Funções, funcionais e operadores na física matemática, com índices matriciais

With the advance of mathematical methods throughout the centuries, in particular with respect to the differential calculus, the notion of fractional derivative emerged with Leibniz and later developed by several well known scientists. Today that formalism is well used in the study of diffusion phenomena among other areas. We extend the fractional indices to matricial indices and develop a formalism to handle this generalized derivative, as well as other operators, functions and functionals in mathematical physics, originally defined for natural indices. Here we only consider 2x2 hermitian and anti-hermitian matrices. These matrices are associated to the well known Pauli matrices and Hamilton's quaternions. Applications with mathematical physics functions are presented

Integral form of Yang-Mills equations and its gauge invariant conserved charges

Despite the fact that the integral form of the equations of classical electrodynamics is well known, the same is not true for non-Abelian gauge theories. The aim of the present paper is threefold. First, we present the integral form of the classical Yang-Mills equations in the presence of sources and then use it to solve the long-standing problem of constructing conserved charges, for any field configuration, which are invariant under general gauge transformations and not only under transformations that go to a constant at spatial infinity. The construction is based on concepts in loop spaces and on a generalization of the non-Abelian Stokes theorem for two-form connections. The third goal of the paper is to present the integral form of the self-dual Yang-Mills equations and calculate the conserved charges associated with them. The charges are explicitly evaluated for the cases of monopoles, dyons, instantons and merons, and we show that in many cases those charges must be quantized. Our results are important in the understanding of global properties of non-Abelian gauge theories.

The concept of quasi-integrability for modified non-linear Schrodinger models

We consider modifications of the nonlinear Schrodinger model (NLS) to look at the recently introduced concept of quasi-integrability. We show that such models possess an in finite number of quasi-conserved charges which present intriguing properties in relation to very specific space-time parity transformations. For the case of two-soliton solutions where the fields are eigenstates of this parity, those charges are asymptotically conserved in the scattering process of the solitons. Even though the charges vary in time their values in the far past and the far future are the same. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. Our findings may have important consequences on the applications of these models in several areas of non-linear science. We make a detailed numerical study of the modified NLS potential of the form V similar to (vertical bar psi vertical bar(2))(2+epsilon), with epsilon being a perturbation parameter. We perform numerical simulations of the scattering of solitons for this model and find a good agreement with the results predicted by the analytical considerations. Our paper shows that the quasi-integrability concepts recently proposed in the context of modifications of the sine-Gordon model remain valid for perturbations of the NLS model.

Sharp estimates for eigenvalues of integral operators generated by dot product kernels on the sphere

We obtain explicit formulas for the eigenvalues of integral operators generated by continuous dot product kernels defined on the sphere via the usual gamma function. Using them, we present both, a procedure to describe sharp bounds for the eigenvalues and their asymptotic behavior near 0. We illustrate our results with examples, among them the integral operator generated by a Gaussian kernel. Finally, we sketch complex versions of our results to cover the cases when the sphere sits in a Hermitian space.

Some aspects of self-duality and generalised BPS theories

If a scalar eld theory in (1+1) dimensions possesses soliton solutions obeying rst order BPS equations, then, in general, it is possible to nd an in nite number of related eld theories with BPS solitons which obey closely related BPS equations. We point out that this fact may be understood as a simple consequence of an appropriately generalised notion of self-duality. We show that this self-duality framework enables us to generalize to higher dimensions the construction of new solitons from already known solutions. By performing simple eld transformations our procedure allows us to relate solitons with di erent topological properties. We present several interesting examples of such solitons in two and three dimensions.

A Skyrme-like model with an exact BPS bound

We propose a new Skyrme-like model with fields taking values on the sphere S3 or, equivalently, on the group SU(2). The action of the model contains a quadratic kinetic term plus a quartic term which is the same as that of the Skyrme-Faddeev model. The novelty of the model is that it possess a first order Bogomolny type equation whose solutions automatically satisfy the second order Euler-Lagrange equations. It also possesses a lower bound on the static energy which is saturated by the Bogomolny solutions. Such Bogomolny equation is equivalent to the so-called force free equation used in plasma and solar Physics, and which possesses large classes of solutions. An old result due to Chandrasekhar prevents the existence of finite energy solutions for the force free equation on the entire three- dimensional space R3. We construct new exact finite energy solutions to the Bogomolny equations for the case where the space is the three-sphere S3, using toroidal like coordinates.

Heterogeneous multicore systems for signal processing

This thesis explores the capabilities of heterogeneous multi-core systems, based on multiple Graphics Processing Units (GPUs) in a standard desktop framework. Multi-GPU accelerated desk side computers are an appealing alternative to other high performance computing (HPC) systems: being composed of commodity hardware components fabricated in large quantities, their price-performance ratio is unparalleled in the world of high performance computing. Essentially bringing “supercomputing to the masses”, this opens up new possibilities for application fields where investing in HPC resources had been considered unfeasible before. One of these is the field of bioelectrical imaging, a class of medical imaging technologies that occupy a low-cost niche next to million-dollar systems like functional Magnetic Resonance Imaging (fMRI). In the scope of this work, several computational challenges encountered in bioelectrical imaging are tackled with this new kind of computing resource, striving to help these methods approach their true potential. Specifically, the following main contributions were made: Firstly, a novel dual-GPU implementation of parallel triangular matrix inversion (TMI) is presented, addressing an crucial kernel in computation of multi-mesh head models of encephalographic (EEG) source localization. This includes not only a highly efficient implementation of the routine itself achieving excellent speedups versus an optimized CPU implementation, but also a novel GPU-friendly compressed storage scheme for triangular matrices. Secondly, a scalable multi-GPU solver for non-hermitian linear systems was implemented. It is integrated into a simulation environment for electrical impedance tomography (EIT) that requires frequent solution of complex systems with millions of unknowns, a task that this solution can perform within seconds. In terms of computational throughput, it outperforms not only an highly optimized multi-CPU reference, but related GPU-based work as well. Finally, a GPU-accelerated graphical EEG real-time source localization software was implemented. Thanks to acceleration, it can meet real-time requirements in unpreceeded anatomical detail running more complex localization algorithms. Additionally, a novel implementation to extract anatomical priors from static Magnetic Resonance (MR) scansions has been included.

Variations of the FEAST Eigenvalue Algorithm

FEAST is a recently developed eigenvalue algorithm which computes selected interior eigenvalues of real symmetric matrices. It uses contour integral resolvent based projections. A weakness is that the existing algorithm relies on accurate reasoned estimates of the number of eigenvalues within the contour. Examining the singular values of the projections on moderately-sized, randomly-generated test problems motivates orthogonalization-based improvements to the algorithm. The singular value distributions provide experimentally robust estimates of the number of eigenvalues within the contour. The algorithm is modified to handle both Hermitian and general complex matrices. The original algorithm (based on circular contours and Gauss-Legendre quadrature) is extended to contours and quadrature schemes that are recursively subdividable. A general complex recursive algorithm is implemented on rectangular and diamond contours. The accuracy of different quadrature schemes for various contours is investigated.

Eigenvalue estimates for non-selfadjoint Dirac operators on the real line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with complex mass and non-Hermitian potential V lie in the disjoint union of two disks, provided that the L1-norm of V is bounded from above by the speed of light times the reduced Planck constant. The result is sharp; moreover, the analogous sharp result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on V implies the absence of non-real eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.

Medidas autosemejantes en el plano, momentos y matrices de Hessenberg

La tesis MEDIDAS AUTOSEMEJANTES EN EL PLANO, MOMENTOS Y MATRICES DE HESSENBERG se enmarca entre las áreas de la teoría geométrica de la medida, la teoría de polinomios ortogonales y la teoría de operadores. La memoria aborda el estudio de medidas con soporte acotado en el plano complejo vistas con la óptica de las matrices infinitas de momentos y de Hessenberg asociadas a estas medidas que en la teoría de los polinomios ortogonales las representan. En particular se centra en el estudio de las medidas autosemejantes que son las medidas de equilibrio definidas por un sistema de funciones iteradas (SFI). Los conjuntos autosemejantes son conjuntos que tienen la propiedad geométrica de descomponerse en unión de piezas semejantes al conjunto total. Estas piezas pueden solaparse o no, cuando el solapamiento es pequeño la teoría de Hutchinson [Hut81] funciona bien, pero cuando no existen restricciones falla. El problema del solapamiento consiste en controlar la medida de este solapamiento. Un ejemplo de la complejidad de este problema se plantea con las convoluciones infinitas de distribuciones de Bernoulli, que han resultado ser un ejemplo de medidas autosemejantes en el caso real. En 1935 Jessen y A. Wintner [JW35] ya se planteaba este problema, lejos de ser sencillo ha sido estudiado durante más de setenta y cinco años y siguen sin resolverse las principales cuestiones planteadas ya por A. Garsia [Gar62] en 1962. El interés que ha despertado este problema así como la complejidad del mismo está demostrado por las numerosas publicaciones que abordan cuestiones relacionadas con este problema ver por ejemplo [JW35], [Erd39], [PS96], [Ma00], [Ma96], [Sol98], [Mat95], [PS96], [Sim05],[JKS07] [JKS11]. En el primer capítulo comenzamos introduciendo con detalle las medidas autosemejante en el plano complejo y los sistemas de funciones iteradas, así como los conceptos de la teoría de la medida necesarios para describirlos. A continuación se introducen las herramientas necesarias de teoría de polinomios ortogonales, matrices infinitas y operadores que se van a usar. En el segundo y tercer capítulo trasladamos las propiedades geométricas de las medidas autosemejantes a las matrices de momentos y de Hessenberg, respectivamente. A partir de estos resultados se describen algoritmos para calcular estas matrices a partir del SFI correspondiente. Concretamente, se obtienen fórmulas explícitas y algoritmos de aproximación para los momentos y matrices de momentos de medidas fractales, a partir de un teorema del punto fijo para las matrices. Además utilizando técnicas de la teoría de operadores, se han extendido al plano complejo los resultados que G. Mantica [Ma00, Ma96] obtenía en el caso real. Este resultado es la base para definir un algoritmo estable de aproximación de la matriz de Hessenberg asociada a una medida fractal u obtener secciones finitas exactas de matrices Hessenberg asociadas a una suma de medidas. En el último capítulo, se consideran medidas, μ, más generales y se estudia el comportamiento asintótico de los autovalores de una matriz hermitiana de momentos y su impacto en las propiedades de la medida asociada. En el resultado central se demuestra que si los polinomios asociados son densos en L2(μ) entonces necesariamente el autovalor mínimo de las secciones finitas de la matriz de momentos de la medida tiende a cero. ABSTRACT The Thesis work “Self-similar Measures on the Plane, Moments and Hessenberg Matrices” is framed among the geometric measure theory, orthogonal polynomials and operator theory. The work studies measures with compact support on the complex plane from the point of view of the associated infinite moments and Hessenberg matrices representing them in the theory of orthogonal polynomials. More precisely, it concentrates on the study of the self-similar measures that are equilibrium measures in a iterated functions system. Self-similar sets have the geometric property of being decomposable in a union of similar pieces to the complete set. These pieces can overlap. If the overlapping is small, Hutchinson’s theory [Hut81] works well, however, when it has no restrictions, the theory does not hold. The overlapping problem consists in controlling the measure of the overlap. The complexity of this problem is exemplified in the infinite convolutions of Bernoulli’s distributions, that are an example of self-similar measures in the real case. As early as 1935 [JW35], Jessen and Wintner posed this problem, that far from being simple, has been studied during more than 75 years. The main cuestiones posed by Garsia in 1962 [Gar62] remain unsolved. The interest in this problem, together with its complexity, is demonstrated by the number of publications that over the years have dealt with it. See, for example, [JW35], [Erd39], [PS96], [Ma00], [Ma96], [Sol98], [Mat95], [PS96], [Sim05], [JKS07] [JKS11]. In the first chapter, we will start with a detailed introduction to the self-similar measurements in the complex plane and to the iterated functions systems, also including the concepts of measure theory needed to describe them. Next, we introduce the necessary tools from orthogonal polynomials, infinite matrices and operators. In the second and third chapter we will translate the geometric properties of selfsimilar measures to the moments and Hessenberg matrices. From these results, we will describe algorithms to calculate these matrices from the corresponding iterated functions systems. To be precise, we obtain explicit formulas and approximation algorithms for the moments and moment matrices of fractal measures from a new fixed point theorem for matrices. Moreover, using techniques from operator theory, we extend to the complex plane the real case results obtained by Mantica [Ma00, Ma96]. This result is the base to define a stable algorithm that approximates the Hessenberg matrix associated to a fractal measure and obtains exact finite sections of Hessenberg matrices associated to a sum of measurements. In the last chapter, we consider more general measures, μ, and study the asymptotic behaviour of the eigenvalues of a hermitian matrix of moments, together with its impact on the properties of the associated measure. In the main result we demonstrate that, if the associated polynomials are dense in L2(μ), then necessarily follows that the minimum eigenvalue of the finite sections of the moments matrix goes to zero.

Zeta functions and Eisenstein series on classical groups

We construct an Euler product from the Hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a CM field and the eigenform is holomorphic. We also prove analytic continuation of an Eisenstein series on another unitary group, containing the group just mentioned defined with such an eigenform. As an application of our methods, we prove an explicit class number formula for a totally definite hermitian form over a CM field.

A Lanczos-powered implementation of the Faber polynomial quantum time propagator for reaction probabilities

Complementing our recent work on subspace wavepacket propagation [Chem. Phys. Lett. 336 (2001) 149], we introduce a Lanczos-based implementation of the Faber polynomial quantum long-time propagator. The original version [J. Chem. Phys. 101 (1994) 10493] implicitly handles non-Hermitian Hamiltonians, that is, those perturbed by imaginary absorbing potentials to handle unwanted reflection effects. However, like many wavepacket propagation schemes, it encounters a bottleneck associated with dense matrix-vector multiplications. Our implementation seeks to reduce the quantity of such costly operations without sacrificing numerical accuracy. For some benchmark scattering problems, our approach compares favourably with the original. (C) 2004 Elsevier B.V. All rights reserved.

Various topological excitations in the SO(4) gauge field in higher dimensions

Following the original analysis Of Zhang and Hu for the 4-dimensional generalization of Quantum Hall effect, there has been much work from different viewpoints on the higher dimensional condensed matter systems. In this paper, we discuss three kinds of topological excitations in the SO(4) gauge field of condensed matter systems in 4-dimension-the instantons and anti-instantons, the 't Hooft-Polyakov monopoles, and the 2-membranes. Using the phi-mapping topological theory, it is revealed that there are 4-, 3-, and 2-dimensional topological currents inhering in the SO (4) gauge field, and the above three kinds of excitations can be directly and explicitly derived from these three kinds of currents, respectively. Moreover, it is shown that the topological charges of these excitations are characterized by the Hopf indices and Brouwer degrees of phi-mapping. (c) 2005 Elsevier Inc. All rights reserved.

A Bethe ansatz solvable model for superpositions of Cooper pairs and condensed molecular bosons

We introduce a general Hamiltonian describing coherent superpositions of Cooper pairs and condensed molecular bosons. For particular choices of the coupling parameters, the model is integrable. One integrable manifold, as well as the Bethe ansatz solution, was found by Dukelsky et al. [J. Dukelsky, G.G. Dussel, C. Esebbag, S. Pittel, Phys. Rev. Lett. 93 (2004) 050403]. Here we show that there is a second integrable manifold, established using the boundary quantum inverse scattering method. In this manner we obtain the exact solution by means of the algebraic Bethe ansatz. In the case where the Cooper pair energies are degenerate we examine the relationship between the spectrum of these integrable Hamiltonians and the quasi-exactly solvable spectrum of particular Schrodinger operators. For the solution we derive here the potential of the Schrodinger operator is given in terms of hyperbolic functions. For the solution derived by Dukelsky et al., loc. cit. the potential is sextic and the wavefunctions obey PT-symmetric boundary conditions. This latter case provides a novel example of an integrable Hermitian Hamiltonian acting on a Fock space whose states map into a Hilbert space of PE-symmetric wavefunctions defined on a contour in the complex plane. (c) 2006 Elsevier B.V. All rights reserved.

Appearance of bound states in random potentials with applications to soliton theory

We analyze the stochastic creation of a single bound state (BS) in a random potential with a compact support. We study both the Hermitian Schrödinger equation and non-Hermitian Zakharov-Shabat systems. These problems are of special interest in the inverse scattering method for Korteveg–de-Vries and the nonlinear Schrödinger equations since soliton solutions of these two equations correspond to the BSs of the two aforementioned linear eigenvalue problems. Analytical expressions for the average width of the potential required for the creation of the first BS are given in the approximation of delta-correlated Gaussian potential and additionally different scenarios of eigenvalue creation are discussed for the non-Hermitian case.