Binary trees, coproducts, and integrable systems

We provide a unified framework for the treatment of special integrable systems which we propose to call "generalized mean field systems". Thereby previous results on integrable classical and quantum systems are generalized. Following Ballesteros and Ragnisco, the framework consists of a unital algebra with brackets, a Casimir element, and a coproduct which can be lifted to higher tensor products. The coupling scheme of the iterated tensor product is encoded in a binary tree. The theory is exemplified by the case of a spin octahedron.Comment: 15 pages, 6 figures, v2: minor correction in theorem 1, two new appendices adde

Cosmological diagrammatic rules

A simple set of diagrammatic rules is formulated for perturbative evaluation of ``in-in" correlators, as is needed in cosmology and other nonequilibrium problems. These rules are both intuitive, and efficient for calculational purposes.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

The Ljapunov-Schmidt reduction for some critical problems

This is a survey about the application of the Ljapunov-Schmidt reduction for some critical problems

Gaudin Models and Bending Flows: a Geometrical Point of View

In this paper we discuss the bihamiltonian formulation of the (rational XXX) Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued spins. In particular, we focus on the homogeneous models. We find a pencil of Poisson brackets that recursively define a complete set of integrals of the motion, alternative to the set of integrals associated with the 'standard' Lax representation of the Gaudin model. These integrals, in the case of su(2), coincide wih the Hamiltonians of the 'bending flows' in the moduli space of polygons in Euclidean space introduced by Kapovich and Millson. We finally address the problem of separability of these flows and explicitly find separation coordinates and separation relations for the r=2 case.Comment: 27 pages, LaTeX with amsmath and amssym

Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra U(\g) of a semisimple Lie algebra \g. This family is parameterized by collections of pairwise distinct complex numbers $z_1,...,z_n$. We obtain some new commutative subalgebras in U(\g)^{\otimes n} as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.Comment: 11 pages, references adde

Effect of interchain separation on the photoinduced absorption spectra of polycarbazolyldiacetylenes

The photoinduced absorption spectra of a novel polycarbazolyldiacetylene with long aliphatic chains on the carbazolyl side groups are measured and compared with those of the unsubstituted polyDCHD. The two polymers in the blue form exhibit very similar electronic absorption spectra and Raman frequencies. This fact indicates that the conjugation length of the polydiacetylene backbone is not too affected by the long substituents. In contrast, the near steady-state photoinduced absorption spectra show that different photogeneration mechanisms are involved in the two polymers. This result can be ascribed to the role played by the interchain distance in the dynamics of the relaxation processes in polydiacetylenes

Non-coboundary Poisson-Lie structures on the book group

All possible Poisson-Lie (PL) structures on the 3D real Lie group generated by a dilation and two commuting translations are obtained. Its classification is fully performed by relating these PL groups with the corresponding Lie bialgebra structures on the corresponding "book" Lie algebra. By construction, all these Poisson structures are quadratic Poisson-Hopf algebras for which the group multiplication is a Poisson map. In contrast to the case of simple Lie groups, it turns out that most of the PL structures on the book group are non-coboundary ones. Moreover, from the viewpoint of Poisson dynamics, the most interesting PL book structures are just some of these non-coboundaries, which are explicitly analysed. In particular, we show that the two different q-deformed Poisson versions of the sl(2,R) algebra appear as two distinguished cases in this classification, as well as the quadratic Poisson structure that underlies the integrability of a large class of 3D Lotka-Volterra equations. Finally, the quantization problem for these PL groups is sketched.Comment: 15 pages, revised version, some references adde