Comptes Rendus de l'Académie des Sciences - Series I - Mathematics最新文献

Let G be a connected solvable Lie group with abelian derived group and μ be a spread out probability measure on G. We give an explicit description of the Poisson boundary in terms of almost sure convergence of the right random walk of law μ. We characterize the Poisson boundary by an integral criterion for some matricial groups.

In this Note, we give necessary and sufficient conditions for the generator of a $\text{C}{}_0$ semigroup to be semiregular. We also prove that the spectral inclusion for semigroups holds for the singular spectrum. By applying the preceding results, we establish some stability results for semigroups.

In this paper we give the Karhunen–Loève expansion of a centered Gaussian process whose covariance function depends on two parameters, including the covariance function of the Brownian bridge as a special case. A statistical application, related to the weighted uniform empirical process is provided.

The notion of integer ambiguity resolution is associated with the problem of finding the point of  $\text{Z}{}^{\mathrm{n}}$ closest to a point of  $\text{R}{}^{\mathrm{n}}$, the distance being the one induced by a given quadratic form. This problem is solved with the aid of computational techniques currently used in algebra of numbers. The notion of reduced basis then plays a key role. In interferometry (more precisely in phase closure imaging) and in GPS (Global positioning system), the statement of these problems also appeals to the notion of a graph, and thereby, to the related linear algebra. The corresponding approach, which is very attractive from a conceptual point of view, finally leads to very efficient solution methods.

In this Note a result on the strict concavity of the harmonic radius in open convex domains of $\text{R}{}^{\mathrm{N}}$, for N⩾3, is given. It implies the strict convexity of the Robin function and the uniqueness of the harmonic center in bounded convex domains.

We study the dynamical system defined by the map $\text{Φ:}\text{]0,1]→}\text{]0,1]}$, where Φ(x)=px−1 on ]1/p,1/q] if q and p are consecutive prime numbers. We relate the existence of an absolutely continuous invariant measure to ergodicity of a Markov chain $\text{P}$ on the union of orbits stemming from numbers 1/p (p prime). We prove that ergodicity of $\text{P}$ implies ergodicity of Φ. We establish a link between transfer probabilities of order n and some sets of sequences of the symbolic dynamic. This leads to a way of computing these coefficients using Monte Carlo method. We propose an algorithm which leads to a density indicating a good experimental fit with a typical orbit.

In this work we show the role which plays a recent theorem on the strong asymptotic stability [17,14,1] in the analysis of the strong stabilizability problem in Hilbert spaces. We consider a control system with skew-adjoint operator and one-dimensional control. We examine in details the property for a linear feedback control to stabilize such a system. A robustness analysis of stabilizing controls is also given.

By using the pseudo-differential and para-differential calculus introduced by J.-M. Bony [1], we study the incompressible isotropic Navier–Stokes equations. We prove the short-time existence and uniqueness of solutions for arbitrary data with supercritical regularity. We exploit pseudo-differential calculus to extend the analysis to compact Riemannian manifolds.

This paper deals with a generalization of a result due to Brascamp and Lieb which states that in the space of probabilities with log-concave density with respect to a Gaussian measure on $\text{R}{}^{\mathrm{n}}$, this Gaussian measure is the one which has strongest moments. We show that this theorem remains true if we replace x^α by a general convex function. Then, we deduce a correlation inequality for convex functions quite better than the one already known. Finally, we prove a result concerning stochastic analysis on Wiener space through the notion of approximate limit.

Let $\text{P}{}_{\mathrm{n}}\text{(z)=∑}{}_{\mathrm{k=0}}{}^{\mathrm{n}}\text{a}{}_{\mathrm{k,n}}\text{z}{}^{\mathrm{k}}\text{∈}\text{C}\text{[z]}$ be a sequence of unimodular polynomials (|a_k,n|=1 for all k, n) which is ultraflat in the sense of Kahane, i.e., $\text{lim}\text{n→∞}\text{max}\text{|z|=1}\text{|(n+1)}{}^{\mathrm{-1/2}}\text{|P}{}_{\mathrm{n}}\text{(z)|−1|=0.}$ We prove the following conjecture of Saffari (1991): ∑_k=0ⁿa_k,na_n−k,n=o(n) as n→∞, that is, the polynomial P_n(z) and its “conjugate reciprocal” $\text{P}{}_{\mathrm{n}}{}^{\mathrm{\ast }}\text{(z)=∑}{}_{\mathrm{k=0}}{}^{\mathrm{n}}\text{a}{}_{\mathrm{n-k,n}}\text{z}{}^{\mathrm{k}}$ become “nearly orthogonal” as n→∞. To this end we use results from [3] where (as well as in [5]) we studied the structure of ultraflat polynomials and proved several conjectures of Saffari.