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Consider the braid group $B_3=< a,b| aba=bab>$ and the nearest neighbor random walk defined by a probability $��$ with support $\{a,a^{-1},b,b^{-1}\}$. The rate of escape of the walk is explicitly expressed in function of the unique solution of a set of eight polynomial equations of degree three over eight indeterminates. We also explicitly describe the harmonic measure of the induced random walk on $B_3$ quotiented by its center. The method and results apply, mutatis mutandis, to nearest neighbor random walks on dihedral Artin groups., Published at http://dx.doi.org/10.1214/105051606000000754 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This paper is an appendix to the paper ``Randomly Growing Braid on Three Strands and the Manta Ray'' by J. Mairesse and F. Math��us (to appear in the Annals of Applied Probability). It contains the details of some computations, and the proofs of some results concerning the examples treated there, as well as some extensions., 12 pages, 4 figures. The parent paper ``Randomly Growing Braid on Three Strands and the Manta Ray'' was incorporated in the previous versions. It has been accepted for publication in the Annals of Applied Probability

This paper is an appendix to the paper "Random walks on free products of cyclic groups" by J.Mairesse and F.Math��us. It contains the details of the computations and the proofs of the results concerning the examples treated there., 15 pages, 9 figures. The parent paper "Random walks on free products of cyclic groups" is available at http://arxiv.org/abs/math.PR/0509211

Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbour random walk on G. We give a new proof of the fact that the harmonic measure is a special Markovian measure entirely determined by a finite set of polynomial equations. We show that in several simple cases of interest, the polynomial equations can be explicitely solved, to get closed form formulas for the drift. The examples considered are the modular group Z/2Z*Z/3Z, Z/3Z*Z/3Z, Z/kZ*Z/kZ, and the Hecke groups Z/2Z*Z/kZ. We also use these various examples to study Vershik's notion of extremal generators, which is based on the relation between the drift, the entropy, and the volume of the group., 20 pages, 3 figures. Added a paragraph on Free products of countable groups. Added an example in the Extremal generators section

Journal version: "Extremal throughput in free-choice nets", Discrete Event Dynamic Systems, vol. 16, n. 3, p. 327-352, 2006.; International audience; We give a method to compute the throughput in a timed live and bounded free-choice Petri net under a total allocation. We also characterize the conflict-solving policies that achieve the smallest throughput in the special case of a 1-bounded net. It does not correspond to a total allocation, but "almost".

International audience; In a heap model, solid blocks, or pieces, pile up according to the Tetris game mechanism. An optimal schedule is an infinite sequence of pieces minimizing the asymptotic growth rate of the heap. In a heap model with two pieces, we prove that there always exists an optimal schedule which is balanced, either periodic or Sturmian. We also consider the model where the successive pieces are chosen at random, independently and with some given probabilities. We study the expected growth rate of the heap. For a model with two pieces, the rate is either computed explicitly or given as an infinite series. We show an application for a system of two processes sharing a resource, and we prove that a greedy schedule is not always optimal.

Regenerative systems are able to overcome significant perturbations, and maintain autonomously their functionality in dynamic and uncertain environments. More and more this ability of biological systems plays a role in designing technical systems, e.g., in sensor networks, as well. Important properties of regenerative systems are their dynamic structures and their operation on different spatial and temporal scales. Those propel the development of new modeling, simulation, and visualization methods. Among them, variants of the π-calculus formalism, a portfolio of Gillespie related spatial simulation algorithms, means for automatically configuring simulators, and the integrated visualization methods, that make use of innovative layouts and linked and coordinated views target challenges in analyzing biological regenerative systems. They provide a basis for analyzing regenerative systems in general by means of simulation.