Šurov komplement i teorija H-matrica

This thesis studies subclasses of the class of H-matrices and their applications, with emphasis on the investigation of the Schur complement properties. The contributions of the thesis are new nonsingularity results, bounds for the maximum norm of the inverse matrix, closure properties of some matrix classes under taking Schur complements, as well as results on localization and separation of the eigenvalues of the Schur complement based on the entries of the original matrix.Докторска дисертација изучава поткласе класе Х-матрица и њихове примене, првенствено у истраживању својстава Шуровог комплемента. Оригиналан допринос тезе представљају нови услови за регуларност матрица, оцене максимум норме инверзне матрице, резултати о затворености појединих класа матрица на Шуров комплемент, као и резултати о локализацији и сепарацији карактеристичних корена Шуровог комплемента на основу елемената полазне матрице.Doktorska disertacija izučava potklase klase H-matrica i njihove primene, prvenstveno u istraživanju svojstava Šurovog komplementa. Originalan doprinos teze predstavljaju novi uslovi za regularnost matrica, ocene maksimum norme inverzne matrice, rezultati o zatvorenosti pojedinih klasa matrica na Šurov komplement, kao i rezultati o lokalizaciji i separaciji karakterističnih korena Šurovog komplementa na osnovu elemenata polazne matrice

Accurate inverses of Nekrasov Z-matrices

We present a parametrization of a Nekrasov Z-matrix that allows us to compute its inverse with high relative accuracy. Numerical examples illustrating the accuracy of the method are included

Numerical methods and accurate computations with structured matrices

Esta tesis doctoral es un compendio de 11 artículos científicos. El tema principal de la tesis es el Álgebra Lineal Numérica, con énfasis en dos clases de matrices estructuradas: las matrices totalmente positivas y las M-matrices. Para algunas subclases de estas matrices, es posible desarrollar algoritmos para resolver numéricamente varios de los problemas más comunes en álgebra lineal con alta precisión relativa independientemente del número de condición de la matriz. La clave para lograr cálculos precisos está en el uso de una parametrización diferente que represente la estructura especial de la matriz y en el desarrollo de algoritmos adaptados que trabajen con dicha parametrización.Las matrices totalmente positivas no singulares admiten una factorización única como producto de matrices bidiagonales no negativas llamada factorización bidiagonal. Si conocemos esta representación con alta precisión relativa, se puede utilizar para resolver ciertos sistemas de ecuaciones y para calcular la inversa, los valores propios y los valores singulares con alta precisión relativa. Nuestra contribución en este campo ha sido la obtención de la factorización bidiagonal con alta precisión relativa de matrices de colocación de polinomios de Laguerre generalizados, de matrices de colocación de polinomios de Bessel, de clases de matrices que generalizan la matriz de Pascal y de matrices de q-enteros. También hemos estudiado la extensión de varias propiedades óptimas de las matrices de colocación de B-bases normalizadas (que en particular son matrices totalmente positivas). En particular, hemos demostrado propiedades de optimalidad de las matrices de colocación del producto tensorial de B-bases normalizadas.Si conocemos las sumas de filas y las entradas extradiagonales de una M-matriz no singular diagonal dominante con alta precisión relativa, entonces podemos calcular su inversa, determinante y valores singulares también con alta precisión relativa. Hemos buscado nuevos métodos para lograr cálculos precisos con nuevas clases de M-matrices o matrices relacionadas. Hemos propuesto una parametrización para las Z-matrices de Nekrasov con entradas diagonales positivas que puede utilizarse para calcular su inversa y determinante con alta precisión relativa. También hemos estudiado la clase denominada B-matrices, que está muy relacionada con las M-matrices. Hemos obtenido un método para calcular los determinantes de esta clase con alta precisión relativa y otro para calcular los determinantes de las matrices de B-Nekrasov también con alta precisión relativa. Basándonos en la utilización de dos matrices de escalado que hemos introducido, hemos desarrollado nuevas cotas para la norma infinito de la inversa de una matriz de Nekrasov y para el error del problema de complementariedad lineal cuando su matriz asociada es de Nekrasov. También hemos obtenido nuevas cotas para la norma infinito de las inversas de Bpi-matrices, una clase que extiende a las B-matrices, y las hemos utilizado para obtener nuevas cotas del error para el problema de complementariedad lineal cuya matriz asociada es una Bpi-matriz. Algunas clases de matrices han sido generalizadas al caso de mayor dimensión para desarrollar una teoría para tensores extendiendo la conocida para el caso matricial. Por ejemplo, la definición de la clase de las B-matrices ha sido extendida a la clase de B-tensores, dando lugar a un criterio sencillo para identificar una nueva clase de tensores definidos positivos. Hemos propuesto una extensión de la clase de las Bpi-matrices a Bpi-tensores, definiendo así una nueva clase de tensores definidos positivos que puede ser identificada en base a un criterio sencillo basado solo en cálculos que involucran a las entradas del tensor. Finalmente, hemos caracterizado los casos en los que las matrices de Toeplitz tridiagonales son P-matrices y hemos estudiado cuándo pueden ser representadas en términos de una factorización bidiagonal que sirve como parametrización para lograr cálculos con alta precisión relativa.<br /

Prednosti generalizovane dijagonalne dominacije

This theses is dedicated to the study of generalized diagonal dominance and its various beneflts. The starting point is the well known nonsingularity result of strictly diagonally dominant matrices, from which generalizations were formed in difierent directions. In theses, after a short overview of very well known results, special attention was turned to contemporary contributions, where overview of already published original material is given, together with new obtained results. Particulary, Ger•sgorin-type localization theory for matrix pencils is developed, and application of the results in wireless sensor networks optimization problems is shown.Ova teza je posvećena izučavanju generalizovane dijagonalne dominacije i njenih brojnih prednosti. Osnovu čini poznati rezultat o regularnosti strogo dijagonalnih matrica, čija su uopštenja formirana u brojnim pravcima. U tezi, nakon kratkog pregleda dobro poznatih rezultata, posebna pažnja je posvećena savremenim doprinosima, gde je dat i pregled već objavljenih autorovih rezultata, kao i detaljan tretman novih dobijenih rezultata. Posebno je razvijena teorija lokalizacije Geršgorinovog tipa generalizovanih karakterističnih korena i pokazana je primena rezultata u problemima optimizacije bežičnih senzor mreža

Relative Gromov-Witten theory and vertex operators

In this thesis, we report on two projects applying representation theoretic techniques to solve enumerative and geometric problems, which were carried out by the author during his pursuit of Ph.D. at Columbia. We first study the relative Gromov-Witten theory on T*P¹ x P¹ and show that certain equivariant limits give relative invariants on P¹ x P¹. By formulating the quantum multiplications on Hilb(T*P¹) computed by Davesh Maulik and Alexei Oblomkov as vertex operators and computing the product expansion, we demonstrate how to get the insertion operator computed by Yaim Cooper and Rahul Pandharipande in the equivariant limits. Brenti proves a non-recursive formula for the Kazhdan-Lusztig polynomials of Coxeter groups by combinatorial methods. In the case of the Weyl group of a split group over a finite field, a geometric interpretation is given by Sophie Morel via weight truncation of perverse sheaves. With suitable modifications of Morel's proof, we generalize the geometric interpretation to the case of finite and affine partial flag varieties. We demonstrate the result with essentially new examples using sl₃ and sl₄.

On Algebraic Singularities, Finite Graphs and D-Brane Gauge Theories: A String Theoretic Perspective

In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes. We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NS-NS B-field, as well as the T-duals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay's Correspondence will also be considered. The present work is a transcription of excerpts from the first three volumes of the author's PhD thesis which was written under the direction of Prof. A. Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student.Comment: 513 pages, 71 figs, Edited Excerpts from the first 3 volumes of the author's PhD Thesi

On algebraic singularities, finite graphs and D-brane gauge theories: A String theoretic perspective

In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes. We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NS-NS B-field, as well as the T-duals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay's Correspondence will also be considered. The present work is a transcription of excerpts from the first three volumes of the author's PhD thesis which was written under the direction of Prof. A. Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student

Eisenstein series and automorphic representations

We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adeles A, thereby also paving the way for connections to number theory, representation theory and the Langlands program. Most of the results we present are already scattered throughout the mathematics literature but our exposition collects them together and is driven by examples. Many interesting aspects of these functions are hidden in their Fourier coefficients with respect to unipotent subgroups and a large part of our focus is to explain and derive general theorems on these Fourier expansions. Specifically, we give complete proofs of the Langlands constant term formula for Eisenstein series on adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic spherical Whittaker function associated to unramified automorphic representations of G(Q_p). In addition, we explain how the classical theory of Hecke operators fits into the modern theory of automorphic representations of adelic groups, thereby providing a connection with some key elements in the Langlands program, such as the Langlands dual group LG and automorphic L-functions. Somewhat surprisingly, all these results have natural interpretations as encoding physical effects in string theory. We therefore also introduce some basic concepts of string theory, aimed toward mathematicians, emphasising the role of automorphic forms. In particular, we provide a detailed treatment of supersymmetry constraints on string amplitudes which enforce differential equations of the same type that are satisfied by automorphic forms. Our treatise concludes with a detailed list of interesting open questions and pointers to additional topics which go beyond the scope of this book.Comment: 326 pages. Detailed and example-driven exposition of the subject with highlighted applications to string theory. v2: 375 pages. Substantially extended and small correction

Twisted Holography: The Examples of 4d and 5d Chern-Simons Theories

Twisted holography is a duality between a twisted supergravity, and a twisted supersymmetric gauge theory living on the D-branes in the supergravity. The main objectives of this duality is the comparison between the algebra of observables in the bulk twisted supergravity and the algebra of observables in the boundary twisted supersymmetric gauge theory. In this thesis, two example of the twisted holography duality are explored. The bulk theory for the first example is the 4d topological-holomorphic Chern-Simons theory, which is expected to be dual to 2d BF theory with line defects. The algebra of observables in the 2d BF theory is computed by two methods: perturbation theory (Feynman diagrams), and phase space quantization. By holography duality this algebra is expected to be isomorphic to the algebra of bulk-boundary scattering process, and the latter is computed in this thesis using perturbative method. The bulk theory for the second example is the 5d topological-holomorphic Chern-Simons theory, which is expected to be dual to the large-N limit of a family of 1d quantum mechanics built from the ADHM quivers. The generators and relations of the large-N limit algebra of observables in the 1d quantum mechanics are studied from algebraic point view. By holography duality, this algebra is expected to be the algebra of observables on the universal line defect coupled to the 5d Chern-Simons theory, and some nontrivial relations of the latter algebra are computed in this thesis using perturbative method. The surface defects and various fusion process between line and surface defects are also explored

クラスS理論における欠陥演算子への幾何学的アプローチ

学位の種別: 課程博士審査委員会委員 : （主査）東京大学准教授 松尾 泰, 東京大学教授 浅井 祥仁, 東京大学教授 国場 敦夫, 東京大学准教授 渡利 泰山, 東京大学教授 押川 正毅University of Tokyo(東京大学