Georgia Institute of Technology College of Sciences

In the simplest form, our result gives a characterization of bounded,divergence-free vector fields on the plane such that the Cauchyproblem for the associated continuity equation has a unique boundedsolution (in the sense of distribution).Unlike previous results in this directions (Di Perna-Lions, Ambrosio,etc.), the proof does not rely on regularization, but rather on adimension-reduction argument which allows us to prove uniqueness usingwell-known one-dimensional results (it is indeed a variant of theclassical method of characteristics).Note that our characterization is not given in terms of functionspaces, but using a qualitative property which is completelynon-linear in character, namely a suitable weak formulation of theSard property.This is a joint work with Giovanni Alberti (University of Pisa) andStefano Bianchini (SISSA, Trieste).

Solutions of differential equations with singular source terms easily becomenon-smooth or even discontinuous. High order approximations of suchsolutions yield the Gibbs phenomenon. This results in the deterioration ofhigh order accuracy. If the problem is nonlinear and time-dependent it mayalso destroy the stability. In this presentation, we focus on thedevelopment of high order methods to obtain high order accuracy rather thanregularization methods. Regularization yields a good stability condition,but may lose the desired accuracy. We explain how high order collocationmethods can be used to enhance accuracy, for which we will adopt severalmethods including the Green’s function approach and the polynomial chaosmethod. We also present numerical issues associated with the collocationmethods. Numerical results will be presented for some differential equationsincluding the nonlinear sine-Gordon equation and the Zerilli equation.

The Faber-Krahn problem for the cube deals with understanding the function, Lambda(t) = the maximal eigenvalue of an induced t-vertex subgraph of the cube (maximum over all such subgraphs). We will describe bounds on Lambda(t), discuss connections to isoperimetry and coding theory, and present some conjectures.

We will discuss about the paper "An efficient algorithm for large-scale detection of protein families" by A Enright, S Van Dongen and C Ouzounis