Courants du type résiduel attachés à une intersection complète

We construct in complete intersection's case, elementary currents which describe the local ideal, and give a decomposition in it for holomorphic function.     

Sur la répartition des puissances modulo 1

For almost all x>1$x>1$, (xⁿ)$(x^n)$(n=1,2,…)$(n=1,2,\dots )$ is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (b_n)$(b_n)$ in [0,1[$[0,1[$ and ε>0$\epsilon >0$, the x  -set such that |xⁿ−b_n|<ε$|x^n-b_n|<\epsilon$ modulo 1 for n   large enough has dimension 1. However, its intersection with an interval [1,X]$[1,X]$ has a dimension <1, depending on ε and X. Some results are given and a question is proposed.     

Interprétation bijective d'une récurrence des nombres de Motzkin

The purpose of this note is to give a combinatorial proof of the three-term linear recurrence for Motzkin numbers. The present work is inspired by Rémy's combinatorial proof of the linear recurrence for Catalan numbers (RAIRO Inform. Theor. 19(2) (1985) 179) and the more recent proof given by Foata and Zeilberger (J. Combin. Theory Ser. A 80(2) (1997) 380) for Schröder numbers (Z. Math. Phys. 15 (1870) 361).     

Oscillations fortes sur un champ linéairement dégénéré

Transparencies happen for quasilinear waves associated to linearly degenerate eigenvalues, leading to linear geometric optics. To reach nonlinear phenomena, we have to consider larger amplitudes. We show that approximate solutions can still be constructed. We find intrinsic conditions allowing to justify the approximation. In the absence of these conditions, instabilities are created. We analyse the different types of the underlying instabilities.     

Équivalence rationnelle sur les hypersurfaces cubiques de mauvaise réduction

This article consists of two independent, but related, parts. The first one proves the vanishing of the Chow group of classes of zero-cycles of degree zero modulo rational equivalence for a cubic hypersurface of dimension ?10 on a p-adic or C₂ field (and, in fact, the R-triviality of such a hypersurface). This is done without the assumption of good reduction (or even smoothness). The second part goes in the other direction and gives an explicit example of a smooth cubic hypersurface of dimension 3 (necessarily of bad reduction) on a field such as C((ν,t)) (or C((ν))((t))) whose Chow group of classes of zero-cycles of degree zero modulo rational equivalence does not vanish.     

Principe local-global pour les zéro-cycles sur les surfaces réglées

Let be a number field, a smooth projective curve, and a smooth projective surface which is a conic bundle over . Let be the relative Chow group, which is the kernel of the projection map on Chow groups of zero-cycles. For each place of , one may consider the relative Chow group . Under minor assumptions, we identify the diagonal image of in the product of all as the kernel of the natural pairing with the Brauer group of . When is an elliptic curve with finite Tate-Shafarevich group, under minor assumptions, we show that the Brauer-Manin obstruction to the existence of a zero-cycle of degree one on is the only obstruction.

     

Quelques aspects sur la géométrie des surfaces algébriques réelles

Let f be a real polynomial of degree n?3 in two variables. It is known that its hessian is a real polynomial in two variables of degree at most 2n−4. In 1876, A. Harnack prove that the number of connected components of an algebraic plane curve of degree m embedded in is at most (m−1)(m−2)/2+1. So, by A. Harnack, the number of compact connected components of the parabolic curve of the graph of f is at most (2n−5)(n−3)+1. Until now, we do not know if this bound is optimal for n?4.In this note we give a class of real polynomials of degree n?3 in two variables such that the parabolic curve of the graph of each polynomial being to this class has exactly (n−1)(n−2)/2 connected components and exactly n(n−2) special parabolic points. Moreover, all these parabolic curves are smooth and compact.It is known that for each smooth algebraic surface of degree n?3 embedded in , the maximal number of connected components of its parabolic curve is 2n(n−2)(5n−12)+2. Until now, we do not know if this bound is optimal. In this note we give a family of smooth algebraic surfaces of degree n?3 embedded in . The parabolic curve of each surface being to this family is smooth and it has at least n(n−1)/(n−2)2 connected components and at least n²(n−2) special parabolic points.     

Sur les distributions covariantes dans les algèbres de Lie réductives

We study the possibility of factoring a covariant distribution on reductive Lie algebras as finite sum of products of an invariant distribution by a covariant polynomial.     

Sur le changement de base stable des intégrales orbitales pondérées

Let G^′ be a quasi-split connected reductive group over a p-adic field F. Let E be a cyclic extension of F. In the context of cyclic base change, we can attach to G^′ and E a twisted space G^* (in the sense of Labesse). Let G be an inner form of G^*. If G^′ is GL(n), SL(n) or more generally a group which we call L-stable, we define and prove the existence of a non-invariant transfer between the weighted orbital integrals of G and those of G^′. For GL(n), such a transfer has been conjectured by Labesse. The proof is based on previous results of harmonic analysis on Lie algebras and on a generalization of a result of Waldspurger concerning Arthur's (G,M)-families.     

Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes

In the first part of the paper, we prove the existence of a unique global solution to the axisymmetric Navier-Stokes system with initial data and external force with . This improves the result obtained by S. Leonardi, J. Málek, J. Necǎs and M. Pokorný [S. Leonardi, J. Málek, J. Necǎs, M. Pokorný, On axially symmetric flows in R³, Zeitschrift für analysis und ihre anwendungen, J. Anal. Appl. 18 (3) (1999) 639-649], where H²(R³) regularity was required. In the second part, we state global existence and uniqueness for the axisymmetric Navier-Stokes system with initial data in W2,p(R³) and external force in with 1<p<2. This also improves [S. Leonardi, J. Málek, J. Necǎs, M. Pokorný, On axially symmetric flows in R³, Zeitschrift für analysis und ihre anwendungen, J. Anal. Appl. 18 (3) (1999) 639-649] because less integrability is required on v₀ and on f.     

Filtration de certains espaces de fonctions sur un espace symétrique réductif

We introduce a filtration of a -module of some space of functions on a reductive symmetric space G/H, and compute the associated grading as a direct sum of induced representations. As an application of this result to the reductive groups viewed as symmetric spaces, we are able to realize any Harish-Chandra module as a subquotient of a direct sum of induced representations from parabolic subgroups, the inducing representations being trivial on the unipotent radical.