The recurrence coefficients of a generalised symmetric Freud weight are positive solutions of a discrete equation in the discrete Painlevé I hierarchy. They also satisfy a coupled system of nonlinear differential equations. Such orthogonality weights also arise in the context of Hermitian matrix models and random symmetric matrix ensembles. In this talk I will report on properties of such special solutions of this integrable system of equations in the dP-I hierarchy, explaining the connections to other areas of mathematics.

The problem of solving Fredholm integral equations of the first kind is a prototype of an ill-posed problem of the form $T(x) =y$, where $T$ is a compact operator between Hilbert spaces. Regularization and discretization of such equations is necessary for obtaining stable approximate solutions for such problems.  For ill-posed integral equations, a quadrature based collocation method has been considered by Nair (2012) for obtaining discrete regularized approximations.  As a generalization, a projection collocation method has been proposed by the author in 2016.

We consider approximate solutions of the following  Fredholm integral equation $$x (s) - \int_0^1 \kappa (s, t) x (t) d t  = f(s), \; s \in [0, 1],$$ using projection methods.

Resumo: Nesta palestra apresentaremos condições para assegurar a existência global das soluções fraca limitadas do problema de Cauchy da equação parabólica com o termo difusivo dado pelo operador p-Laplaciano (p>2) e condição inicial u_0\in L^\infty (\R^m) \cap L^1(\R^n). Considera-se o caso em que o termo advectivo da equação estimula o crescimento da solução em certas regiões (ou mesmo no espaço todo), de modo a competir com a tendência de decaimento devido ao termo difusivo.

Abstract:  see attached file