Une preuve de la conjecture de Mahler-Manin

Sans résumé Oblatum 7-IV-1995     

Regular modes in rotating stars

Despite more and more observational data, stellar acoustic oscillation modes are not well understood as soon as rotation cannot be treated perturbatively. In a way similar to semiclassical theory in quantum physics, we use acoustic ray dynamics to build an asymptotic theory for the subset of regular modes which are the easiest to observe and identify. Comparisons with 2D numerical simulations of oscillations in polytropic stars show that both the frequency and amplitude distributions of these modes can accurately be described by an asymptotic theory for almost all rotation rates. The spectra are mainly characterized by two quantum numbers; their extraction from observed spectra should enable one to obtain information about stellar interiors.     

Statistical mechanics of two-dimensional shuffled foams: prediction of the correlation between geometry and topology

We propose an analytical model for the statistical mechanics of shuffled two-dimensional foams with moderate bubble size polydispersity. It predicts without any adjustable parameters the correlations between the number of sides n of the bubbles (topology) and their areas A (geometry) observed in experiments and numerical simulations of shuffled foams. Detailed statistics show that in shuffled cellular patterns n correlates better with √A (as claimed by Desch and Feltham) than with A (as claimed by Lewis and widely assumed in the literature). At the level of the whole foam, standard deviations Δn and ΔA are in proportion. Possible applications include correlations of the detailed distributions of n and A, three-dimensional foams, and biological tissues.     

Catégories de Möbius et fonctorialités: un cadre général pour l'inversion de Möbius

This paper proposes a general setting for Möbius inversion which includes the cases of locally finite partially ordered sets and monoids with the finite decomposition property (f.d.p.). The unifying concept is that of categories with the f.d.p., called Möbius categories, which we characterize in four different ways. The established theory of incidence algebras, Möbius functions, product formulas, reduced algebras, etc., is carried over. Furthermore, the study of functors between Möbius categories yields results in two directions: firstly, inspired from the substitution of formal power series is the construction of a homomorphism between incidence algebras which allows the transfer of Möbius inversions; secondly, surjective functors often give rise to a reduced incidence algebra, thus shedding new light on many important reduced algebras in combinatorial theory.     

Neural languages

In order to provide the theoretical framework necessary to study the neural mechanisms underlying languages, we present here a mathematical formalization of some neural behaviors. In such a context: (i) the neuron is defined as a coupling of automata, dealing with the transduction and codifying processes; (ii) the coupling between neurons is measured by a membership relation defined as the ratio between the transmitted and the system entropies; (iii) the attention given to the messages arriving at the system is considered as the difference between the couplings of the excitatory and inhibitory pools of neurons; (iv) the graphs in the neural systems are described by means of these couplings and, finally (v) the semantic productions are described by the fuzzy formal languages accepted by the automata formalized on these graphs. In such a way, both the verbal and neural semantics can now be correlated and experimentally investigated.     

Sur les difféomorphismes d'Anosov affines à feuilletages stable et instable différentiables

Résumé Nous étudions les difféomorphismes d'Anosov topologiquement transitifs sur une variété compacte dont les feuilletages stable et instable sont de classeC . Nous montrons que si un tel difféomorphisme préserve une structure symplectique ou une connexion, il est alorsC conjugué à un infranilautomorphisme hyperbolique.Oblatum 18-III-1992     

A geometric model for active contours in image processing

Summary We propose a new model for active contours based on a geometric partial differential equation. Our model is intrinsec, stable (satisfies the maximum principle) and permits a rigorous mathematical analysis. It enables us to extract smooth shapes (we cannot retrieve angles) and it can be adapted to find several contours simultaneously. Moreover, as a consequence of the stability, we can design robust algorithms which can be engineed with no parameters in applications. Numerical experiments are presented.     

Polyhedral hyperbolic metrics on surfaces

Let S be a topologically finite surface, and g be a hyperbolic metric on S with a finite number of conical singularities of positive singular curvature, cusps and complete ends of infinite area. We prove that there exists a convex polyhedral surface P in hyperbolic space and a group G of isometries of such that the induced metric on the quotient P/G is isometric to g. Moreover, the pair (P, G) is unique among a particular class of convex polyhedra.