S1 : Lavault ---------------------------------------------------------- S2 : Younès Bennani Mustapha Lebbah (Paper 4) ---------------------------------------------------------- S3 : Gérard H. E. Duchamp to Penson Karol , Poinsot Laurent , Goodenough Silvia (Paper 2) ---------------------------------------------------------- S4 : Draief, Moez M ---------------------------------------------------------- S5 : Damien Jamet ---------------------------------------------------------- S6 : gaubert@enib.fr (Paper 3) ---------------------------------------------------------- S7: Farida Benmakrouha, Christiane Hespel , Monnier (Paper 5) ---------------------------------------------------------- S8 : Farida Benmakrouha, Christiane Hespel (Paper 6) ---------------------------------------------------------- S9 : On approximation of nonlinear generating series by rational series Mikhail V. Foursov, Christiane Hespel (Paper 7) ---------------------------------------------------------- S10 : Krob Daniel ---------------------------------------------------------- S11 : Mhalla et Simon Perdrix ---------------------------------------------------------- S12 : samuel.vidal@online.fr (Paper 1) ---------------------------------------------------------- S13 Hoang Ngoc Minh Vincel ---------------------------------------------------------- %%%%%%%%%%%%%%%%%%%%%%%%%%% S1) Title: "Asymptotic analysis and convergence of some leader election algorithms" Abstract: We start with a set of $n$ players. With some probability $P(n,k)$, we kill $n-k$ players; the other ones stay alive, and we repeat with them. What is the distribution of the number $X_n$ of rounds before getting only one player? We present a probabilistic analysis of this very general algorithm (expectation and moments of $X_n$), under some conditions on the probability distributions $P(n,k)$ (if any), including stochastic monotonicity and the assumption that roughly a fixed proportion of the players survive in each round. Applications of the general result include asymptotic analysis of leader election algorithms where players are eliminated by independent coin tosses. %%%%%%%%%%%%%%%%%%%%%%%%%%%% S2) 3M-SOM : Learning Self Organizing Maps as a Markov Mixture Models Mustapha Lebbah and Younès Bennani Abstract: This paper describes a new algorithm to learn Self-Organizing Maps (SOM) as a Markov Mixture Models The new model that we present is valid for all structure of graphical model. 3M-SOM realizes an unsupervised learning using unlabelled evolutionary data sets, namely those that describe sequential data. We use E-M (Expectation-Maximisation) standard algorithm to maximize the likelihood. The graph structure is integrated in the parameter estimation of markov model using a neighborhood function to learn a topographic clustering. The 3M-SOM model provides a self-organizing markov model using an original learning algorithm. %%%%%%%%%%%%%%%%%%%%%%%%%%%% S3) Auteurs : Laurent Poinsot, Silvia Goodenough, Karol A. Penson and GHED Titre : Gases of graphs, exponential formula and Combinatorial Physics In 1953, the Chemical Physicists R. J. Ridell and G. E. Uhlenbeck gave a statement of the "Exponential formula" (sometimes called "isomer expansion"). This formula is essentially based on a (mixed) bivariate stastistics relating to one-parameter groups. A quick review of some models is given here as well as some contemporary open problems in Combinatorial Physics. %%%%%%%%%%%%%%%%%%%%%%%%%%%%% S4) Exponential random graphs as models of overlay networks Moez Draief (Imperial College London) In this talk, we describe a Metropolis-Hastings type algorithm to construct optimised overlay networks in the context of Peer-to-Peer systems to provide load balancing and minimise communication costs. We show that the graphs obtained are distributed according to a Gibbs like distribution $?(G) = \frac{1}{Z}e^{-\beta sum_i d_i^2} $ on a subset of of the set of connected graphs with $n$ nodes, $d_i$ being the degree of node i. Using analogies with the configuration model for graph generation we derive a number of asymptotic results for the properties of such graphs. More precisely we show that the degrees are concentrated around their mean value, derive asymptotic results on the number of edges crossing a graph cut and use these results (i) to compute the graph expansion and conductance, and (ii) to analyse the graph resilience to random failures. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S5) Damien Jamet Coding rhombus tilings by multidimensional words: a first attempt Since the discovery of quasicrystals, rhombus tilings have provided well adapted models for quasicrystalline alloys. In the present work, we focus on a particular class of rhombus tilings, namely the ones one can code with a 2D-sequence over the three-letter alphabet {1,2,3}. Among these tilings, we deeply investigate the ones coding radomized rhombus tilings and the ones coding the trajectories of an element of the torus R/Z under the action of two irrational rotations. Firstly, we study the recognition problem of 2D-sequences coding a rhombus tiling: given a 2D-sequence, does it represent a rhombus tiling ? We then show that the set of such sequences is of finite type, that is, a sequence coding a rhombus tiling is entirely characterized by a finite set of its configurations. In the second part of this talk, we investigate the combinatoric properties of such tilings such as, for instance, the enumeration of local configurations. We will end this talk by stating several perspectives and challenges of these researchs, in computer science as well as in mathematics. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S6) Laurent Gaubert We address the question of frequencies locking in coupled differential systems, related to the existence of quasi-periodic solutions of differential systems. Our tool is what we call ``cellular systems'', quite general as it deals with countable number of coupled systems in some general Banach spaces. Moreover, the inner dynamics of each subsystem does not have to be specified. We reach some general results about how the frequencies locking phenomenon is related to the structure of the coupling map, and therefore about the localization of quasi-periodic solutions of some differential systems that may be seen as cellular systems. This paper gives some explanations about how and why synchronized behaviors naturally occur it a wide variety of complex systems. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S7) Farida Benmakrouha, Christiane Hespel, Edouard Monnier : Drawing approximated solution curve of differential equation Abstract We develop a method for drawing approximated solution curves of differential equations. This method is based on the juxtaposition of local approximating curves on successive intervals [ti, ti+1]0in-1. The differential equation, considered as a dynamical system, is described by its state equations and its initial value at t = t0. A generic expression of its generating series Gt truncated at any order k, of the output and its derivatives y(j)(t) expanded at any order k, can be calculated. These expressions are obtained from the vector fields, from the observation of the state at time t, in the state equations [2, 3]. More precisely, the coefficients of Gt are expressed in terms of the vector fields and the observation. The output and its derivatives y(j)(t) are expressed in terms of the coefficients of the series Gt and of the Chen series [1]. At every initial point of the present interval, we specify the previous expressions of Gt and y(j)(t) for t = ti. Then we obtain an approximated output y(t) at order k in every interval [ti, ti+1]0in-1. By using Maple system, we have developed a package corresponding to the creation of the generic expression of Gt and y(j)(t) at order k and to the drawing of the local curves on every interval [ti, ti+1]0in-1, by iterations on the initial points t = (ti)0in-1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S8) Generating series : a combinatorial computation F.Benmakrouha, C Hespel We study, in this paper, the reliability and the quality of a model for non-linear black-box identification. This modelling is of an unknown dynamical system () by a family (Bk) of bilinear systems. Two formal power series in noncommutative variables are used for describing () : the generating series for the system’s behavior (G) and the Chen series for the system’s input. The family (Bk) of bilinear systems is described by its rational generating series (Gk) such that the coefficients of (G) and (Gk) coincide up to order k. We investigate the quality of the model throughout two criteria : a convergence’s measure, an amplitude’s overestimation of the outputs (¯yk) of systems (Bk). We provide, by a symbolic computation, a valuation relating to the convergence of the family (Bk). This computation is a sum of differential monomials in the input functions and behavior system. We identify each differential monomial with its colored multiplicity and analyse our computation in the light of the free differential calculus. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S9) On approximation of nonlinear generating series by rational series Mikhail V. Foursov IRISA/Universit´e de Rennes–I Campus Universitaire de Beaulieu 35042 Rennes Cedex France Christiane Hespel IRISA/INSA de Rennes 20, avenue des Buttes de Co¨esmes 35043 Rennes Cedex France %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S10) How to model Complex Systems? Daniel Krob Abstract---We propose a general mathematical definition of a system jointly with system transformation operators that model respectively system integration and ''grain'' level description mechanisms. We deduce then from it a generic notion of system model consisting in an exhaustive series of views allowing to represent completely any system. We finally explain how to use practically this formalism which leads us to the concept of system architecture. All these notions are illustrated by a case study. Index Terms---Architecture, Model, System, View. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S11) Auteurs: Mehdi Mhalla (CNRS, Laboratoire d’Informatique de Grenoble) et Simon Perdrix (LFCS, University of Edinburgh et PPS, Universit´e Paris Diderot ) Titre: Graph Theory for quantum computing Abstract: Graph theory plays an important role in quantum computing. The connections between graph theory and quantum computing have mainly emerged from the development of the graph state formalism [1] and the one-way model of quantum computation [4]. A graph state is a compact representation of quantum states which consists in representing a quantum state using a graph underlying entanglement properties. Graph states have several applications in quantum information theory: quantum protocols (e.g. quantum secret sharing), quantum correcting codes and models of quantum computing (one-way model). The one-way model, introduced by Briegel and Raussendorf is one of the most promising model of quantum computing. We present here a new result about the universality for quantum computing of a family of graphs, and summarize some previous results [2, 3, 5] that ex- hibits the strong connection between graph theory and quantum computing answering the folowing questions: What are the graph properties that characterize the complexity of preparing a graph state? Which graphs can be used for quantum computing? How to characterize the depth complexity of a computation? Which families of graphs are universal for quantum computing? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S12) samuel.vidal@online.fr Counting unrooted triangulations A triangulation of a compact oriented surface is described by two permutations satifying simple relations. We propose to present a counting of triangulations up to isomorphism using cycle index generating series. The conjugacy classes of subgroups in the modular groups SL(2,Z) can be counted using similar techniques. We have writen a program to generate an exhaustive list of those triagulations. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S13) About a group of Drinfel'd associators Hoang Ngoc Minh1 Extended abstract The biological activivity of DNA molecule depends mainly on the way it is arranged in space and the way in which it is twisted { things which fall within the province of the mathematical theory of knots [5]. There is a real parallel between mathematical transformations of knots (see the Conway's ip and uncrossing elementary operations) and enszymatic mechanisms (the topo-isomerases) [9].