Sur la courbure d’une série de surfaces et de lignes

Annali di Matematica Pura ed Applicata (1858-1865) -     

Dérivations de Peirce et structure d’algèbre

Dans ce travail nous analysons ia structure d’une algèbre, sous la condition d’existence d’un espace, suffisamment grand, de dé qui inversent la décomposition de Peirce.     

Groupes résolubles de difféomorphismes de l’intervalle,du cercle et de la droite

We give a complete classification (from the algebraic and dynamical points of view) of solvable groups of orientation preserving C²-diffeomorphisms of the interval, the circle and the real line. We also give some complementary results in the real-analytic case.     

Grandes valeurs du nombre de factorisations d’un entier en produit ordonné de facteurs premiers

Among various functions used to count the factorizations of an integer n, we consider here the number of ways of writing n as an ordered product of primes, which, if , is equal to the multinomial coefficient . The function P(s)=∑ _{p prime} p ^−s , sometimes called the prime zeta function, plays an important role in the study of the function h. We denote by λ=1.399433… the real number defined by P(λ)=1. The mean value of the function h satisfies . In this paper, we study how large h(n) can be. We prove that there exists a constant C ₁>0 such that, for all n≥3, holds. We also prove that there exists a constant C ₂ such that, for all n≥3, there exists m≤n satisfying . Let us call h-champion an integer N such that M<N implies h(M)<h(N). S. Ramanujan has called highly composite a τ-champion number, where τ(n)=∑ _d∣n 1 is the number of divisors of n. We give several results about the number of prime factors of an h-champion number N, about the exponents in the standard factorization into primes of such an N and about the number Q(X) of h-champion numbers N≤X. At the end of the paper, several open problems are listed. Recherche partiellement financée par le CNRS, Institut Camille Jordan, UMR 5208 et par l’action de coopération franco-algérienne 01 MDU 514, Arithmétique, Géométrie Algébrique et Applications.     

Action hamiltonienne d’un tore et formule de localisation en cohomologie équivariante

Using an idea of Witten (see [8]), we give a localization formula in equivariant cohomology in the case of an Hamiltonian torus action on a compact symplectic manifold χ. The integral over χ of an equivariant closed form can be written as an integral over the submanifold of critical points of the square of the moment map.     

Temps de sortie d’un intervalle borné pour l’intégrale du mouvement brownien

Let (B_t)_t≥0 be a standard Brownian motion starting at y, X_t = x+ ∫₀^tB_s, ds, x ∈ (a, b). Let us set T_ab = inf{t > 0 : X_t ∉ (a,b)}. In this paper, we compute the moments of the random variable B_Ta,b, and deduce the probability law of B_Ta,b. We show how to obtain the expectation E_(x,y)(T_ab^mB_Tabⁿ). We also explicitly determine the probabilities P_(x,y){X_Tab = a} and P_(x,y) { X_Tab = b}.     

Distances entre exponentielles et puissances d’éléments de certaines algèbres de Banach

Let A be a Banach algebra which does not contain any nonzero idempotent element, let γ > 0, and let . We show that if then . We also show, assuming a suitable spectral condition on x, that if , then Received: 12 July 2006 Revised: 31 January 2007     

Nombre de branchement d’un pseudogroupe

Lyons has defined an average number of branches per vertex of an infinite locally finite rooted tree. This number has an important role in several probabilistic processes such as random walk and percolation. In this paper, we extend the notion of branching number to any measurable graphed pseudogroup of finite type acting on a probability space. We prove that such a pseudogroup is Liouvillian (i.e. almost every orbit does not admit non-constant bounded harmonic functions) if its branching number is equal to 1. In order to prove that this actually generalizes results of C. Series and V. Kaimanovich on equivalence relations with polynomial and subexponential growth, we describe an example of minimal lamination whose holonomy pseudogroup has exponential growth and branching number equal to 1.