Differential Galois Theory of Linear Difference Equations

We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hoelder's Theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric functions.Comment: 50 page

Spatial modelling for mixed-state observations

In several application fields like daily pluviometry data modelling, or motion analysis from image sequences, observations contain two components of different nature. A first part is made with discrete values accounting for some symbolic information and a second part records a continuous (real-valued) measurement. We call such type of observations "mixed-state observations". This paper introduces spatial models suited for the analysis of these kinds of data. We consider multi-parameter auto-models whose local conditional distributions belong to a mixed state exponential family. Specific examples with exponential distributions are detailed, and we present some experimental results for modelling motion measurements from video sequences.Comment: Published in at http://dx.doi.org/10.1214/08-EJS173 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Parameterized generic Galois groups for q-difference equations, followed by the appendix "The Galois D-groupoid of a q-difference system" by Anne Granier

We introduce the parameterized generic Galois group of a q-difference module, that is a differential group in the sense of Kolchin. It is associated to the smallest differential tannakian category generated by the q-difference module, equipped with the forgetful functor. Our previous results on the Grothendieck conjecture for q-difference equations lead to an adelic description of the parameterized generic Galois group, in the spirit of the Grothendieck-Katz's conjecture on p-curvatures. Using this description, we show that the Malgrange-Granier D-groupoid of a nonlinear q-difference system coincides, in the linear case, with the parameterized generic Galois group introduced here. The paper is followed by an appendix by A. Granier, that provides a quick introduction to the D-groupoid of a non-linear q-difference equation.Comment: The content of this paper was previously included in arXiv:1002.483

The CLAWAR project

In Europe, there are two main thematic groups focusing on robotics, the Climbing and Walking Robots (CLAWAR) project (http://www.clawar.net) and the European Robotics Network (EURON) project (http://www.euron.org). The two networks are complementary: CLAWAR is industrially focused on the immediate needs, and EURON is focused more on blue skies research. This article presents the activities of the CLAWAR project

Tannakian categories, linear differential algebraic groups, and parameterized linear differential equations

We provide conditions for a category with a fiber functor to be equivalent to the category of representations of a linear differential algebraic group. This generalizes the notion of a neutral Tannakian category used to characterize the category of representations of a linear algebraic group.Comment: 26 pages; corrected misprints; simplified Definition 2; more references adde

Iterative qq-Difference Galois Theory

Initially, the Galois theory of $q$-difference equations was built for $q$ unequal to a root of unity. This choice was made in order to avoid the increase of the field of constants to a transcendental field. Inspired by the work of B.H. Matzat and M. van der Put, we consider in this paper a family of iterative difference operators instead of considering just one difference operator, and in this way we stop the increase of the constant field and succeed in setting up a Picard-Vessiot theory for $q$-difference equations where $q$ is a root of unity that extend the Galois theory of difference equations of Singer and van der Put. The theory we obtain is quite the exact translation of the iterative differential Galois theory developed by B.H. Matzat and M. van der Put to the $q$-difference world

Influence d'une contamination initiale sur une dynamique spatiale non itérative

International audiencen consommateurs répartis sur un réseau spatial S choisissent tour à tour entre deux standards A et B suivant des régles locales. Un unique balayage du réseau est effectué, c'est-à-dire que la dynamique est non itérative. Dans ce cas, et contrairement aux dynamiques itératives ergodiques, les caractéristiques de la configuration spatiale finale du réseau dépendent de la configuration initiale et ne peuvent pas être évaluées mathématiquement. Nous en faisons l'étude empirique par simulation, pour un certain nombre de règles d'adoption bien spécifiées. L'objectif central de ce travail est de voir quel est l'effet d'une contamination initiale, ou effet de dumping, sur le standard A au taux τ sur la répartition spatiale finale. On évaluera en particulier de manière empirique la fréquence finale du standard A, la corrélation spatiale, ainsi que des mesures d'aggrégation et de connexité. Pour chacun de ces indicateurs, on constate que l'effet du dumping est d'autant plus important que le taux de contamination initial est faible

Duality and interval analysis over idempotent semirings

In this paper semirings with an idempotent addition are considered. These algebraic structures are endowed with a partial order. This allows to consider residuated maps to solve systems of inequalities $A \otimes X \preceq B$. The purpose of this paper is to consider a dual product, denoted $\odot$, and the dual residuation of matrices, in order to solve the following inequality $A \otimes X \preceq X \preceq B \odot X$. Sufficient conditions ensuring the existence of a non-linear projector in the solution set are proposed. The results are extended to semirings of intervals