Hamiltonian flows on null curves

The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the trajectories of a certain particle model on null curves described by a Lagrangian linear in the curvature. In addition, it is shown that the curvature of a null curve which evolves by similarities can be computed in terms of the solutions of the second Painlev\'e equation.Comment: 14 pages, v2: final version; minor changes in the expositio

Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

An infinite family of quasi-maximally superintegrable Hamiltonians with a common set of (2N-3) integrals of the motion is introduced. The integrability properties of all these Hamiltonians are shown to be a consequence of a hidden non-standard quantum sl(2,R) Poisson coalgebra symmetry. As a concrete application, one of this Hamiltonians is shown to generate the geodesic motion on certain manifolds with a non-constant curvature that turns out to be a function of the deformation parameter z. Moreover, another Hamiltonian in this family is shown to generate geodesic motions on Riemannian and relativistic spaces all of whose sectional curvatures are constant and equal to the deformation parameter z. This approach can be generalized to arbitrary dimension by making use of coalgebra symmetry.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Superintegrable Deformations of the Smorodinsky-Winternitz Hamiltonian

A constructive procedure to obtain superintegrable deformations of the classical Smorodinsky-Winternitz Hamiltonian by using quantum deformations of its underlying Poisson sl(2) coalgebra symmetry is introduced. Through this example, the general connection between coalgebra symmetry and quasi-maximal superintegrability is analysed. The notion of comodule algebra symmetry is also shown to be applicable in order to construct new integrable deformations of certain Smorodinsky-Winternitz systems.Comment: 17 pages. Published in "Superintegrability in Classical and Quantum Systems", edited by P.Tempesta, P.Winternitz, J.Harnad, W.Miller Jr., G.Pogosyan and M.A.Rodriguez, CRM Proceedings & Lecture Notes, vol.37, American Mathematical Society, 200

On the Reliability of the Langevin Pertubative Solution in Stochastic Inflation

A method to estimate the reliability of a perturbative expansion of the stochastic inflationary Langevin equation is presented and discussed. The method is applied to various inflationary scenarios, as large field, small field and running mass models. It is demonstrated that the perturbative approach is more reliable than could be naively suspected and, in general, only breaks down at the very end of inflation.Comment: 7 pages, 3 figure

Étude comparée des communautés benthiques et ripicoles endogées d'un réseau méditerranéen perturbé : l'Arc (Bouches-du-Rhône, France)

L'utilisation des méthodes biologiques et plus particulièrement l'étude des communautés benthiques pour l'appréciation de la qualité des eaux est actuellement d'un usage général. L'impact des perturbations des milieux peut être déduit de la composition et de la structure des peuplements qu'ils hébergent. En est-il de même pour les peuplements d'invertébrés ripicoles endogés humides, soumis moins directement à t'influence du milieu aquatique? Pour répondre à cette question une étude comparée (par analyse factorielle des correspondances) a été entreprise sur le réseau hydrographique d'une rivière de la région méditerranéenne, l'Arc, présentant des situations hydrochimiques variées.L'A.F.C., à partir de l'axe I, sépare les prélèvements d'invertébrés benthiques des prélèvements d'invertébrés ripicoles. Sur l'axe II les couples, prélèvement benthique - prélèvement ripicole de chaque station, possèdent sensiblement la même valeur sur l'axe et s'ordonnent selon un gradient de trophie croissant de l'amont vers l'aval. Les communautés ripicoles endogées humides peuvent donc être considérées comme des bio-indicateurs de la chimie de l'eau de la même manière que les communautés benthiques. Par contre le facteur temporarité (axe III) a une influence plus importante sur les communautés benthiques que sur les communautés ripicoles.L'examen de la répartition des taxons sur les axes factoriels a permis de déterminer un certain nombre d'espèces qui, tout en étant représentées dans les 2 milieux, sont caractéristiques de l'habitat rivulaire endogé humide. La prise en compte des peuplements ripicoles, malgré des difficultés d'ordre taxonomique, pourrait être envisagée lors de la détermination de la qualité des eaux.Utilization of biological methods and particularly through the study of benthic communities in order to estimate the quality of the water is for the present time very common. The impact of the disturbances of the medium may be deducted from the composition and the structure of the populations living there. But is it altogether the same for endogenous riparian wet invertebrate populations less directly submitted to the influence of the aquatic medium?To answer to this question, a comparative study (Factorial Analysis of Correspondences) has been carried out on the hydrographic network of the Arc River (Mediterranean region) which exhibits diversified hydrochemical situations. From and after axis I, the F.A.C. separates the benthic invertebrate samples from the riparian invertebrate ones. On axis II, the couples "benthic samples-riparian samples" of each station appreciably posses the same value on the axis and are arranged according to a trophic gradient ascending from upstream to downstream. The wet endogenous riparian communities can therefore be considered as bioindicators of the chemistry of the water in the same way as benthic communities. In contrast, the temporality factor (axis III) has a more important influence on the benthic communities than on the riparian ones.Examination of the distribution of the taxons on the factorial axis leads to identify a certain number of species which, although present in both mediums are characteristic of the wet endogenous riverside habitat. Taking into account of the riparian populations, in spite of taxonomic difficulties, should be considered as a valuable method for the determination of the quality of the waters