Une justification du modèle de coques de Naghdi

We consider a thin linearly elastic loaded shell allowing non-zero inextensional displacements. Under some assumptions on the loads, we prove that the tangential and normal parts of the stress tensor are small compared with the transverse pan, when the thickness of the shell goes to zero. Besides, the displacement vector and the transverse pan of the stress tensor are of the same order of magnitude with respect to the thickness when the material constituting the shell is Isotropic and homogeneous. The limit model, which is a flexural model, can also be obtained from Naghdi's model but not from Koiter's model. In some cases of anisotropic materials, the displacement vector is of a larger order of magnitude than the stress tensor, when the thickness goes to zero.     

Limite asymptotique pour le modèle de BGK

We present, for the BGK equation, asymptotic limits leading to various equations of incompressible and compressible fluid mechanics: the Navier-Stokes equations, the linearized Navier-Stokes equations, the Euler equation, the linearized Euler equation, and the compressible Euler equation. We state a convergence theorem for the nonlinear Navier-Stokes, as well as a result for the linear Navier-Stokes case, and for the compressible Euler equation.     

Une méthode de Γ-convergence pour un modèle de membrane non linéaire

We consider a cylindrical three-dimensional nonlinear hvperelastic body whose stored energy goes to infinity when the jacobian of the deformation gradient goes to zero. We show, under appropriate hypotheses on the applied loads, that the three-dimensional energies Γ-converge to the energy of a nonlinear membrane when the thickness of the cylinder goes to zero. The Γ-convergence takes place in the weak topology of a Sobolev space. As a consequence, we show that the sequence of minimizers also converges toward a minimizer of the limit energy.     

Singularités rationnelles du modèle entier d'un affinoïde de la droite projective

We consider an affinoid of the projective line over a complete discrete valuation field. We show that its affine normal model over the ring of integers has only rational singularities.     

Orbite homocline dans un modèle à une infinité de particules. Une approche variationnelle

We study by a variational method, the infinite lattices of particles. We show, under some hyperbolic assumptions, the existence of an orbit homoclinic to the equilibrium.     

Estimation spatio-temporelle d'un modèle de système de particules

Let X be a discrete time contact process (CP) on ${\mathbb{Z}}^2$ as defined by Durrett and Levin (1994). We study the estimation of the model based on space–time evolution of X, that is, $T+1$ successive observations of X on a finite subset S of sites. We consider the maximum marginal pseudo-likelihood (MPL) estimator and show that, when $T\to \infty$, this estimator is consistent and asymptotically normal for a non vanishing supercritical CP. To cite this article: X. Guyon, B. Pumo, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Un modèle du calcul de Dirac dans la théorie relative des ensembles

The aim of this Note is a new justification of the pertinence of the Dirac calculus (see [1]). We give a model of it in a relatively consistent set theory,the Relative Set Theory (see [2]). We have obtained this result by introducing a derivation of step functions.     

Une histoire du théorème de Poincare-Birkhoff-Witt

A story of the so-called Poincaré-Birkhoff-Witt theorem Since the publication of Bourbaki's book on the Lie algebra in 1960, a theorem giving information on the structure of the enveloping algebra of a Lie algebra is called a Poincaré-Birkhoff-Witt theorem. The purpose of this article is to understand the contribution of Poincaré, Birkhoff and Witt to the definition and the study of properties of this enveloping algebra. In the second part of this article, one takes an interest to the generalizations of this theorem during the second half of the twentieth century to eventually result in its extension to the Leibniz algebra.     

Un modèle de Rotenberg avec des vitesses de maturité nulles

The Note [1] is devoted to the study of the Rotenberg's [8] model describing the growth of cell population when the maturation velocity belongs to a, b, with 0 < a < b < ∞. In this Note we bring new techniques for treating the case 0 = a < b < ∞.     

Existence de solutions pour le modèle dérive-diffusion avec terme d'avalanche

The aim of this Note is the study of a system of partial differential equations arising in semiconductors in the presence of the avalanche term and nonconstant mobilities. We prove an existence of variational solution using Schauder's fixed point theorem.     

Une approche polynomiale du théorème de Faltings

We present here an elementary proof of a quantitatively improved version of the Mordell's Conjecture which is now Faltings's Theorem. For the count of the ‘large points’ of $C(K)$ (see below for the notations) we use Vojta's method which was simplified by Bombieri and then by T. de Diego, G. Rémond. To count the points of small heights of $C(K)$, we use a result of S. David and P. Philippon, allowing to us estimate the number of points of small height of the bigger set $C(\overline{K})\cap J(K)$. To cite this article: B. Farhi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

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.     

Une approche intrinsèque d'un modèle non linéaire de la théorie des coques

We consider the model of a nonlinearly elastic “shallow” shell proposed by L.H. Donnell, V.Z. Vlasov, K.M. Mushtari & K.Z. Galimov, and W.T. Koiter. We show that the linearized change of curvature and nonlinear strain tensor fields appearing in the energy of this model can be taken as the sole unknowns of the problem, instead of the displacement field as is customary. In order to justify this “intrinsic approach” to this nonlinear model, we identify nonlinear compatibility conditions that these new unknowns must satisfy. These conditions are of Donati type, in the sense that they take the form of integral orthogonality relations against divergence-free tensor fields.     

Une interprétation du théorème de colonnetti

Summary Considering an elastic-plastic continuum, the plastic deformations of which are given, it is prooved in a general way thatColonnetti's function (4), where ( _x , ..., _xy , ...) is the stress tensor, ( ) the plastic-strain tensor and the elastic potential energy, differs only by a constant value from the fictitious strain energy evaluated from the total strain as if it were purely elastic. Consequently, both functions are simultaneously minima andColonnetti's theorem can be rephrased in terms of fictitious (elastic) strain energy.     

Visibilité dans le modèle Booléen

The aim of this Note is to give an estimate on the tail probability of the visibility function in a Boolean model: this function is defined as the length of the longest ray starting at the origin that does not intersect an obstacle. Convergence results are added, when the size of obstacles goes to zero and when the distance between the origin and the closest obstacle goes to infinity. To cite this article: P. Calka et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Une version feuilletée du théorème de translation de Brouwer

The Brouwers plane translation theorem asserts that for a fixed point free orientation preserving homeomorphism f of the plane, every point belongs to a proper topological imbedding C of R, disjoint from its image and separating $f(C)$ and $f^{-1}(C)$. Such a curve is called a Brouwer line. We prove that we can construct a foliation of the plane by Brouwer lines.      

Un modèle scalaire analogue aux équations de Navier–Stokes

We discuss on a new scalar model whose properties are similar to the NS equations (it preserves scaling, the antisymmetry of the bilinear term and is invariant by translations and dilations) but contains a singular integral operator. We construct a global-in-time weak solution when initial data is in a critical Morrey–Campanato space and show that it also satisfies a local energy inequality comparable to Scheffer?s one.     

Convergence d'un modèle à vitesses discrètes pour l'équation de Boltzmann-BGK

We consider a conservative and entropie discrete-velocity model for the Bathnagar-Gross-Krook (BGK) equation. In this model, the approximation of the Maxwellian is based on a discrete entropy minimization principle. First, we prove a consistency result for this approximation. Then, we demonstrate that the discrete-velocity model possesses a unique solution. Finally, the model is written in a continuous equation form, and we prove the convergence of its solution toward a solution of the BGK equation.