Sur la théorie des demi-groupes

Sans résumé Dédié à Monsieur S. Bays pour son soixante-dixième anniversaire     

Filtration perverse et théorie de Hodge

The compatibility of the perverse filtration with Hodge theory with coefficients in an admissible variation of a mixed Hodge structure on the complement of a normally crossing divisor is established using a logarithmic complex, with a view to obtaining a new proof of the decomposition theorem.     

Diviseur thêta et formes différentielles

This article concerns the geometry of algebraic curves in characteristic p > 0. We study the geometric and arithmetic properties of the theta divisor Q{\Theta} associated to the vector bundle of locally exact differential forms of a curve. Among other things, we prove that, for a generic curve of genus ≥ 2, this theta divisor Q{\Theta} is always geometrically normal. We give also some results in the case where either p or the genus of the curve is small. In the last part, we apply our results on Q{\Theta} to the study of the variation of fundamental group of algebraic curves. In particular, we refine a recent result of Tamagawa on the specialization homomorphism between fundamental groups at least when the special fiber is supersingular.     

On 2n—th—Order Eigenvalue Problem

In [1], Lu discussed the existence of positive solution for the BVPwhere f: [0, 1] × [0, ∞) → R is continuous and satisfies the following conditions(H1) There exists an M > 0 such that(H2) limx→∞ f(t, x)/x =∞uniformly for t [α,β] (0, 1) for some α < β and λ > 0 is a parameter.In this paper, the result of [1] is generalized to the following 2n-th-order BVPThe result of this paper isTheorem 1 Assume (H1) and (H2) hold. Then the BVP (3)-(4) has a positive solutionifλ > 0 is sm…     

Sur un théorème de Choquet

Résumé Le but de ce travail est de caractériser les éspaces topologiques qui possèdent la propriété suivant: pour toute famille(f _t ) _{t I} de fonctions numériques définies dansX il existe une partie dénombrableI ₀ de l'ensemble d'indicesI telle que les fonctions inf _tI f _i et inf _tI0 f _i ont la meme régularisation inférieurement semi-continue.On démontre que l'espace topologiqueX possède cette propriété si et seulement si pour tout sous-ensembleA deX il existe une sous-ensemble dénombrableA ₀,A ₀ A tel queA ₀.On généralise un résultat topologique de G. Choquet.

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Deux remarques sur la théorie des martingales

Résumé Ce travail se compose de deux parties entièrement distinctes. Dans la première partie, on montre que le théorème des martingales (décroissantes) peut Être considéré comme un cas particulier limite d'un théorème ergodique de E. Hopf. La seconde partie énonce un principe général simple de construction de super-martingales qui ne semble jamais avoir été énoncé, bien qu'il ait été utilisé sous une forme plus particulière en théorie du potentiel par J. L. Doob.Ce travail a fait l'objet d'une conférence à la Joint European Conference of the Institute of Mathematical Statistics and the Econometric Society qui eut lieu à Copenhague du 8 au 12 juillet 1963.     

Un théorème de dualité

Sans résumé Herrn H. Hopf zum sechzigsten Geburtstag gewidmet     

Tracing the early history of algebra: Testimonies on Diophantus in the Greek-speaking world (4th–7th century CE)

The transmission and reception of the mathesis carried by Diophantus' Arithmetica has not attracted much attention among historians of Greek mathematics, who have devoted their scholarly activity almost exclusively to questions about the proper understanding of the character of the mathematical undertaking of the Alexandrian mathematician. As a result, the common belief is that Diophantus' Arithmetica is presented as an isolated, and thus uncontextualized phenomenon in the history of ancient Greek mathematics. The aim of this paper is to investigate testimonies and other piece of evidence suggesting that Diophantus' heritage was present in intellectual milieus of the Greek-speaking world during the late antique and early medieval times. Special emphasis is given to a number of scholia to the arithmetical epigrams of the Palatine Anthology which witness the persistence of the method of problem solving taught by Diophantus in the late antique world.     

Une histoire du théorème de Poincare-Birkhoff-Witt

A story of the so-called Poincaré-Birkhoff-Witt theorem Since the publication of Bourbaki's book on the Lie algebra in 1960, a theorem giving information on the structure of the enveloping algebra of a Lie algebra is called a Poincaré-Birkhoff-Witt theorem. The purpose of this article is to understand the contribution of Poincaré, Birkhoff and Witt to the definition and the study of properties of this enveloping algebra. In the second part of this article, one takes an interest to the generalizations of this theorem during the second half of the twentieth century to eventually result in its extension to the Leibniz algebra.     

Une identité remarquable en théorie des partitions

We prove an identity about partitions, previously conjectured in the study of shifted Jack polynomials. The proof given is using -ring techniques. It would be interesting to obtain a bijective proof.

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Received January 18, 2000 / Revised January 23, 2000 / Published online July 20, 2000     

Un théorème de Vitali-Hahn-Saks pour les semimartingales

Summary Let (, , P) be a complete probability space; let _t0 be an increasing right-continuous family of -complete sub--fields of ; let be a sequence of semimartingales. Assume that for all positive t and for all bounded predictable processes H, the r.v.'s converge in probability to a limit J(t, H) when n tends to infinity. Then there exists a semimartingale X such that, for all t and H, J(t, H)= .