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We study the structure of the set of extreme points of the compact convex set of matrix-valued holomorphic functions with positive real part on a finitely-connected planar domain 𝐑 normalized to have value equal to the identity matrix at some prescribed point t₀ ∈ 𝐑. This leads to an integral representation for such functions more general than what would be expected from the result for the scalar-valued case. After Cayley transformation, this leads to a integral Agler decomposition for the matrix Schur class over 𝐑 (holomorphic contractive matrix-valued functions over 𝐑). Application of a general theory of abstract Schur-class generated by a collection of test functions leads to a transfer-function realization for the matrix Schur-class over 𝐑, extending results known up to now only for the scalar case. We also explain how these results provide a new perspective for the dilation theory for Hilbert space operators having 𝐑 as a spectral set. / Ph. D.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2015-05-18T18:27:17Z No. of bitstreams: 2 Dissertação - Jefferson dos Santos e Silva - 2015.pdf: 741852 bytes, checksum: 8cb431ec9a186100784d60268a133fcf (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Erika Demachki (erikademachki@gmail.com) on 2015-05-18T18:28:34Z (GMT) No. of bitstreams: 2 Dissertação - Jefferson dos Santos e Silva - 2015.pdf: 741852 bytes, checksum: 8cb431ec9a186100784d60268a133fcf (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-05-18T18:28:34Z (GMT). No. of bitstreams: 2 Dissertação - Jefferson dos Santos e Silva - 2015.pdf: 741852 bytes, checksum: 8cb431ec9a186100784d60268a133fcf (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-03-19 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this work, based on [9], describes an effective method for computing a consistent polycyclic presentation for the nonanbeian tensor square G G of a group G given by a consistent polycyclic presentation. / Este trabalho, baseado em [9], determina um efetivo método para calcular uma apresentação policíclica consistente para o quadrado tensorial não abeliano G G de um grupo G dado por uma apresentação policíclica consistente.

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. / In the following text, we are interested in the study of Lie-groups valued random variables. We give a deformation of the Weingarten calculus introduced by Benoît Collins and Piotr Sniady. We study the asymptotic behavior of Brownian motion on compact Lie groups in high dimensions and obtain new fluctuations results. Two new objects called the planar gaussian master field and the planar oriented master field are introduced here to describe the asymptotic behavior of the Yang-Mills measure as the dimension of the structure group is large.

O multiplicador de Schur parcial de um grupo G é um semigrupo inverso comutativo pM(G) que, no estudo de representações parciais projetivas de grupos, desempenha um papel análogo ao do multiplicador de Schur clássico M(G). Há uma descrição de pM(G) como uma união de grupos abelianos, em que cada componente pM_D(G) é formada por classes de equivalência de certas funções parciais (chamadas de conjuntos fatores parciais), as quais assumem valores em um corpo e têm como domínio um subconjunto D de G × G. Os domínios D formam um reticulado e foram caracterizados como os subconjuntos T-invariantes de G × G, em que T é um monoide específico atuando em G × G. A componente total pM_{G × G}(G), que corresponde aos conjuntos fatores totalmente definidos, é particularmente interessante pois contém M(G) como um de seus subgrupos e, além disso, qualquer outra componente é uma imagem epimorfa da componente total. Um dos objetivos deste trabalho é determinar a componente total do multiplicador de Schur parcial para algumas classes importantes de grupos, como os grupos diedrais, os grupos dicíclicos e os produtos de grupos cíclicos. Outro tópico que será abordado é a estrutura do reticulado dos domínios dos conjuntos fatores parciais, destacando-se propriedades daqueles que correspondem às representações parciais ditas elementares, as quais possuem um papel relevante na teoria. Provaremos que todo domínio pode ser representado em uma forma única como uma reunião de certos domínios indecomponíveis, que consistem de peças estruturais chamadas de blocos e domínios minimais. Também será determinada a estrutura dos domínios elementares e serão obtidos alguns invariantes numéricos do conjunto parcialmente ordenado dos domínios elementares. Como uma consequência dos resultados obtidos, serão caracterizados os grupos para os quais todos os domínios elementares são indecomponíveis. Além disso será feita uma aplicação da teoria de álgebras de semigrupos à álgebra parcial de grupo, que é uma álgebra responsável pelas representações parciais de grupos. / The partial Schur multiplier of a group G is a commutative inverse semigroup pM(G) which, in the study of partial projective representations, plays a role analogous to the classical Schur multiplier M(G). There is a description of pM(G) as a union of abelian groups, in which each component pM_D(G) is formed by the equivalence classes of certain partial functions (called partial factor sets), taking values in a field and having as its domain a subset D of G × G. The domains D form a lattice and were characterized as the T-invariant subsets of G × G, where T is a specific monoid acting on G × G. The total component pM_{G × G}(G), which corresponds to the totally defined factor sets, is particularly interesting because it contains M(G) as one of its subgroups and, moreover, any other component is an epimorphic image of the total component. One of the objectives of this work is to determine the total component of the partial Schur multiplier for some important classes of groups, such as the dihedral groups, the dicyclic groups and the products of cyclic groups. Another topic which will be considered is the structure of the lattice of domains of partial factor sets, emphasizing properties of those domains that correspond to the so-called elementary partial representations, which play a relevant role in the theory. We shall prove that each domain can be represented in a unique way as a union of certain indecomposable domains, where the latter consists of the so-called blocks and minimal domains. The structure of the elementary domains also will be determined, and some numerical invariants of the partially ordered set of the elementary domains will be given. As a consequence of the obtained facts, the groups whose elementary domains are indecomposable will be characterized. We will also give an application of the theory of semigroup algebras to the partial group algebra, an algebra which is responsible for partial group representations.

Dans le premier chapitre, nous commençons par définir certains produits tensoriels et identifions leur dual. Nous donnons ensuite quelques propriétés des classes de Schatten. La fin du chapitre est dédiée à l’étude des espaces de Bochner à valeurs dans l'espace des opérateurs factorisables par un espace de Hilbert. Le deuxième chapitre est consacré aux multiplicateurs de Schur linéaires. Nous caractérisons les multiplicateurs bornés sur B(Lp, Lq) lorsque p est inférieur à q puis appliquons ce résultat pour obtenir de nouvelles relations d'inclusion entre espaces de multiplicateurs. Dans le troisième chapitre, nous caractérisons, au moyen de multiplicateurs de Schur linéaires, les multiplicateurs de Schur bilinéaires continus à valeurs dans l'espace des opérateurs à trace. Dans le quatrième chapitre, nous donnons divers résultats concernant les opérateurs intégraux multiples. En particulier, nous caractérisons les opérateurs intégraux triples à valeurs dans l'espace des opérateurs à trace puis nous donnons une condition nécessaire et suffisante pour qu'un opérateur intégral triple définisse une application complètement bornée sur le produit de Haagerup de l'espace des opérateurs compacts. Enfin, le cinquième chapitre est dédié à la résolution des problèmes de Peller. Nous commençons par étudier le lien entre opérateurs intégraux multiples et théorie de la perturbation pour le calcul fonctionnel des opérateurs autoadjoints pour finir par la construction de contre-exemples à ces problèmes. / In the first chapter, we define some tensor products and we identify their dual space. Then, we give some properties of Schatten classes. The end of the chapter is dedicated to the study of Bochner spaces valued in the space of operators that can be factorized by a Hilbert space.The second chapter is dedicated to linear Schur multipliers. We characterize bounded multipliers on B(Lp, Lq) when p is less than q and then apply this result to obtain new inclusion relationships among spaces of multipliers.In the third chapter, we characterize, by means of linear Schur multipliers, continuous bilinear Schur multipliers valued in the space of trace class operators. In the fourth chapter, we give several results concerning multiple operator integrals. In particular, we characterize triple operator integrals mapping valued in trace class operators and then we give a necessary and sufficient condition for a triple operator integral to define a completely bounded map on the Haagerup tensor product of compact operators. Finally, the fifth chapter is dedicated to the resolution of Peller's problems. We first study the connection between multiple operator integrals and perturbation theory for functional calculus of selfadjoint operators and we finish with the construction of counter-examples for those problems.

La complexification de la modélisation multi-physique conduit d’une part à devoir simuler des systèmes d’équations différentielles ordinaires et d’équations différentielles algébriques de plus en plus grands en nombre d’inconnues et sur des temps de simulation longs. D’autre part l’évolution des architectures de calcul parallèle nécessite d’autres voies de parallélisation que la décomposition de système en sous-systèmes. Dans ce travail, nous proposons de concevoir des méthodes de décomposition de domaine pour la résolution d’EDO en temps. Nous reformulons le problème à valeur initiale en un problème aux valeurs frontières sur l’intervalle de temps symétrisé, sous l’hypothèse de réversibilité du flot. Nous développons deux méthodes, la première apparentée à une méthode de complément de Schur, la seconde basée sur une méthode de type Schwarz dont nous montrons la convergence pouvant être accélérée par la méthode d’Aitken dans le cadre linéaire. Afin d’accélérer la convergence de cette dernière dans le cadre non-linéaire, nous introduisons les techniques d’extrapolation et d’accélération de la convergence des suites non-linéaires. Nous montrons les avantages et les limites de ces techniques. Les résultats obtenus nous conduisent à développer l’accélération de la méthode de type Schwarz par une méthode de Newton. Enfin nous nous intéressons à l’étude de conditions de raccord non-linéaires adaptées à la décomposition de domaine de problèmes non-linéaires. Nous nous servons du formalisme hamiltonien à ports, issu du domaine de l’automatique, pour déduire les conditions de raccord dans le cadre l’équation de Saint-Venant et de l’équation de la chaleur non-linéaire. Après une étude analytique de la convergence de la DDM associée à ces conditions de transmission, nous proposons et étudions une formulation de Lagrangien augmenté sous l’hypothèse de séparabilité de la contrainte. / Complexification of multi-physics modeling leads to have to simulate systems of ordinary differential equations and algebraic differential equations with increasingly large numbers of unknowns and over large times of simulation. In addition the evolution of parallel computing architectures requires other ways of parallelization than the decomposition of system in subsystems. In this work, we propose to design domain decomposition methods in time for the resolution of EDO. We reformulate the initial value problem in a boundary values problem on the symmetrized time interval, under the assumption of reversibility of the flow. We develop two methods, the first connected with a Schur complement method, the second based on a Schwarz type method for which we show convergence, being able to be accelerated by the Aitken method within the linear framework. In order to accelerate the convergence of the latter within the non-linear framework, we introduce the techniques of extrapolation and of acceleration of the convergence of non-linear sequences. We show the advantages and the limits of these techniques. The obtained results lead us to develop the acceleration of the method of the type Schwarz by a Newton method. Finally we investigate non-linear matching conditions adapted to the domain decomposition of nonlinear problems. We make use of the port-Hamiltonian formalism, resulting from the control field, to deduce the matching conditions in the framework of the shallow-water equation and the non-linear heat equation. After an analytical study of the convergence of the DDM associated with these conditions of transmission, we propose and study a formulation of augmented Lagrangian under the assumption of separability of the constraint.

設H是一個實數正交的矩陣，我們要求它的特徵值以及特徵向量。H可以表示成Schur參數的形式。根據Ammar，Gragg及Reichel的論文，我們把H的特徵問題轉換成兩個元素由Schur參數決定的二對角矩陣的奇異值(奇異向量)的問題。我們用這個方法寫成程式並且與CLAPACK的程式比較準確度及速度。最後列出一些數值的結果作為結論。 / Let H be an orthogonal Hessenberg matrix whose eigenvalues, and possibly eigenvectors, are to be determined. Then H can be represented in Schur parametric form [2]. Following Ammar, Gragg, and Reichel's paper [1], we compute the eigenproblem of H by finding the singular values (and vectors) of two bidiagonal matrices whose elements are explicitly known functions of the Schur parameters. We compare the accuracy and speed of our programs using the method described aboved with those in CLAPACK. Numerical results conclude this thesis.

Hierarchical Markovian Models (HMMs) are composed of multiple low level models (LLMs) and high level model (HLM) that defines the interaction among LLMs. The essence of the HMM approach is to model the system at hand in the form of interacting components so that its (larger) underlying continous-time Markov chain (CTMC) is not generated but implicitly represented as a sum of Kronecker products of (smaller) component matrices. The Kronecker structure of an HMM induces nested block partitionings in its underlying CTMC. These partitionings may be used in block versions of classical iterative methods based on splittings, such as block SOR (BSOR), to solve the underlying CTMC for its stationary vector. Therein the problem becomes that of solving multiple nonsingular linear systems whose coefficient matrices are the diagonal blocks of a particular partitioning. This paper shows that in each HLM state there may be diagonal blocks with identical off-diagonal parts and diagonals differing from each other by a multiple of the identity matrix. Such diagonal blocks are named candidate blocks. The paper explains how candidate blocks can be detected and how the can mutually benefit from a single real Schur factorization. It gives sufficient conditions for the existence of diagonal blocks with real eigenvalues and shows how these conditions can be checked using component matrices. It describes how the sparse real Schur factors of candidate blocks satisfying these conditions can be constructed from component matrices and their real Schur factors. It also demonstrates how fill in- of LU factorized (non-candidate) diagonal blocks can be reduced by using the column approximate minimum degree algorithm (COLAMD). Then it presents a three-level BSOR solver in which the diagonal blocks at the first level are solved using block Gauss-Seidel (BGS) at the second and the methods of real Schur and LU factorizations at the third level. Finally, on a set of numerical experiments it shows how these ideas can be used to reduce the storage required by the factors of the diagonal blocks at the third level and to improve the solution time compared to an all LU factorization implementation of the three-level BSOR solver.

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.