A Schur ring is a subring of the group algebra with a basis that is formed by a partition of the group. These subrings were initially used to study finite permutation groups, and classifications of Schur rings over various finite groups have been studied. Here we investigate Schur rings over various infinite groups, including free groups. We classify Schur rings over the infinite cyclic group.

P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary results for 1D case, condition number estimates and some inequalities for 2D reference element. Part II is devoted to the derivation of the Schur complement preconditioner and conditionality number estimates for the p-version finite element matrixes. Also DD preconditioning is considered.

We study finite groups $G$ having a nontrivial subgroup $H$ and $D \subset G \setminus H$ such that (i) the multiset $\{ xy^{-1}:x,y \in D\}$ has every element that is not in $H$ occur the same number of times (such a $D$ is called a {\it relative difference set}); (ii) $G=D\cup D^{(-1)} \cup H$; (iii) $D \cap D^{(-1)} =\emptyset$. We show that $|H|=2$, that $H$ has to be normal, and that $G$ is a group with a single involution. We also show that $G$ cannot be abelian. We give examples of such groups, including certain dicyclic groups, by using results of Schmidt and Ito. We describe an infinite family of dicyclic groups with these relative difference sets, and classify which groups of order up to $72$ contain them. We also define a relative difference set in dicyclic groups having additional symmetries, and completely classify when these exist in generalized quaternion groups. We make connections to Schur rings and prove additional results.

Sequences of matrices arise in many applications in science and engineering. In this thesis we consider matrices that are closely related (or closely related in groups), and we take advantage of the small differences between them to efficiently solve sequences of linear systems and eigenproblems. Recycling techniques, such as recycling preconditioners or subspaces, are popular approaches for reducing computational cost. In this thesis, we introduce two novel approaches for recycling previously computed information for a subsequent system or eigenproblem, and demonstrate good results for sequences arising in several applications. Preconditioners are often essential for fast convergence of iterative methods. However, computing a good preconditioner can be very expensive, and when solving a sequence of linear systems, we want to avoid computing a new preconditioner too often. Instead, we can recycle a previously computed preconditioner, for which we have good convergence behavior of the preconditioned system. We propose an update technique we call the sparse approximate map, or SAM update, that approximately maps one matrix to another matrix in our sequence. SAM updates are very cheap to compute and apply, preserve good convergence properties of a previously computed preconditioner, and help to amortize the cost of that preconditioner over many linear solves. When solving a sequence of eigenproblems, we can reduce the computational cost of constructing the Krylov space starting with a single vector by warm-starting the eigensolver with a subspace instead. We propose an algorithm to warm-start the Krylov-Schur method using a previously computed approximate invariant subspace. We first compute the approximate Krylov decomposition for a matrix with minimal residual, and use this space to warm-start the eigensolver. We account for the residual matrix when expanding, truncating, and deflating the decomposition and show that the norm of the residual monotonically decreases. This method is effective in reducing the total number of matrix-vector products, and computes an approximate invariant subspace that is as accurate as the one computed with standard Krylov-Schur. In applications where the matrix-vector products require an implicit linear solve, we incorporate Krylov subspace recycling. Finally, in many applications, sequences of matrices take the special form of the sum of the identity matrix, a very low-rank matrix, and a small-in-norm matrix. We consider convergence rates for GMRES applied to these matrices by identifying the sources of sensitivity. / Doctor of Philosophy / Problems in science and engineering often require the solution to many linear systems, or a sequence of systems, that model the behavior of physical phenomena. In order to construct highly accurate mathematical models to describe this behavior, the resulting matrices can be very large, and therefore the linear system can be very expensive to solve. To efficiently solve a sequence of large linear systems, we often use iterative methods, which can require preconditioning techniques to achieve fast convergence. The preconditioners themselves can be very expensive to compute. So, we propose a cheap update technique that approximately maps one matrix to another in the sequence for which we already have a good preconditioner. We then combine the preconditioner and the map and use the updated preconditioner for the current system. Sequences of eigenvalue problems also arise in many scientific applications, such as those modeling disk brake squeal in a motor vehicle. To accurately represent this physical system, large eigenvalue problems must be solved. The behavior of certain eigenvalues can reveal instability in the physical system but to identify these eigenvalues, we must solve a sequence of very large eigenproblems. The eigensolvers used to solve eigenproblems generally begin with a single vector, and instead, we propose starting the method with several vectors, or a subspace. This allows us to reduce the total number of iterations required by the eigensolver while still producing an accurate solution. We demonstrate good results for both of these approaches using sequences of linear systems and eigenvalue problems arising in several real-world applications. Finally, in many applications, sequences of matrices take the special form of the sum of the identity matrix, a very low-rank matrix, and a small-in-norm matrix. We examine the convergence behavior of the iterative method GMRES when solving such a sequence of matrices.

Cette thèse présente quelques résultats d’analyse sur les espaces Lp le plus souvent non commutatifs.La première partie exhibe de large classes de contractions sur des espaces Lp non commutatifsqui vérifient l’analogue non commutatif de la conjecture de Matsaev. De plus, cette partie fournitune comparaison entre certaines normes apparaissant naturellement dans ce domaine. La deuxièmepartie traite des fonctions carrées. Le premier résultat principal énonce que si T est un opérateurR-Ritt sur un espace Lp alors les fonctions carrées associées sont équivalentes. Le second résultatprincipal est une caractérisation de certaines estimations carrées utilisant les dilatations. La troisièmepartie de cette thèse introduit de nouvelles fonctions carrées pour les opérateurs de Ritt définis surdes espaces Lp non commutatifs. Le résultat principal est qu’en général ces fonctions carrées ne sontpas équivalentes. Cette partie contient aussi un résultat d’équivalence entre la norme usuelle et unecertaine fonction carrée. La quatrième partie introduit un analogue non commutatif de l’algèbre deFigà-Talamanca-Herz Ap(G) sur le prédual naturel de l’espace d’opérateurs Mp,cb des multiplicateursde Schur complètement bornées sur l’espace de Schatten Sp. / This thesis presents some results of analysis in Lp-spaces, especially often noncommutative. Thefirst part exhibits large classes of contractions on noncommutative Lp-spaces which satisfy the noncommutativeanalogue of Matsaev’s conjecture. Moreover, this part gives a comparison between variousnorms arising naturally from this field. The second part is devoted to square functions. The firstmain result states that if T is an R-Ritt operator on a Lp-space then the involved square functionsare equivalent. The second principal result is a characterization of some square functions estimatesin terms of dilations. In the third part of this thesis, we introduce some new square functions forRitt operators defined on noncommutative Lp-spaces. The main result is that these square functionsare generally not equivalent. This part also contains a result of equivalence between the usual normand some special square function. The fourth part introduces a noncommutative analogue of theFigà-Talamanca-Herz algebra Ap(G) on the natural predual of the operator space Mp,cb of completelybounded Schur multipliers on the Schatten space Sp.

Les catégories de fusion pointées sont des catégories de fusion pour lesquelles les objets simples sont inversibles. Nous développons des méthodes basés par ordinateur pour classifier les catégories pointées à équivalence de Morita près, et les appliquons aux catégories pointées de dimensions comprises entre 2 et 32. Nous prouvons qu'il existe 1126 classes de Morita pour de telles catégories. Aussi, nous prouvons que les indicateurs de Frobenius-Schur du centre d'une catégorie pointée de dimension inférieure à 32, accompagnés de structure enrubannée de ce centre, déterminent sa classe de Morita. Ceci est faux en général: les données modulaires, et donc a fortiori les indicateurs et structures enrubannées, ne distinguent pas les catégories modulaires. Nous donnons une famille d'exemples ; en réalité, il existe un nombre arbitrairement grand de catégories modulaires deux-à-deux non équivalentes qui peuvent partager les mêmes données modulaires. / Pointed fusion categories are fusion categories in which all simple objects are invertible. We develop computer-based methods to classify pointed categories up to Morita equivalence, and apply them to pointed fusion categories of dimension from 2 to 31. We prove that there are 1126 Morita classes of such categories. Also, we prove that the Frobenius-Schur indicators of the centers of a pointed category of dimension less than 32, along with its ribbon twist, determine its Morita class. This is not true in general: the modular data, and a fortiori the indicators and the ribbon twists, do not distinguish modular categories. We give a family of examples; in fact, arbitrarly many pairwise non-equivalent modular categories can share the same modular data.

On s'intéresse dans cette thèse à l'étude de variables aléatoires sur les groupes de Lie compacts classiques. On donne une déformation du calcul de Weingarten tel qu'il a été introduit par B. Collins et P. Sniady. On fait une étude asymptotique du mouvement brownien sur les groupes de Lie compacts de grande dimension en obtenant des nouveaux résultats de fluctuations. Deux nouveaux objets, que l'on appelle champ maître gaussien planaire et champ maître orienté planaire, sont introduits pour décrire le comportement asymptotique des mesures de Yang-Mills pour des groupes de structure de grande dimension.

Made available in DSpace on 2014-07-29T16:02:20Z (GMT). No. of bitstreams: 1 Dissertacao Flavio P Vieira.pdf: 631827 bytes, checksum: b4c3dfd45d41b0313e21e87b61f0c94e (MD5) Previous issue date: 2012-04-12 / The main goal of this work is to establish a primary bound for the dimension of the second homology group of a group G, with coefficients in a field k of characteristic p, H2(G;k), using the operating system GAP. It will be presented with several examples, where in some cases, will be calculated the exact dimensions and in other cases only an upper bound. / O trabalho tem por objetivo principal estabelecer uma cota superior para a dimensão do segundo grupo de homologia de um grupo G, com coeficientes em um corpo k de característica p, H2(G;k), usando o sistema operacional GAP. Será apresentado uma gama de exemplos, onde em alguns casos, calcularemos exatamente a dimensão e em outras somente uma cota superior.

Les calculs de réactivité constituent une brique fondamentale dans la simulation des coeurs des réacteurs nucléaires. Ceux-ci conduisent à la résolution de problèmes aux valeurs propres généralisées résolus par l'algorithme de la puissance inverse. A chaque itération, on est amené à résoudre un système linéaire de manière approchée via un algorithme d'itérations imbriquées. Il est difficile de traiter les modélisations très fines avec le solveur développé à EDF, au sein de la plate-forme Cocagne, en raison de la consommation mémoire et du temps de calcul. Au cours de cette thèse, on étudie une méthode de décomposition de domaine de type Schur dual. Plusieurs placements de l'algorithme de décomposition de domaine au sein du système d'itérations imbriquées sont envisageables. Deux d'entre eux ont été implémentés et les résultats analysés. Le deuxième placement, utilisant les spécificités des éléments finis de Raviart-Thomas et de l'algorithme des directions alternées, conduit à des résultats très encourageants. Ces résultats permettent d'envisager l'industrialisation de la méthodologie associée. / The reactivity computations are an essential component for the simulation of the core of a nuclear plant. These computations lead to generalized eigenvalue problems solved by the inverse power iteration algorithm. At each iteration, an algebraic linear system is solved through an inner/outer process. With the solver Cocagne developed at EDF, it is difficult to take into account very fine discretisation, due to the memory requirement and the computation time. In this thesis, a domain decomposition method based on the Schur dual technique is studied. Several placement in the inner/outer process are possible. Two of them are implemented and the results analyzed.The second one, which uses the specificities of the Raviart Thomas finite element and of the alternating directions algorithm, leads to very promising results. From these results the industrialization of the method can be considered.

Kronecker structured representations are used to cope with the state space explosion problem in Markovian modeling and analysis. Currently an open research problem is that of devising strong preconditioners to be used with projection methods for the computation of the stationary vector of Markov chains (MCs) underlying such representations. This paper proposes a block SOR (BSOR) preconditioner for hierarchical Markovian Models (HMMs) that are composed of multiple low level models and a high level model that defines the interaction among low level models. The Kronecker structure of an HMM yields nested block partitionings in its underlying continuous-time MC which may be used in the BSOR preconditioner. The computation of the BSOR preconditioned residual in each iteration of a preconditioned projection method becoms the problem of solving multiple nonsingular linear systems whose coefficient matrices are the diagonal blocks of the chosen partitioning. The proposed BSOR preconditioner solvers these systems using sparse LU or real Schur factors of diagonal blocks. The fill-in of sparse LU factorized diagonal blocks is reduced using the column approximate minimum degree algorithm (COLAMD). A set of numerical experiments are presented to show the merits of the proposed BSOR preconditioner.