Complexity spectrum of some discrete dynamical systems

We first study birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of $3 \times 3$ matrices. We introduce a semi-numerical analysis which enables to compute the Arnold complexities for all the $9!$ possible birational transformations. These complexities correspond to a spectrum of eighteen algebraic values. We then drastically generalize these results, replacing permutations of the entries by homogeneous polynomial transformations of the entries possibly depending on many parameters. Again it is shown that the associated birational, or even rational, transformations yield algebraic values for their complexities.Comment: 1 LaTex fil

Separation, for Analytical Purpose, of Np Traces from different Solutions of Fuel Reprocessing

AbstractFour separation methods were developed for performance control of hydrometallurgical extraction processes as COEX™ or advanced PUREX. These methods used implemented the operations of radionuclides oxidation state adjustment and chromatographic separation using TEVA resin. Concerning FP raffinate, the method consisted in reducing Np traces to the valence IV by a mixture of ferrous sulfamate and ascorbic acid, to fix Np(IV) on “TEVA” resin and to eluate it by a nitrohydrofluoric acid solution. The Np recovery yield is 100%. The decontamination of Np is sufficiently high to allow its analysis by FXL (Zr/Np < 1). The study also showed that in presence of Zr and Tc, Pu behaved like Np. The mixture of ferrous sulfamate and ascorbic acid had surprisingly no action on Pu(IV). Concerning plutonium solution ([Pu] > 10g/L) and uranium solution ([U] > 100g/L), the same method used for Np recovery from FP raffinate led to an eluate containing 100% of the initial Np ([Np]: 10mg/L). The low concentration of U and Pu (< 100mg/L) allows the determination of Np by FXL. Concerning Pu(III)-U(IV) solution, the method, included 2 redox stages, the first one to oxidize all actinides to oxidation state VI et the second one to reduce Np and Pu respectively to IV and III oxidation state. Then Np(IV) was fixed on TEVA resin. The eluate contains 100% of the initial Np ([Np]: 10mg/L) and a low concentration of U and Pu ([U] < 20mg/L, [Pu] < 10mg/L). The next experiments will consist in consolidating these good results by working with real solutions of fuel reprocessing

Birational Mappings and Matrix Sub-algebra from the Chiral Potts Model

We study birational transformations of the projective space originating from lattice statistical mechanics, specifically from various chiral Potts models. Associating these models to \emph{stable patterns} and \emph{signed-patterns}, we give general results which allow us to find \emph{all} chiral $q$-state spin-edge Potts models when the number of states $q$ is a prime or the square of a prime, as well as several $q$-dependent family of models. We also prove the absence of monocolor stable signed-pattern with more than four states. This demonstrates a conjecture about cyclic Hadamard matrices in a particular case. The birational transformations associated to these lattice spin-edge models show complexity reduction. In particular we recover a one-parameter family of integrable transformations, for which we give a matrix representationComment: 22 pages 0 figure The paper has been reorganized, splitting the results into two sections : results pertaining to Physics and results pertaining to Mathematic

A classification of four-state spin edge Potts models

We classify four-state spin models with interactions along the edges according to their behavior under a specific group of symmetry transformations. This analysis uses the measure of complexity of the action of the symmetries, in the spirit of the study of discrete dynamical systems on the space of parameters of the models, and aims at uncovering solvable ones. We find that the action of these symmetries has low complexity (polynomial growth, zero entropy). We obtain natural parametrizations of various models, among which an unexpected elliptic parametrization of the four-state chiral Potts model, which we use to localize possible integrability conditions associated with high genus curves.Comment: 5 figure

Symmetry, complexity and multicritical point of the two-dimensional spin glass

We analyze models of spin glasses on the two-dimensional square lattice by exploiting symmetry arguments. The replicated partition functions of the Ising and related spin glasses are shown to have many remarkable symmetry properties as functions of the edge Boltzmann factors. It is shown that the applications of homogeneous and Hadamard inverses to the edge Boltzmann matrix indicate reduced complexities when the elements of the matrix satisfy certain conditions, suggesting that the system has special simplicities under such conditions. Using these duality and symmetry arguments we present a conjecture on the exact location of the multicritical point in the phase diagram.Comment: 32 pages, 6 figures; a few typos corrected. To be published in J. Phys.

On the complexity of some birational transformations

Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines, and relies mainly on univariate polynomial algebra, the second approach is a singularity analysis, and the third method is more numerical, using integer arithmetics. Each method has its own domain of application, but they give corroborating results, and lead us to a conjecture on the complexity of a class of maps constructed from matrix inversions

Determination of Total Antioxidant Content in Various Drinks by Amperometry

In the present work the total content of phenolic antioxidants in juice of some fruit and vegetables, in wines, water extracts of tea and herb were measured by amperometry. Efficiency of the method allowed determining the total antioxidant content in their binary and multimixes, including processes of frosting-defrosting and juice diluting as well. The deviation of experimentally received values of the total antioxidant content in some drink mixes from the values calculated proceeding from the additivity principle of the antioxidant content in separate drinks has been revealed

Random Matrix Theory and higher genus integrability: the quantum chiral Potts model

We perform a Random Matrix Theory (RMT) analysis of the quantum four-state chiral Potts chain for different sizes of the chain up to size L=8. Our analysis gives clear evidence of a Gaussian Orthogonal Ensemble statistics, suggesting the existence of a generalized time-reversal invariance. Furthermore a change from the (generic) GOE distribution to a Poisson distribution occurs when the integrability conditions are met. The chiral Potts model is known to correspond to a (star-triangle) integrability associated with curves of genus higher than zero or one. Therefore, the RMT analysis can also be seen as a detector of ``higher genus integrability''.Comment: 23 pages and 10 figure