Integrable mappings and polynomial growth

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on $q \times q$ matrices: the inversion of the $q \times q$ matrix and an (involutive) permutation of the entries of the matrix. We concentrate on the case where these permutations are elementary transpositions of two entries. In this case the birational transformations fall into six different classes. For each class we analyze the factorization properties of the iteration of these transformations. These factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties. Some mappings yield a polynomial growth of the complexity of the iterations. For three classes the successive iterates, for $q=4$, actually lie on elliptic curves. This analysis also provides examples of integrable mappings in arbitrary dimension, even infinite. Moreover, for two classes, the homogeneous polynomials are shown to satisfy non trivial non-linear recurrences. The relations between factorizations of the iterations, the existence of recurrences on one or several variables, as well as the integrability of the mappings are analyzed.Comment: 45 page

Singularity, complexity, and quasi--integrability of rational mappings

We investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models, by exploiting their nature of birational transformations of projective spaces. We give an algorithmic analysis of the structure of invariants of such mappings. We discuss some characteristic conditions for their (quasi)--integrability, and in particular its links with their singularities (in the 2--plane). Finally, we describe some of their properties {\it qua\/} dynamical systems, making contact with Arnol'd's notion of complexity, and exemplify remarkable behaviours.Comment: Latex file. 17 pages. To appear in CM

The relationships between organic farming and agroecology.

While acknowledging an extension of agroecology in the organic sector and a growing influence of agroecology in the academic world, we explore their relationships. These relationships cannot be reduced to an opposition between a scientific field and a practical domain. A Brazilian case study based on the analysis of researchers and social actors trajectories exemplifies the diversity of existing relations, whether inclusive or exclusive. With a literature review, this allows characterising the specific attributes of both organic agriculture and agroecology. We discuss them in the light of current challenges for organic farming research and development

Analysis of the spatial distribution of organic farming in the Provence-Alpes-Côte d’Azur (PACA) region of France: building a methodology for observation and prospective

Analysis of the distribution of organic farming and its determinants is a major challenge for the construction of scenarios related to the evolution of organic farming that take the set of factors that limit or facilitate conversions and their sustainability into account. In order to support different development scenarios for organic farming in view of environmental and marketing challenges, an econometric and geostatistic study was initiated in the PACA region. The approach consists of three stages. The first stage involves the construction, over a 20-year period, of a longitudinal spatial database concerning certified organic producers and distributors. This database is georeferenced at the plot level. It will then be necessary to statistically analyse the trends and the determinants of the organic farming distribution process over time and space. Finally, this knowledge will be used to build scenarios for the development of organic farming production on the medium term. Intermediary results presented within this framework well be based on the georeferencing of producers and distributors at the community level in 2007, without drawing on farm data issued by the MRA (French farmers mutual insurance system), and on the results of an exhaustive survey of all organic distributors implemented in 2008 in the PACA region

The liquid-glass transition of silica

We studied the liquid-glass transition of $SiO_2$ by means of replica theory, utilizing an effective pair potential which was proved to reproduce a few experimental features of silica. We found a finite critical temperature $T_0$, where the system undergoes a phase transition related to replica symmetry breaking, in a region where experiments do not show any transition. The possible sources of this discrepancy are discussed.Comment: 14 pages, 6 postscript figures. Revised version accepted for pubblication on J.Chem.Phy

Thermal noise properties of two aging materials

In this lecture we review several aspects of the thermal noise properties in two aging materials: a polymer and a colloidal glass. The measurements have been performed after a quench for the polymer and during the transition from a fluid-like to a solid-like state for the gel. Two kind of noise has been measured: the electrical noise and the mechanical noise. For both materials we have observed that the electric noise is characterized by a strong intermittency, which induces a large violation of the Fluctuation Dissipation Theorem (FDT) during the aging time, and may persist for several hours at low frequency. The statistics of these intermittent signals and their dependance on the quench speed for the polymer or on sample concentration for the gel are studied. The results are in a qualitative agreement with recent models of aging, that predict an intermittent dynamics. For the mechanical noise the results are unclear. In the polymer the mechanical thermal noise is still intermittent whereas for the gel the violation of FDT, if it exists, is extremely small.Comment: to be published in the Proceedings of the XIX Sitges Conference on ''Jammming, Yielding and Irreversible Deformation in Condensed Matter'', M.-C.Miguel and M. Rubi eds.,Springer Verlag, Berli

Baxterization, dynamical systems, and the symmetries of integrability

We resolve the `baxterization' problem with the help of the automorphism group of the Yang-Baxter (resp. star-triangle, tetrahedron, \dots) equations. This infinite group of symmetries is realized as a non-linear (birational) Coxeter group acting on matrices, and exists as such, {\em beyond the narrow context of strict integrability}. It yields among other things an unexpected elliptic parametrization of the non-integrable sixteen-vertex model. It provides us with a class of discrete dynamical systems, and we address some related problems, such as characterizing the complexity of iterations.Comment: 25 pages, Latex file (epsf style). WARNING: Postscript figures are BIG (600kB compressed, 4.3MB uncompressed). If necessary request hardcopy to [email protected] and give your postal mail addres

Out-of-equilibrium dynamics in a gaussian trap model

The violations of the fluctuation-dissipation theorem are analyzed for a trap model with a gausssian density of states. In this model, the system reaches thermal equilibrium for long times after a quench to any finite temperature and therefore all aging effect are of a transient nature. For not too long times after the quench it is found that the so-called fluctuation-dissipation ratio tends to a non-trivial limit, thus inicating the possibility for the definition of a time scale dependent effective temperature. However, different definitions of the effective temperature yield distinct results. In particular plots of the integrated response versus the correlation function strongly depend on the way they are constructed. Also the definition of effective temperatures in the frequency domain is not unique for the model considered. This may have some implications for the interpretation of results from computer simulations and experimental determinations of effective temperatures.Comment: Proceedings of the workshop on non-equilibrium phenomena in supercooled fluids, glasses and amorphous materials (17-22 September, Pisa

Raisonner l'utilisation sylvo-pastorale du chêne vert

Dans la région Languedoc-Roussillon, des éleveurs utilisent des terrains boisés en chêne vert, notamment des taillis. L'offre pastorale de ces végétations complexes est présentée. Des pratiques combinant les deux activités, production ligneuse et élevage, permettent de réaménager ces espaces boisés : la présence de troupeaux conduit à des interventions originales sur les bois, c'est-à-dire à des sylvicultures diversifiées. Un exemple de scénario de gestion sylvo-pastorale pluriannuelle est développé