Quelques proprietes des systemes dynamiques qui se decomposent en un produit de deux systemes dont l’un est un schema de Bernoulli

The purpose of this paper is to investigate some properties of dynamical systems that split into the product of two systems, one of them being a Bernoulli shift. With this view in mind, a relative version of the main results of D. S. Ornstein’s theory of Bernoulli shifts is developed here.     

Représentation de −1 comme somme de carrés dans un corps cyclotomique quelconque

Let m ≥ 3 be an odd integer, and let K^(m) = Q(e^2πi/m) be the cyclotomic field of the m-th roots of unity. Then s(K^(m)) (the “stufe” of K^(m), that is to say, the smallest number of squares necessary to represent ?1 in K^(m) is equal to 2 or to 4 depending on whether the multiplicative order of 2 modulo m is even or odd.     

Le nombre minimal de colorations d'un hypergraphe k-chromatique

A (0,λ)-graph (λ?2) is a connected graph in which any two vertices have λ common neighbours or none at all. Such a graph is regular. When the diameter is at least four, a lower bound for the degree can be obtained. This lower bound is attained only by Hadamard graphs and n-cubes.     

Une méthode de détermination du temps de transfert dans un système de deux zones multiplicatrices de neutrons

With a simplified delay time distribution function in the kinetic equations of a binode system, the neutron evolution is found to follow a two mode exponential decay. In the non-excited core, the evolution curve presents a maximum which time of occurrence takes place in a relation giving the interzonal transfer time. Values obtained from pulsed neutron experiments are discussed.     

Écoulement d'un fluide de Bingham dans un domaine non-cylindrique

We obtain an existence theorem for Bingham flows in a noncylindrical domain, as a simple application of a previous result on Bingham flows with a monotone perturbation.     

Approximation simultanée d'un nombre et de son carré

In 1969, H. Davenport and W.M. Schmidt established a measure of the simultaneous approximation for a real number ξ and its square by rational numbers with the same denominator, assuming only that ξ is not rational nor quadratic over $\text{Q}$. Here, we show by an example, that this measure is optimal. We also indicate several properties of the numbers for which this measure is optimal, in particular with respect to approximation by algebraic integers of degree at most three. To cite this article: D. Roy, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

La résolution minimale de l’idéal d’un arrangement général d’un grand nombre de points dans Pn

Sans résumé Oblatum 13-XII-1994 & 3-IV-1996     

Normalisation d'un processus de branchement critique dans un environnement aléatoire

Let (Z_n) be a critical branching process in an independent and identically distributed (i.i.d.) random environment. For each fixed environment ω, let C_n=E^ω[Z_n∣Z_n>0] be the conditional expectation of Z_n given Z_n>0. We prove an analogue of Yaglom's law: as n→∞, the conditional law of Z_n/C_n, conditional on Z_n>0, converges to a non-degenerate law on [0,∞). We give also an analogue of Kolmogorov's law, as well as a local limit theorem for the semi-group of probability generating functions. To cite this article: Y. Guivarc'h et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Comportement des semi-martingales dans un grossissement de filtration

Sans résumé     

Écoulement d'un fluide de Bingham dans un domaine non-cylindrique avec singularité

We consider a Bingham flow on a noncylindrical domain with a singularity at t = 0. Using the well known technique of Galerkin approximations, we give an existence theorem. For the bidimensional case we also show uniqueness of the solution of our problem.     

Borne explicite du nombre de tores plats isospectraux à un tore donné

One gives an explicit bound of the number of isometry classes of flat tori isospectral to a given flat torus. The bound depends on the geometry of the given flat torus. These results hold in any dimension. In these case where dimension is two or three, one obtains bounds which only depend on the dimension.

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Cette recherche a été faite en partie dans le cadre du contrat C.E.E. “GADGET” SC1-0105-C au laboratoire associé au C.N.R.S., n^o188     

Étude de la consistance d'un algorithme stochastique dans un cadre mélangeant

This Note is devoted to the study of the consistency of a stochastic algorithm having for object the estimates of a parameter under strong mixing assumption. To cite this article: A. Ait Saidi, A. Dahmani, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Approximation dans un Problème de Viscoélasticité Périodique

We are concerned with equations which derive from a quasistatic periodic problem of viscoelasticity. We give a condition which yields to existence and uniqueness of a periodic solution. Then we prove a finite element method based on equilibrium elements for the space approximation and on the explicit Euler scheme for the time approximation.     

Existence de deux solutions du type front progressif pour un modèle de combustion avec pertes de chaleur

This Note deals with the existence of planar flames, in the case of a single-step chemical reaction with volumetric heat losses. We prove the existence of two distinct solutions, for small values of the heat loss rate parameter. We also give upper bounds for the flame speed and for the heat loss rate parameter. To cite this article: L. Roques, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Polynômes de Degré Minimum connectant deux projections dans une Algébre de Banach

Etant données deux projections p et q dans une algèbre de Banach A réelle ou complexe, qui appartiennent à la même composante connexe de l'ensemble P des projections de A, il existe un polynôme joignant p et q dans P. On cherche le degré minimum d'un tel polynôme. On démontre que si p ? q est inversible, alors le degré minimum est inférieur ou égal à 3. On établit que ce résultat reste vrai, sans l'hypothése que p ? q est inversible, dans le cas où A = M_n(K) (K = R ou C), et on donne une interprétation géométrique de ce dernier résultat pour A = M₂(K).