à propos de l’indice de composition des nombres

 We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,

À propos de l’indice de composition des nombres

 We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,

Uniformisation de l’espace des feuilles de certains feuilletages de codimension un

This paper deals with codimension one (may be singular) foliations on compact Kälher manifoldswhose conormal bundle is assumed to be pseudo-effective. Using currents with minimal singularities, we show that one can endow the space of leaves with a metric of constant non positive curvature wich may degenerate on a “rigidly” embedded invariant hypersurface.     

Limites approximatives dans l’espace de Wiener

We show that a family of square integrable random variables defined on the Wiener space possess an approximate limit with respect to quadratic norms and that some variables of the second Wiener chaos possess an approximate limit with respect to measurable norms.     

Conjecture de l’inertie modérée de Serre

Résumé On considère K un corps complet pour une valuation discrète, de caractéristique nulle et dont le corps résiduel est supposé parfait de caractéristique p. On appelle e l’indice de ramification absolue de K, son anneau des entiers, et une cl?ture algébrique. Soit X _K un schéma propre et lisse sur K admettant un modèle propre et semi-stable X sur . Généralisant les résultats de Breuil (valables pour e=1), on démontre dans cet article un isomorphisme de périodes reliant le r-ième groupe de cohomologie étale de à coefficients dans ℤ/p ⁿ ℤ et un r-ième groupe de cohomologie log-cristalline de la fibre spéciale de X. Nous avons toutefois les restrictions er<p-1 et e(r+1)<p-1 si n>1. On en déduit une preuve complète de la conjecture de Serre sur l’inertie modérée (voir [Ser72]).     

Étude de l’autocorrélation multiplicative de la fonction ‘partie fractionnaire’

A first encounter with 0$$" align="middle" border="0"> , a continuous function with a strict local maximum at every rational point.2000 Mathematics Subject Classification: Primary—11M26, 11M41     

Unicité de la solution faible à support compact de l’équation de Vlasov-Poisson

We study here the 3-dimensional Vlasov-Poisson equation of stellar dynamics. It is well known that this equation has weak solutions for every bounded initial density with finite kinetic energy. In [6], Lions and Perthame prove a uniqueness result under a Lipschitz continuity assumption on the initial datum. Using the moment estimates of [6], we can easily see that if the initial datum is compactly supported, the solution will remain compactly supported for ever. We prove here the uniqueness of the compactly supported weak solution. Our proof is an adaptation of that of Youdovitch (see [8]) for the 2-dimensional Euler equation.     

Une remarque sur l’inégalité de McKean

Résumé Pour une classe de variétés riemanniennes équipées de métriques tordues, nous donnons dans ce travail une borne inférieure optimale pour la première valeur propre du laplacien dans le problème de Dirichlet. Linégalité présentée, qui est obtenue très simplement, étend celle de McKean à des variétés dont la courbure peut être de signe variable. Cest le principal intérêt de linégalité. Elle répond aussi partiellement à une question de Schoen et Yau.      

Sur l’itération des opérations de Steenrod

Sans résumé A Monsieur Heinz Hopf, en témoignage de profonde admiration     

Principes de grandes déviations pour l’estimateur à noyau de la densité de probabilité

We establish pointwise and uniform large deviations principles for nonparametric kernel density estimators. The estimates are based on sequences of independent and identically distributed real random variables.     

Groupes résolubles de difféomorphismes de l’intervalle,du cercle et de la droite

We give a complete classification (from the algebraic and dynamical points of view) of solvable groups of orientation preserving C²-diffeomorphisms of the interval, the circle and the real line. We also give some complementary results in the real-analytic case.     

Compactification de Chabauty de l’espace des sous-groupes de Cartan de {\text{ SL}}_n(\mathbb{R })

Let $G$ be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of $G$ , which can be also seen as the space of maximal flats of the symmetric space of $G$ . We define its Chabauty compactification as the closure in the space of closed subgroups of $G$ , endowed with the Chabauty topology. We show that when the real rank of $G$ is 1, or when $G={\text{ SL}}_3(\mathbb{R })$ or ${\text{ SL}}_4(\mathbb{R })$ , this compactification is the set of all closed connected abelian subgroups of dimension the real rank of $G$ , with real spectrum. And in the case of ${\text{ SL}}_3(\mathbb{R })$ , we study its topology more closely and we show that it is simply connected.     

Distances entre exponentielles et puissances d’éléments de certaines algèbres de Banach

Let A be a Banach algebra which does not contain any nonzero idempotent element, let γ > 0, and let . We show that if then . We also show, assuming a suitable spectral condition on x, that if , then Received: 12 July 2006 Revised: 31 January 2007     

Plans dʼexpérience pour lʼestimation de la courbe de concentration et de lʼAUC

The problem of interest is to estimate the concentration curve and the area under the curve (AUC) by estimating the parameters of a linear regression model with autocorrelated error process. We introduce a simple linear nonparametric unbiased estimator of the concentration curve and the AUC. We show that this estimator constructed from an appropriate regular sampling design is asymptotically optimal.     

Doubles mélanges des polylogarithmes multiples aux racines de l’unité

Abstract. – The values at positive integers of the polyzeta functions are solutions of the polynomial equations arising from Drinfeld’s associators, which have numerous applications in quantum algebra. As iterated integrals they are periods of the motivic fundamental groupoid of P¹?{0,1,∞}. From this arises a fundamental, yet no more explicit, system of algebraic relations; it implies the system of associators.¶We focus here on the combinatorial properties of another system of relations, the “double shuffles”, which expresses some elementary series and integrals manipulations. We show that it shares an important property with associators and “motivic” relations, is implied by the latter and defines a polynomial algebra over Q (Écalle’s theorem). We obtain these results for more general numbers: values of Goncharov’s multiple polylogarithms at roots of unity.     

Temps de sortie d’un intervalle borné pour l’intégrale du mouvement brownien

Let (B_t)_t≥0 be a standard Brownian motion starting at y, X_t = x+ ∫₀^tB_s, ds, x ∈ (a, b). Let us set T_ab = inf{t > 0 : X_t ∉ (a,b)}. In this paper, we compute the moments of the random variable B_Ta,b, and deduce the probability law of B_Ta,b. We show how to obtain the expectation E_(x,y)(T_ab^mB_Tabⁿ). We also explicitly determine the probabilities P_(x,y){X_Tab = a} and P_(x,y) { X_Tab = b}.