Répartition de Suites et Équations Fonctionnelles Associées

LetX be a compact metric space, a probability measure onX, T a measure preserving transformation ofX into itself. Furthermore, letZ be a compact abelian group, andS the skew-product defined onX×Z by the formula:S(x,z)=(T x,z+(x)) where is some map fromX toZ. It is known that under the hypothesis of unique ergodicity for the triple (X, ,T) and some suitable conditions of continuity onT and , the triple (X×Z,m,S) wherem is the product measure of by the Haar measure onZ is uniquely ergodic if and only if an associated functional equation has no non trivial solution.—We introduce here a generalization of unique ergodicity (which appears quite naturally when studying well distributed sequences) and prove about it the same kind of results.     

Sur la répartition des valeurs de la fonction d'Euler

Let (x) denote the number of those integers n with (n) x, where denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that (x)-Ax R(x), where A=(2)(3)/(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.     

Résultats combinatoires sur les modules de Demazure

In this paper, we introduce the notion of polytopes with integral structure, which generalizes the notion of usual polytopes whose vertices have integral coordinates. Then we consider a symmetrizable Kac-Moody algebra. For every dominant weight λ and every element of the Weyl group w, we construct a polytope with integral structure such that its Ehrhart polynomial describes the dimension of the Demazure module of highest weight nλ associated to w.     

Différentiabilité de la composition et complétitude de certains espaces fonctionnels

Sans résumé Ce travail a été partiellement subventionné par le Fonds National Suisse pour la Recherche scientifique.     

Espace des modules de certains polyèdres projectifs miroirs

A projective mirror polyhedron is a projective polyhedron endowed with reflections across its faces. We construct an explicit diffeomorphism between the moduli space of a mirror projective polyhedron with fixed dihedral angles in (0,\fracp2]{(0,\frac{\pi}{2}]}, and the union of n copies of \mathbbR^d{\mathbb{R}^{d}}, when the polyhedron has the combinatorics of an ecimahedron, an infinite class of combinatorial polyhedra we introduce here. Moreover, the integers n and d can be computed explicitly in terms of the combinatorics and the fixed dihedral angles.     

Estimation sous biais de sélection et avec fonction de poids inconnue

We consider the problem of estimating the cumulative distribution function (cdf) G of a non-negative random variable (r.v.) X from the observation of a biased r.v. Y with cdf $F_w=\int w(x)\mathrm{d}G(x)/{\mu }_w$, where w is an unknown weighting function. We assume moreover that the random sample with common cdf $F_w$ is right-censored. We construct an estimator $\stackrel{?}{G}$ for the cdf G and state its strong consistency and weak convergence. To cite this article: A. Guilloux, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Equirépartition des coefficients de Fourier des fonctions L de carrés et de cubes symétriques

Dans ce travail, on étudie l’équirépartition et l’équirépartition harmonique des coefficients de Fourier des fonctions L de carrés et de cubes symétriques d’une forme primitive.     

Régulateurs supérieurs,périodes et valeurs spéciales de la fonction L de degré 4 de GSp(4)

We present some results in direction of Beilinson's conjecture for the degree four L-function of an automorphic representation π of the symplectic group GSp(4). Our main result relates some motivic cohomology classes over powers of the universal abelian scheme over the Shimura variety of GSp(4) to the product of an archimedean integral, an occult period defined by Harris, a Deligne period and the special value of the L-function predicted by Beilinson's conjecture. We assume that π is stable, of multiplicity one, and has a Bessel model of a specific type. To cite this article: F. Lemma, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Un choix de fenêtre optimal en estimation polynomiale locale de la fonction de répartition conditionnelle

We propose an optimal choice for the bandwidth parameter used in the local polynomial estimation of the conditional distribution function. This choice is approximated by the bandwidth which minimizes the asymptotic weighted mean integrated squared error (MISE). To cite this article: S. Ferrigno, G.R. Ducharme, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Sur la fonction sommatoire de la fonction de von Mangoldt généralisée

We give an identity for the summatory function of the generalized von Mangoldt function Received: 19 February 2007     

Points entiers de certains schémas de matrices

Let R be an Euclidean domain, M _n (R) the R-module of matrices with n rows and n columns, V an intersection of n?1 affine hyperplanes of M _n (R). Assume V is not empty. Then, for all a in R, there exists a matrix in V with determinant equal to a.     

Estimation de la fonction de régression pour variable explicative et réponse fonctionnelles dépendantes

We consider a kernel estimate of the regression when the response variable is in a Banach space and the explanatory variable takes its values in a semi-metric space. Our main result states the almost complete convergence (with rate) of the constructed estimate when the sample considered is a β-mixing sequence.     

Classification analytique d'équations différentielles ydy +...=0 et espace de modules

Résumé Dans ce travail, on s'intéresse à la classification analytique de certains types d'équations différentielles de (²,0) de la forme =ydy+...=O Cette classification est en général donnée par celle de l'holonomie projective apparaissant dans la résolution de . Dans un cas spécial la classification est donnée par celles de l'holonomie associée à l'unique séparatrice de . On précise l'espace de modules et on prouve la rigidité générique de .

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In this paper we study the analytic classification of class of differential equations =ydy+...=O in (²,0). We prove that generically they are rigid. We also give the moduli spaces in special cases.

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L'auteur remercie l'IRMAR pour l'accueil qu'il a rencontré durant ses séjours (long et courts) à l'institut.