Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation

According to Dirac's ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative part of the spectrum of the free Dirac operator $D^0$. In the presence of an external field, these virtual particles react and the vacuum becomes polarized. In this paper, following Chaix and Iracane ({\it J. Phys. B}, 22, 3791--3814, 1989), we consider the Bogoliubov-Dirac-Fock model, which is derived from no-photon QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum is defined to be a minimizer of this energy. If it exists, such a minimizer is solution of a self-consistent equation. We show the existence of a unique minimizer of the BDF-energy in the presence of an external electrostatic field, by means of a fixed-point approach. This minimizer is interpreted as the polarized vacuum.Comment: final version, to appear in Commun. Math. Phy

Normal Forms for Semilinear Quantum Harmonic Oscillators

We consider the semilinear harmonic oscillator i\psi_t=(-\Delta +\va{x}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d, t\in \R where $M$ is a Hermite multiplier and $g$ a smooth function globally of order 3 at least. We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on $M$ related to the non resonance of the linear part, this normal form is integrable when $d=1$ and gives rise to simple (in particular bounded) dynamics when $d\geq 2$. As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions

Innovation et développement dans les systèmes agricoles et alimentaires

L’innovation est souvent présentée comme l’un des principaux leviers pour promouvoir un développement plus durable et plus inclusif. Dans les domaines de l’agriculture et de l’alimentation, l’innovation est marquée par des spécificités liées à sa relation à la nature, mais aussi à la grande diversité d’acteurs concernés, depuis les agriculteurs jusqu’aux consommateurs, en passant par les services de recherche et de développement. L’innovation émerge des interactions entre ces acteurs, qui mobilisent des ressources et produisent des connaissances dans des dispositifs collaboratifs, afin de générer des changements. Elle recouvre des domaines aussi variés que les pratiques de production, l’organisation des marchés, ou les pratiques alimentaires. L’innovation est reliée aux grands enjeux de développement : innovation agro-écologique, innovation sociale, innovation territoriale, etc. Cet ouvrage porte un regard sur l’innovation dans les systèmes agricoles et alimentaires. Il met un accent particulier sur l’accompagnement de l’innovation, en interrogeant les méthodes et les organisations, et sur l’évaluation de l’innovation au regard de différents critères. Il s’appuie sur des réflexions portées par différentes disciplines scientifiques, sur des travaux de terrain conduits tant en France que dans de nombreux pays du Sud, et enfin sur les expériences acquises en accompagnant des acteurs qui innovent. Il combine des synthèses sur l’innovation et des études de cas emblématiques pour illustrer les propos. L’ouvrage est destiné aux enseignants, professionnels, étudiants et chercheurs

Solutions of the Dirac-Fock equations without projector

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Rainfall regime over the Sahelian climate gradient in the Gourma region, Mali

The Sahelian zone is characterized by low and highly variable rainfall, which strongly affects the hydrology and the climate of the region and creates severe constraints for agriculture and water management. This study provides the first characterization of the rainfall regime in a poorly described region of Central Sahel, Gourma region (14.5° to 17.5° N and 2° to 1° S) in Mali, as obtained from long-term daily precipitation time series covering the period 1950-2007 and from high-frequency dataset between 2005 and 2008. First, raingauge data collected since the middle of the 20th century were analysed both in terms of interannual variability and spatial distribution. Second, we investigate the diurnal cycle of precipitation and the nature of the rainfall using automatic raingauges during the period 2005-2008. This study also completes previous analyses conducted in Sahelian areas located further South, where the direct continental influence of the Saharan heat low is expected to be less pronounced in summer. The Gourma region rainfall presents a succession of wet (1950-1969) and dry decades (1970-2007). The decrease of the summer cumulative rainfall is explained by a reduction of the number of the rainy days in the southern Gourma region and by both a decrease of the number of rainy days and of the mean precipitation amount per rainy day in the North and the Centre. This difference may be related to their respective distances from the intertropical discontinuity, which is closer to northern stations. The length of the rainy season has varied since the 1950s with two episodes of shorter rainy seasons: during the drought of the 1980s and also, since 2000. However, this second episode is characterized by an increase of the mean rainfall per rainy day, which suggests an intensification of the rain events in the more recent years. High-frequency data reveal that most of the rainfall is produced by intense rain events occurring mostly in the late evening and the early morning during the rainy season (July/August/September). Conversely, rainfall amounts are the weakest at noon, and this mid-day damping is more pronounced in the North. All these characteristics have strong implications for agriculture and water resources management

La crue de 2012 à Niamey : un paroxysme du paradoxe du Sahel ?

International audienceLa crue de 2012 a Niamey : un paroxysme du paradoxe du Sahel ? E n août 2012, le fleuve Niger a connu une crue exceptionnelle dans le bassin du Niger moyen. Plusieurs episodes pluvieux parfois exceptionnels se sont succ ed e dans ce bassin et notamment sur le bassin-versant des affluents burkinab es. Elle a provoqu e de tr es graves inondations dans la r egion de Niamey. L'Autorit e du bassin du Niger (ABN) qui coordonne les activit es de suivi du fleuve, doit eclairer les d ecisions a prendre, tant au niveau r egional que dans les pays concern es. Le bilan humain est terrible puisque 81 personnes ont perdu la vie au Niger ; les d egâts mat eriels sont enormes. Cette crue fai