Minorations pour l'équation de Catalan

We consider Catalan's equation x^p − y^q = 1 (with p, q prime and |x|, |y| > 1). We show that, besides the obvious solution 3² − 2³ = 1, min {p; q} > 10⁵ and max {p; q} > 10⁶.     

Inégalités de concentration pour les processus empiriques de classes de parties

We propose new concentration inequalities for maxima of set-indexed empirical processes. Our approach is based either on entropy inequalities or on martingale methods. The improvements we get concern the rate function which is exactly the large deviations rate function of a binomial law in most of the cases. Received: 11 January 2000 / Revised version: 12 May 2000 / Published online: 14 December 2000     

Décomposition de domaine pour un calcul hybride de l'équation de helmholtz

We prove an estimate for the Dirichlet-Neumann operator, and for the H¹ local norm for solutions of Helmholtz equation outside an obstacle without trapping rays. We give an algorithm solving Helmholtz equation outside a union of such obstacles. Convergence follows from this estimate. At each step of the resolution, only one obstacle is considered for itself; this defines a decomposition domain technique fitting this equation. One can use different numerical schemes, one at each step, adapted to the considered component of the obstacle; therefore, this algorithm is a hybrid computation. The results are given for two obstacles, and the generalization is straightforward     

Quelques nouvelles identités de fluctuation pour les processus de Lévy

Let τ and H be the ladder time and ladder height processes of a Lévy process X. We give an identity in law between (τ,H) and (X,H^*), H^* being the right continuous inverse of the process H. The later allows us to get a relationship between the entrance law of X and the entrance law of the excursion measure away from 0 of the reflected process (X_t- inf_s≤ X_s >- 0). In the stable case, some explicit calculations are provided.     

Inégalités isosystoliques conformes pour la bouteille de Klein

We prove sharp isosystolic inequalities for all conformal equivalence classes of continuous Riemannian metrics on the Klein bottle, i.e.

Sur le test de maximum de vraisemblance pour le mélange de populations

In [2] a theory is developed which under some regularity conditions, gives the asymptotical distribution of the likelihood ratio for testing the number of populations in a mixture. We give here two results. First, we exhibit a simple example where the likelihood ratio converges to +∞ with probability ½. Up to our knowledge, it is the first non-trivial example of a non-convergent likelihood test in a compact parameter space. Also, we give some enough regularity conditions allowing to apply the results of [2].     

Existence et unicité pour le modéle de Nagdhi pour une coque peu réguliére

We give a simple proof of existence and uniqueness for Nagdhi 's model for shells whose midsurface can have discontinuous curvatures. This improves earlier proofs by Ciarlet, Miara and Bernadou, and Ciarlet and Miara for C³ -midsurfaces.     

L'inégalité de Bohr pour les séries de Dirichlet

We extend to the setting of Dirichlet series previous results of Bohr for Taylor series in one variable, themselves generalized by Paulsen, Popescu and Singh or extended to several variables by Aizenberg, Boas and Khavinson. We show in particular that, if $f(s)={\sum }_{n=1}^{\infty }{a}_n{n}^{?s}$, with $6f{6}_{\infty }:={\sup }_{\mathfrak{R}s>0}|f(s)|<\infty$, then ${\sum }_{n=1}^{\infty }|a_n|n^{\mathrm{?2}}?6f{6}_{\infty }$ and even slightly better, and ${\sum }_{n=1}^{\infty }|a_n|n^{?1/2}?C6f{6}_{\infty }$, C being an absolute constant. To cite this article: R. Balasubramanian et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Une correspondance de modules pour les blocs à groupes de défaut abéliens

Let b be a block of a finite group G with an abelian defect group P and an inertial quotient E. Let us denote by L the semi-direct product of P and E. If E is cyclic and acts freely on P−{1}, we prove that the stable categories of Gb- and L-modules are equivalent, as a consequence of a more general result, without any hypothesis on E, about partial covering exomorphisms relating Gb with a suitable twisted group algebra _*

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à Jacques Tits à l'aube de sa sexagénie     

Un modéle pseudo-tridimensionnel d'écoulement de fluides parfaits et généralisation de l'équation pour la fonction de courant

Résumé Dans une région des l'espace on considère l'écoulement d'un fluide et on fait l'hypothèse qu'il est pseudotridimensionnel, c'est-à-dire que son mouvement est stationnaire et qu'il existe un système de coordonnées curvilignes (y ¹,y ²,y ³) tel que les surfaces coordonnéesy ³=constante sont des surfaces de courant. Cette hypothèse entraîne l'existence d'une fonction de courant (y ¹,y ²,y ³). Si le fluide est parfait et barotrope on obtient que ³/ le long des lignes de courant oú ³ est la troisième composante contravariante du rotationnel de la vitesse et la densité de masse. Cette propriété permet de formuler une équation pour la fonction de courant, généralisant l'équation habituelle de la mécanique des fluides plans.

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Let us consider a flow motion in a given special domain. We make the assumption that the movement of the fluid is pseudo-tridimensional, i.e., the flow motion is steady and there exists a system of curvilinear coordinates (y ¹,y ²,y ³) such that stream surfaces are defined byy ³=constant. Under this assumption there exists a stream function .Whenever the fluid is perfect and barotropic, we show that ³/= constant on stream lines (where ³ is the 3rd contravariant component of the speed's curl and is the density).Using this property we are able to write an equation for the stream function, which embeds the usual equation for two-dimensional fluid movements.

     

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.     

Analyse numérique de la méthode des variables adiabatiques pour l'approximation de l'hamiltonien nucléaire

Many problems in quantum chemistry deal with the computation of fundamental or excited states of molecules and lead to the resolution of eigenvalue problems. One of the major difficulties in these computations lies in the very large dimension of the systems to be solved. Indeed, these eigenfunctions depend on 3n variables, where n stands for the number of particles (electrons or protons) in the molecule. In order to diminish the size of the systems to be solved, the chemists have proposed many interesting ideas. Among those stands the adiabatic variable method; we present in this Note a mathematical analysis of this approximation and propose, in particular, an a posteriori estimate that might allow for verifying the adiabaticity hypothesis that is done on some variables.     

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.     

Construction de solutions ramifiées autour de deux hypersurfaces caractéristiques pour des opérateurs quasi-linéaires

Let P(x,D)u=v be a partial differential equation, where P(x,D) is a quasi-linear operator of second order with simple characteristic hypersurfaces; we get solutions u which are ramified around two simple characteristic hypersurfaces if v is ramified function and D_t1v, D_t2,v are bounded.     

Une méthode de point fixe pour le p-laplacien

Due to the compactness of the operator (-Δ_p)¹ N_ƒ, 1 < p < ∞, where Δ_p is the p-Laplacian and N_ƒ is the Nemytskii operator corresponding to a Caratheodory function ƒ: Ω × R → R, which satisfies a particular growth condition, the homotopy invariance of Leray-Schauder degree can be used in order to prove the existence of a W₀^1,p (Ω)-solution for the equation -Δ_p u = N_ƒ u.     

Un résultat de densité pour les équations de Maxwell

We prove the density of regular fielclx in the space of square-integrahle fields with square-integrable curls and square-integrable tangential traces on the boundary. This space is involved in one of the variational formulations of the electmmagnetism equations.     

Théorème central limite pour l'intersection de deux saucisses de Wiener indépendantes

Summary J.F. Le Gall [4] proved thatn ² times the volume of the intersection of two independent Wiener sausages in ³, with radius 1/n, converges inL ², asn, towards a multiple of the intersection local time at 0, for the underlying Brownian motions.We complete this result by proving a corresponding central limit theorem.