We study a recent model of random networks based on the presence of an intrinsic character of the vertices called fitness. The vertices fitnesses are drawn from a given probability distribution density. The edges between pair of vertices are drawn according to a linking probability function depending on the fitnesses of the two vertices involved. We study here different choices for the probability distribution densities and the linking functions. We find that, irrespective of the particular choices, the generation of scale-free networks is straightforward. We then derive the general conditions under which scale-free behavior appears. This model could then represent a possible explanation for the ubiquity and robustness of such structures.

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We give a not trivial upper bound on the velocity of disturbances in an infinitely extended anharmonic system at thermal equilibrium. The proof is achieved by combining a control on the non equilibrium dynamics with an explicit use of the state invariance with respect to the time evolution.

We study the time evolution of an incompressible Euler fluid with planar symmetry when the vorticity is initially concentrated in small disks. We discuss how long this concentration persists, showing that in some cases this happens for quite long times. Moreover, we analyze a toy model that shows a similar behavior and gives some hints on the original problem.2010 Mathematics Subject Classification. 76B47, 37N10, 70K65.

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a "soft" repulsion from the boundary. We finally show how a "hard" repulsion can be obtained by an extra diffusive scaling.

Abstract. We consider a conservative system of stochastic PDE's, namely a weakly coupled, one dimensional phase field model with additive noise. We study the fluctuations of the front proving that, in a suitable scaling limit, the front evolves according to a non-Markov process, solution of a linear stochastic equation with long memory drift. Part I. Introduction 1 General setting, model, and resultsThe term "sharp interface limit" denotes a scaling procedure aimed at the derivation of interfaces as geometric objects, e.g. surfaces of codimension one with bounded variation, that is, enough regular for the area measure to be well defined. Of course this makes only sense in the context of systems which undergo phase transitions and of states where different phases coexist. In the limit the other degrees of freedom are lost and we are left with the interface alone. Rigorous proofs are hard, yet a great variety of models has been successfully worked out. The mathematics involved is correspondingly rich, e.g. the theory of Γ-convergence (to study the sharp interface limit of Ginzburg-Landau like free energy functionals in relation with the equilibrium shape of the interface, as in the Wulff problem) and correspondingly the theory of Gibbsian large deviations (to study the same problems at the more microscopic level of statistical mechanics); singular limit in PDE's, like in the Allen-Cahn, Cahn-Hilliard and phase field equations, and correspondingly, at the microscopic level, hydrodynamic limits of spin or particle systems.This paper deals with fluctuations. Here again the questions are, first, whether in a sharp interface limit the system is described by a [fluctuating] interface with closed equations of motion and, secondly, the nature of such equations. The problem greatly simplifies in one space dimension where the limit interface is represented by a point which separates the two phases (one to its left and the other one *

The computation of the structured pseudospectral abscissa and radius (with respect to the Frobenius norm) of a Toeplitz matrix is discussed and two algorithms based on a low rank property to construct extremal perturbations are presented. The algorithms are inspired by those considered in [GO11] for the unstructured case, but their extension to structured pseudospectra and analysis presents several difficulties. Natural generalizations of the algorithms, allowing to draw significant sections of the structured pseudospectra in proximity of extremal points are also discussed. Since no algorithms are available in the literature to draw such structured pseudospectra, the approach we present seems promising to extend existing software tools (Eigtool [Wri02], Seigtool [KKK10]) to structured pseudospectra representation for Toeplitz matrices. We discuss local convergence properties of the algorithms and show some applications to a few illustrative examples.2010 Mathematics Subject Classification. 65F15, 65L07.

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