In a variety of problems in engineering and applied science, the goal is to design or control a network of dynamical agents so as to achieve some desired asymptotic behaviour. Examples include consensus and rendez-vous problems in robotics, synchronization of generator angles in power grids or coordination of oscillations in bacterial populations. A pressing challenge in all of these problems is to derive appropriate analytical tools to prove convergence towards the target behaviour. Such tools are not only invaluable to guarantee the desired performance, but can also provide important guidelines for the design of decentralized control strategies to steer the collective behaviour of the network of interest in a desired manner. During this talk, a methodology for analysis and design of convergence in networks will be presented which is based on the use of a classical, yet not fully exploited, tool for convergence analysis: contraction theory. As opposed to classical methods for stability analysis, the idea is to look at convergence between trajectories of a system of interest rather that at their asymptotic convergence towards some solution of interest. After introducing the problem, a methodology will be derived based on the use of matrix measures induced by non-Euclidean norms that will be exploited to design strategies to control the collective behaviour of networks of dynamical agents. Representative examples will be used to illustrate the theoretical results.
The self-consistent analytic theory of the wind-driven sea can be developed due to the presence of small parameter, ratio of atmospheric and water densities. Because of low value of this parameter the sea is "weakly nonlinear" and the average steepness of sea surface is also relatively small. Nevertheless, the weakly nonlinear four-wave resonant interaction is the dominating process in the energy balance. The wind-driven sea can be described statistically in terms of the Hasselmann kinetic equation.
This equation has a rich family of Kolmogorov-type solutions perfectly describing "rear faces" of wave spectra right behind the spectral peak.
More short waves are described by steeper Phillips spectrum formed by ensemble of microbreakings. From the practical view-point the most important question is the spatial and temporal evolution of spectral peaks governed by self-similar solutions of the Hasselmann equation. This analytic theory is supported by numerous experimental data and computer
simulations.
(Note change in time and location)
The purpose of this talk is to give an introduction to the theory and
practice of integer factorization. More precisely, I plan to talk about the
p-1 method, the elliptic curve method, the quadratic sieve, and if time
permits the number field sieve.