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

We prove a conjecture of I. Korec [4] on decidability of some fragments of arithmetic equipped with a pairing function; as consequence, we give an axiomatization of the fragment of arithmetic equipped with Cantor pairing function, precising a result of [5].

We propose an adaptive MCMC method consisting of a set of parallel but non-Markovian and non i.i.d. discrete time processes, where each marginal uses the Hastings–Metropolis dynamic based on proposal densities depending on the other processes and learning from their past. We prove the geometric convergence of each marginal with a rate a.s. better than any arbitrary independent Hastings–Metropolis algorithm.

We prove the Zahariuta's conjecture, which itself solves a Kolmogorov's problem on the ε-entropy of classes of analytic functions. For a given holomorphically convex compact subset K in a bounded pseudoconvex domain D in $\text{C}{}^{\mathrm{n}}$, the Zahariuta's conjecture consists in approximating uniformly on any compact subset of D⧹K, the relative extremal function u_K,D by a sequence of pluricomplex Green functions on D with logarithmic poles in the compact set K.

In this Note, we determine the set of algebraic points of degree at most 3 on the Klein quartic curve. This result extends a previous result given by Hurwitz, who described the set of rational points.

We study the stabilization of the one-dimensional wave equation with a boundary or locally distributed feedback that is positive-negative or on-off. We also give results of controllability in “arbitrarily short time” for the one-dimensional wave equation with a locally distributed control.

A general class of conditional U-statistics was introduced by W. Stute [3] as a generalization of the Nadaraya–Watson estimates of a regression function. It was shown that such statistics are universally consistent (Stute [3]). Also, universal consistency of the window and k_n-nearest neighbor estimators (as two special cases of the conditional U-statistics) were proved by Stute [3]. In this paper, we extend these results from the independent case to dependent case.

In this Note we prove that if {G_n} is a sequence of connected k-regular graphs in which the length of odd cycles approaches infinity as n→∞, then the $\text{lim}\text{sup}$ of the smallest eigenvalue of G_n greater than −k is at most $\text{−2}\text{k−1}$ as n tends to infinity.

The motion of point vortices in a 2-dimensional ideal fluid is treated as a Hamiltonian system. We describe an infinite family of periodic solutions for an even number of vortices on the sphere, the ellipsoid, the torus, and other surfaces, as well as some of their relative variants.

From a uni-dimensional model for acoustic waves in a flow duct, an analysis of the exact controllability of initial perturbations is suggested using the so-called HUM method of J.-L. Lions. Several restrictive hypothesis are required and the minimum delay necessary for an exact control is a function of the three following velocities: the one of the steady flow, and the two waves celerities in the flexible structure and in the flow. Then we discuss the possibility to use actuators in order to control the noise perturbations. The goal of the study is to evaluate a new technology for a noise insulator in an air conduct.

This Note deals with a semi-parametric model for Hilbertian random variables. The model is said semi-parametric by analogy with the finite dimensional case since the model involves a composition of any measurable mapping with a linear mapping which represents the “parametric” part. Under mild conditions, we derive a way for estimating this linear component in a particular case. We show that this method is actually a generalization of Li's Sliced Inverse Regression. However, in the Hilbertian context, SIR requires some adaptations of the estimation procedure and results concerning the consistency of the proposed estimates are given.