Abstract:
Let $F$ be a non--Archimedean locally compact field (${\rm car}(F)\geq 0$), ${\bf G}$ be a connected reductive group defined over $F$, $\theta$ be an $F$--automorphism of ${\bf G}$, and $\omega$ be a character of ${\bf G}(F)$. We fix a Haar measure $dg$ on ${\bf G}(F)$. For a smooth irreducible $(\theta,\omega)$--stable complex representation $\pi$ of ${\bf G}(F)$, that is such that $\pi\circ \theta\simeq \pi\otimes \omega$, the choice of an isomorphism $A$ from $\pi\otimes \omega$ to $\pi\circ \theta$ defines a distribution $\Theta_\pi^A$, called the \og ($A$--)twisted character of $\pi$\fg: for a compactly supported locally constant function $f$ on ${\bf G}(F)$, we put $\Theta_\pi^A(f)={\rm trace}(\pi(fdg)\circ A)$. In this paper, we study these distributions $\Theta_\pi^A$, without any restrictive hypothesis on $F$, ${\bf G}$ or $\theta$. We prove in particular that the restriction of $\Theta_\pi^A$ on the open dense subset of ${\bf G}(F)$ formed of those elements which are $\theta$--quasi--regular is given by a locally constant function, and we describe how this function behaves with respect to parabolic induction and Jacquet restriction. This leads us to take up again the Steinberg theory of automorphisms of an algebraic group, from a rationnal point of view.

Abstract:
Let ${\boldsymbol{G}}$ be a connected reductive group defined over a non--Archimedean local field $F$. Put $G={\boldsymbol{G}}(F)$. Let $\theta$ be an $F$--automorphism of ${\boldsymbol{G}}$, and let $\omega$ be a smooth character of $G$. This paper is concerned with the smooth complex representations $\pi$ of $G$ such that $\pi^\theta=\pi\circ\theta$ is isomorphic to $\omega\pi=\omega\otimes\pi$. If $\pi$ is admissible, in particular irreducible, the choice of an isomorphism $A$ from $\omega\pi$ to $\pi^\theta$ (and of a Haar measure on $G$) defines a distribution $\Theta_\pi^A={\rm tr}(\pi\circ A)$ on $G$. The twisted Fourier transform associates to a compactly supported locally constant function $f$ on $G$, the function $(\pi,A)\mapsto \Theta_\pi^A(f)$ on a suitable Grothendieck group. Here we describe its image (Paley--Wiener theorem), and we reduce the description of its kernel (spectral density theorem) to a result on the discrete part of the theory.

Abstract:
We prove the fundamental lemma for twisted endoscopy, for the unit elements of the spherical Hecke algebras, in the case of a non ramified elliptic endo- scopic datum whose underlying group is a torus. This implies that the fundamental lemma for twisted endoscopy is now proved, for all elements in the spherical Hecke algebras, in characteristic zero and any residue characteristic.

Abstract:
We show here that the fundamental lemma for twisted endoscopy, now proved for the unit elements in the spherical Hecke algebras, implies the fundamental lemma for all elements of these Hecke algebras. The proof, whose idea is due to Arthur, uses the transfer, which is known as a consequence of the fundamental lemma for the units.

Abstract:
Chrysotile asbestos is formed by densely packed bundles of multiwall hollow nanotubes. Each wall in the nanotubes is a cylindrically wrapped layer of $Mg_3 Si_2 O_5 (OH)_4$. We show by experiment and theory that the infrared spectrum of chrysotile presents multiple plasmon resonances in the Si-O stretching bands. These collective charge excitations are universal features of the nanotubes that are obtained by cylindrically wrapping an anisotropic material. The multiple plasmons can be observed if the width of the resonances is sufficiently small as in chrysotile.

Abstract:
grasslands have to be considered not only as a mean for providing foods for domestic herbivore but also as an important biome of terrestrial biosphere. this function of grasslands as an active component of our environment requires specific studies on the role and impact of this ecosystem on soil erosion and soil quality, quality and quantity of water resources, atmosphere composition and greenhouse gas emission or sequestration, biodiversity dynamics at different scales from field plot to landscape. all these functions have to be evaluated in conjunction with the function of providing animal products for increasing human population. so multifunctionality of grasslands become a new paradigm for grassland science. environmental and biodiversity outputs require long term studies, being the long term retro-active processes within soil, vegetation and micro-organism communities in relation to changes in management programme. so grassland science needs to carry on long term integrated experimentation for studying all the environmental outputs and ecological services associated to grassland management systems.

Abstract:
I outline the development of four generations of kinetic models, starting with Chamberlain's solar breeze exospheric model. It is shown why this first kinetic model did not give apposite supersonic evaporation velocities, like early hydrodynamic models of the solar wind. When a self-consistent polarization electric potential distribution is used in the coronal plasma, instead of the Pannekoek-Rosseland's one, supersonic bulk velocities are readily obtained in the second generation of kinetic models. It is outlined how the third and fourth generations of these models have improved the agreement with observations of slow and fast speed solar wind streams.

Abstract:
Chapman's (1957) conductive model of the solar corona is characterized by a temperature varying as r**(-2/7) with heliocentric distance r. The density distribution in this non-isothermal hydrostatic model has a minimum value at 123 RS, and increases with r above that altitude. It is shown that this hydrostatic model becomes convectively unstable above r = 35 RS, where the temperature lapse rate becomes superadiabatic. Beyond this radial distance heat conduction fails to be efficient enough to keep the temperature gradient smaller than the adiabatic lapse rate. We report the results obtained by Lemaire (1968) who showed that an additional mechanism is then required to transport the energy flux away from the Sun into interplanetary space. He pointed out that this additional mechanism is advection: i.e. the stationary hydrodynamic expansion of the corona. In other words the corona is unable to stay in hydrostatic equilibrium. The hydrodynamic solar wind expansion is thus a physical consequence of the too steep (superadiabatic) temperature gradient beyond the peak of coronal temperature that can be determined from white light brightness distributions observed during solar eclipses. The thermodynamic argument for the existence of a continuous solar wind expansion which is presented here, complements Parker's classical argument based on boundary conditions imposed to the solutions of the hydrodynamic equations for the coronal expansion: i.e. the inability of the mechanical forces to hold the corona in hydrostatic equilibrium. The thermodynamic argument presented here is based on the energy transport equation. It relies on the temperature distribution which becomes super-adiabatic above a certain altitude in the inner corona.

Abstract:
We propose a new scheme for the long time approximation of a diffusion when the drift vector field is not globally Lipschitz. Under this assumption, regular explicit Euler scheme --with constant or decreasing step-- may explode and implicit Euler scheme are CPU-time expensive. The algorithm we introduce is explicit and we prove that any weak limit of the weighted empirical measures of this scheme is a stationary distribution of the stochastic differential equation. Several examples are presented including gradient dissipative systems and Hamiltonian dissipative systems.

Abstract:
The aim of this paper is to study the behavior of the weighted empirical measures of the decreasing step Euler scheme of a one-dimensional diffusion process having multiple invariant measures. This situation can occur when the drift and the diffusion coefficient are vanish simultaneously. As a first step, we give a brief description of the Feller's classification of the one-dimensional process. We recall the concept of attractive and repulsive boundary point and introduce the concept of strongly repulsive point. That allows us to establish a classification of the ergodic behavior of the diffusion. We conclude this section by giving necessary and sufficient conditions on the nature of boundary points in terms of Lyapunov functions. In the second section we use this characterization to study the decreasing step Euler scheme. We give also an numerical example in higher dimension.