Limit laws of entrance times for homeomorphisms of the circle

Given a homeomorphismf of the circle with irrational rotation number and a descending chain of renormalization intervalsj _n off, we consider for each interval the point process obtained by marking the times for the orbit of a point in the circle to enterJ _n. Assuming the point is randomly chosen by the unique invariant probability measure off, we obtain necessary and sufficient conditions which guarantee convergence in law of the corresponding point process and we describe all the limiting processes. These conditions are given in terms of the convergent subsequences of the orbit of the rotation number off under the Gauss transformation and under a certain realization of its natural extension. We also consider the case when the point is randomly chosen according to Lebesgue measure,f being a diffeomorphism which isC ¹-conjugate to a rotation, and we show that the same necessary and sufficient conditions guarantee convergence in this case.     

Asymptotics for statistical functionals of long-memory sequences

We present two general results that can be used to obtain asymptotic properties for statistical functionals based on linear long-memory sequences. As examples for the first one we consider L- and V-statistics, in particular tail-dependent L-statistics as well as V-statistics with unbounded kernels. As an example for the second result we consider degenerate V-statistics. To prove these results we also establish a weak convergence result for empirical processes of linear long-memory sequences, which improves earlier ones.     

Uniform null controllability of the heat equation with rapidly oscillating periodic density

We consider the heat equation with fast oscillating periodic density, and an interior control in a bounded domain. First, we prove sharp convergence estimates depending explicitly on the initial data for the corresponding uncontrolled equation; these estimates are new, and their proof relies on a judicious smoothing of the initial data. Then we use those estimates to prove that the original equation is uniformly null controllable, provided a carefully chosen extra vanishing interior control is added to that equation. This uniform controllability result is the first in the multidimensional setting for the heat equation with oscillating density. Finally, we prove that the sequence of null controls converges to the optimal null control of the limit equation when the period tends to zero. To cite this article: L. Tebou, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Statistical approximation properties of high order operators constructed with the Chan-Chyan-Srivastava polynomials

In this paper, by including high order derivatives of functions being approximated, we introduce a general family of the linear positive operators constructed by means of the Chan-Chyan-Srivastava multivariable polynomials and study a Korovkin-type approximation result with the help of the concept of A-statistical convergence, where A is any non-negative regular summability matrix. We obtain a statistical approximation result for our operators, which is more applicable than the classical case. Furthermore, we study the A-statistical rates of our approximation via the classical modulus of continuity.     

l-solutions and stability analysis of difference equations using the Kummer’s test

We address the p-summability and asymptotic stability properties in nonautonomous linear difference equations. We focus our discussion on two kind of difference equations. The first one is a first order system of linear nonautonomous difference equations, and our discussion involves the use of Kummer’s convergence test. The second one is a linear nonautonomous scalar higher order difference equation. In this case our discussion is based on a recently introduced transformation of a higher order system into a first-step recursion, where the companion matrices are well treatable from our point of view. We give insight on our ideas that are behind our methods, prove new results, and show applications.     

On the convergence of finite state mean-field games through Γ-convergence

In this study, we consider the long-term convergence (trend toward an equilibrium) of finite state mean-field games using Γ-convergence. Our techniques are based on the observation that an important class of mean-field games can be viewed as the Euler–Lagrange equation of a suitable functional. Therefore, using a scaling argument, one can convert a long-term convergence problem into a Γ-convergence problem. Our results generalize previous results related to long-term convergence for finite state problems.     

Inexact-Newton method for solving operator equations in infinite-dimensional spaces

In this paper, we develop an inexact-Newton method for solving nonsmooth operator equations in infinite-dimensional spaces. The linear convergence and superlinear convergence of inexact-Newton method under some conditions are shown. Then, we characterize the order of convergence in terms of the rate of convergence of the relative residuals. The present inexact-Newton method could be viewed as the extensions of previous ones with same convergent results in finite-dimensional spaces.     

Smale's α-theory for inexact Newton methods under the γ-condition

The present paper is concerned with the convergence problem of inexact Newton methods. Assuming that the nonlinear operator satisfies the γ-condition, a convergence criterion for inexact Newton methods is established which includes Smale's type convergence criterion. The concept of an approximate zero for inexact Newton methods is proposed in this paper and the criterion for judging an initial point being an approximate zero is established. Consequently, Smale's α-theory is generalized to inexact Newton methods. Furthermore, a numerical example is presented to illustrate the applicability of our main results.     

Integration by parts on Bessel Bridges and related stochastic partial differential equationsIntegration par parties sur Ponts de Bessel et EDPS correspondantes

We prove integration by parts formulae with respect to the law of Bessel Bridges of dimension δ?3. For δ=3 we have an infinite-dimensional boundary measure, and for δ>3 a singular logarithmic derivative. We give applications to SPDEs with additive space-time white noise and singular drifts, whose solutions are non-negative. To cite this article: L. Zambotti, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 209–212.     

Compact inverses of first-order normal differential operators

In this work, firstly we describe all normal extensions of a minimal operator generated by linear differential-operator expression of first order in the Hilbert space of vector functions in finite interval. Later on, we investigate discreteness of spectrum and asymptotical behavior of s-numbers of the inverses of these normal extensions.     

Asymptotics of Plancherel measures for the infinite-dimensional unitary group

We study a two-dimensional family of probability measures on infinite Gelfand-Tsetlin schemes induced by a distinguished family of extreme characters of the infinite-dimensional unitary group. These measures are unitary group analogs of the well-known Plancherel measures for symmetric groups.We show that any measure from our family defines a determinantal point process on Z₊×Z, and we prove that in appropriate scaling limits, such processes converge to two different extensions of the discrete sine process as well as to the extended Airy and Pearcey processes.     

Two new q-exponential operator identities and their applications

In this paper, we use the q-Chu–Vandermonde formula to prove two new operator identities, which are the extensions of Liu's results. These two q-exponential operator identities are used to derive some q-summation formulas and q-integrals.     

Γ-convergence and homogenization of functionals in Sobolev spaces with variable exponents

This paper is devoted to homogenization and minimization problems for variational functionals in the framework of Sobolev spaces with continuous variable exponents. We assume that the sequence of exponents converges in the uniform metric and that the Lagrangian has a periodic microstructure. Then under natural coerciveness assumptions we prove a Γ-convergence result and, as a consequence, the convergence of minimizers (solutions to the corresponding Euler equations).     

Deviation property of periodic measures in C 1 non-uniformly hyperbolic systems with limit domination

In a C1 non-uniformly hyperbolic systems with limit domination, we consider the periodic measures that supported on the Pesin set and keep a distance at least δ to a hyperbolic ergodic measure μ given before. And then, we bound from top the exponential growth rate of such periodic measures by the supremum of measure theoretic entropy on a closed set.     

Convergence de l'estimateur à noyau des k plus proches voisins en régression fonctionnelle non-paramétrique

This note focuses on the k nearest neighbor method when one regresses a real random variable on a functional random variable (i.e. valued in an infinite-dimensional space). More precisely, we consider a kernel estimator of the regression based on a local bandwidth using exactly the k nearest neighbors. Although it is frequently used in functional data analysis, this method has not given any theoretical result so far. The aim of this Note is to show the pointwise almost-complete convergence of the k nearest neighbor kernel estimator in nonparametric functional regression. To cite this article: F. Burba et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A new simultaneous iterative method with a parameter for solving the extended split equality problem and the extended split equality fixed point problem

In this article, we first propose an extended split equality problem which is an extension of the convex feasibility problem, and then introduce a parameter w to establish the fixed point equation system. We show the equivalence of the extended split equality problem and the fixed point equation system. Based on the fixed point equation system, we present a simultaneous iterative algorithm and obtain the weak convergence of the proposed algorithm. Further, by introducing the concept of a G-mapping of a finite family of strictly pseudononspreading mappings \(\{T_{i}\}_{i = 1}^{N}\), we consider an extended split equality fixed point problem for G-mappings and give a simultaneous iterative algorithm with a way of selecting the stepsizes which do not need any prior information about the operator norms, and the weak convergence of the proposed algorithm is obtained. We apply our iterative algorithms to some convex and nonlinear problems. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithms.     

Factorization of delta-monotone linear mappings

For a delta-monotone linear mapping we prove that the factors in the polar decomposition are delta-monotone. Also, we prove that every delta-monotone linear mapping can be factored into a product of (1-ε)-monotone mappings for any ε∈(0,1). As an application in nonlinear case, we give a new proof of the following fact: the quasiconformality constant K(δ,n) of a δ-monotone mapping can be chosen such that K(δ,n) tends to 1 as δ tends to 1.