Quantification de la dualité de Grassmann

Let V be a complex vector space of dimension n, %plane1D;53E; (resp. %plane1D;53E;^*) the Grassmann manifold of p-dimensional (resp. (n — p)-dimensional) subspaces of V, and of Ω the relation of transversality in %plane1D;53E;*%plane1D;53E;^*. We announced in [6] equivalences between derived categories of sheaves and of D-modules on %plane1D;53E; and %plane1D;53E; defined by the integral transforms associated to Ω. We show here that these transforms exchange the D-modules associated to the holomorphic line bundles on %plane1D;53E; and %plane1D;53E;^*. This is equivalent to “quantizing” the underlying contact transformation between certain open dense subsets of the cotangent bundles. In the case p = 1, we recover already known results for the projective duality (see [1] and [5]).     

Une nouvelle famille de multiplicateurs et applications à la contrôlabilité exacte de l'équation d'ondes

Two new choices in the multiplier method to study the exact controllability of the wave equation are introduced: a new family of multipliers with rotated direction and multipliers with multiple directions. New geometrical examples with explicit estimates of the observability constant are found.     

Une nouvelle approche pour la stabilisation des systèmes distribués non dissipatifs

We propose a new approach to prove the nonlinear (internal or boundary) stabilization of certain nondissipative distributed evolutionary systems (the usual energy is not decreasing). This approach is direct and very flexible; it leads to decay estimates (known in the dissipative case) when the integral inequalities method due to Haraux (1985) and Komornik (1994) cannot be applied due to the lack of dissipativity.     

Une nouvelle propriété d'approximation diophantienne

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

Remarque sur la régularité locale de l'équation de Kadomtsev-Petviashvili-II

We give a short proof of a part of the Bourgain's result from [3]. More precisely, we prove that the Cauchy problem for the Kadomtsev-Petviashvili equation is locally well posed in Sobolev spaces with positive index.     

Une nouvelle source d'énergie: l'énergie d'instabilité de l'atmosphère

Summary The author summarises his ideas about how to use the latent energy of a humid and unstable air mass, an integral form of solar energy. With a small addition of thermal energy, it should be possible to exert a concerted influence on meteorological phenomena. Artificial convection, either using areas of solar heating or by sudden strong combustion, should permit recovery of part of this latent energy, in the form of rain or in more concentrated forms, of which the limit-type would be ball lightning.     

Du théorème de décomposition à la pureté locale

A new relation is established between the decomposition theorem and a version of the local purity theorem of Deligne and Gabber adapted to complex algebraic varieties. A reduction to the case of a fibred morphism by relative normal crossing divisors on the strata of a stratification is established.     

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.     

Une démonstration simple de la fidélité de la représentation de Lawrence–Krammer–Paris

The theorem of linearity of the Artin–Tits groups of spherical type and the theorem of injectivity of any Artin–Tits monoid in its group are essentially based on the faithfulness of the Lawrence–Krammer–Paris representation restricted to the monoid. We prove this faithfulness using neither the normal forms of the elements of the monoid nor the closed subsets of the associated root system; only very elementary notions are needed.     

Estimation locale linéaire de la densité conditionnelle pour des données fonctionnelles

In this Note, we introduce the local linear estimation of the conditional density of a scalar response variable given a random variable taking values in a semi-metric space. Under some general conditions, we establish the pointwise and uniform almost complete convergences with rates of this estimator. Moreover, as an application, we use the obtained results to derive some asymptotic properties for the local linear estimator of the conditional mode.     

Une variante de la méthode de majoration de Cauchy

Sans résumé Conférence faite au Congrès des mathématiciens scandinaves à Copenhague en ao?t 1964. 10-652932     

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).     

Une q-déformation de la transformation de Bargmann vraie-polyanalytique

We introduce a q-analog of the true-polyanalytic Bargmann transform on $\mathbb{C}$.     

Une propriété de continuité associée aux classes de cohomologie de Hodge

The obstructions, for a closed smooth differential form of bidimension $(p,p)$ on a projective manifold, to be cohomologous to an algebraic cycle with complex coefficients, are calculated in terms of the Chow transformation. They can be expressed as an orthogonality condition, on the manifold itself, with families parametrized by the Grassmannian of currents that are completely determined. Each of these currents is $\mathrm{d}{\mathrm{d}}^{\mathrm{c}}$-closed and with support in the intersection of the manifold and of the projective subspace associated with the parameter. By the theory of harmonic forms, a period is thus associated with that differential form for each parameter. We study the set of periods, obtained when the parameter varies, and we arrive at a continuity on the Grassmannian, when the cohomology class is rational. The same property can be obtained by going to the space of divisors of the Grassmannian and by using a characterization of Chow forms. We proceed here directly, by calculating the periods by means of the Atiyah–Hirzebruch theorem. This global continuity implies the orthogonality for all parameter.     

Caractérisation de la correspondance de Langlands locale par les facteursε de paires

Résumé SoitF un corps local non archimédien à corps résiduel fini. La conjecture de Langlands locale (prouvée par Laumon, Rapoport et Stuhler quandF est de caractéristique non nulle) associe à une classe d'isomorphisme de représentations irréductibles du groupe de Weil deF une classe d'isomorphisme () de représentations admissibles irréductibles supercuspidales d'un groupe linéaire surF, le groupe GL _n (F) si est de dimensionn. Nous prouvons dans cet appendice qu'il existe au plus une telle correspondance qui préserve les facteurs de paires et coïncide, en dimension 1, avec l'application donnée par la théorie du corps de classes.

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Summary LetF be a non-archimedean local field with finite residue field. The local Langlands conjecture (proved by Laumon, Rapoport and Stuhler when the characteristic ofF is positive) associates to an isomorphism class of irreducible representations of the Weil group ofF an isomorphism class () of admissible irreducible supercuspidal representations of some linear group overF, viz. GL _n (F) if dim ()=n. In this appendix we prove that there is at most one such correspondence which extends local class field theory in dimension 1, and preserves -factors for pairs.

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Oblatum 10 XII 1991

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Un complément à l'article -elliptic sheaves and the Langlands correspondence par G. Laumon, M. Rapoport et U. Stuhler.     

Dérivabilité globale et locale des fonctionnelles de Wiener

In this Note we give sufficient conditions which imply the global Sobolev differentiability of locally differentiable Wiener functionals.