Polynômes hyperboliques et combinaisons linéaires de certains polynômes hyperboliques

We give some results in the theory of hyperbolic polynomials and we study the hyperbolicity of some linear combinations of hyperbolic polynomials.Note présentée par Philippe G. Ciarlet.     

Sous-Solutions de Problèmes aux Limites en Dimension 1

We consider a boundary value problem where is a maximal monotone operator in and E is a multivalued operator in ² with non decre asing resolvent. We introduce a condition on E which insures that the operator inL ¹ (0,1) associated to this problem has non decreasing resolvent, and caracterise the subsolutions of the problem. We give different examples of operatorsE satisfying this condition.     

Une classe d'algèbres de Fréchet à caractères continus

We prove in a simpler and direct manner using the spectral theory that every Fréchet algebra which is rationnally generated by a real or orthogonal subset is functionally continuous. These results include as a special case the results of R. Arens, H. Goldmann, E. A. Michael and W. Zelazko.

     

Un D-groupoïde de Galois local pour les systèmes aux q-différences fuchsiens

We compute the Galois D-groupoid of a constant linear q-difference system and we define, giving its realisations, a local Galois D-groupoid for fuchsian q-difference systems.     

Un théorème de Torelli infinitésimal à coefficients

We generalize an infinitesimal Torelli theorem for hypersurfaces of high degree of M. Green (1985) [1], for twisted coefficients.     

Solutions à variations bornées pour certains systèmes hyperboliques de lois de conservation

Blake Temple (Trans. Amer. Math. Soc.280 (1983), 781–795) has described the hyperbolic systems of two conservation laws whose shock and rarefaction curves coincide. In this note, we prove the global existence of weak solutions for such systems, with any bounded variation initial condition. The proof is based upon standard numerical schemes, as well as upon parabolic regularization. The key is that the total variation of the Riemann invariants is decreasing in time. At least, in the case of the initial condition with compact support, we prove by using the Glimm scheme that the system is decoupling in two conservation laws in one unknown, in finite time.     

Valeurs propres relatives à un cône convexe : caractérisation et résultats de cardinalité

Let A be an n×n real matrix, and $\text{K?}\text{R}{}^{\mathrm{n}}$ be a closed convex cone. The spectrum of A relative to K, denoted by σ(A,K), is the set of all $\text{λ∈}\text{R}$ for which the linear complementarity problem

$\text{x∈K,}\text{Ax?λx∈K}{}^{\mathrm{+}}\text{,}\text{〈x,Ax?λx〉=0}$

admits a nonzero solution $\text{x∈}\text{R}{}^{\mathrm{n}}$. The aim of this Note is to study the main properties of the set-valued mapping $\text{σ(·}\text{,K)}$, and discuss some structural differences existing between the polyhedral case (i.e., K is finitely generated) and the non-polyhedral case. To cite this article: A. Seeger, M. Torki, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Applications des EDSR à horizon infini à un problème de contrôle impulsionnel

We study an impulse control problem in infinite horizon. To solve this problem, we extend to the infinite horizon case results of double barrier reflected backward stochastic differential equations. The properties of the Snell envelope reduce our problem to the existence of a pair of continuous processes.     

Sur la distance à l'instabilité de polynômes matriciels quadratiques

A bisection method is developed for computing the distance to instability of quadratic matrix polynomials. The computation takes rounding errors into account.     

Problèmes elliptiques à frontière libre axi-symétriques: Estimation du diamètre de la section au moyen de la capacité

Résumé Le but de cet article est d'estimer la capacité d'un pétit segment [a(1–), a(1+)]×{z=0} du démi-plan := {x = (r, z)|r > 0, z } de ² par rapport à la capacité définie par la norme . Ce résultat est ensuite utilisé pour estimer le diamètre de la surface libre de certains problèmes elliptiques non linéaires.     

Sur un problème de potentiel avec conditions aux limites mixtes: une conjecture de A. Acker et sa démonstration à l'aide de problèmes auxiliaries à une dimension

Résumé On étudie quelques problèmes relatifs aux fonctions harmoniques satisfaisant des conditions aux limites de type radiation, par des méthodes de variation. On démontre notamment une conjecture de A. Acker [1, 2] à l'aide de problèmes auxiliaires à une dimension.

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Summary Some problems for harmonic functions with radiation-type boundary conditions are studied, and a conjecture of A. Acker [1, 2] is proved by using one-dimensional auxiliary problems.

     

Configurations combinatoires des polynômes fibrés de degré 2

In this Note, we are interested in quadratic fibered polynomials which are skew products of the kind P_c : X × C → X × C, P_c(x, z) = (ƒ(x),z² + c(x)), where X is a compact set, ƒ is a continuous map from X into X, and c is a continuous complex valued function on X considered as a parameter. We construct a compact connected configuration space which gives a combinatorial model of a subset of the parameters space. Then, we explain in which way an abstract configuration can be realized by a quadratic fibered polynomial.     

Les SurfacesW (Weingarten) à Lignes de Courbure planes dans les deux Systèmes

Sans résumé     

Théorèmes du type Ingham et application à la théorie du contrôle

Ingham [4] improved a previous result of Wiener [10] on nonharmonic Fourier series. Modifying his weight function we obtain optimal results improving several earlier theorems of Kahane [7], Castro and Zuazua [2] and of Jaffard, Tucsnak, and Zuazua [5]. Then we apply these results to simultaneous observability problems.     

Un algorithme de Schur pour les fonctions de transfert des systèmes surdéterminés invariants dans une direction

We define an analogue of the Schur algorithm for transfer functions of lossless 2D systems which are invariant with respect to one of the variables. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Transmission de la chaleur à un fluide visqueux se mouvant entre deux sphères concentriques tournantes

Summary The problem investigated is the temperature distribution in an incompressible viscous fluid moving between two concentric rotating spheres whose temperatures are maintained constant, i.e. the inner sphere temperature isT _a and that of the outer sphereT _b .