Reconstruction and subgaussian processes

This Note presents a randomized method to approximate any vector v from some set $T\subset {\mathbb{R}}^n$. The data one is given is the set T, and k scalar products ${(〈X_i,v〉)}_{i=1}^k$, where ${(X_i)}_{i=1}^k$ are i.i.d. isotropic subgaussian random vectors in ${\mathbb{R}}^n$, and $k\ll n$. We show that with high probability any $y\in T$ for which ${(〈X_i,y〉)}_{i=1}^k$ is close to the data vector ${(〈X_i,v〉)}_{i=1}^k$ will be a good approximation of v, and that the degree of approximation is determined by a natural geometric parameter associated with the set T. This extends and improves recent results by Candes and Tao. To cite this article: S. Mendelson et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Scaling and non-standard matching theorems

Consider the standard Gaussian measure μ on ${\mathbb{R}}^2$. Consider independent r.v.s ${(X_i)}_{i\le N}$ distributed according to μ, and an independent copy ${(Y_i)}_{i\le N}$ of these r.v.s. We prove that, for some number C and N large, we have

(1)$\frac{{(\log ?N)}^2}{C}\le \mathsf{E}\underset{\pi }{\inf }?\sum _{i\le N}d{(X_i,Y_{\pi (i)})}^2\le C{(\log ?N)}^2,$

where the infimum is over all permutations π of $\{1,\dots ,N\}$. The striking point of this result is the factor ${(\log ?N)}^2$. Indeed, if instead of μ we consider the uniform distribution on the unit square, it is well known that the proper factor is $\log ?N$. The upper bound was proved by Michel Ledoux (2017) [3].     

Slow convergence in generalized central limit theorems

We study the central limit theorem in the non-normal domain of attraction to symmetric α-stable laws for $0<\alpha \le 2$. We show that for i.i.d. random variables $X_i$, the convergence rate in $L^{\infty }$ of both the densities and distributions of ${{\sum }_i}^n{X}_i/(n^{1/\alpha }L(n))$ is at best logarithmic if L is a non-trivial slowly varying function. Asymptotic laws for several physical processes have been derived using convergence of ${{\sum }_{i=1}}^n{X}_i/\sqrt{n\log ?n}$ to Gaussian distributions. Our result implies that such asymptotic laws are accurate only for exponentially large n.     

Global convergence method for singularly perturbed boundary value problems

We give uniformly convergent splines difference scheme for singularly perturbed boundary value problems(1)$-\epsilon u^{″}+p(x)u^{\prime }+q(x)u=f(x),u(a)={\alpha }_0,u(b)={\alpha }_1,$by using splines fitted with delta sequence due to the very stiff nature of the problem under consideration. We prove the $\mathrm{O}(\min (h^2,{\epsilon }^2))$ order of uniform convergence with respect to small parameter $\epsilon$ at nodes on uniform mesh and $\mathrm{O}(\min (h,\epsilon ))$ order of uniform global convergence with respect to the approximate solution given by $S(x)={\sum }_{i=1}^N{S}_{{\Delta }_i}(x)H(x_i-x)$ where H is the Heaviside function, which is the approximation for the closed form of the exact solution.     

On parabolic Kazhdan–Lusztig R-polynomials for the symmetric group

Parabolic R-polynomials were introduced by Deodhar as parabolic analogues of ordinary R-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic R-polynomials for the symmetric group. Let $S_n$ be the symmetric group on $\{1,2,\dots ,n\}$, and let $S=\{s_i|1\le i\le n?1\}$ be the generating set of $S_n$, where for $1\le i\le n?1$, $s_i$ is the adjacent transposition. For a subset $J?S$, let ${(S_n)}_J$ be the parabolic subgroup generated by J, and let ${(S_n)}^J$ be the set of minimal coset representatives for $S_n/{(S_n)}_J$. For $u\le v\in {(S_n)}^J$ in the Bruhat order and $x\in \{q,?1\}$, let $R_{u,v}^{J,x}(q)$ denote the parabolic R-polynomial indexed by u and v. Brenti found a formula for $R_{u,v}^{J,x}(q)$ when $J=S?\{s_i\}$, and obtained an expression for $R_{u,v}^{J,x}(q)$ when $J=S?\{s_{i?1},s_i\}$. In this paper, we provide a formula for $R_{u,v}^{J,x}(q)$, where $J=S?\{s_{i?2},s_{i?1},s_i\}$ and i appears after $i?1$ in v. It should be noted that the condition that i appears after $i?1$ in v is equivalent to that v is a permutation in ${(S_n)}^{S?\{s_{i?2},s_i\}}$. We also pose a conjecture for $R_{u,v}^{J,x}(q)$, where $J=S?\{s_k,s_{k+1},\dots ,s_i\}$ with $1\le k\le i\le n?1$ and v is a permutation in ${(S_n)}^{S?\{s_k,s_i\}}$.     

A note on stronger forms of sensitivity for dynamical systems

Let (X,d) be a compact metric space and (κ(X),d_H) be the space of all non-empty compact subsets of X equipped with the Hausdorff metric d_H. The dynamical system (X,f) induces another dynamical system $(\kappa (X)\text{,}\overline{f})$, where f:X → X is a continuous map and $\overline{f}:\kappa (X)\to \kappa (X)$ is defined by $\overline{f}(A)=\{f(a):a\in A\}$ for any A ∈ κ(X). In this paper, we introduce the notion of ergodic sensitivity which is a stronger form of sensitivity, and present some sufficient conditions for a dynamical system (X,f) to be ergodically sensitive. Also, it is shown that $\overline{f}$ is syndetically sensitive (resp. multi-sensitive) if and only if f is syndetically sensitive (resp. multi-sensitive). As applications of our results, several examples are given. In particular, it is shown that if a continuous map of a compact metric space is chaotic in the sense of Devaney, then it is ergodically sensitive. Our results improve and extend some existing ones.     

Explicit formulas for C1,1 and Cconv1,ω extensions of 1-jets in Hilbert and superreflexive spaces

Given X a Hilbert space, ω a modulus of continuity, E an arbitrary subset of X, and functions $f:E\to \mathbb{R}$, $G:E\to X$, we provide necessary and sufficient conditions for the jet $(f,G)$ to admit an extension $(F,\mathrm{?}F)$ with $F:X\to \mathbb{R}$ convex and of class $C^{1,\omega }(X)$, by means of a simple explicit formula. As a consequence of this result, if ω is linear, we show that a variant of this formula provides explicit $C^{1,1}$ extensions of general (not necessarily convex) 1-jets satisfying the usual Whitney extension condition, with best possible Lipschitz constants of the gradients of the extensions. Finally, if X is a superreflexive Banach space, we establish similar results for the classes $C_{\mathrm{conv}}^{1,\alpha }(X)$.     

On the Erdös–Lax inequality concerning polynomials

Let $P(z)$ be a polynomial of degree n and for any complex number α, let $D_{\alpha }P(z):=nP(z)+(\alpha ?z)P^{\prime }(z)$ denote the polar derivative of $P(z)$ with respect to α. In this paper, we present an integral inequality for the polar derivative of a polynomial. Our theorem includes as special cases several interesting generalisations and refinements of Erdöx–Lax theorem.     

An integrable connection on the configuration space of a Riemann surface of positive genus

Let X be a Riemann surface of positive genus. Denote by $X^{(n)}$ the configuration space of n distinct points on X. We use the Betti–de Rham comparison isomorphism on $H^1(X^{(n)})$ to define an integrable connection on the trivial vector bundle on $X^{(n)}$ with fiber the universal algebra of the Lie algebra associated with the descending central series of ${\pi }_1$ of $X^{(n)}$. The construction is inspired by the Knizhnik–Zamolodchikov system in genus zero and its integrability follows from Riemann period relations.     

On the topology of valuation-theoretic representations of integrally closed domains

Let F be a field. For each nonempty subset X of the Zariski–Riemann space of valuation rings of F, let $A(X)={?}_{V\in X}V$ and $J(X)={?}_{V\in X}{\mathfrak{M}}_V$, where ${\mathfrak{M}}_V$ denotes the maximal ideal of V. We examine connections between topological features of X and the algebraic structure of the ring $A(X)$. We show that if $J(X)\ne 0$ and $A(X)$ is a completely integrally closed local ring that is not a valuation ring of F, then there is a space Y of valuation rings of F that is perfect in the patch topology such that $A(X)=A(Y)$. If any countable subset of points is removed from Y, then the resulting set remains a representation of $A(X)$. Additionally, if F is a countable field, the set Y can be chosen homeomorphic to the Cantor set. We apply these results to study properties of the ring $A(X)$ with specific focus on topological conditions that guarantee $A(X)$ is a Prüfer domain, a feature that is reflected in the Zariski–Riemann space when viewed as a locally ringed space. We also classify the rings $A(X)$ where X has finitely many patch limit points, thus giving a topological generalization of the class of Krull domains, one that includes interesting Prüfer domains. To illustrate the latter, we show how an intersection of valuation rings arising naturally in the study of local quadratic transformations of a regular local ring can be described using these techniques.     

On the Banach ⁎-algebra crossed product associated with a topological dynamical system

Given a topological dynamical system $\Sigma =(X,\sigma )$, where X is a compact Hausdorff space and σ a homeomorphism of X, we introduce the Banach ^?-algebra crossed product ${?}^1(\Sigma )$ most naturally associated with Σ and initiate its study. It has a richer structure than its well investigated $C^{?}$-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. We link its ideal structure to the dynamics, determining when the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. A structure theorem is obtained when X consists of one finite orbit, and the algebra is shown to be Hermitian if X is finite. The key lies in analysing the commutant of $C(X)$ in the algebra, which is shown to be a maximal abelian subalgebra with non-zero intersection with each non-zero closed ideal.