Search

We provide a complete and rigorous description of phase transitions for kinetic models of self-propelled particles interacting through alignment. These models exhibit a competition between alignment and noise. Both the alignment frequency and noise intensity depend on a measure of the local alignment. We show that, in the spatially homogeneous case, the phase transition features (number and nature of equilibria, stability, convergence rate, phase diagram, hysteresis) are totally encoded in how the ratio between the alignment and noise intensities depend on the local alignment. In the spatially inhomogeneous case, we derive the macroscopic models associated to the stable equilibria and classify their hyperbolicity according to the same function., Archive for Rational Mechanics and Analysis (2014) accepted

In view of applications to conformal field theory or to other branches of theoretical physics and mathematics, new examples of character tables for Drinfeld doubles of finite groups (modular data) are made available on a website., Comment: 7 pages, 1 figure, 7th International Conference on Mathematical Methods in Physics, Rio de Janeiro, Brazil, April 2012. Version 2: a misleading sentence was removed from section 2. http://pos.sissa.it/archive/conferences/175/024/ICMP%202012_024.pdf

In this paper, we introduce a novel inference method for dynamic genetic networks which makes it possible to face a number of time measurements n that is much smaller than the number of genes p. The approach is based on the concept of a low order conditional dependence graph that we extend here in the case of dynamic Bayesian networks. Most of our results are based on the theory of graphical models associated with the directed acyclic graphs (DAGs). In this way, we define a minimal DAG G which describes exactly the full order conditional dependencies given in the past of the process. Then, to face with the large p and small n estimation case, we propose to approximate DAG G by considering low order conditional independencies. We introduce partial qth order conditional dependence DAGs G(q) and analyze their probabilistic properties. In general, DAGs G(q) differ from DAG G but still reflect relevant dependence facts for sparse networks such as genetic networks. By using this approximation, we set out a non-Bayesian inference method and demonstrate the effectiveness of this approach on both simulated and real data analysis. The inference procedure is implemented in the R package 'G1DBN' freely available from the R archive (CRAN).

We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain ${\mathcal D} \subset {\mathbb R}^d$ until it hits the boundary and bounces randomly inside according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord "picked at random" in ${\mathcal D}$, and we study the angle of intersection of the process with a $(d-1)$-dimensional manifold contained in ${\mathcal D}$., Comment: 50 pages, 10 figures; To appear in: Archive for Rational Mechanics and Analysis; corrected Theorem 2.8 (induced chords in nonconvex subdomains)

We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a $\C^*$-action. For varieties with an isolated singularity, we show that the presence of sufficiently many rational curves outside the singular point strongly affects the character of the singularity. This provides an explanation of classical results due to H.A. Schwartz and G.H. Halphen on polynomial solutions of the generalized Fermat equation., Comment: 24 pages; uses Paul Taylor's diagrams.tex, see e.g. ftp://ftp.dante.de/tex-archive/macros/generic/diagrams/taylor/V3,88.tex