Seminario MATESCO    

En esta charla estudiamos el valor esperado de la energía logarítmica de las soluciones del polynomial eigenvalue problem. Esto extiende resultados ya conocidos para dos casos importantes: los polinomios de Shub-Smale y el spherical ensamble, los cuales son ejemplos extremos del problema.

Los resultados obtenidos muestran que la energía del polynomial eigenvalue problem se encuentra entre la de estos dos ejemplos y obteniendose la mínima en el caso de los polinomios de Shub-Smale.

(Trabajo en conjunto con D. Armentano y M. Fiori)

In this talk we consider some well known quotients related with either eigenvalue problems,
Sobolev or Hardy's inequality. We consider the infimum of these quotients and their discrete
analogous in a finite element subspace. We estimate the difference between the best constants
above as the discretization parameter goes to zero and obtain sharp convergence rates.

Robust estimation of models to multivariate circular data (torus data) is considered based on the weighted likelihood methodology. Robust model fitting is achieved by a set of estimating equations based on the computation of data dependent weights aimed to down-weight anomalous values, such as unexpected directions that do not share the main pattern of the bulk of the data. To solve these equations an algorithm based on a data augmentation approach and a suitable modification of the Expectation-Maximization (EM) algorithm is proposed. Advantages and disadvantages of a Classification EM algorithm is also discussed. Asymptotic properties and robustness features of the estimators under study are presented, whereas their finite sample behavior has been investigated by Monte Carlo numerical experiment and real data examples.

First, we will introduce the notion of equidistribution of a finite point set W on the sphere and a way to measure its quality of dispersion: the discrepancy of W. We give a brief motivation of why this might be important by mentioning Koksma-Hlawka type inequalities and show some real-world realizations of equidistributed sets. We then proceed to give a construction of a sequence on the sphere for which we can bound the discrepancy. This talk is based on the paper: "Uniform distribution via lattices: from point sets to sequences".

The aim of the talk is to present myself as a new member of the department and give an overview of my previous work and stations. This is intended to be a rather informal talk without too much reliance on rigorous presentation, or proofs. Thematically, I will cover my research in various areas of randomness, foremost quasi Monte Carlo methods, but also others. While the (mathematical) applications lie rather in the area of numerical mathematics, specifically highly multivariate numerical integration and statistics, the background comes from diverse areas, such as combinatorics, coding theory but also algebraic geometry. The industrial applications include simulation, image processing and financial mathematics. Further topics that may be touched are pseudo random numbers and cryptology, i.e., the construction and analysis of deterministically produced point sets under the requirement of various paradigms of randomness.

In biomechanics, local phenomena, such as tissue perfusion, are strictly related to the global features of the whole blood circulation. We propose a heterogeneous model where a local, accurate, 3D description of tissue perfusion by means of poroelastic equations is coupled with a systemic 0D lumped model of the remainder of the circulation. This represents a multiscale strategy, which couples an initial boundary value problem to be used in a specific tissue region with an initial value problem in the rest of the circulatory system. We discuss well-posedness analysis for this multiscale model, as well as solution methods focused on a detailed comparison between functional iterations and an energy-based operator splitting method and how they handle the interface conditions.

p-adic integrable systems and symplectic manifolds

Luis Crespo Ruiz (Universidad de Cantabria)

The notions of symplectic manifold and integrable system are usually formulated over the real field. However, some discoveries in mathematical physics (by B. Dragovich and others) lead to the question of whether these notions can be extended to p-adic fields. A. Pelayo, V. Voevodsky and M. Warren laid the foundations for these definitions a decade ago. I will present new formulations and results for p-adic symplectic geometry, centered at the p-adic version of the Jaynes-Cummings system. Some aspects of this system are familiar to us, such as the fact that the fiber of (-1,0) is the only one that may contain an isolated point (depending on p), but unlike the real case, there are more points in that fiber, and they are not isolated. This is joint work with Alvaro Pelayo.

Stable discontinuous stationary solutions to reaction-diffusion systems

Grzegorz Karch (University of Wroclaw, Poland)

I shall review results, obtained jointly with Anna Marciniak-Czochra, Kanako Suzuki, and Szymon Cygan, on a certain class of reaction-diffusion systems from Mathematical Biology, where ordinary differential equations are coupled with one reaction-diffusion equation. Such systems may have regular (i.e. sufficiently smooth) stationary solutions, however, all of them are unstable. We showed that solutions to such problems may behave in a singular way for large values of time and converge towards discontinuous stationary solutions.