Showing papers in "Bulletin Des Sciences Mathematiques in 2001"

Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one

Abstract: We show some integral representations of the heat kernels and explicit expressions of the Green functions for the Laplace–Beltrami operators on three series of hyperbolic spaces.

Open problems in algebraic geometry

Abstract: The open problems presented here were collected on the occasion of a workshop on Arithmetic Geometry at the University ofUtrecht, 26{30 June, 2000. This workshop was organized by the editors of the present article, and was made possible by support of: | NWO, the Netherlands Organization for Scientic Research, | KNAW, the Royal Netherlands Academy of Arts and Sciences, | MRI, the Mathematical Research Institute, | the Faculty of Mathematics and Computer Science of the University of Utrecht, and | the Department of Mathematics of the University of Utrecht. We thank these organizations heartily for their support. All problems in this list have been reviewed by at least one referee. In this process many useful suggestions and new references have come up. We thank all referees for their valuable comments.

Malliavin calculus and martingale expansion

Abstract: We proved the validity of the asymptotic expansion for the distribution of a martingale with jumps. A sufficient condition is presented in terms of the decay of certain integrations of Fourier type. In order to estimate such Fourier type integrals, we use the Malliavin calculus and show that it becomes a key to our program.

Perturbations of the non-generic quadratic Hamiltonian vector fields with hyperbolic segment

Abstract: The paper deals with the non-generic quadratic Hamiltonian vector fields with hyperbolic segment. It is proved that in this situation the cyclicity of period annulus under quadratic perturbation is equal to two.

Planar analytic vector fields with generalized rational first integrals

Abstract: The main purpose of this paper is to characterize a germ of planar holomorphic vector field at an elementary singular point having a generalized rational first integral. Our results generalize a result due to Poincare on a necessary condition of the existence of a rational first integral for planar polynomial systems. As two applications of our main result, we give the necessary and sufficient conditions on the existence of rational first integral for planar quadratic systems having either a weak nondegenerate singular point, or a degenerate elementary singular point.

Toda systems and exponents of simple lie groups

Abstract: Results on the finite nonperiodic An Toda lattice are extended to the Bogoyavlesky Toda systems of typeBn and Cn. The areas investigated, include master symmetries, recursion operators, higher Poisson brackets and invariants. The results are presented both in Flaschka coordinates .a;b/ as well as in the natural .q;p/ coordinates. A conjecture which relates the degrees of higher Poisson brackets and the exponents of the corresponding Lie group is verified for these systems. © 2001 Editions scientifiques et medicales Elsevier SAS RESUME. - On etend des resultats sur le systeme TodaAn, fini et non-periodique, aux systemes Toda de Bogoyavlesky de typesBn et Cn. Les sujets traites comprennent les master symmetries, les operateurs de recursion, les crochets de Poisson et les invariants. Les resultats sont presentes en coordonnees .a;b/ de Flaschka et aussi en coordonnees naturelles .q;p/. Une conjecture qui met en rapport les degres des crochets de Poisson avec les exponents des groupes de Lie qui leur correspondent est verifiee pour ces systemes. © 2001 Editions scientifiques et medicales Elsevier SAS Mathematics Subject Classification: 34A05, 58F05, 58F07

Intersection of minimal surfaces of bounded curvature

Abstract: Resume Soient M1 et M2 deux surfaces minimales completes dans l'espace Euclidien R 3 de dimension trois, ou dans une variete N de dimension trois, de courbure de Ricci bornee inferieurement par un nombre strictement positif. Nous montrons que M1 rencontre M2 dans N si leurs courbures de Gauss sont bornees. Si elles sont dans R 3 et que leurs courbures de Gauss sont bornees, nous montrons qu'elles se rencontrent ou sont des plans paralleles. Nous montrons egalement qu'une surface minimale complete dans N, de courbure bornee, et injectivement immergee, est compacte.

Re-filtering and exactness of the gelfand-kirillov dimension

Abstract: We prove that any multi-filtered algebra with semi-commutative associated graded algebra can be endowed with a locally finite filtration keeping up the semi-commutativity of the associated graded algebra. As consequences, we obtain that Gelfand–Kirillov dimension is exact for finitely generated modules and that the algebra is finitely partitive. Our methods apply to algebras of current interest like the quantized enveloping algebras, iterated differential operators algebras, quantum matrices or quantum Weyl algebras.

Quadratic Wiener functionals and dynamics on Grassmannians

Abstract: We introduce a new framework to study the structure of the probabilistic laws of quadratic Wiener functionals in terms of dynamics on Grassmannians. The key point is to identify the Jacobi equation derived by an analogue of Van Vleck formula with the space of its solutions according to M. Sato's fundamental idea.

Positive solutions of Ambrosetti–Prodi problems involving the critical Sobolev exponent

Abstract: Resume Sur une variete Riemannienue compacte ( M , g ) de dimension n ⩾3, avec ou sans bord, nous etudions le probleme d'Ambrosetti–Prodi avec exposant critique de Sobolev et r(x)⩾0 (r≢0) . Il s'agit de resoudre Eq. (0,1). Nous supposous K >0 et Δ + a coercif. Il existe λ ∗ >0 tel que Eq. (0,1) admet une solution (minimale) si et seulement si 0 ∗ . Pour t=λ ∗ cette solution est unique, et il existe au moins deux solutions pour 0 ∗ . Certaines hypotheses supplementaires sont requises suivant les valeurs de n . Le cas ou Δ + a est le laplacien conforme est etudie.

A Chevalley–Kostant presentation of basic modules for sl(2)̂ and the associated affine KPRV determinants at q=1☆

Abstract: Let g be a semisimple Lie algebra with triangular decomposition g = n − ⊕ h ⊕ n + and ρ the half-sum of the roots of n + . It is a remarkable fact that up to a shift by ρ the weights of the exterior algebra of n − are exactly those of the simple highest weight module V ( ρ ) with highest weight ρ . Moreover a construction of Chevalley and Kostant realizes V ( ρ ) in the Clifford algebra over n − . With a little care this construction can be generalized to the central extensions of the corresponding loop algebra and in particular for sl (2) . In this paper a Chevalley–Kostant construction is given for the basic modules of sl (2) . This has no analogue for semisimple g and indeed is dependent on the peculiarities of bounded modules. It is used to construct a certain endomorphism of a basic module and to compute its trace on weight subspaces, which relates to the corresponding KPRV determinant at q =1.

Quasi-invariant measures and their characterization by conditional probabilities

Abstract: Based on quasi-invariance properties of the Gaussian process and the gamma process, we give certain formal expressions of their laws on infinite-dimensional spaces They are also consistent with some identities of conditional probabilities which are shown to be equivalent to the quasi-invariance properties

Formes normales et problème du centre

Abstract: Resume Soit X=(x ∂ ∂y −y ∂ ∂x )+f(x,y) ∂ ∂x +g(x,y) ∂ ∂y un champ de vecteurs polynomial de degre n du plan R 2 . En utilisant la theorie des formes prenormales de J. Ecalle, on caracterise les perturbations donnant naissance a un centre. On decrit ensuite la structure des monomes resonnants, en utilisant une caracterisation geometrique (l'invariance sous une action de C ∗ ) de ces monomes. On obtient un theoreme de decomposition qui permet d'ecrire chacun de ces monomes comme produit de monomes universels. On resout alors le probleme de lecture dans le cas des conditions algebriques du centre reversible, pour une perturbation polynomiale de degre quelconque.

Relevements continus de champs de vecteurs

Abstract: We improve the known theorems on the continuous lifting of vector fields def ined on as tratumX of a regular stratification of a closed set of a smooth manifold by introducing the notion of canonical distributionDX: a continuous l-bundle near X (l = dim X) such that the lifting on DX of each vector field defined on X gives a canonical continuous stratified extension.  2001 Editions scientifiques et medicales Elsevier SAS

A simple example of black noise

Abstract: By a noise in continuous time t∈R, we mean a family {Fs,t, s⩽t} of sub σ-fields of events on a separable probability space with properties given in Definition 1.2, a property of independence, in particular. A noise is black if there does not exist any nontrivial Levy process ξt such that ξt−ξs is Fs,t-measurable. We give an example of black noise for which the proof that it is black may be simpler than any other known examples of black noises.

Quelques résultats sur les algèbres de Lie symétriques

Abstract: Resume On etudie comment certains resultats de la theorie des algebres de Lie semi-simples se modifient dans le cas des algebres de Lie semi-simples symetriques.

Large deviation for pinned covering diffusion

Abstract: We shall establish large deviations for the pinned motions of a periodic diffusion process on n dimensional Euclidean space and the Brownian motion on 2 dimensional Lobachevsky space. It will be shown that the rate functions for the large deviations are corresponding to the infinitesimal generators of the diffusion processes obtained from the above processes through a kind of harmonic transform by positive principal eigenfunctions.

Commuting matrix differential operators and loop algebras

Abstract: We consider for fixed positive integers p and q which are coprime the space of all pairs (P,Q) of commuting matrix differential operators (of a fixed size n), where P is monic of order p and Q is normalized of order q. We use the vector valued Sato Grassmannian to construct a natural bijection to an affine subspace of the loop algebra gl (nq)((λ −1 )) . In the scalar case (n=1) the KP flows on the Grassmannian, which are known to trace out Jacobians, lead to commuting flows on this affine space. These flows are Hamiltonian with respect to a family of Poisson structures which are obtained from a family of Lie brackets on the loop algebra.

Sur le système de Gauss–Manin d'un polynôme modéré

Abstract: Resume Given a cohomologically tame polynomial we compute the determinant of a period matrix and, when the Gauss–Manin connexion is described by a fuchsian system, we give a formula for the trace of the residue matrices.

Stricte convexité des domaines bornés et unicité des géodésiques complexes

Abstract: Resume Pour un domaine convexe borne D d'un espace de Banach reflexif E , j'etudie les relations entre les propositions suivantes : (a) D est un domaine borne strictement convexe ; (b) pour tout a ∈ D , pour tout r >0, la boule B c ( a , r ) pour la distance de Caratheodory c D est strictement convexe ; (c) pour tout a ∈ D , pour tout b ∈ D , les geodesiques complexes passant par a et b sont uniques (a composition par un automorphisme analytique de Δ pres). Je montre en particulier dans cet article que (a) implique (b) et que (b) implique (c). Je montre aussi que les reciproques sont fausses.

Problème de Cauchy non linéaire à données initiales lentement oscillantes

Abstract: We solve the Cauchy problem with weakly oscillating data, in the space of holomorphic functions with respect to time, and Gevrey class with respect to the other variables. We show the stability of Cauchy problem's solution with nul initial data. In the stationary case, we give estimates of the solution's lifespan with respect to the speed's size of the oscillations of initial data.

Méthodes à ε-près et applications à des problèmes aux limites

Abstract: Resume La minimisation de fonctionnelles definies sur un espace vectoriel est etudiee en encadrant a e -pres differentes de leurs caracteristiques : gradient, borne inferieure ou contraintes Une relation entre formule de Ito et methode variationnelle est presentee Des classes de problemes aux limites illustrent chaque resultat

Ground state estimations in gauge theory

Abstract: The lowest eigenvalue λ(F) of a vector bundle F over a compact Riemannian manifold M is estimated in terms of the curvature of M and of the connection Γ on F.

Decomposition des varietes affines

Abstract: Resume Soit ( M ,∇) une variete affine compacte et connexe telle que Aff ( M ,∇) 0 n'est pas resoluble. On associe a un revetement fini ( M ′,∇′) de ( M ,∇), un corps K ( K est le corps des reels, des complexes ou des quaternions) et un K -feuilletage affine transverse F U . Soit p=codim K F U , on montre que si p>1, (M′,∇′) ou le K -eclatement de ( M ′,∇′) le long d'une sous-variete affine, est l'espace total d'une fibration au-dessus de P 1 K ( K p ). On donne une classification partielle des varietes affines ( M ′,∇′) si p =1. Puis on determine la structure differentiable (a revetement fini pres) des 3 et 4-varietes affines compactes telles que Aff ( M ,∇) 0 n'est pas resoluble.

Substitution to parametrized generalized Wiener functionals and pinned Brownian local times

Abstract: We discuss how to substitute Wiener functionals for parameters in generalized Wiener functionals: Let {Φ(x);x∈ R N } be generalized Wiener functionals parametrized by x∈ R N and F be a non-degenerate Wiener functional in Malliavin's sense. Then, formally, it holds that Φ(F)=∫ R N Φ(x)δ x (F) d x . We define the substitution by justification of the formal representation above. Under this definition we show the relationship between pinned Brownian local times and Brownian local times.

Singularités génériques des distributions de codimension 2 et des structures bihamiltonniennes

Abstract: Resume Nous construisons des stratifications dans l'espace des couples de 2-formes qui permettent de donner une description des courbes anormales pour des distributions generiques de codimension 2 en toute dimension. Ces stratifications permettent aussi de decrire les singularites des feuilletages de Weinstein associes a un couple de tenseurs de Poisson compatibles intervenant dans le contexte des varietes bihamiltoniennes.

La table de caractères pour une algèbre de Hopf

Abstract: Resume Les caracteres irreductibles d'un groupe fini G forment un outil tres puissant dans l'etude de G. On considere leurs valeurs comme des invariants numeriques de G, invariants qui doivent satisfaire a plusieurs conditions fortes comme les relations d'orthogonalite. Ces invariants numeriques qu'on appelle table de caracteres, peuvent etre interpretes comme une matrice de passages entre les idempotents orthogonaux minimaux du centre de kG et la base B={∑g∈Cjg}j=1s du centre de kG, avec Cj : designent les classes de conjugaison de G. Cependant dans le cas d'une algebre de Hopf semi-simple cette interpretation parait moins precise car la notion des classes de conjugaison n'apparait pas d'une facon naturelle, ainsi pour esquisser cette difficulte, A. Joseph a suggere une definition elegante, utilisant la theorie des representations et plus precisemment les D(H)-modules ou D(H) designe le double de Drinfeld de H. Ainsi, on pourra definir la table de caracteres, par la connaissance des D(H)-modules simples de H de dimension finie via l'action adjointe et le coproduit et des H-modules simples de dimension finie.

Sur les pinceaux de courbes définis par une fonction régulière

Abstract: Resume Dans cet article nous generalisons beaucoup de resultats sur la topologie d'une fonction polynomiale, f : C 2 → C dans le cas d'une fonction reguliere f :U→ C .