Рассматриваются задачи наилучшего одностороннего приближения (снизу и сверху) в пространстве функций, суммируемых с весом на (-1,1), характеристической функции интервала (a,b) из (-1,1 ) множеством алгебраических многочленов степени не выше заданной. Приведено решение задач в случае, когда a и b - узлы положительной квадратурной формулы при некоторых условиях на ее алгебраическую точность, а также в случае симметричного интервала для четного веса. / We consider the problems of the best one-sided approximation (from below and from above) in the space of functions integrable on (-1,1) with a weight to the characteristic function of an interval (a,b) from (-1,1) by the set of algebraic polynomials of degree not exceeding a given number. We solve the problems in the case when a and b are nodes of a positive quadrature formula under some conditions on its degree of precision as well as in the case of a symmetric interval for an even weight.

This thesis is concerned with convex optimization problems over matrix polynomials that are constrained to be positive semidefinite on the unit circle. Problems of this form appear in signal processing and can often be solved as semidefinite programs (SDPs). Interior-point solvers for these SDPs scale poorly, and this thesis aims to design first-order methods that are more efficient. We propose methods based on a generalized proximal operator defined in terms of a Bregman divergence. Empirical results on three applications in signal processing demonstrate that the proposed methods scale much better than interior-point solvers. As an example, for sparse estimation of spectral density matrices, Douglas--Rachford splitting with the generalized proximal operator is about 1000 times faster and scales to much larger problems. The ability to solve larger problems allows us to perform functional connectivity analysis of the brain by constructing a sparse estimate of the inverse spectral density matrix.

The 3-satisfiability Problem (3-SAT) is a demanding combinatorial problem that is of central importance among the nondeterministic polynomial (NP) complete problems, with applications in circuit design, artificial intelligence, and logistics. Even with optimized algorithms, the solution space that needs to be explored grows exponentially with the increasing size of 3-SAT instances. Thus, large 3-SAT instances require excessive amounts of energy to solve with serial electronic computers. Network-based biocomputation (NBC) is a parallel computation approach with drastically reduced energy consumption. NBC uses biomolecular motors to propel cytoskeletal filaments through nanofabricated networks that encode mathematical problems. By stochastically exploring possible paths through the networks, the cytoskeletal filaments find possible solutions. However, to date, no NBC algorithm for 3-SAT has been available. Herein, an algorithm that converts 3-SAT into an NBC-compatible network format is reported and four small 3-SAT instances (with up to three variables and five clauses) using the actin–myosin biomolecular motor system are experimentally solved. Because practical polynomial conversions to 3-SAT exist for many important NP complete problems, the result opens the door to enable NBC to solve small instances of a wide range of problems.

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We propose a new framework for understanding block Krylov subspace methods, which hinges on a matrix-valued inner product. We can recast the ``classical" block Krylov methods, such as O'Leary's block conjugate gradients, global methods, and loop-interchange methods, within this framework. Leveraging the generality of the framework, we develop an efficient restart procedure and error bounds for the shifted block full orthogonalization method (Sh-BFOM(m)). Regarding BFOM as the prototypical block Krylov subspace method, we propose another formalism, which we call modified BFOM, and show that block GMRES and the new block Radau-Lanczos method can be regarded as modified BFOM. In analogy to Sh-BFOM(m), we develop an efficient restart procedure for shifted BGMRES with restarts (Sh-BGMRES(m)), as well as error bounds. Using this framework and shifted block Krylov methods with restarts as a foundation, we formulate block Krylov subspace methods with restarts for matrix functions acting on multiple vectors f(A)B. We obtain convergence bounds for \bfomfom (BFOM for Functions Of Matrices) and block harmonic methods (i.e., BGMRES-like methods) for matrix functions. With various numerical examples, we illustrate our theoretical results on Sh-BFOM and Sh-BGMRES. We also analyze the matrix polynomials associated to the residuals of these methods. Through a variety of real-life applications, we demonstrate the robustness and versatility of B(FOM)^2 and block harmonic methods for matrix functions. A particularly interesting example is the tensor t-function, our proposed definition for the function of a tensor in the tensor t-product formalism. Despite the lack of convergence theory, we also show that the block Radau-Lanczos modification can reduce the number of cycles required to converge for both linear systems and matrix functions. / Mathematics

We study the equivariant cohomology of moduli spaces of objects in the derived category of elliptic surfaces in order to find new examples of infinite dimensional quantum integrable systems and their geometric representation theoretic interpretation in enumerative geometry. This problem is related to a program to understand the cohomological and K-theoretic Hall algebras of holomorphic symplectic surfaces and to understand how it related to the Donaldson-Thomas theory of threefolds fibered in those surfaces. We use the theory of noncommutative deformations of Poisson surfaces and especially van den Berg’s noncommutative P1 bundles as well as Rains’s analysis of moduli theory for quasi-ruled noncommutative surfaces as well as the theory of Bridgeland stability conditions and their relative versions to understand equivariant deformations and birational transformations of Hilbert schemes of points on equivariant elliptic surfaces. The moduli spaces are closely related to elliptic versions of classical integrable systems. We also use these moduli spaces to construct vertex algebra representations of extensions of toroidal extended affine algebras on their equivariant cohomology, building on work of Eswara-Rao–Moody–Yokonuma, of Billig, and of Chen–Li–Tan on vertex representations of toroidal algebras, full toroidal algebras, and toroidal extended affine algebras. Using Fourier-Mukai transforms and their relative action on families of dg-categories we study the relationship between automorphisms of toroidal extended affine algebras and families of derived equivalences on dg categories, in particular finding a relativistic (difference) generalization of the Laumon-Rothstein deformation of the Fourier-Mukai duality. Finally, using the above analysis we extend the construction of Maulik–Okounkov’s stable envelopes to moduli of framed torsionfree sheaves on noncommutative surfaces in some cases and use this to study coproducts on associated algebras assigned to elliptic surfaces with applications to understanding new representation theoretic structures in the Donaldson-Thomas theory of local curves.

This dissertation explores an analogue of the prime number theorem for polynomials over finite fields as well as its connection to the necklace factorization algorithm T-transform and the string complexity measure T-complexity. Specifically, a precise asymptotic expansion for the prime polynomial counting function is derived. The approximation given is more accurate than previous results in the literature while requiring very little computational effort. In this context asymptotic series expansions for Lerch transcendent, Eulerian polynomials, truncated polylogarithm, and polylogarithms of negative integer order are also provided. The expansion formulas developed are general and have applications in numerous areas other than the enumeration of prime polynomials. A bijection between the equivalence classes of aperiodic necklaces and monic prime polynomials is utilized to derive an asymptotic bound on the maximal T-complexity value of a string. Furthermore, the statistical behaviour of uniform random sequences that are factored via the T-transform are investigated, and an accurate probabilistic model for short necklace factors is presented. Finally, a T-complexity based conditional string complexity measure is proposed and used to define the normalized T-complexity distance that measures similarity between strings. The T-complexity distance is proven to not be a metric. However, the measure can be computed in linear time and space making it a suitable choice for large data sets. / Graduate / 0544 0984 0405 / nrebenich@gmail.com

Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2011. / Includes bibliography. / ENGLISH ABSTRACT: Latimer & MacDuffee showed in 1933 that there is a one-to-one correspondence between equivalence classes of matrices with a given minimum polynomial and equivalence classes of ideals of a certain ring. In the case where the matrices are taken over the integers, Behn and Van der Merwe developed an algorithm in 2002 to produce a representative in each equivalence class. We extend this algorithm to matrices taken over the ring Fq[T] of polynomials over a finite field and prove a modified version of the Latimer-MacDuffee theorem which holds for proper equivalence classes of matrices. / AFRIKAANSE OPSOMMING: Latimer & MacDuffee het in 1933 bewys dat daar 'n een-tot-een korrespondensie is tussen ekwivalensieklasse van matrikse met 'n gegewe minimumpolinoom en ekwivalensieklasse van ideale van 'n sekere ring. In die geval waar die matrikse heeltallige inskrywings het, het Behn en Van der Merwe in 2002 'n algoritme ontwikkel om verteenwoordigers in elke ekwivalensieklas voort te bring. Ons brei hierdie algoritme uit na die geval van matrikse met inskrywings in die ring Fq[T] van polinome oor 'n eindige liggaam en ons bewys 'n gewysigde weergawe van die Latimer-MacDuffee stelling wat geld vir klasse van streng ekwivalente matrikse.

Travail réalisé en cotutelle avec l'université Paris-Diderot et le Commissariat à l'Energie Atomique sous la direction de John Harnad et Bertrand Eynard. / Cette thèse traite des aspects géométriques et d'intégrabilité associés aux modèles de matrices aléatoires. Son but est de présenter diverses applications des modèles de matrices aléatoires allant de la géométrie algébrique aux équations aux dérivées partielles des systèmes intégrables. Ces différentes applications permettent en particulier de montrer en quoi les modèles de matrices possèdent une grande richesse d'un point de vue mathématique. Ainsi, cette thèse abordera d'abord l'étude de la jonction de deux intervalles du support de la densité des valeurs propres au voisinage d'un point singulier. On montrera plus précisément en quoi ce régime limite particulier aboutit aux équations universelles de la hiérarchie de Painlevé II des systèmes intégrables. Ensuite, l'approche des polynômes (bi)-orthogonaux, introduite par Mehta pour le calcul des fonctions de partition, permettra d'énoncer des problèmes de Riemann-Hilbert et d'isomonodromies associés aux modèles de matrices, faisant ainsi le lien avec la théorie de Jimbo-Miwa-Ueno. On montrera en particulier que le cas des modèles à deux matrices hermitiens se transpose à un cas dégénéré de la théorie isomonodromique de Jimbo-Miwa-Ueno qui sera alors généralisé. La méthode des équations de boucles avec ses notions centrales de courbe spectrale et de développement topologique permettra quant à elle de faire le lien avec les invariants symplectiques de géométrie algébrique introduits récemment par Eynard et Orantin. Ce dernier point fera également l'objet d'une généralisation aux modèles de matrices non-hermitien (beta quelconque) ouvrant ainsi la voie à la ``géométrie algébrique quantique'' et à la généralisation de ces invariants symplectiques pour des courbes ``quantiques''. Enfin, une dernière partie sera consacrée aux liens étroits entre les modèles de matrices et les problèmes de combinatoire. En particulier, l'accent sera mis sur les aspects géométriques de la théorie des cordes topologiques avec la construction explicite d'un modèle de matrices aléatoires donnant le dénombrement des invariants de Gromov-Witten pour les variétés de Calabi-Yau toriques de dimension complexe trois utilisées en théorie des cordes topologiques. L'étendue des domaines abordés étant très vaste, l'objectif de la thèse est de présenter de façon la plus simple possible chacun des domaines mentionnés précédemment et d'analyser en quoi les modèles de matrices peuvent apporter une aide précieuse dans leur résolution. Le fil conducteur étant les modèles matriciels, chaque partie a été conçue pour être abordable pour un spécialiste des modèles de matrices ne connaissant pas forcément tous les domaines d'application présentés ici. / This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of integrable systems. The variety of these applications shows why matrix models are important from a mathematical point of view. First, the thesis will focus on the study of the merging of two intervals of the eigenvalues density near a singular point. Specifically, we will show why this special limit gives universal equations from the Painlevé II hierarchy of integrable systems theory. Then, following the approach of (bi) orthogonal polynomials introduced by Mehta to compute partition functions, we will find Riemann-Hilbert and isomonodromic problems connected to matrix models, making the link with the theory of Jimbo, Miwa and Ueno. In particular, we will describe how the hermitian two-matrix models provide a degenerate case of Jimbo-Miwa-Ueno's theory that we will generalize in this context. Furthermore, the loop equations method, with its central notions of spectral curve and topological expansion, will lead to the symplectic invariants of algebraic geometry recently proposed by Eynard and Orantin. This last point will be generalized to the case of non-hermitian matrix models (arbitrary beta) paving the way to ``quantum algebraic geometry'' and to the generalization of symplectic invariants to ``quantum curves''. Finally, this set up will be applied to combinatorics in the context of topological string theory, with the explicit computation of an hermitian random matrix model enumerating the Gromov-Witten invariants of a toric Calabi-Yau threefold. Since the range of the applications encountered is large, we try to present every domain in a simple way and explain how random matrix models can bring new insights to those fields. The common element of the thesis being matrix models, each part has been written so that readers unfamiliar with the domains of application but familiar with matrix models should be able to understand it.

Un circuit arithmétique dont les entrées sont des entiers ou une variable x et dont les portes calculent la somme ou le produit représente un polynôme univarié. On assimile la complexité de représentation d'un polynôme par un circuit arithmétique au nombre de portes multiplicatives minimal requis pour cette modélisation. Et l'on cherche à obtenir une borne inférieure à cette complexité, et cela en fonction du degré d du polynôme. A une chaîne additive pour d, correspond un circuit arithmétique pour le monôme de degré d. La conjecture de Strassen prétend que le nombre minimal de portes multiplicatives requis pour représenter un polynôme de degré d est au moins la longueur minimale d'une chaîne additive pour d. La conjecture de Strassen généralisée correspondrait à la même proposition lorsque les portes du circuit arithmétique ont degré entrant g au lieu de 2. Le mémoire consiste d'une part en une généralisation du concept de chaînes additives, et une étude approfondie de leur construction. On s'y intéresse d'autre part aux polynômes qui peuvent être représentés avec très peu de portes multiplicatives (les d-gems). On combine enfin les deux études en lien avec la conjecture de Strassen. On obtient en particulier de nouveaux cas de circuits vérifiant la conjecture. / An arithmetic circuit with inputs among x and the integers which has product gates and addition gates represents a univariate polynomial. We define the complexity of the representation of a polynomial by an arithmetic circuit as the minimal number of product gates required for this modelization. And we seek a lower bound to this complexity, with respect to the degree d of the polynomial. An addition chain for d corresponds to an arithmetic circuit computing the monomial of degree d. Strassen's conjecture states that the minimal number of product gates required to represent a polynomial of degree d is at least the minimal length of an addition chain for d. The generalized Strassen conjecture corresponds to the same statement where the indegree of the gates of the arithmetic circuit is g instead of 2. The thesis consists, on the one hand, of the generalization of the concept of addition chains, and a study of the subject. On the other hand, it is concerned with polynomials which can be represented with very few product gates (d-gems). Both studies related to Strassen's conjecture are combined. In particular, we get new classes of circuits verifying the conjecture.