Monotone thematic factorizations of matrix functions

We continue the study of the so-called thematic factorizations of admissible very badly approximable matrix functions. These factorizations were introduced by V.V. Peller and N.J. Young for studying superoptimal approximation by bounded analytic matrix functions. Even though thematic indices associated with a thematic factorization of an admissible very badly approximable matrix function are not uniquely determined by the function itself, R.B. Alexeev and V.V. Peller showed that the thematic indices of any monotone non-increasing thematic factorization of an admissible very badly approximable matrix function are uniquely determined. In this paper, we prove the existence of monotone non-decreasing thematic factorizations for admissible very badly approximable matrix functions. It is also shown that the thematic indices appearing in a monotone non-decreasing thematic factorization are not uniquely determined by the matrix function itself. Furthermore, we show that the monotone non-increasing thematic factorization gives rise to a great number of other thematic factorizations.Comment: To appear in Journal of Approximation Theor

Unitary interpolants and factorization indices of matrix functions

For an $n\times n$ bounded matrix function $\Phi$ we study unitary interpolants $U$, i.e., unitary-valued functions $U$ such that $\hat U(j)=\hat\Phi(j)$, $j<0$. We are looking for unitary interpolants $U$ for which the Toeplitz operator $T_U$ is Fredholm. We give a new approach based on superoptimal singular values and thematic factorizations. We describe Wiener--Hopf factorization indices of $U$ in terms of superoptimal singular values of $\Phi$ and thematic indices of $\Phi-F$, where $F$ is a superoptimal approximation of $\Phi$ by bounded analytic matrix functions. The approach essentially relies on the notion of a monotone thematic factorization introduced in [AP]. In the last section we discuss hereditary properties of unitary interpolants. In particular, for matrix functions $\Phi$ of class H^\be+C we study unitary interpolants $U$ of class $QC$.Comment: 20 page

Algorithms for curve design and accurate computations with totally positive matrices

Esta tesis doctoral se enmarca dentro de la teoría de la Positividad Total. Las matrices totalmente positivas han aparecido en aplicaciones de campos tan diversos como la Teoría de la Aproximación, la Biología, la Economía, la Combinatoria, la Estadística, las Ecuaciones Diferenciales, la Mecánica, el Diseño Geométrico Asistido por Ordenador o el Álgebra Numérica Lineal. En esta tesis nos centraremos en dos de los campos que están relacionados con matrices totalmente positivas.This doctoral thesis is framed within the theory of Total Positivity. Totally positive matrices have appeared in applications from fields as diverse as Approximation Theory, Biology, Economics, Combinatorics, Statistics, Differential Equations, Mechanics, Computer Aided Geometric Design or Linear Numerical Algebra. In this thesis, we will focus on two of the fields that are related to totally positive matrices.<br /

Superfiltered A∞A_\infty-deformations of the exterior algebra, and local mirror symmetry

The exterior algebra $E$ on a finite-rank free module $V$ carries a $\mathbb{Z}/2$-grading and an increasing filtration, and the $\mathbb{Z}/2$-graded filtered deformations of $E$ as an associative algebra are the familiar Clifford algebras, classified by quadratic forms on $V$. We extend this result to $A_\infty$-algebra deformations $\mathcal{A}$, showing that they are classified by formal functions on $V$. The proof translates the problem into the language of matrix factorisations, using the localised mirror functor construction of Cho-Hong-Lau, and works over an arbitrary ground ring. We also compute the Hochschild cohomology algebras of such $\mathcal{A}$. By applying these ideas to a related construction of Cho-Hong-Lau we prove a local form of homological mirror symmetry: the Floer $A_\infty$-algebra of a monotone Lagrangian torus is quasi-isomorphic to the endomorphism algebra of the expected matrix factorisation of its superpotential.Comment: v3 Improved exposition and corrected some mistakes in light of referee comments. Accepted version, to appear in JLM

Numerical Linear Algebra applications in Archaeology: the seriation and the photometric stereo problems

The aim of this thesis is to explore the application of Numerical Linear Algebra to Archaeology. An ordering problem called the seriation problem, used for dating findings and/or artifacts deposits, is analysed in terms of graph theory. In particular, a Matlab implementation of an algorithm for spectral seriation, based on the use of the Fiedler vector of the Laplacian matrix associated with the problem, is presented. We consider bipartite graphs for describing the seriation problem, since the interrelationship between the units (i.e. archaeological sites) to be reordered, can be described in terms of these graphs. In our archaeological metaphor of seriation, the two disjoint nodes sets into which the vertices of a bipartite graph can be divided, represent the excavation sites and the artifacts found inside them. Since it is a difficult task to determine the closest bipartite network to a given one, we describe how a starting network can be approximated by a bipartite one by solving a sequence of fairly simple optimization problems. Another numerical problem related to Archaeology is the 3D reconstruction of the shape of an object from a set of digital pictures. In particular, the Photometric Stereo (PS) photographic technique is considered

Stability of traveling waves in partly parabolic systems

Abstract. We review recent results on stability of traveling waves in partly parabolic reactiondiffusion systems with stable or marginally stable equilibria. We explain how attention to what are apparently mathematical technicalities has led to theorems that allow one to convert spectral calculations, which are used in the sciences and engineering to study stability of a wave, into detailed, theoretically-based information about the behavior of perturbations of the wave