Sur Une Classe D’Espaces D’Interpolation

Sans résumé Les A. ont été soutenus par l’Interpol.     

Une remarque sur l’inégalité de McKean

Résumé Pour une classe de variétés riemanniennes équipées de métriques tordues, nous donnons dans ce travail une borne inférieure optimale pour la première valeur propre du laplacien dans le problème de Dirichlet. Linégalité présentée, qui est obtenue très simplement, étend celle de McKean à des variétés dont la courbure peut être de signe variable. Cest le principal intérêt de linégalité. Elle répond aussi partiellement à une question de Schoen et Yau.      

Dégénérescence de séries d’Eisenstein hyperboliques

In this work, we deal with hyperbolic Eisenstein series and more particularly, given a degenerating family S _l (l ≥ 0) of Riemann Surfaces with their canonical hyperbolic metrics, we work out the degeneration of hyperbolic Eisenstein series associated to the pinching geodesics in S _l . Our principal results are theorems 2.2, 4.1 et 4.2.     

Une interprétation byzantine de Diophante

In this paper, we develop a new interpretation, based on the notion of proportion, of Diophantus's procedure, which is applicable to all cases of equations with two unknowns that Diophantus (ca.A.D. 250) solves. This interpretation was proposed for the first time by Maximus Planudes (ca.A.D. 1255—1305) in his commentary on problem II 8 of theArithmetica.Copyright 1998 Academic Press.Dans cet article on propose une nouvelle interprétation, basée sur la notion de la proportion, de la méthode utilisée par Diophante à la résolution des équations à deux inconnues. L'interprétation par proportion avait été proposée pour la première fois par Maxime Planude dans son commentaire au problème II 8 desArithmétiques.Copyright 1998 Academic Press.Στην ϵργασια προτϵινϵται μια νϵα ϵρμηνϵια, μϵ χρηση αναλογιων, της μϵθοδου πουχρησιμοποιϵι ο Διοϕαντος για να ϵπιλυσϵι τις ϵξισωσϵις μϵ δυο αγνωστους. H ϵρμηνϵια αυτη ϵιχϵ προταθϵι για πρωτη ϕορα απο τον Mαξιμο Πλανουδη στο σχολιο του στο προβλημα B 8 τωνAριθμητικων.Copyright 1998 Academic Press.AMS 1991 subject classification: 01 A 20     

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.     

Une réciproque générique du théorème Sunada

Let G a compact group of isometries of a closed riemannian manifold(X,m). Sunada proved that if are twofinite almost-conjugated subgroups of G, then and are isospectral. We prove that if G is finite, there exists an open dense set in the set of G-invariant metrics for which the converse ofthis resukt is true. If G is infinite, the situations is more complicated and we obtain some partial results.     

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).     

Une nouvelle propriété d'approximation diophantienne

We prove a new diophantine approximation property which improve a recent result of M. Laurent and D. Roy.     

Une série génératrice des formes paraboliques de Maass

A certain periodic function F_μ,ν(z), an eigenfunction of the Laplacian on the upper half-plane with respect to z depending on some parameters μ, ν, is not automorphic, but the function μ → F_μ,ν(z)−F_μ,ν(−1/0 extends to a larger domain than the function F_μ,ν(z) itself. Consequently, at the poles of this latter function of μ, the coefficients of the polar parts provide non-analytic modular forms: all Maass cusp forms are finite linear combinations of forms obtained in this way, allowing the second parameter ν to vary     

Une généralisation du théorème de Kobayashi-Ochiai

We establish relations between the growth of non-degenerate holomorphic maps \(\varphi\) from \({\mathbb{C}}^n\) to X, an n-dimensional compact Kähler manifold, and the positivity of the canonical bundle of X. The general principle is that the growth increases with this positivity. In the extreme case where X is of general type, such maps do not exist, a result of Kobayashi-Ochiai. In the other extreme, if the growth is sufficiently slow (see theorem 1), we show that X is uniruled if projective. K. Kodaira obtained the weaker property that X has no nonzero pluricanonical forms. Our results interpolate between, and include, these two extreme cases.     

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.     

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.     

Une caractérisation Riemannienne du groupe de Heisenberg

We study the left-invariant Riemannian metrics on a class of models of nilpotent Lie groups. In particular we prove that the Heisenberg groups are, up to local isomorphism, the only nilpotent non-decomposable Lie groups endowed with a homogeneous Riemannian naturally reductive space for every left invariant metric.

Membre du G.N.S.A.G.A., G.N.R. d'Italie et du groupe national Geometria delle varietà differenziabili, 40%, M.P.I. Italie.     

Une estimation de type Aronson–Bénilan

We consider the equation u_t=Δϕ(u), where ϕ∈C³(0,∞) is increasing. Under the condition ν·sϕ″(s)/ϕ′(s)⩾γ for some γ>0 and ν∈{−1;1}, we prove the estimate ν·du/dt⩾−u/γt. This result improves the estimates given by M.G. Crandall and M. Pierre (in J. Funct. Anal. 45 (1982) 194–212) for this equation. To cite this article: E. Chasseigne, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Une réponse négative au problème de Noether différentiel

According to a theorem first stated by Clifford, Noether [8] and Rosanes [12], which later received a complete proof by Castelnuovo [2], the Cremona group of the plane is generated by de Jonquières transformations. J.F. Ritt [10] asked whether such a result could be generalized to the differential plane. We give a negative answer, relying on a flat system of Rouchon.     

Une caractérisation des variétés complexes compactes parallélisables admettant des structures affines

We classify complex compact parallelizable manifolds which admit flat torsion free holomorphic affine connections. We exhibit complex compact manifolds admitting holomorphic affine connections, but no flat torsion free holomorphic affine connections. To cite this article: S. Dumitrescu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).