Viscoélastodynamique monodimensionnelle avec conditions de SignoriniOne-dimensional viscoelastodynamics with Signorini boundary conditions

Let α be a positive number. The one-dimensional viscoelastic problem

$\text{u}{}_{\mathrm{tt}}\text{?u}{}_{\mathrm{xx}}\text{?αu}{}_{\mathrm{xxt}}\text{=f,}\text{x∈(?∞,0],}\text{t∈[0,+∞),}$

with unilateral boundary conditions

$\text{u(0,·)?0,}\text{(u}{}_{\mathrm{x}}\text{+αu}{}_{\mathrm{xt}}\text{)(0,·)?0,}\text{(u(u}{}_{\mathrm{x}}\text{+αu}{}_{\mathrm{xt}}\text{))(0,·)=0,}$

can be reduced to the following variational inequality:

$\text{λ}{}_1\text{1w=g+b,}\text{w?0,}\text{b?0,}\text{〈w,b〉=0.}$

Here $\text{λ}\text{?}{}_1\text{(ω)}$ is the causal determination of $\text{i}\text{ω}\text{1+}\text{i}\text{αω}$. We show that the energy losses are purely viscous; this result is a consequence of the relation $\text{〈}\text{w}\text{?}\text{,b〉=0}$; since a priori, b is a measure and $\text{w}\text{?}$ is defined only almost everywhere, this relation is not trivial. To cite this article: A. Petrov, M. Schatzman, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 983–988.     

Structures réticulées renforcées en élasticité

The asymptotic study of a plate periodically perforated by square holes leads to a loss of “ellipticity” of the problem. We show how to reinforce this structure in order to regain the ellipticity. We then present a general method to study the reticulated structures with oblique bars not necessarily crossing the structure completely.     

Méthodes quantiques en topologie algébrique

Using quantum methods, we introduce here the notion of “neo-algebra” which generalizes the notion of a commutative differential graded algebra. Under some mild finiteness conditions, we can associate functorially to a space a neo-algebra over the finite field F_p: its quasi-isomorphism's class determines the p-adic-homotopy type of X. As a matter of fact, from this data, we can describe in a simple way Steenrod operations in the cohomology of X, as well as the p primary part of its homotopy groups. This point of view extends to finite characteristics the well-known rational homotopy theory of D. Quillen [9] and D. Sullivan [11]. It is deeply related to previous works of P. May, I. Kriz [5] and M.A. Mandell [7], [8].     

Schémas en groupe et modules filtrés

Let k be a perfect field of characteristic p > 0. When p ≥ 3, Fontaine has classified p-divisible groups and finite flat commutative p-groups over the Witt vectors W(k) in terms of filtered modules ([9]+[6],[7]). Still assuming p ≥ 3, we extend these classifications over an arbitrary complete discrete valuation ring of unequal characteristic and residue field k by using “generalized” filtered modules. In particular, there is no restriction on the ramification index.     

Les résonateurs supra-conducteurs en onde métrique

Summary It is possible to construct a superconducting resonator with a high quality coefficient (up to 10⁷) and very good stability in a frequency band from 250 to 300 Mc/s. For the construction of such a resonator a number of conditions must be fulfilled, but nevertheless the construction presents no great difficulties.     

La dérivée schwarzienne en dynamique unimodale

The first return map of a C³ unimodal map to a neighbourhood of the critical point is conjugated by a real-analytic diffeomorphism to a map with negative Schwarzian derivative. Consequently, we obtain the classification of the metric attractors for smooth unimodal maps with non-degenerate critical point.     

Décomposition d"une théorie du potentiel en des sous-espaces elliptiques et paraboliques

Let V be a proper kernel on a measurable space (X, B), and E _V the cone of excessive functions generated by V. We give a necessary and sufficient condition to decompose the space (X, E _V ) in an ordered and countable family of subspaces (X _n, E _{V n} ), which are elliptic or parabolic. The X _i 's are finely open and measurable and form a partition of X. The kernel W = V _i on X _i is subordinate to V, and has a triangular matrix. Résumé. Soit V un noyau propre sur un espace mesurable (X, B), et E _V le cône des fonctions excessives engendré par V. Nous donnons une condition nécéssaire et suffisante pour décomposer l"espace (X, E _V ) en une famille dénombrable et ordonnée (X _n , E _{V n} ) de sous-espaces elliptiques ou paraboliques. Les X _i sont des ouverts fins mesurables et forment une partition de X. Le noyau W = V _i sur X _i est subordonné à V, et sa matrice est triangulaire.     

Résolutions de singularités non-abéliennes en dimension trois

Let E be an elliptic curve and R be a real three dimensional root system. Let W be the Weyl group associated to R. Denote $W_{+}=W\cap {\mathrm{SL}}_3(\mathbb{C})$. The quotient $E?Q(\mathrm{R})/W_{+}$ admits two natural crepant resolutions. One is the result of a Jung process of desingularization of singularities, the other the equivariant Hilbert scheme. To compare these resolutions, we calculate the fibres over any singular point in both cases. We exhibit a McKay correspondence phenomenon for the Hilbert scheme resolution and construct a new family of vector bundles on E parametrized by the $W_{+}$-Hilbert scheme. To cite this article: S. Térouanne, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Opérateurs de projection en théorie des primitives aréolaires bornées

Résumé On donne des propriétés générales des primitives aréolaires bornées et l'on met en évidence dans leur représentation intégralé deux opérateurs de projection: C de Cauchy et T de Pompeiu. à. M. Enrico Bompiani pour son Jubilé scientifique.     

Sur les critères d'équivalence en géométrie algébrique

Sans résumé     

Dualité de Spanier–Whitehead en géométrie algébrique

In this Note, we study finitely presented objects in the stable ${\mathbb{A}}^1$-homotopy category introduced by Morel and Voevodsky and we use a theorem by Voevodsky and Ayoub on the four functors to get Spanier–Whitehead duality in ${\mathbb{A}}^1$-homotopy theory. To cite this article: J. Riou, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Inégalité d’Ahlfors en dimension supérieure

We prove an Ahlfors’ like inequality for the holomorphic curves with boundary of a complex compact Kobayashi-hyperbolic manifold.     

Propriétés asymptotiques des processus de branchement en environnement aléatoire

Let (Z_n) be a branching process in a random environment. If the process in sub-critical or critical, we study the convergence rate of the survival probability P(Z_n>0) as n→∞; if the process is supercritical, we give a necessary and sufficient condition for the convergence in L^p of the natural martingale, where p>1 is given.     

Exposants de Lyapounov pour opérateurs de Schrödinger discrètes quasi-périodiquesLyapounov exponents for discrete,quasi-periodic Schrödinger operators

We consider quasi-periodic Schrödinger operators H on $\text{Z}$ of the form H=H_λ,x,ω=λv(x+nω)δ_n,n′+Δ where v is a non-constant real analytic function on the d-torus $\text{T}{}^{\mathrm{d}}\text{(d?1)}$ and Δ denotes the discrete lattice Laplacian on $\text{Z}$. Denote by L_ω(E) the Lyapounov exponent, considered as function of the energy E and the rotation vector $\text{ω∈}\text{T}{}^{\mathrm{d}}$. It is shown that for |λ|>λ₀(v), there is the uniform minoration $\text{L}{}_{\mathrm{\omega }}\text{(E)>}\text{1}\text{2}\text{log}\text{|λ|}$ for all E and ω. For all λ and ω, L_ω(E) is a continuous function of E. Moreover, L_ω(E) is jointly continuous in (ω,E), at any point $\text{(ω}{}_0\text{,E}{}_0\text{)∈}\text{T}{}^{\mathrm{d}}\text{×}\text{R}$ such that k·ω₀≠0 for all $\text{k∈}\text{Z}{}^{\mathrm{d}}\text{?{0}}$. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 529–531.     

Bandes phononiques interdites en élasticité linéarisée

In this Note we rigorously justify the existence of elastic band gaps in three-dimensional periodic composite materials with strong heterogeneities. In particular, we show how to compute these bands. To cite this article: A. Ávila et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Résolution d'EDP par un schéma en temps «pararéel »

The purpose of this Note is to propose a time discretization of a partial differential evolution equation that allows for parallel implementations. The method, based on an Euler scheme, combines coarse resolutions and independent fine resolutions in time in the same spirit as standard spacial approximations. The resulting parallel implementation is done in the non standard time direction. Its main goal concerns real time problems, hence the proposed terminology of “parareal” algorithm.     

Sur quelques résultats de convergence dans l'inférence d'une classe de processus ponctuelsConvergence results for point processes estimators

Our aim is to generalize some results obtained for a Poisson point process in [7], to a general point process. Those results are in field of complete convergence of two like Parzen–Rosenblatt estimates of density of mean measure function and regression curves. Those estimates are defined from the superposition of n i.i.d. point processes as:

$\text{f}{}_{\mathrm{n}}\text{(x)=}\text{1}\text{nh}\text{∑}\text{i=1}\text{m}\text{K}\text{x?X}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{h(n)}\text{and}\text{Ψ}{}_{\mathrm{n}}\text{(x)=}\text{∑}{}_{\mathrm{i=1}}{}^{\mathrm{m}}\text{Y}{}_{\mathrm{i}}\text{K}\text{x?X}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{h(n)}\text{∑}{}_{\mathrm{i=1}}{}^{\mathrm{m}}\text{K}\text{x?X}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{h(n)}\text{,}$

where m is the number of seem generics points of the superposition. We give some sufficient conditions for the convergence of those kernel-like estimators. To cite this article: A. Diakhaby, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 597–602.     

Noethérianité et privilège en géométrique analytique p-adique

Let k be a non-Archimedean field, X a k-affinoid space and f an analytic function over X. We describe precisely how the geometric connected components of the spaces $\{x\in X||f(x)|?\epsilon \}$ behave with regards to ε. We also obtain a result concerning privileged neighbourhoods and adapt a theorem from complex analytic geometry about Noetherianity for germs of analytic functions. To cite this article: J. Poineau, C. R. Acad. Sci. Paris, Ser. I 343 (2006).