Abstract:
We consider populations structured by a phenotypic trait and a space variable, in a non-homogeneous environment. In the case of sex- ual populations, we are able to derive models close to existing mod- els in theoretical biology, from a structured population model. We then analyze the dynamics of the population using a simplified model, where the population either propagates through the whole space or it survives but remains confined in a limited range. For asexual pop- ulations, we show that the dynamics are simpler. In this case, the population cannot remain confined in a limited range, i.e. the popu- lation, if it does not get extinct, propagates through the whole space.

Abstract:
We are concerned with travelling wave solutions arising in a reaction diffusion equation with bistable and nonlocal nonlinearity, for which the comparison principle does not hold. Stability of the equilibrium $u\equiv 1$ is not assumed. We construct a travelling wave solution connecting 0 to an unknown steady state, which is "above and away", from the intermediate equilibrium. For focusing kernels we prove that, as expected, the wave connects 0 to 1. Our results also apply readily to the nonlocal ignition case.

Abstract:
We study a generalized system of ODE's modeling a finite number of biological populations in a competitive interaction. We adapt the techniques in two previous articles to prove the convergence to a unique stable equilibrium.

Abstract:
Invasion fronts in ecology are well studied but very few mathematical results concern the case with variable motility (possibly due to mutations). Based on an apparently simple reaction-diffusion equation, we explain the observed phenomena of front acceleration (when the motility is unbounded) as well as other quantitative results, such as the selection of the most motile individuals (when the motility is bounded). The key argument for the construction and analysis of traveling fronts is the derivation of the dispersion relation linking the speed of the wave and the spatial decay. When the motility is unbounded we show that the position of the front scales as $t^{3/2}$. When the mutation rate is low we show that the canonical equation for the dynamics of the fittest trait should be stated as a PDE in our context. It turns out to be a type of Burgers equation with source term.

Abstract:
A new theorem on random walks suggest some possible revisions of the foundations of Quantum Mechanics. This is presented below in the simplified framework of the description of the evolution of a material point in space. Grossly speaking, it is shown that the probabilities generated by normalizing the square modulus of a sum of probability amplitudes, in the setup of Quantum Mechanics, becomes asymptotically close (under the appropriate limiting conditions) to the probabilities generated by the usual causal processes of Classical Mechanics. This limiting coincidence has a series of interesting potential applications. In particular it allows us to reintroduce the concept of causality within the core of Quantum Mechanics. Moreover, it suggests, among other consequences, that gravitational interaction may not even exist. Even though the interpretations of Quantum Mechanics which follow from this mathematical result may seem to bring some unexpected innovations in the context of theoretical physics, there is an obvious necessity to study its theoretical impact on Quantum Mechanics. The first steps toward this aim are taken in the present article.

Abstract:
We have proposed a prequantum physics, itself founded on classical mechanics completed by the existence of an universal cloud of tiny particles noted U. These U particle command the mass, variable, of electron, neutron, proton, and atom particles noted M. The “shocks” between U and M particles in the cloud, with screen effect, give birth to electrical forces among charged particles with very small differences between attractive and repulsive forces, and to certain gravitational forces. This cloud with the electromagnetic waves propagating thus recalls an ether, yet much different regarding its effects on the inertial mass of any particle within it. The electromagnetic wave and the photon look like if they were born from a statistical mechanics induced by the universal cloud, and their status, in this regard, may be compared to the status conferred by atomics to a temperature or a pressure. The wave transversality is explained. By the same token, one understands why the photon, a vectorial bearer of a statistical information, may thus describe a particle as well as a wave.

Abstract:
We have proposed, thanks to a new model of the hydrogen atom [1], some explanation of the lines observed by Lyman in the spectrographic analysis of this atom. The model is based on a prequantum physics, itself founded on classical mechanics completed by the existence of a universal cloud of tiny particles called U. This cloud induces simultaneously and similarly electromagnetic and gravitational effects. This common origin creates a narrow link between how planets are arranged in a solar system, say the Titus-Bode law, and how the electrons are arranged in an atom, say the lines of Lyman. We describe what this link is in the following text and, more generally, what is the preferred orbit of an isolated celestial body.

Abstract:
We exhibit a pseudogroup of smooth local transformations of the real line which is compactly generated, but not realizable as the holonomy pseudogroup of a foliation of codimension 1 on a compact manifold. The proof relies on a description of all foliations with the same dynamic as the Reeb component.

Abstract:
We prove the existence of a minimal (all leaves dense) foliation of codimension one, on every closed manifold of dimension at least 4 whose Euler characteristic is null, in every homotopy class of hyperplanes distributions, in every homotopy class of Haefliger structures, in every differentiability class, under the obvious embedding assumption. The proof uses only elementary means, and reproves Thurston's existence theorem in all dimensions. A parametric version is also established.

Abstract:
We consider singular foliations of codimension one on 3-manifolds, in the sense defined by A. Haefliger as being Gamma_1-structures. We prove that under the obvious linear embedding condition, they are Gamma_1-homotopic to a regular foliation carried by an open book or a twisted open book. The latter concept is introduced for this aim. Our result holds true in every regularity C^r, r at least 1. In particular, in dimension 3, this gives a very simple proof of Thurston's 1976 regularization theorem without using Mather's homology equivalence.