividing ℓ is certainly at most ℓ − 1. This establishes our claim.
+The lemma now follows by applying this observation to the polynomials appearing in
+Frobenius’s formula for the character values χλ
+µ and χλ
+ν (see Chapter 4 of [4]).
+□
+Proof of Lemma 2.4. The proof is essentially identical to that of Proposition 1 of [16], but
+we include the short argument for completeness. Since every partition of n is a t-core for
+t > n, we may naturally assume that t ≤ n. From Lemma 5 of [14], we know that at most
+(t + 1)p(n − t) partitions of n are not t-cores. By the asymptotic formula
+p(m) ∼
+1
+4
+√
+3m exp
+� 2π
+√
+6
+√m
+�
+for the partition function, we have
+(t + 1)p(n − t) ≪
+t + 1
+n − t + 1 exp
+� 2π
+√
+6
+√
+n − t
+�
+≤
+t + 1
+n − t + 1 exp
+� 2π
+√
+6
+√n −
+πt
+√
+6n
+�
+.
+
+DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP
+5
+In the range n ≥ t ≥ (1 + 1/L)
+√
+6
+2π
+√n log n, the right-hand side above is maximized at the
+lower endpoint t = (1 + 1/L)
+√
+6
+2π
+√n log n. It follows that the number of partitions of n that
+are not t-cores is
+≪ log n
+√n n−(1+1/L)/2 exp
+� 2π
+√
+6
+√n
+�
+≪ p(n) log n
+n1/2L ,
+where the last step uses again the asymptotic for the partition function.
+□
+4. Partitions and Abaci
+The proof of Theorem 2.2 requires the machinery of the abacus associated to a partition.
+A good reference for this theory is Section 2.7 of of [6], and we recall some salient facts
+below.
+4.1. The notion of an abacus. An abacus is a bi-infinite sequence of 0’s and 1’s beginning
+with an infinite sequence of 1’s and ending with an infinite sequence of 0’s.
+More formally, let
+S := {s : Z → {0, 1} : there exists a k ≥ 0 such that s(−i) = 1 and s(i) = 0 for all i ≥ k}
+denote the set of all sequences of 0’s and 1’s indexed using the integers, that begin with an
+infinite sequence of 1’s and end with an infinite sequence of 0’s. For example,
+. . . , 1, . . . , 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, . . ., 0, . . .
+is in S. We consider two sequences s and s′ in S to be equivalent if there is some integer j
+such that s(i) = s′(i − j) for all i, that is, if s′ can be produced by shifting the terms in s by
+j. This is an equivalence relation, and an abacus refers to an equivalence class in S under
+this relation. We denote by A the set of such abaci, so that by an element a of A we mean
+the equivalence class consisting of some sequence s ∈ S together with all its shifts.
+4.2. The abacus associated to a partition. We now show how abaci are in one-to-one
+correspondence with partitions of integers. Starting with an integer partition λ, we construct
+an abacus aλ ∈ A as follows. Draw the Young diagram of λ, and trace out the boundary of
+the diagram, moving from the lower left-hand corner to the upper right-hand corner, writing
+a 0 for each horizontal move and a 1 for each vertical move. Then prepend an infinite string of
+1’s and append an infinite string of 0’s to find a representative of the corresponding element
+aλ of A.
+This procedure is easily reversed, and starting with an abacus a in A we obtain a Young
+diagram, which corresponds to a partition λ. If s ∈ S is a representative of a, then the
+partition λ is a partition of the integer n(a) which counts the number of pairs of indices (i, j)
+with i < j such that s(i) = 0 and s(j) = 1.
+To illustrate, consider the partition (6, 5, 3, 1, 1, 1), whose Young diagram is pictured in
+Figure 1.
+If we start in the lower left-hand corner of this diagram and move along the
+boundary to the upper right-hand corner, we move right, up three times, right twice, up,
+right twice, up, right, and up. The correspondence described above produces the string
+(4.1)
+0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1,
+which we can turn into a bi-infinite sequence by adding an infinite sequence of 1’s to the
+beginning and an infinite sequence of 0’s to the end:
+(4.2)
+. . . , 1, . . . , 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, . . ., 0, . . . .
+
+6
+SARAH PELUSE AND KANNAN SOUNDARARAJAN
+Figure 1. The Young diagram of (6, 5, 3, 1, 1, 1)
+The equivalence class of this sequence is the abacus associated to (6, 5, 3, 1, 1, 1).
+4.3. Hooks and border strips. Let λ be a partition. The hook h associated to a box b
+in the Young diagram of λ consists of the box b together with all the boxes directly to its
+right and directly below it. The hook-length of h, denoted by ℓ(h), is the number of boxes
+contained in the hook. The height of the hook h, denoted by ht(h), is one less than the
+number of rows in the Young diagram of λ that contain a box of h. Associated to each hook
+is a border strip (also known as a skew hook), denoted bs(h), which is the connected region
+of boundary boxes of the Young diagram running from the rightmost to the bottommost box
+of h. Removing such a border strip leaves behind a smaller Young diagram. These notions
+play a prominent role in the representation theory of the symmetric group, and in particular
+feature in the Murnaghan–Nakayama rule for computing character values, which we next
+recall (see Theorem 2.4.7 of [6], and also Chapter 4 of [4]).
+Theorem 4.1 (The Murnaghan–Nakayama rule). Let n and t be positive integers, with
+t ≤ n. Let σ ∈ Sn be of the form σ = τ · ρ, where ρ is a t-cycle, and τ is a permutation of
+Sn with support disjoint from ρ. Let λ be a partition of n. Then
+(4.3)
+χλ(σ) =
+�
+h∈λ
+ℓ(h)=t
+(−1)ht(h)χλ\bs(h)(τ).
+Above, χλ(σ) denotes the value of the character of the irreducible representation of Sn
+corresponding to the partition λ, evaluated on the conjugacy class of σ, λ \ bs(h) denotes
+the partition of n − t obtained by removing the border strip bs(h) from the Young diagram
+of λ, and χλ\bs(h)(τ) denotes the character value of the irreducible representation of Sn−t
+corresponding to the partition λ \ bs(h) evaluated on the conjugacy class of τ.
+The abacus notation helps with thinking about hook lengths and border strips. Let λ be a
+partition, let aλ denote the corresponding abacus, and let s be a representative in S for the
+abacus aλ. Each hook h in the Young diagram of λ is in natural one-to-one correspondence
+with a pair of indices (i, j), i < j, with s(i) = 0 and s(j) = 1. The length of the hook h is
+j − i. In particular, the partition λ contains no hooks of length t (that is, λ is a t-core) if
+and only if there is no pair of indices (i, i + t) with s(i) = 0 and s(i + t) = 1. The height of
+the hook h equals the number of 1’s in the sequence s lying strictly between the 0 at index
+i and the 1 at index j:
+ht(h) = # {i < k < j : s(k) = 1} .
+
+DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP
+7
+11 7
+6
+4
+3
+1
+9
+5
+4
+2
+1
+6
+2
+1
+3
+2
+1
+Figure 2. Hook-lengths for (6, 5, 3, 1, 1, 1)
+,
+Figure 3. The Young diagram of (6, 2, 1, 1, 1, 1)
+Further, the abacus notation gives an easy description of the result of removing a border
+strip from a partition. Define, for any pair of distinct integers (i, j) the operator Tij : S → S
+that swaps the terms indexed by i and j in a bi-infinite sequence s ∈ S and leaves all other
+entries fixed. Thus for s ∈ S
+(Tijs)(k) =
+
+
+
+
+
+s(k)
+k ̸= i, j
+s(j)
+k = i
+s(i)
+k = j.
+With this notation in place, suppose λ is a partition, and s ∈ aλ is a representative of the
+abacus of λ. Let h be a hook of λ, corresponding to the pair of indices (i, j) (with i < j) in
+s. Then Tijs is a representative of the abacus associated to λ \ bs(h).
+Returning to our example of the partition (6, 5, 3, 1, 1, 1), Figure 2 contains its Young
+diagram again, but now with each box filled in with the corresponding hook-length. The
+unique hook h of length 5 in the diagram corresponds to the pair of indices (5, 10) of the
+sequence (4.1). If we remove the corresponding border strip, we obtain the diagram pictured
+in Figure 3, which corresponds to the partition (6, 5, 3, 1, 1, 1) \ bs(h) = (6, 2, 1, 1, 1, 1) and
+the bi-infinite sequence
+. . . , 1, . . . , 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0 . . ., 0, . . .
+of 0’s and 1’s.
+Note that if we swap the 0 and 1 corresponding to the hook h in the
+representative (4.2) of a(6,5,3,1,1,1), then we get an equivalent bi-infinite sequence.
+4.4. Removing several hooks in succession. In our work below, we will need to remove
+several hooks (more precisely, the border strips corresponding to those hooks) in succession
+
+8
+SARAH PELUSE AND KANNAN SOUNDARARAJAN
+from a partition. By removing a sequence of hooks h1, . . ., hR from a partition λ, we mean
+the following: h1 is a hook of λ, h2 is a hook of λ \ bs(h1), h3 is a hook of λ \ bs(h1) \ bs(h2),
+and so on, until we arrive at hR which is a hook of λ \ bs(h1) . . . \ bs(hR−1), and when this
+is removed we obtain the final partition λ′ = λ \ bs(h1) . . . \ bs(hR).
+Let s be a representative of the abacus aλ associated to λ. Let (i1, j1) denote the pair of
+indices in s corresponding to the hook h1, (i2, j2) the corresponding pair to h2 (which,
+recall, is a hook of λ \ bs(h1) corresponding to the bi-infinite sequence Ti1,j1s), and so
+on.
+Thus, the sequence of hooks h1, . . ., hR may be encoded by the R-tuple of pairs
+((i1, j1), (i2, j2), . . . , (iR, jR)), and the process of removing these hooks results in the sequence
+s′ = TiR,jRTiR−1,jR−1 · · · Ti1,j1s.
+The sequence s′ is a representative of the abacus aλ′ associated to the partition λ′.
+Of particular interest for us will be the situation where all the hooks have the same length,
+m say. Here jk = ik + m for all 1 ≤ k ≤ R, and we may encode the sequence of hooks by
+simply the R-tuple (i1, . . . , iR). Note that the indices i1, . . ., iR may contain repeats, but
+there are also constraints, such as i2 ̸= i1 (since (i1, i1 + m) is a hook in s and so it cannot
+be a hook in Ti1,i1+ms).
+5. Plan of the proof of Theorem 2.2
+We begin by restating Theorem 2.2 in terms of values of irreducible characters at elements
+of Sn, which will make the notation involved in its proof cleaner.
+Theorem 5.1 (An equivalent formulation of Theorem 2.2). Let n, m1, . . . , mr be distinct
+positive integers. Let σ ∈ Sn be a permutation of the form
+σ = τ ·
+r�
+i=1
+pr−1
+�
+j=1
+ρ(j)
+i ,
+where each ρ(j)
+i
+is a cycle of length mi, the supports of all the cycles ρ(j)
+i
+are disjoint, and
+τ ∈ Sn is a permutation with support disjoint from those of the ρ(j)
+i ’s. Suppose that λ is a
+(�r
+i=1 kimi)-core partition of n for all r-tuples (k1, . . . , kr) of integers 0 ≤ k1, . . . , kr ≤ pr−1
+for which some ki = pr−1. Then
+pr | χλ(σ).
+The proof of Theorem 5.1 rests on the following crucial proposition, which is based on
+applying the Murnaghan–Nakayama rule pr−1 times.
+Proposition 5.2. Let r, m and n be positive integers. Let σ ∈ Sn be of the form
+σ = τ ·
+pr−1
+�
+j=1
+ρ(j),
+where each ρ(j) is an m-cycle, with all the cycles ρ(j) being disjoint, and with τ ∈ Sn being
+a permutation whose support is disjoint from all the cycles ρ(j). Denote by L the set of
+partitions of n − pr−1m that can be obtained from λ by removing, in succession, pr−1 border
+strips of length m. If λ is a pr−1m-core partition of n, then
+χλ(σ) = p
+�
+λ′∈L
+ǫλ′χλ′(τ),
+
+DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP
+9
+where each ǫλ′ is an integer.
+We will quickly deduce Theorem 5.1 (and hence Theorem 2.2) from Proposition 5.2 and
+the following simple observation.
+Lemma 5.3. Let n, t and m be positive integers. Let λ be a partition of n which is both a
+t-core and a (t + m)-core. Let λ′ be a partition of n − m that can be obtained by removing a
+border strip of length m from λ. Then λ′ is a t-core.
+Proof. If λ has no hook (and thus no border strip) of length m then the lemma holds
+vacuously. Suppose that λ′ arises from removing the border strip corresponding to the hook
+h of length m in λ. Let aλ be the abacus of λ, and s be a representative bi-infinite sequence
+in aλ. Suppose the hook h corresponds to the pair of indices (i, i + m) with s(i) = 0 and
+s(i + m) = 1, so that the partition λ′ corresponds to the abacus containing s′ = Ti,i+ms.
+If λ′ is not a t-core, then there must exist a pair of indices (j, j + t) with s′(j) = 0 and
+s′(j + t) = 1. Since the entries of s and s′ differ only at the indices i and i + m, and since
+λ is a t-core, we must have either j = i + m, or j + t = i. If j = i + m, then s(i) = 0 and
+s(i + t + m) = s′(j + t) = 1 which contradicts the assumption that λ is a (t + m)-core. If
+j = i−t, then s(i−t) = s′(j) = 0 and s(i+m) = 1, which again contradicts the assumption
+that λ is a (t + m)-core.
+□
+Deducing Theorem 5.1 from Proposition 5.2. Apply Proposition 5.2 first with m = mr to
+obtain
+χλ(σ) = p
+�
+λ′∈L
+ǫλ′χλ′�
+τ
+r−1
+�
+i=1
+pr−1
+�
+j=1
+ρ(j)
+i
+�
+.
+If t is any number of the form t = �r−1
+i=1 kimi where the ki lie in [0, pr−1] with at least one
+of them being pr−1, then λ is a (t + krmr)-core for all 0 ≤ kr ≤ pr−1. Since any λ′ ∈ L
+arises from λ by removing pr−1 border strips of length mr, it follows by pr−1 applications of
+Lemma 5.3 that λ′ is a t-core.
+We may now repeat this argument, applying Proposition 5.2 to each λ′ ∈ L and now
+removing pr−1 border strips of length mr−1. Applications of Lemma 5.3 show that the new
+partitions λ′′ that arise are (�r−2
+i=1 kimi)-cores for all choices of 0 ≤ ki ≤ pr−1 with some
+ki = pr−1.
+Carrying this argument out r times, we obtain the desired result.
+□
+The proof of Proposition 5.2 depends on the following two lemmas, which we shall prove
+in the next two sections.
+Lemma 5.4. Let λ be a partition, and let λ′ be obtained from λ by removing a sequence of
+R border strips of the same length m. Let h1, . . ., hR be a sequence of R hooks of length m
+which may be removed from the initial partition λ to result in the final partition λ′. Then
+(−1)ht(h1)+...+ht(hR) = ǫ(λ, λ′)
+where the sign ǫ(λ, λ′) = ±1 depends only on the initial and final partitions λ and λ′ and is
+the same for all such possible sequences of hooks.
+Lemma 5.5. Let λ be a pr−1m-core partition, and let λ′ be a partition that can be obtained
+from λ by removing R = pr−1 border strips of length m. The number of tuples (i1, . . . , iR)
+such that
+s′ = TiR,iR+mTiR−1,iR−1+m · · ·Ti1,i1+ms
+
+10
+SARAH PELUSE AND KANNAN SOUNDARARAJAN
+is a multiple of p. Here s is a representative of the abacus of λ, and the partition λ′ corre-
+sponds to the abacus containing s′
+Once Lemmas 5.4 and 5.5 are in place, it is a simple matter to deduce Proposition 5.2.
+Deducing Proposition 5.2. We apply the Murnaghan–Nakayama rule repeatedly while re-
+moving in succession R = pr−1 hooks of length m from λ. This will result in an expression
+for χλ(σ) of the form �
+λ′∈L cλ′χλ′(τ), for suitable integers cλ′ which we must show are
+multiples of p. Now
+cλ′ =
+�
+(i1,...,iR)
+(−1)ht(h1)+...+ht(hR)
+where the sum is over all R-tuples (i1, . . . , iR) corresponding to hooks h1, . . ., hR, which
+when removed from λ in order result in the partition λ′. Lemma 5.4 tells us that the sign
+(−1)ht(h1)+...+ht(hR) is the same for all suitable tuples (i1, . . . , iR), and Lemma 5.5 tells us that
+the number of such R-tuples is a multiple of p.
+□
+6. Parity of heights of hooks: Proof of Lemma 5.4
+Let λ be a partition, and s a representative of the abacus aλ associated to λ. Augment s
+by coloring a finite number N of 1’s in s with distinct colors, taking care to color all the 1’s
+appearing to the right of the first zero in s. The 1’s appearing to the left of the first 0 are
+unimportant, but we allow the flexibility of coloring some of them since this situation may
+arise at an intermediate step when we remove hooks from λ. Note that the number of 1’s
+appearing to the right of the first zero equals the number of rows in the partition λ. Thus
+N is at least the number of rows in λ. Color these 1’s in the order of their appearance in s
+using the colors c1, . . ., cN. Call the augmented sequence �s.
+We begin with a general observation on removing hooks. Suppose (i, j) is a pair of indices
+corresponding to a hook h in s (at the moment the hook can have any length j−i). Removing
+this hook produces the sequence Ti,js.
+Considering the augmented sequence �s, we have
+the corresponding augmented sequence Ti,j�s after removing this hook. If we consider the
+sequence of colors among the 1’s in this sequence, we obtain a permutation πij, say, of the
+original sequence of colors (c1, . . . , cN) — the 1 appearing in (Ti,j�s)(i) has the color of the 1
+in �s(j), and all other 1’s in Tij(�s) retain their color in �s. If the height of the hook removed
+is k, then note that �s had k colored 1’s between s(i) = 0 and s(j) = 1 and the permutation
+πij can be obtained by making k-transpositions, each time swapping the color of the 1 at
+position j by the color immediately preceding it. Thus (−1)k = (−1)ht(h) equals the sign of
+the permutation πij.
+If we remove hooks h1, . . ., hℓ in succession (again, their lengths could be arbitrary), then
+the associated permutations of colors multiply, and therefore so do the signs of these permu-
+tations. Thus, after removing these hooks in succession we would arrive at a permutation π
+of the sequence of colors (c1, . . . , cN) and
+(−1)ht(h1)+ht(h2)+...+ht(hℓ) = sgn(π).
+We now turn to the situation of the lemma, where a sequence h1, . . ., hR of R hooks
+is removed all of length m.
+Our observation above shows that removing these hooks in
+order leads to the sequence �s ′ where the color of the 1’s is given by a permutation π of the
+original sequence of colors c1, . . ., cN. Further the sign of this permutation sgn(π) equals
+(−1)ht(h1)+...+ht(hR).
+
+DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP
+11
+To complete the proof, we will show that every way of removing R hooks of length m
+that leads to the partition λ′ results in the same permutation of colors π. Consider the
+subsequence of �s obtained by restricting to a progression (mod m): namely, (�s(a + ℓm))ℓ∈Z.
+There are m such subsequences corresponding to a = 1, . . ., m. Since the hooks removed all
+have length m, each removal of a hook affects only the terms within one of these subsequences,
+leaving all the other subsequences unaltered. Further within any particular subsequence
+(�s(a + ℓm))ℓ∈Z, it is impossible to alter the original sequence of colors by removing any
+sequence of hooks of length m.
+Therefore we can determine uniquely the color of any
+element in �s ′: the 1’s appearing in this sequence in the progression a (mod m) have colors
+determined by their order of appearance in the original sequence s.
+7. Proof of Lemma 5.5
+Let λ be a pr−1m-core partition, and let s be a representative of its abacus. Let s′ be the
+sequence obtained by removing a sequence of R = pr−1 border strips of length m from λ,
+and let λ′ be the partition associated to s′. Our goal is to show that the number of ways of
+reaching λ′ starting from λ is a multiple of p.
+Let us first note that when r = 1, it is impossible to remove a border strip of length m
+from λ, since λ is m-core by assumption. Thus the number of ways here is 0, and the lemma
+holds (vacuously). Henceforth, assume that r ≥ 2.
+For each a = 1, . . ., m, consider the subsequences of s and s′ obtained by restricting to
+the progression a (mod m): thus, set
+s(a; m) = (s(a + ℓm))ℓ∈Z,
+s′(a; m) = (s′(a + ℓm))ℓ∈Z.
+We may think of s(a; m) and s′(a; m) as corresponding to partitions λ(a; m) and λ′(a; m), and
+note that a hook of length m in the partition λ corresponds to a hook of length 1 (or simply
+a border square) in the partition λ(a; m) (for some choice of a). Since λ′(a; m) arises from
+λ(a; m) by removing some number of hooks of length 1, the Young diagram of the partition
+λ′(a; m) is contained in the Young diagram of the partition λ(a; m) (that is, λi(a; m) ≥
+λ′
+i(a; m) for all i). The difference between the Young diagram of λ(a; m) and λ′(a; m) (in
+other words, the boxes in λ(a; m) that are not in λ′(a; m)) is a skew diagram, which we
+denote by λ(a; m)/λ′(a; m). Let ℓa denote the size of this skew diagram |λ(a; m)/λ′(a; m)|,
+so that ℓa hooks of length 1 must be removed from λ(a; m) to reach λ′(a; m). Since a total
+of R = pr−1 hooks of length m are removed to go from λ to λ′, note that
+R = pr−1 =
+m
+�
+a=1
+ℓa.
+The number of ways to go from λ(a; m) to λ′(a; m) by removing successively ℓa hooks
+of length 1 equals the number of standard Young tableaux of skew shape λ(a; m)/λ′(a; m),
+which we denote (in the usual notation) by fλ(a;m)/λ′(a;m). Recall that a standard Young
+tableau of this skew shape is a numbering of the boxes in the skew diagram using the
+numbers 1 to ℓa such that the entries are increasing from left to right in each row, and
+increasing down each column. Each such tableau corresponds to a way of removing hooks,
+by removing boxes in descending order of their entries.
+We can now count the number of ways of going from λ to λ′ by removing R hooks of
+length m. Note that removing a hook from one subsequence s(a; m) has no impact on the
+
+12
+SARAH PELUSE AND KANNAN SOUNDARARAJAN
+hooks in any of the other subsequences. Therefore the desired number of ways to proceed
+from λ to λ′ equals
+�
+pr−1
+ℓ1, ℓ2, . . . , ℓm
+� m
+�
+a=1
+fλ(a;m)/λ′(a;m).
+The multinomial coefficient
+�
+pr−1
+ℓ1,ℓ2,...,ℓm
+�
+is a multiple of p, except in the situation where
+ℓa = pr−1 for some a (and all other ℓj are 0). Thus we are left with the case when all the
+hooks of length m in going from λ to λ′ are confined to one subsequence s(a; m). So far, we
+have not made use of the condition that λ is a pr−1m-core, and it is only in this case that
+we need this assumption. The assumption implies that λ(a; m) is pr−1-core, and so the skew
+diagram λ(a; m)/λ′(a; m) (which has size ℓa = pr−1) cannot be a border strip of λ(a; m). In
+this situation, it turns out that fλ(a;m)/λ′(a;m) is a multiple of p. This is implied by our next
+lemma, which is perhaps of independent interest.
+Lemma 7.1. Let π and τ be two partitions, with the Young diagram of π containing the
+Young diagram of τ (thus πi ≥ τi for all i). Suppose the skew diagram π/τ is not a border
+strip of the partition π (equivalently, either π/τ is disconnected, or it contains a 2×2 square),
+and that |π/τ| = pt is a prime power. Then the number of standard Young tableaux of skew
+shape π/τ, denoted fπ/τ, is a multiple of p.
+Proof. First suppose that π/τ is disconnected, and is composed of k ≥ 2 maximally connected
+skew shapes S1, . . ., Sk, with |Sj| = sj ≥ 1. Then
+fπ/τ =
+�
+pt
+s1, . . . , sk
+�
+fS1 · · · fSk,
+is clearly a multiple of p.
+Now suppose that π/τ is a connected skew shape, but contains a 2 × 2 square so that it
+is not a border strip of π. Since fπ/τ depends only on the shape π/τ, we may assume that π
+is minimal, having only as many rows and columns as needed for the skew shape π/τ. Then
+the maximal hook length of π equals the number of border squares of π, which is strictly
+smaller than |π/τ| = pt (since π/τ is not a border strip by assumption).
+It is a basic fact (see Section I.9 of [8], for example — the identity below follows from
+equation (9.1) of [8] by taking the Hall inner product of both sides with the symmetric
+function ept
+1 ) that
+fπ/τ =
+�
+ν⊢pt
+fνcπ
+τν,
+where the sum is over partitions ν of |π/τ| = pt, fν = χν
+(1,...,1) is the degree of the irreducible
+character corresponding to ν and the cπ
+τν are the Littlewood–Richardson coefficients (which
+are integers). By Lemma 2.1, fν ≡ χν
+(pt) (mod p), so that p | fν unless ν is a hook of length
+pt. Suppose now that ν is a hook of length pt. Here we use that the Littlewood–Richardson
+coefficient cπ
+τν is zero unless the Young diagram of the partition ν is contained in that of π
+(see Section I.9 of [8] once again). But all the hooks of π have length < pt, and therefore π
+cannot contain a hook ν of length pt. Thus either cπ
+τν = 0 or p|fν, and therefore the lemma
+follows.
+□
+
+DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP
+13
+8. Preliminaries for the proof of Proposition 2.3
+As in [16], let �p(k) denote the number of partitions of a nonnegative integer k into powers
+of p, with the convention that �p(0) = 1. Denote by Fp(t) the associated generating function
+Fp(t) :=
+∞
+�
+k=0
+�p(k)e−k/t =
+∞
+�
+j=0
+�
+1 − e−pj/t�−1
+,
+where t > 0 is a real number. We begin by recalling some estimates from our prior work [16].
+Lemma 8.1 (Lemma 2 of [16]). When 0 < t ≤ 1, we have Fp(t) = O(1), and when t ≥ 1,
+we have
+(log t)2
+2 log p + 1
+2 log t + O(1) ≤ log Fp(t) ≤ (log t)2
+2 log p + 1
+2 log t + 1
+8 log p + O(1).
+More precise results are known for fixed primes p, as partitions into prime powers have
+been studied extensively since the work of Mahler [9] and de Bruijn [1]. We will only require
+the estimates of Lemma 8.1, which are cruder but uniform in p.
+Given a partition µ of k into powers of p, let �µ denotes the partition obtained by repeatedly
+replacing every occurrence of pr parts of the same size pj by pr−1 parts of size pj+1 until no
+part appears more than pr − 1 times. For every nonnegative integer s, define �p(k; s) to be
+the number of partitions µ of k into powers of p such that �µ does not contain (at least) pr−1
+parts of the same size pj for any j ≥ s. The second lemma of this section gives a useful lower
+bound for the difference between �p(k) and �p(k; s).
+Lemma 8.2. For all s ≥ 2 and k ≥ pr+s−1(1 + 4/s), we have
+�p(k) − �p(k; s) ≥ ps(s−1)/2
+(s − 1)s−1.
+Proof. We will construct at least ps(s−1)/2/(s − 1)s−1 partitions of k counted in �p(k) but not
+in �p(k; s). For each 1 ≤ i ≤ s − 1, pick an integer ai in the range
+0 ≤ ai ≤ ps−i
+s − 1.
+Each choice of a1, . . . , as−1 gives a partition µ counted in �p(k) by using ai copies of pi for
+1 ≤ i ≤ s − 1 and k − �s−1
+i=1 aipi copies of 1. The number of such partitions is
+s−1
+�
+i=1
+� ps−i
+s − 1
+�
+≥
+s−1
+�
+i=1
+ps−i
+s − 1 =
+ps(s−1)/2
+(s − 1)s−1.
+Note that if i > s − log(s − 1)/ log p, then ai must be zero, so that all of these partitions
+have largest part at most
+ps
+s−1.
+We must check that each such µ is not counted in �p(k; s); that is, that the corresponding
+�µ contains at least pr−1 copies of some part pj with j ≥ s. Suppose that this is not the
+case. Notice that, by construction, the number of times any part appears in µ is congruent
+modulo pr−1 to the number of times it appears in �µ. Since no part can appear more than
+pr − 1 times in �µ, it follows that any part that appears fewer than pr−1 times or more than
+pr − pr−1 times in �µ must have appeared in the original partition µ. Since all the parts of µ
+are below ps/(s − 1), we conclude that �µ can contain (i) at most pr − 1 copies of any part pj
+
+14
+SARAH PELUSE AND KANNAN SOUNDARARAJAN
+with pj ≤ ps/(s −1), (ii) at most pr −pr−1 copies of any part pj with ps/(s −1) < pj ≤ ps−1,
+and (iii) no parts of size pj with j ≥ s. But these constraints imply that
+k = |�µ| ≤ (pr − 1)
+�
+pj≤ps/(s−1)
+pj + (pr − pr−1)
+�
+ps/(s−1)<pj≤ps−1
+pj
+< (pr−1 − 1)
+ps
+(s − 1)
+�
+1 − 1
+p
+�−1
++ (pr − pr−1)ps−1�
+1 − 1
+p
+�−1
+< pr+s−1�
+1 + 4
+s
+�
+,
+which contradicts our assumption on the size of k.
+□
+9. Proof of Proposition 2.3
+Let L be a set of positive integers coprime to p, and define p(n; L, s) to be the number of
+partitions µ of n for which �µ contains fewer than pr−1 parts of the same size ℓpj for every
+ℓ ∈ L and j ≥ s. We will prove Proposition 2.3 by obtaining an upper bound for p(n; L, s)
+for well-chosen L and s.
+Lemma 9.1. Suppose that n is large and pr ≤ 10−3 log n/ log log n. Put
+(9.1)
+x =
+√
+6n
+π ,
+s =
+�log √n
+epr
+�
+,
+and let L be the set of integers in the interval [L, L + x/pr+s−1] that are coprime to p, where
+L is a parameter lying in the range
+(9.2)
+√
+6n
+2πpr+s−1 ≤ L ≤
+�
+1 + 1
+5pr
+�
+√
+6n
+2πpr+s−1 log n.
+Then
+p(n; L, s) ≪ p(n)n
+3
+4 exp(−n
+1
+16pr ).
+Before proving the lemma, let us see how Proposition 2.3 would follow. Choose r distinct
+values Lj (with 1 ≤ j ≤ r) all in the range
+�
+1 + 1
+6pr
+�
+√
+6n
+2πpr+s−1 log n ≤ Lj ≤
+�
+1 + 1
+5pr
+�
+√
+6n
+2πpr+s−1 log n,
+such that the corresponding sets Lj are all disjoint. A partition µ for which �µ does not
+contain r distinct parts m1, . . ., mr each appearing at least pr−1 times and suitably large as
+desired in the proposition, must be counted among some p(n; Lj, s) with 1 ≤ j ≤ r. Thus
+by Lemma 9.1 the number of such bad partitions µ is
+≤
+r
+�
+j=1
+p(n; Lj, s) ≪ rp(n)n
+3
+4 exp(−n
+1
+16pr ) ≪ p(n)n exp(−n
+1
+16pr ) ≪ p(n) exp(−n
+1
+20pr ),
+as claimed.
+Proof of Lemma 9.1. Consider the process of going from a partition µ to �µ by combining pr
+parts of the same size m into pr−1 parts of size pm. Suppose that ℓ is coprime to p, and that
+the sum of all parts of the form ℓpj appearing in µ equals ℓk. Restricting our attention to
+these parts, we may think of µ as giving rise to a partition of k into powers of p, and then �µ
+correspondingly gives a partition of k into powers of p obtained by repeatedly combining pr
+
+DIVISIBILITY OF CHARACTER VALUES OF THE SYMMETRIC GROUP
+15
+parts of size pj into pr−1 parts of size pj+1. It follows that p(n; L, s) is the coefficient of zn
+in the generating function
+�
+ℓ/∈L
+(ℓ,p)=1
+∞
+�
+j=0
+�
+1 − zℓpj�−1 �
+ℓ∈L
+�
+∞
+�
+k=0
+�p(k; s)zℓk�
+,
+which equals
+∞
+�
+i=1
+�
+1 − zi�−1 �
+ℓ∈L
+��∞
+k=0 �p(k; s)zℓk
+�∞
+k=0 �p(k)zℓk
+�
+.
+Since all of the coefficients in the generating function for p(n; L, s) are nonnegative, we must
+have, for any 0 < z < 1,
+(9.3)
+p(n; L, s) ≤ 1
+zn
+∞
+�
+i=1
+�
+1 − zi�−1 �
+ℓ∈L
+��∞
+k=0 �p(k; s)zℓk
+�∞
+k=0 �p(k)zℓk
+�
+.
+Recall that x =
+√
+6n/π, and take z = e−1/x in the bound (9.3). Then, by the asymptotic
+formula for the partition function and basic estimates for the generating function of the
+number of partitions (see Section VIII.6 of [3]), we obtain
+(9.4)
+p(n; L, s) ≪ n3/4p(n)
+�
+ℓ∈L
+��∞
+k=0 �p(k; s)zℓk
+�∞
+k=0 �p(k)zℓk
+�
+≪ n3/4p(n) exp(−∆),
+where
+∆ :=
+�
+ℓ∈L
+1
+Fp(x/ℓ)
+∞
+�
+k=0
+(�p(k) − �p(k; s))e−ℓk/x.
+Our work so far applies to any set L of integers that are coprime to p, and we now proceed
+to the situation at hand. The lower bound on L and our choice of x give, for all ℓ ∈ L, the
+bound
+Fp
+�x
+ℓ
+�
+≤ Fp
+� x
+L
+�
+≤ Fp
+�2pr+s−1
+log n
+�
+.
+From this estimate, our choice of L, and Lemma 8.2 it follows that
+(9.5)
+∆ ≥
+1
+Fp(2pr+s−1/ log n)
+�
+L≤ℓ≤L+x/pr+s−1
+(ℓ,p)=1
+�
+k≥pr+s−1(1+4/s)
+ps(s−1)/2
+(s − 1)s−1e−ℓk/x.
+For ℓ in the range L ≤ ℓ ≤ L + x/pr+s−1, we have
+�
+k≥pr+s−1(1+4/s)
+e−ℓk/x ≥ exp
+�
+− ℓpr+s−1
+x
+�
+1 + 4
+s
+��
+e−ℓ/x
+1 − e−ℓ/x
+≥ x
+2L exp
+�
+−
+�Lpr+s−1
+x
++ 1
+��
+1 + 4
+s
+��
+.
+Inserting this into the right-hand side of (9.5) and noting that (since pr+s−1 is small in
+comparison to x)
+|L| ≥
+�
+1 − 1
+p
+�
+x
+pr+s−1 − 2 ≥
+x
+3pr+s−1,
+
+16
+SARAH PELUSE AND KANNAN SOUNDARARAJAN
+we obtain (using our choice of x and the range for L)
+∆ ≥
+1
+Fp(2pr+s−1/ log n) · ps(s−1)/2
+(s − 1)s−1 ·
+x
+3pr+s−1 · x
+2L exp
+�
+−
+�Lpr+s−1
+x
++ 1
+��
+1 + 4
+s
+��
+≥ 1
+6
+1
+Fp(2pr+s−1/ log n) · ps(s−1)/2
+(s − 1)s−1 ·
+x
+log n · exp
+�
+−
+�Lpr+s−1
+x
++ 1
+��
+1 + 4
+s
+��
+.
+Using Lemma 8.1 and the bound pr ≤ log √n, it follows that
+log Fp
+�2pr+s−1
+log n
+�
+≤
+1
+2 log p
+�
+log pr+s−1
+log √n
+�2
++ 1
+2 log pr+s−1
+log √n + 1
+8 log p + O(1)
+≤
+1
+2 log p
+�
+log pr+s−1
+log √n
+�2
++ s
+2 log p + O(1).
+Therefore
+log
+ps(s−1)/2
+Fp(2pr+s−1/ log n)(s − 1)s−1 ≥ s2
+2 log p −
+1
+2 log p
+�
+log pr+s−1
+log √n
+�2
+− s log ps + O(1)
+≥ s log log √n
+prs
+− (log log √n)2
+2 log p
++ O(1).
+Recalling our choice of s, we conclude that
+log ∆ ≥ s log log √n
+prs
++ log √n − (log log √n)2
+2 log p
+− log log n − Lpr+s−1
+x
+�
+1 + 4
+s
+�
++ O(1)
+≥
+�
+1 + 1
+epr
+�
+log √n − Lpr+s−1
+x
+− log log n − (log log √n)2
+2 log p
++ O(1)
+≥
+� 1
+epr − 1
+5pr
+�
+log √n − (log log n)2 + O(1) ≥ log n
+15pr − (log log n)2 + O(1),
+upon using the upper bound on L in (9.2). In the range pr ≤ 10−3 log n/(log log n)2 we find
+log ∆ ≥ log n
+16pr + O(1),
+which when used in (9.4) yields the lemma.
+□
+References
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+

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