On the Equivalence Between Type I Liouville Dynamical Systems in the Plane and the Sphere

Producción CientíficaSeparable Hamiltonian systems either in sphero-conical coordinates on an S2 sphere or in elliptic coordinates on a R2 plane are described in a unified way. A back and forth route connecting these Liouville Type I separable systems is unveiled. It is shown how the gnomonic projection and its inverse map allow us to pass from a Liouville Type I separable system with a spherical configuration space to its Liouville Type I partners where the configuration space is a plane and back. Several selected spherical separable systems and their planar cousins are discussed in a classical context

Le corps igné de Nefza (Tunisie septentrionale) : caractéristiques géophysiques et discussion du mécanisme de sa mise en place

La présence de roches ignées a été mise en évidence dans le Nord-Ouest de la Tunisie dans les régions de Nefza et des Mogod. La répartition géographique de ces indices montre qu'ils sont alignés suivant la direction Est-Ouest (N80°). L'interprétation des données aéromagnétiques et gravimétriques, conjuguée avec une modélisation géophysique du type 2D 1/2 ont permis de caractériser le corps igné de la région de Nefza. Les résultats montrent qu'il s'agit d'une roche intra-sédimentaire de densité supérieure à 2.8, peu profonde (700 à 800 m), allongée dans la direction N80° et de susceptibilité magnétique de l'ordre de 500 E-6 (cgs). Les corps d'origine magmatique de la région, en particulier celui de Nefza, sont associés à un gradient géothermique anormalement élevé. Compte tenu de la géométrie de ce corps et du contexte géologique, il semble que sa mise en place est liée à une zone de faiblesse de la lithosphère qui correspond à une fracturation profonde de direction proche de l'Est-Ouest. (Résumé d'auteur

Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities

We examine in detail the relative equilibria in the four-vortex problem where two pairs of vortices have equal strength, that is, \Gamma_1 = \Gamma_2 = 1 and \Gamma_3 = \Gamma_4 = m where m is a nonzero real parameter. One main result is that for m > 0, the convex configurations all contain a line of symmetry, forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for all m but the isosceles trapezoid case exists only when m is positive. In fact, there exist asymmetric convex configurations when m < 0. In contrast to the Newtonian four-body problem with two equal pairs of masses, where the symmetry of all convex central configurations is unproven, the equations in the vortex case are easier to handle, allowing for a complete classification of all solutions. Precise counts on the number and type of solutions (equivalence classes) for different values of m, as well as a description of some of the bifurcations that occur, are provided. Our techniques involve a combination of analysis and modern and computational algebraic geometry

A comparison of surface fitting algorithms for geophysical data

Cet article présente les résultats d'une comparaison de différents algorithmes d'approximation de surface. Pour chacun de ces algorithmes (approximation polynomiale, combinaison spline-laplace, krigeage, approximation aux moindres carrés, méthode des éléments finis) la pertinence pour différents ensembles de données et les limites d'application sont discutée

Projective dynamics and classical gravitation

Given a real vector space V of finite dimension, together with a particular homogeneous field of bivectors that we call a "field of projective forces", we define a law of dynamics such that the position of the particle is a "ray" i.e. a half-line drawn from the origin of V. The impulsion is a bivector whose support is a 2-plane containing the ray. Throwing the particle with a given initial impulsion defines a projective trajectory. It is a curve in the space of rays S(V), together with an impulsion attached to each ray. In the simplest example where the force is identically zero, the curve is a straight line and the impulsion a constant bivector. A striking feature of projective dynamics appears: the trajectories are not parameterized. Among the projective force fields corresponding to a central force, the one defining the Kepler problem is simpler than those corresponding to other homogeneities. Here the thrown ray describes a quadratic cone whose section by a hyperplane corresponds to a Keplerian conic. An original point of view on the hidden symmetries of the Kepler problem emerges, and clarifies some remarks due to Halphen and Appell. We also get the unexpected conclusion that there exists a notion of divergence-free field of projective forces if and only if dim V=4. No metric is involved in the axioms of projective dynamics.Comment: 20 pages, 4 figure

The Lie-Poisson structure of the reduced n-body problem

The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. As a result we obtain a reduced system with a Lie-Poisson structure which is isomorphic to sp(2n-2), independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. Hence we proceed to construct a Poisson integrator for the reduced n-body problem using a splitting method.Comment: 26 pages, 2 figure

Projected impacts of climate warming on the functional and phylogenetic components of coastal Mediterranean fish biodiversity

Climate warming affects biodiversity distribution across all ecosystems. However, beyond changes in species richness, impacts on other biodiversity components are still overlooked, particularly in the marine realm. Here we forecasted the potential effect of climate warming on the phylogenetic and functional components of coastal Mediterranean fish biodiversity. To do so, we used species distribution models to project the potential distribution of 230 coastal fish species by the end of the 21st century based on the IPCC A2 scenario implemented with the Mediterranean climatic model NEMOMED8. From these projections, we assessed the changes in phylogenetic (PD) and functional diversity (FD) of fish assemblages at multiple spatial scales using a dated molecular phylogeny and an extensive functional trait database. At the scale of the entire Mediterranean Sea, the projected extinctions of 40 coastal fish species would lead to a concomitant erosion of PD and FD (13.6 and 3%, respectively). However, a null model revealed that species loss at this scale would not lead to a disproportionate erosion of PD and FD. Similar results were found when considering fish assemblages at the grid cell scale. Indeed, at this scale, the projected changes in species richness would lead to unexpected losses of PD and FD for localized and small areas only. A disproportionate erosion of PD under climate warming was only forecasted when analysing fish assemblages at an intermediate spatial scale, namely the Mediterranean marine ecoregions. Overall, our results emphasize the importance of considering multiple spatial scales when assessing potential impacts of climate warming on the multiple components marine biodiversity

HIGH SPIN ISOMERIC STATES IN 197Pb

High spin isomeric states in 197Pb have been investigated by in-beam gamma ray technics. These states are produced in (heavy ion, xn) reactions. various experiments performed are summarized in table I

Euler configurations and quasi-polynomial systems

In the Newtonian 3-body problem, for any choice of the three masses, there are exactly three Euler configurations (also known as the three Euler points). In Helmholtz' problem of 3 point vortices in the plane, there are at most three collinear relative equilibria. The "at most three" part is common to both statements, but the respective arguments for it are usually so different that one could think of a casual coincidence. By proving a statement on a quasi-polynomial system, we show that the "at most three" holds in a general context which includes both cases. We indicate some hard conjectures about the configurations of relative equilibrium and suggest they could be attacked within the quasi-polynomial framework.Comment: 21 pages, 6 figure