Cette thèse se situe dans le domaine de la combinatoire algébrique. Autrement dit, l'idée est d'utiliser des structures algébriques, en l'occurence des algèbres de Hopf combinatoires, pour mieux étudier et comprendre les objets combinatoires ainsi que des algorithmes de composition et de décomposition agissant sur ces objets. Ce travail de recherche repose sur la construction et l'étude de structure algébrique sur des objets combinatoires généralisant les permutations. Après avoir rappelé le contexte et les notations des différents objets intervenant dans cette recherche, nous proposons dans la seconde partie l'étude de l'algèbre de Hopf introduite par Aguiar et Orellana indexée par les permutations de blocs uniformes. En se focalisant sur une description de ces objets via d'autres bien connus, les permutations et les partitions d'ensembles, nous proposons une réalisation polynomiale et une étude plus simple de cette algèbre. La troisième partie étudie une deuxième généralisation en interprétant les permutations comme des matrices. Nous définissons et étudions alors des familles de matrices carrées sur lesquelles nous définissons des algorithmes de composition et de décomposition. La quatrième partie traite des matrices à signes alternants. Après avoir définie l'algèbre de Hopf sur ces matrices, nous étudions des statistiques et le comportement de la structure algébrique vis-à-vis de ces statistiques. Tous ces chapitres s'appuient fortement sur l'exploration informatique, et fait l'objet d'une implémentation utilisant le logiciel Sage. Ce dernier chapitre est consacré à la découverte et la manipulation de structures algébriques sur Sage. Nous terminons en expliquant les améliorations apportées pour l'étude de structure algébrique au travers du logiciel Sage / This thesis is in the field of algebraic combinatorics. In other words, the idea is to use algebraic structures, in this case of combinatorial Hopf algebras, to better study and understand the combinatorial objects and algorithms for composition and decomposition about these objects. This research is based on the construction and study of algebraic structure of combinatorial objects generalizing permutations. After recalling the background and notations of various objects involved in this research, we propose, in the second part, the study of the Hopf algebra introduced by Aguiar and Orellana based on uniform block permutations. By focusing on a description of these objects via well-known objects, permutations and set partitions, we propose a polynomial realization and an easier study of this algebra. The third section considers a second generalization interpreting permutations as matrices. We define and then study the families of square matrices on which we define algorithms for composition and decomposition. The fourth part deals with alternating sign matrices. Having defined the Hopf algebra of these matrices, we study the statistics and the behavior of the algebraic structure with these statistics. All these chapters rely heavily on computer exploration, and is the subject of an implementation using Sage software. This last chapter is dedicated to the discovery and manipulation of algebraic structures on Sage. We conclude by explaining the improvements to the study of algebraic structure through the Sage software

A research report submitted to the Faculty of Science, University of the Witwatersrand, in partial fulfillment of the degree of Master of Science. / The Toeplitz form is a most useful and important teo! in many areas of applied. mathematics today including signal processing, time-series analysis and prediction theory. It is even used in quantum mechanics in Ising model correlation functions. (Abbreviation abstract) / AC 2018

by Chau Chun Bun. / Bibliography: leaves 85-86 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1984

Photocopy of typescript. / Digitized by Kansas Correctional Industries

Scattering is a fundamental phenomenon in physics, e.g. large parts of the knowledge about quantum systems stem from scattering experiments. A scattering process can be completely characterized by its K-matrix, also known as the \Wigner reaction matrix" in nuclear scattering or \impedance matrix" in the electromagnetic wave scattering. For chaotic quantum systems it can be modelled within the framework of Random Matrix Theory (RMT), where either the K-matrix itself or its underlying Hamiltonian is taken as a random matrix. These two approaches are believed to lead to the same results due to a universality conjecture by P. Brouwer, which is equivalent to the claim that the probability distribution of K, for a broad class of invariant ensembles of random Hermitian matrices H, converges to a matrix Cauchy distribution in the limit of large matrix dimension of H. For unitarily invariant ensembles, this conjecture will be proved in the thesis by explicit calculation, utilising results about ensemble averages of characteristic polynomials. This thesis furthermore analyses various characteristics of the K-matrix such as the distribution of a diagonal element at the spectral edge or the distribution of an off-diagonal element in the bulk of the spectrum. For the latter it is necessary to know correlation functions involving products and ratios of half-integer powers of characteristic polynomials of random matrices for the Gaussian Orthogonal Ensemble (GOE), which is an interesting and important topic in itself, as they frequently arise in various other applications of RMT to physics of quantum chaotic systems, and beyond. A larger part of the thesis is dedicated to provide an explicit evaluation of the large-N limits of a few non-trivial objects of that sort within a variant of the supersymmetry formalism, and via a related but different method.

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Chan, Ka Kwan Kevin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 84-86). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Preliminaries --- p.6 / Chapter 2.1 --- Eigenvalues of Sums of Hermitian Matrices --- p.6 / Chapter 2.2 --- Highest Weights --- p.12 / Chapter 2.3 --- Schubert Calculus --- p.15 / Chapter 2.4 --- Invariant Factors --- p.18 / Chapter 2.5 --- Singular Values of Sums and Products --- p.19 / Chapter 2.6 --- Relations Among the Problems --- p.25 / Chapter 3 --- Klyachko's Results --- p.27 / Chapter 3.1 --- Klyacho's Proof --- p.27 / Chapter 3.1.1 --- Rayleigh Trace --- p.28 / Chapter 3.1.2 --- Facts from Vector Bundles and Geometric Invariant Theory --- p.31 / Chapter 3.1.3 --- Proof of Klyachko's Results --- p.33 / Chapter 3.2 --- Proof by Symplectic Geometry --- p.42 / Chapter 4 --- Saturation Conjecture --- p.47 / Chapter 4.1 --- Definitions --- p.48 / Chapter 4.2 --- Proof of the Conjecture --- p.52 / Chapter 4.3 --- Remarks --- p.58 / Chapter 5 --- Proof of the Theorems --- p.60 / Chapter 5.1 --- Main Results --- p.60 / Chapter 5.2 --- "Real Symmetric and Quaternionic Hermitian Matrices, Compact Operators" --- p.70 / Chapter 5.3 --- Highest Weights --- p.71 / Chapter 5.4 --- Schubert Calculus --- p.71 / Chapter 5.5 --- Invariant Factors --- p.72 / Chapter 5.6 --- Singular Values of Sums and Products --- p.75 / Chapter 6 --- Further Problems --- p.79 / Bibliography --- p.84

A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand, Johannesburg, 1991. / After an introduction defining basic structutral aspects of near-rings, this report examines how the ring-theoretic notions of generation and cogeneration can be extended from modules over a ring to modules over a near-ring. Chapter four examines matrix near-rings and connections between the J2 and JS radicals of the near-ring and the corresponding matrix near-ring. By extending the concepts of generation and cogeneration from the ring modules to near-ring modules we are investigating how important distribution and an abelian additive structure are to these two concepts. The concept of generation faces the obstacle that the image of a near-ring module homomorphism is not necessarily a subrnodule of the image space but only a subgroup, while the sum of two subgroups need not even be a subgroup. In chapter two, generation trace and socle are defined for near-ring modules and these ideas are linked with those of the essential and module-essential subgroups. Cogeneration, dealing with kernels which are always submodules proved easier to generalise. This is discussed in chapter three together with the concept of the reject, and these ideas are Iinked to the J1/2 and J2 radicals. The duality of the ring theory case is lost. The results are less simple than in the ring theory case due to the different types of near-ring module substructures which give rise to several Jacobson-type radicals. A near-ring of matrices can be obtained from an arbitrary near-ring by regarding each rxr matrix as a mapping from Nr to Nr where N is the near-ring from which entries are taken. The argument showing that the near-ring is 2-semisimple if and only if the associated near-ring of matrices is 2-semisimple is presented and investigated in the case of s-semisimplicity. Questions arising from this report are presented in the final chapter.

Récemment, le développement de systèmes intelligents dédiés pour la compréhension du comportement humain est devenu un axe de recherche très important. En effet, il est très important de comprendre le comportement humain pour rendre les machines capables d'aider et interagir avec les humains. Pour cela, plusieurs approches de l'état de l'art commencent par détecter automatiquement un ensemble de points 2D ou 3D, appelés marqueurs, sur le corps et/ou le visage humain à partir de données visuelles. L’analyse des séquences temporelles de ces marqueurs pose plusieurs défis dus aux erreurs de suivi et aux variabilités temporelles et de pose. Dans cette thèse, nous proposons deux nouvelles représentations spatio-temporelles avec des outils de calcul appropriés pour la compréhension du comportement humain. La première consiste à représenter une séquence temporelle de marqueurs par une trajectoire de matrices de Gram. Les matrices de Gram sont des matrices semi-définies positives de rang fixe et vivent dans un espace non-linéaire dans lequel les outils d’apprentissage automatique conventionnels ne peuvent pas être appliqués directement. Nous évaluons l’efficacité de notre approche dans plusieurs applications, impliquant des marqueurs 2D et 3D de visages et de corps humain, tels que la reconnaissance des émotions à partir des expressions faciales la reconnaissance d’actions et des émotions à partir des données de profondeur 3D. La deuxième représentation proposée dans cette thèse est basée sur les coordonnées barycentriques des marqueurs de visages 2D. Cette représentation permet d’utiliser les outils de calcul et d’apprentissage automatique tels que les techniques d’apprentissage de métrique. Les résultats obtenus en reconnaissance des expressions faciales et en mesure automatique de la sévérité de la dépression à partir du visage montrent tout l’intérêt de la représentation barycentrique combinée à des techniques d’apprentissage automatique. Les résultats obtenus avec les deux méthodes proposées sur des bases de données réelles montrent la compétitivité de nos approches avec les méthodes récentes de l’état de l’art. / Developing intelligent systems dedicated to human behavior understanding has been a very hot research topic in the few recent decades. Indeed, it is crucial to understand the human behavior in order to make machines able to interact with, assist, and help humans in their daily life.. Recent breakthroughs in computer vision and machine learning have made this possible. For instance, human-related computer vision problems can be approached by first detecting and tracking 2D or 3D landmark points from visual data. Two relevant examples of this are given by the facial landmarks detected on the human face and the skeletons tracked along videos of human bodies. These techniques generate temporal sequences of landmark configurations, which exhibit several distortions in their analysis, especially in uncontrolled environments, due to view variations, inaccurate detection and tracking, missing data, etc. In this thesis, we propose two novel space-time representations of human landmark sequences along with suitable computational tools for human behavior understanding. Firstly, we propose a representation based on trajectories of Gram matrices of human landmarks. Gram matrices are positive semi-definite matrices of fixed rank and lie on a nonlinear manifold where standard computational and machine learning techniques could not be applied in a straightforward way. To overcome this issue, we make use of some notions of the Riemannian geometry and derive suitable computational tools for analyzing Gram trajectories. We evaluate the proposed approach in several human related applications involving 2D and 3D landmarks of human faces and bodies such us emotion recognition from facial expression and body movements and also action recognition from skeletons. Secondly, we propose another representation based on the barycentric coordinates of 2D facial landmarks. While being related to the Gram trajectory representation and robust to view variations, the barycentric representation allows to directly work with standard computational tools. The evaluation of this second approach is conducted on two face analysis tasks namely, facial expression recognition and depression severity level assessment. The obtained results with the two proposed approaches on real benchmarks are competitive with respect to recent state-of-the-art methods.