On the convergence of type I Hermite-Padé approximants for rational perturbations of a Nikishin system

We study the convergence of type I Hermite-Padé approximation for a class of meromorphic functions obtained by adding a vector of rational functions with real coefficients to a Nikishin system of functions.Both authors were partially supported by research grant MTM2012-36372-C03-01 of Ministerio de Economía y Competitividad, Spain

Alternating Minimization for Regression with Tropical Rational Functions

We propose an alternating minimization heuristic for regression over the space of tropical rational functions with fixed exponents. The method alternates between fitting the numerator and denominator terms via tropical polynomial regression, which is known to admit a closed form solution. We demonstrate the behavior of the alternating minimization method experimentally. Experiments demonstrate that the heuristic provides a reasonable approximation of the input data. Our work is motivated by applications to ReLU neural networks, a popular class of network architectures in the machine learning community which are closely related to tropical rational functions

The bubbles of matter from multiskyrmions

The multiskyrmions with large baryon number B given by rational map (RM) ansaetze can be described reasonably well within the domain wall approximation, or as spherical bubbles with energy and baryon number density concentrated at their boundary. A special class of profile functions is considered approximating the true profile and domain wall behaviour at the same time. An upper bound is obtained for the masses of RM multiskyrmions which is close to the calculated masses, especially at large B. The gap between rigorous upper and lower bounds for large B multiskyrmions is less than 4%. The basic properties of such bubbles of matter are investigated, some of them being of universal character, i.e. they do not depend on baryon number of configuration and on the number of flavors. As a result, the lagrangian of the Skyrme type models provides field theoretical realization of the bag model of special kind.Comment: 7 pages, no figure

Arithmetic and dynamical systems

In this thesis we look at a number of topics in the area of the interaction between dynamical systems and number theory. We look at two diophantine approximation problems in local �fields of positive characteristic, one a generalisation of the Khintchine{Groshev theorem, another a central limit theorem. We also prove a P�olya{Carlson dichotomy result for a large class of adelicly perturbed rational functions. In particular we prove that for a finite set of primes S that the power series f(z) generated by the Fibonacci series with all primes in S removed has a natural boundary