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Applied Analysis Group
(GAMA)

Francisco Marcellán (pacomarc@ing.uc3m.es)

1. Polinomios ortogonales en espacios de Sobolev con pesos soportados en la recta real. Aplicaciones en problemas de valor inicial y problemas de frontera.

 

In the last years, Sobolev orthogonality has been one of the main generalisations of the classical idea of orthogonality for orthogonal polynomials, with on the real line and on the unit circle. It involves not only the polynomials themselves in the construction of the inner product that defines the orthogonality, but also their derivatives. As such, the construction is very suited for the approximation of solutions of initial value problems and boundary value problems in the theory of differential equations. This generalises the natural application of orthogonal polynomials (in particular classical orthogonal polynomials such as Hermite, Laguerre and Jacobi) in the construction of spectral methods for differential equations.

 

2. Transformaciones espectrales de funcionales lineales hermitianos. Aplicaciones a pares coherentes generalizados de medidas en el círculo unidad.

 

Hermitian linear functionals are natural objects in the study of orthogonal polynomials on the real line and on the unit circle. These linear functionals can be represented by Jacobi matrices, and there is a natural relation between the spectral measure of such Jacobi matrices and the measure of orthogonality of the corresponding sequence of orthogonal polynomials.  When we consider classical factorizations of this Jacobi matrix (for example LU), and then operations with the resulting factors, we obtain modifications (polynomial or rational) of the measure of orthogonality. The main goal of this project is to investigate such transformations in the context of coherent pairs of measures on the unit circle.

José Manuel Rodríguez García (jomaro@math.uc3m.es)

El teorema pequeño de Picard: la prueba de un resultado de variable compleja mediante geometría hiperbólica.

 

The main goal of this project is to prove a classical result in complex analysis, the title Picard theorem, using tools from hyperbolic geometry. An accessible and useful resource for this project is the following publication:

 

https://drive.google.com/file/d/12vCPmnzs9NK7GooVwhGi-p95ajKmwb3-/view?usp=sharing

Alfredo Deaño (alfredo.deanho@uc3m.es)

Special functions are mathematical functions used in Mathematics, Physics and Engineering, that appear in many applications (for example as solutions of differential equations) but cannot be expressed in terms of elementary functions (polynomials, rational functions, sine/cosine, exponential/logarithm). In many cases, we have identities that these functions satisfy (such as power series, integral representations, recurrence relations), and that can be used to approximate them and study their properties. The projects below explore different aspects of this work:

 

 

1. Asymptotic methods for special functions. Asymptotic expansions provide approximations for special functions in different ranges of parameters and variable, and they are a valuable tool in the study of these special functions. They can be obtained from different sources, such as integral representations or differential equations. This project will expand the results seen in the course Orthogonal polynomials and special functions, and some possible topics are WKB approximation, uniform asymptotic expansions or asympotic results obtained from Riemann–Hilbert problems.

 

2. Random matrix theory and special functions. This project is focused on the study of certain ensembles of Hermitian random matrices, with particular interest in their asymptotic properties as their size tends to infinity. The key idea is that some ensembles of random matrices can be studied using the theory of orthogonal polynomials (OPs) on the real line, more precisely a correlation kernel that is constructed from these OPs. This project will explore results on the asymptotic behavior of random matrices (eigenvalues and partition function) using this methodology, as well as related ideas from integrable systems.

 

3. Numerical methods for special functions. Special functions usually satisfy several functional identities, such as differential equations, recurrence relations in parameters or integral representations on the real line and in the complex plane. These identities are useful for numerical evaluation of these functions, but typically they have to be combined in a suitable way in order to get reliable results for different values of the variable and/or parameters. In this project we will explore these ideas, with several possible examples taken from the literature.

Yamilet Quintana (yaquinta@math.uc3m.es)

TFM proposal: Algebraic and analytic properties of degenerate Bernstein polynomials

Among the applications of the Bernstein polynomials in one variable is their use in solving problems associated with stochastic computing. Taking as a starting point the notion of stochastic logic in the sense of Qian-Riedel Rosenberg it is possible to investigate some necessary and sufficient conditions for guaranteeing whether polynomial operations can be implemented with stochastic logic based on multivariate Bernstein polynomials with coefficients in the unit interval. Recent studies suggest that the fundamental properties and identities satisfied by the degenerate Bernoulli polynomials could be used to define a special model of stochastic logic. Under my supervision, Master Thesis on the theoretical/computational   properties of degenerate Bernstein polynomials is offered.

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Complementary information: https://drive.google.com/drive/folders/14A0CzQ9O7pQ49Vc_L2b0w9ay64_p0V8y?usp=sharing

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