Propriétés d'indépendance algébrique de valeurs de séries de Hecke–Mahler

Here we characterize in a complete and explicit way the relations of algebraic dependence over $\mathbb{Q}$ of complex values of Hecke–Mahler series at algebraic points of the multiplicative group ${\mathbb{G}}_{\mathrm{m}}^2(\mathbb{C})$. Our result contains previous results of Loxton and van der Poorten, Mahler, and Masser. To cite this article: F. Pellarin, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Propriétés de l'EMV de rupture de processus de Poisson

This Note concerns the estimation of the parameter of the intensity of an inhomogeneous spatial Poisson process by the maximum likelihood method. We consider an intensity admitting a discontinuity along a closed curve depending of an unknown parameter. The established results show the consistency, the convergence in law and the convergence of the moments of the maximum likelihood estimator.     

Intersection de sous-groupes et de sous-variétés I

We study the intersection of a subvariety X of an abelian variety A over with the union of all the algebraic subgroups of A of given dimension d. Our main result states that if we remove a suitable exceptional subset from X and if d is small enough then the intersection enjoys a Northcott-like property: the points of bounded height on it form a finite set. The condition on d involves only the dimension of X and the structure of A up to isogenies. We show how it can be weakened if we assume certain conjectures in the direction of an abelian version of Lehmer's problem. The theorem is especially meaningful when X is a curve since it is then possible to bound the height and hence to prove finiteness of the set under consideration. This generalises the result of E. Viada on powers of elliptic curves and is analogous to work of E. Bombieri, D. Masser and U. Zannier on tori, whose general strategy we follow.     

Inégalités de Brascamp-Lieb et convexité

This note gives an elementary proof of an inequality of Brascamp-Lieb and of a new dual inequality that has numerous applications to convexity: lower estimates of volumes of projections, of exterior volume ratio and MM^* -estimate in the non symmetric case.     

Lemmes de décomposition appliqués à des inégalités de type Hardy–Sobolev

Decomposition (or concentration-compactness) lemmas have already shown their efficience in order to show existence of minimizers or ground state solutions. The aim of this paper is to apply new version of these lemmas to minimisation problems involving Hardy–Sobolev type inequalities on a specific class of unbounded domains. More precisely, we shall find ground state solution for the following quotient, where value of real numbers ,b,q and are given.

Lemmes de multiplicités et constante de Seshadri

Résumé  On démontre dans cet article un raffinement des lemmes de multiplcités de Philippon, essentiellement dans le cas particulier où l’on dérive dans toutes les directions. L’amélioration est rendue possible grace à un point de vue plus géométrique et notamment par l’apparition nouvelle dans ce contexte de la notion de constante de Seshadri.     

Irrationalité de certaines sommes de séries

Résumé Let with |q| > 1, and a be a rational number such that a ² is not equal to for . In this note, we prove that the sum is irrational.     

Solvariétés projectives de dimension trois

Let V be a hyperbolic 2-torus bundle over the circle, or equivalently, a quotient of the three-dimensional group Sol by a lattice . We study real projective structures on V. We prove that all such structures are left-invariant which means that they are all obtained by the following process: take a representation of Sol in PGL (4, R) with an open orbit in R P ³; it induces a projective structure on Sol and, hence, on the left-quotient \Sol=V. We classify all the real projective structures on V and show that they are in fact all affine. The proof involves a study of codimension one projective subsurfaces, which are tori, and the dynamic of their holonomy group in the whole manifold.     

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     

Sur une caractérisation de la répartition normale de probabilités

Sans résumé Présenté par A. Rényi     

Sur les Propriétés Arithmétiques de la Fonction de Tschakaloff

Let P and Q be non-zero polynomials with integer coefficients; suppose that all the roots of Q are rational numbers, and that Q(n) 0 for every n N. Let q Z, with |q| 2, and let Q^*. We prove that the number is irrational.From this we deduce linear independence results over Q for the values at x = of Tschakaloff's function Ta(x) = , of its derivatives, and of its primitives. The proof uses Loxton and Van der Poorten's extension of Mahler's transcendence method, which leads, in this case, only to irrationality results.     

Dimension de Billingsley d'ensembles saturés

In this Note, the Cajar's formula on the Billingsley dimension of saturated sets is extended to the case of subshifts of finite type for g-measures.     

Sur les séries aléatoires de Dirichlet

The Borel-Cantelli lemma, a method of J.-P. Kahane, and an extension of a lemma of Paley-Zygmund are applied to study random Taylor and Dirichlet series.     

Homologies associées aux variétés de Poisson

In this article, we obtain results on the cyclic homology of Poisson manifolds as defined by Kassel. In particular, we compute the cyclic homology of any symplectic variety. We also show that there exists a family of new operators of the type of the S operator of Connes, defined on the cyclic homology of a Poisson manifold, and describe the associated exact triangles associated to these new operators.

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Received February 23, 1998 / Published online July 20, 2000     

Quelques propriétés des solutions de l'équation de Ginzburg-Landau sur un ouvert de ∝2

For a solution u of –u=u(1–|u|²) on the whole plane, |u|<1 holds everywhere unless u=eⁱ for some ; the derivatives of order k have moduli a constant M _kdepending only on k. For a solution u on an open set ², the moduli of u and its derivatives have upper bounds depending only on the distance to ²\ therefore the set of solutions on a given is compact in C() for the topology of uniform convergence on compact subsets of . For a solution u such that |u|<1, 1–|u| satisfies an estimation similar to the classical Harnack inequality for positive harmonic functions.Finally, if is bounded and |u| has a lim supm at each boundary point, the |u|m in if m1, but if m<1 then |u| admits only a majorant S _m with values in ]m, 1[ and sufficient conditions are given for lim S _m =0 or S _m =O(m) as m0.