Connexions métriquesk-plates

Sans résumé

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This article is the text of a talk given at the Symposium on Differential Geometry in Debrecen, Hungary, on August 28–September 3, 1975.     

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.     

Régularisation spectrale et propriétés métriques des moyennes mobiles

We develop a spectral regularization technique for moving averages , where ϕ is a nondecreasing map andU: H→H is a contraction of a Hilbert space (H, ‖·‖). We obtain a spectral regularization inequality which allows one to evaluate efficiently the increments ‖B _m ^U , ϕ (f)−B _n ^U , ϕ (f)‖,f∈H, by means of where is a properly regularized version of the spectral measure off with respect toU. We apply this inequality to an investigation of metric properties of the sets of moving averages {B _n ^{U, ϕ} (f), n ∈N} with fixedf∈H andN⊂ N. In particular, we obtain estimates of the associated covering numbers as well as of the related Littlewood-Paley-type square functions. This work extends our previous results concerning the case of classical averages (ϕ(n)=0). Since it is well-known that the structure of general moving averages is more complicated, there is no surprise that the general results we obtain are sometimes less complete than their classical counterparts and need suitable moment assumptions on the spectral measure (depending on the growth of the shift function ϕ). Nevertheless, when applied to the classical situation, our estimates still yield optimal bounds.

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Avec pour le premier author, le soutien de la fondation russe pour la recherche fondamentale, subvention 99-01-00112 et INTAS subvention 99-01317.     

Une méthode numérique décentrée pour la résolution d'écoulements diphasiques

This Note focuses on the numerical approximation of two-fluid flow models described by six balance equations. We introduce an original splitting technique especially derived to use the approximate Riemann solvers of the usual gas dynamics and to allow for a straightforward extension to various and detailed exchange source terms. When based on suitable kinetic upwind schemes, the whole scheme preserves the positivity of all the thermodynamic variables under a fairly unrestrictive “CFL like” condition. Several stiff numerical teats, are presented including phase separation, in order to highlight the efficiency of the proposed method.     

Transformations projectives des variétés munies d'une connexion métrique

Résumé Ce travail est consacré à l'étude du groupe P₀(M, ) des transformations projectives d'une variété munie d'une connexion métrique , et du sous-groupe P ₀ ^r (M, ) (resp. P₀(M, )) constitué par celles de ces transformations qui conservent le tenseur de torsion (resp. le co-vecteur trace de torsion). Dans le cas compact on détermine des conditions pour que ces groupes coïncident avec le groupe des transformations affines ou le groupe des isométries. Ces conditions portent sur la courbure de Ricci de la connexion ou de la connexion symétrique associée, ou plus généralement sur la signature d'une forme quadratique dépendant de la courbure scalaire et du tenseur de torsion. On étudie, plus particulièrement, le cas où M est de dimension n, compacte, simplement connexe, munie d'une connexion métrique complète telle que la partie symétrique de la courbure de Ricci de la connexion symétrique associée soit du type d'Einstein: R_(ij)=Cg_ij, où C est une constante positive. Si M admet un champ de vecteurs projectifs propres pour la connexion métrique, laissant le co-vecteur de torsion invariant, alors M est homéomorphe à une n-sphère. On aborde enfin le cas où C est une constante non positive.

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Summary This paper is devoted to the study of the group P₀(M, ) of projective transformations of a manifold endowed with a metric connection and of the subgroup P ₀ ^r (M, ) (resp. P₀(M, )) made of the transformations conserving the torsion tensor (resp. the covector trace of torsion). In the compact case we determine the conditions so that these groups coincide with the group of affine transformations or with the isometry group. More particularly if M is simply connected endowed with a complete metric connection such that the symmetric part of the Ricci curvature of the associated symmetric connection is of Einstein type R_(ij)=Cg_ij, when C is a positive constant and if M admits a projective group leaving invariant the trace of torsion, then M is homeomorphic with a sphere. We examine the case when C is nonpositive constant.

     

Une méthode des sur et sous solutions améliorée

The usual upper and lower solutions method for solving a second order nonlinear elliptic equation is iterative, with the drawback of a tricky, if not sometimes impossible, derivation of the (quite essential) uniform $C^2$ estimate, due to the occurence in the same equation of two successive iteration unknowns. We present here a method free of such a drawback, based on an elementary fixed point argument. To cite this article: P. Delanoë, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Remarques sur un mémoire d'Hermite

Sans résumé     

Désingularisation de métriques d’Einstein. I

We find a new obstruction for a real Einstein 4-orbifold with an A ₁-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point. For asymptotically hyperbolic metrics, with boundary at infinity a conformal metric, we prove that if the obstruction vanishes, one can desingularize Einstein orbifolds with such singularities. The Dirichlet problem consists in finding Einstein metrics with given conformal infinity on the boundary: we prove that our obstruction defines a wall in the space of conformal metrics on the boundary, and that all the Einstein metrics must have their conformal infinity on one side of the wall.     

Rédei-Permutationen auf RestklassenringenZ/(m)

In this paper, Rédeifunctions over residue class ringsZ/(m) are studied. In contrast to earlier investigations, now the modulusm is an arbitrary natural number >1, and the Rédei-functions are defined in dependence of a quadratic polynomial of the general formt(x)=x ² –x–, t (x) irreducible modulo all prime divisorsp _i ofm. A necessary and sufficient condition is formulated that a given Rédei-function induces a permutation ofZ/(m). Given a polynomialt(x), the group structure of the corresponding Rédei-permutations is determined.

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Die vorliegende Arbeit wurde vom österreichischen Fonds zur Förderung der wissenschaftlichen Forschung unter dem FWF-Projekt Nr. P6174P wesentlich unterstützt.     

Contribution à la méthode des équations aux différences

Summary Let a linear boundary value problem or an eigenvalue problem be given, in one or more dimensions. I look for such a construction of a corresponding difference equation that, in the case of ordinary differential equations with constant coefficients, its solutions beexactly those of the continuous problem.—This trivial case is first investigated and several methods are pointed out; these are then applied to: (a) ordinary differential equations with variable coefficients (method of local perturbations); (b) vibration problems for membranes and clamped plates.     

Le flux d'énergie en mécanique: Application aux turbo-machines

Summary The concept of flux of energy, first introduced in electrodynamics byJ. H. Poynting, can be extended to the domain of fluid-mechanics. It conduces to recognize in the flux of energy for a continuous portion of matter two parts: the first one, which can be called ‘transported’, corresponds to the energy localized in the matter and carried with it as a consequence of its displacement; the other results from the work developed by the stresses and can be called ‘transmitted’. The expression of this flux of energy in the general case of variable motion of a compressible fluid introduces the concept of enthalpy. Its application to an extension ofHelmholtz's theorem on the conservation of vorticity in a perfect fluid, according which bound vortexes have a definite relative motion across the fluid, contributes to explain the mechanism by which, even in the case of grids with a large pitch: chord ratio (Kaplan turbines, screw propellers), the energy of all parts of fluid is finally transmitted to the moving blades.      

Une méthode particulaire déterministe pour des équations diffusives non linéaires

We study in this Note a deterministic particle method for heat (or Fokker–Planck) equations or for porous media equations. This method is based upon an approximation of these equations by nonlinear transport equations and we prove the convergence of that approximation. Finally, we present some numerical experiments for the heat equation and for an example of porous media equations.     

L'influence de la géométrie de Bolyai-Lobatchevsky sur le développement de la méthode axiomatique

Sans résumé     

Doubles mélanges des polylogarithmes multiples aux racines de l’unité

Abstract. – The values at positive integers of the polyzeta functions are solutions of the polynomial equations arising from Drinfeld’s associators, which have numerous applications in quantum algebra. As iterated integrals they are periods of the motivic fundamental groupoid of P¹?{0,1,∞}. From this arises a fundamental, yet no more explicit, system of algebraic relations; it implies the system of associators.¶We focus here on the combinatorial properties of another system of relations, the “double shuffles”, which expresses some elementary series and integrals manipulations. We show that it shares an important property with associators and “motivic” relations, is implied by the latter and defines a polynomial algebra over Q (Écalle’s theorem). We obtain these results for more general numbers: values of Goncharov’s multiple polylogarithms at roots of unity.     

Application d'une nouvelle méthode de sommation aux séries trigonométriques et de Dirichlet

Sans résuméPrésenté parP. Turán En commémoration du professeurLéopold Fejér décédée le 15 octobre 1959     

Dimension de la mesure d’équilibre d’applications méromorphes

The Journal of Geometric Analysis - Let ? be a dominating meromorphic self-map of a compact Kähler manifold. Assume that the topological degree of ? is larger than the other...