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)
Several proofs of Picard's small theorem (TFG or TFM)
Picard's small theorem is an important result in Complex Analysis that states that if f is an entire function (a complex function that is holomorphic everywhere in the complex plane) and if f omits two distinct complex values, then f must be a constant function.
The goal of this project is to present several proofs of this theorem, showing the great variety of approaches, ideas and methods that allow obtaining this result.
Each proof highlights different aspects of complex analysis and can be useful depending on the context and the tools available.
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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1) TFG proposal. The Beurling Theorem: An insight into the Invariant Subspace Problem.
The Invariant Subspace Problem is one of the most famous, and yet unsolved, problems in Operator Theory. It asks if every linear bounded operator acting on a complex Hilbert space has non-trivial closed invariant subspace. The main goal of this work is to prove the Beurling Theorem, which characterizes all the closed invariant subspaces of the shift operator acting on l2, that is, the Hilbert space of square-summable sequences. This result involves and connects various areas of Mathematical Analysis, such as Complex Analysis, Functional Analysis or Measure Theory.
2) TFG proposal. Semigroups of analytic functions on the disc: A first step into complex dynamics
In this work, we propose to study the continuous semigroups of holomorphic functions on the unit disc of the complex plane. We will study the main dynamic properties of such semigroups, and will present a classification of them. If time permits, we will analyze the relation of these semigroups with C0-semigroups of operators acting on some Hilbert spaces of analytic functions, and will present some results regarding the spectral properties of the semigroups.
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1) Sobolev orthogonal polynomials for solving boundary value problems.
Spectral methods for solving differential equations are among the most efficient solvers for differential equations [1]. These spectral methods are typically based on classical orthogonal polynomials. In this project the goal is to develop a new class of spectral methods based on Sobolev orthogonal polynomails (SOPs). These SOPs can be generated numerically using linear algebra techniques [2]. In this project we will develop a solver for boundary value problems based on SOPs. Namely, we will generate a basis of SOPs which are closely related to the boundary value problem (BVP) and develop an automatic procedure to take the boundary conditions into account. Planning of the research project:
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1) Study the linear algebra algorithm [2] which is used for constructing SOPs.
2) For some interesting BVPs we will derive the weak formulation, which gives rise to an inner product that can be used to generate a good basis of SOPs.
3) Develop an algorithm that can take boundary conditions into account, this can be done either algebraically or analytically [1].
4) Compare the developed algorithm to other algorithms and perform an analysis of its numerical behavior.
[1] Lloyd N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000. URL: https://doi.org/10.1137/1.9780898719598
[2] Niel Van Buggenhout, On generating Sobolev orthogonal polynomials. Numer. Math. 155, 415–443 (2023). URL: https://doi.org/10.1007/s00211-023-01379-3
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2) Rational quantum algorithms
Rational functions are essential in classical computing, they allow to compute interior eigenvalues of huge matrices and approximate matrix functions which have discontinuities of jumps. Recent advancements in quantum computing have made it possible to compute with rational functions. The computational efficiency of these quantum algorithms for computing with rational functions strongly depend on the location of the poles of the rational function. Due to this dependency, classical rational algorithms cannot be translated directly to a quantum algorithm; it is necessary to rethink how these rational functions can be used in quantum computing. In this project we will develop novel rational quantum algorithms, compare them to existing approaches and quantify their cost.
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