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

Delirium, thrombocytopenia, insomnia, and mild liver damage associated with MAOI withdrawal

International audienc

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

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

On the Symmetries of Integrability

We show that the Yang-Baxter equations for two dimensional models admit as a group of symmetry the infinite discrete group $A_2^{(1)}$. The existence of this symmetry explains the presence of a spectral parameter in the solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetry. Although generalizing naturally the previous one, it is a much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of three-dimensional vertex models. These symmetries are naturally represented as birational projective transformations. They may preserve non trivial algebraic varieties, and lead to proper parametrizations of the models, be they integrable or not. We mention the relation existing between spin models and the Bose-Messner algebras of algebraic combinatorics. Our results also yield the generalization of the condition $q^n=1$ so often mentioned in the theory of quantum groups, when no $q$ parameter is available.Comment: 23 page

Integrability of one degree of freedom symplectic maps with polar singularities

In this paper, we treat symplectic difference equations with one degree of freedom. For such cases, we resolve the relation between that the dynamics on the two dimensional phase space is reduced to on one dimensional level sets by a conserved quantity and that the dynamics is integrable, under some assumptions. The process which we introduce is related to interval exchange transformations.Comment: 10 pages, 2 figure

Foraging is determinant to improve smallholders’ food security in rural areas in Mali, West Africa

Studies on the enabling factors for household food security (HFS) most often used simplified econometric models looking into the links with a selected set of variables. In this research, a livelihood approach of HFS was used and aimed at determining the most significant livelihood assets for HFS in dryland agricultural systems. Elements of the five livelihood assets were assessed through questionnaire surveys with a random sample of 180 households, and six focus group discussions in three communities along the rural-urban continuum, in Southern Mali. The coping strategy index approach was used to evaluate household food security status. Non-parametric and parametric statistical tests were combined, as appropriate, to identify the most significant determinants of HFS status. Findings indicated that most determinant factors of HFS were the diversity of wild and cultivated food plants, and hunting (natural capital); access to clean water and irrigation (infrastructural capital); and off-farm employment (financial capital). HFS also improved along the urban-rural continuum and rural households with high natural capital seemed to be more food secure. Findings call for important investment to expand the natural capital (e.g., domestication of new crops and agricultural diversification) and infrastructural capital (irrigation facilities, clean water) of the rural households