Orientador: Maria Zoraide Martins Costa Soares / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T23:10:07Z (GMT). No. of bitstreams: 1 Campos_LudioEdsondaSilva_M.pdf: 1007258 bytes, checksum: 78308663e7f18b51bedeb284cadca66a (MD5) Previous issue date: 2008 / Resumo: Neste trabalho abordamos algumas fatorações de matrizes, com vistas à resolução de sistemas lineares através de métodos diretos. Enfocamos particularmente as decomposições LU, Cholesky e QR, cujo uso tem sido largamente difundido em implementações computacionais. Nosso objetivo é apresentar um texto didático, acessível a alunos de graduação, que contemple a teoria básica de cada fatoração, incluindo a demonstração dos principais resultados, e que também forneça condições para uma primeira implementação de cada decomposição. Sugerimos alguns algoritmos, que foram implementados no software livre OCTAVE, através dos quais comparamos o tempo gasto para resolução de alguns sistemas lineares, utilizando as fatorações citadas / Abstract: In this work we discuss some matrix factorizations, with a view to the resolution of linear systems through direct methods. We focus particularly the LU, Cholesky and QR decompositions, whose use has been widely spread in computer implementations. Our goal is to present a didactic text, accessible to undergraduate students, which contemplates the basic theory of each factorization, including the demonstration of the main result and that also provide conditions for a first implementation of each decomposition. We suggest some algorithms that were scheduled in the free software OCTAVE, through which we compare the time elapsed for the resolution of a few linear systems, using the factorizations cited. / Mestrado / Algebra linear / Mestre em Matemática

In this thesis a heuristic method for factoring semiprimes by multiagent depth-limited search of PG2N graphs is presented. An analysis of PG2N graph connectivity is used to generate heuristics for multiagent search. Further analysis is presented including the requirements on choosing prime numbers to generate 'hard' semiprimes; the lack of connectivity in PG1N graphs; the counts of spanning trees in PG2N graphs; the upper bound of a PG2N graph diameter and a conjecture on the frequency distribution of prime numbers on Hamming distance. We further demonstrated the feasibility of the HD2 breadth first search of PG2N graphs for factoring small semiprimes. We presented the performance of different multiagent search heuristics in PG2N graphs showing that the heuristic of most connected seedpick outperforms least connected or random connected seedpick heuristics on small PG2N graphs of size N

Orientador: Martinho da Costa Araujo / Dissertação (mestrado profissional) - Universidade Estadual de Campinas. Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T09:34:59Z (GMT). No. of bitstreams: 1 Souza_VeraLuciaGracianide_M.pdf: 1138543 bytes, checksum: 893a12834a41de0bedf2e0e1c71a3fc1 (MD5) Previous issue date: 2009 / Resumo: Este trabalho tem por objetivo fatorar número inteiro utilizando pontos racionais sobre o círculo unitário. Igualmente pretende determinar alguns grupos sobre cônicas. A pesquisa inicia com os conceitos básicos de Álgebra e Teoria dos Números, que fundamentam que o conjunto de pontos racionais sobre o círculo unitário tem uma estrutura de grupo. Desse conjunto é possível estender a idéia de grupo de pontos racionais sobre o círculo para pontos racionais sobre cônicas. Para encontrar os pontos racionais sobre o círculo foi usada uma parametrização do círculo por funções trigonométricas. Para cada ponto sobre o círculo unitário está associado um ângulo com o eixo positivo das abscissas, portanto adicionar pontos sobre o círculo equivale adicionar seus ângulos correspondentes. Com a operação "adição" de pontos sobre o círculo é possível definir uma estrutura de grupo que é utilizada para fatorar números inteiros. Para a cônica, a operação "adição" é determinada algebricamente ao calcular o coeficiente angular da reta que passa por dois pontos dados e o elemento neutro dessa cônica, também justificada geometricamente. No trabalho foram determinados os grupos de pontos racionais sobre cônicas e demonstrado alguns resultados sobre esses grupos usando os resíduos quadráticos e finalizando com a dedução de alguns resultados sobre a soma das coordenadas dos pontos sobre uma cônica. / Abstract: The objective of this paper is to factorize integer number using rational points on the unitary circle. Also, it intends to determinate some groups on the conics. The research begins with the basic concepts of Algebra and Number Theory ensuring that the rational points set on the unitary circle has a structure of group. From this set is possible to extend the idea of rational points on the circle toward rational points on conics. In order to find the rational points on the circle a parametrization by trigonometric function on it was used. For each point on the unitary circle it is associated an angle with abscissa positive axis, therefore adding points on the circle equals to add its corresponding angles. With the operation of "addition" points on the circle it is possible to define a group structure that is used to factorize integer numbers. For the conic, the "addition" operation is algebraically determinated when the angle coeficient of the line is calculated that joins two given points and the neutral element of that conic, which is geometrically justified. In the research the rational points groups on the conics were determined, and some result on these groups using quadratic residues were demonstrated, and it was finalized with the deduction of some results concerning the coordinates sum of points on a conics. / Mestrado / Mestre em Matemática

Cette thèse vise à proposer de nouveaux modèles pour la séparation de sources dans le cadre non linéaire des méthodes à noyaux en apprentissage statistique, et à développer des algorithmes associés. Le domaine d'application privilégié est le démélange en imagerie hyperspectrale. Tout d'abord, nous décrivons un modèle original de la factorisation en matrices non négatives (NMF), en se basant sur les méthodes à noyaux. Le modèle proposé surmonte la malédiction de préimage, un problème inverse hérité des méthodes à noyaux. Dans le même cadre proposé, plusieurs extensions sont développées pour intégrer les principales contraintes soulevées par les images hyperspectrales. Pour traiter des masses de données, des algorithmes de traitement en ligne sont développés afin d'assurer une complexité calculatoire fixée. Également, nous proposons une approche de factorisation bi-objective qui permet de combiner les modèles de démélange linéaire et non linéaire, où les décompositions de NMF conventionnelle et à noyaux sont réalisées simultanément. La dernière partie se concentre sur le démélange robuste aux bandes spectrales aberrantes. En décrivant le démélange selon le principe de la maximisation de la correntropie, deux problèmes de démélange robuste sont traités sous différentes contraintes soulevées par le problème de démélange hyperspectral. Des algorithmes de type directions alternées sont utilisés pour résoudre les problèmes d'optimisation associés / This thesis aims to propose new nonlinear unmixing models within the framework of kernel methods and to develop associated algorithms, in order to address the hyperspectral unmixing problem.First, we investigate a novel kernel-based nonnegative matrix factorization (NMF) model, that circumvents the pre-image problem inherited from the kernel machines. Within the proposed framework, several extensions are developed to incorporate common constraints raised in hypersepctral images analysis. In order to tackle large-scale and streaming data, we next extend the kernel-based NMF to an online fashion, by keeping a fixed and tractable complexity. Moreover, we propose a bi-objective NMF model as an attempt to combine the linear and nonlinear unmixing models. The decompositions of both the conventional NMF and the kernel-based NMF are performed simultaneously. The last part of this thesis studies a supervised unmixing model, based on the correntropy maximization principle. This model is shown robust to outlier bands. Two correntropy-based unmixing problems are addressed, considering different constraints in hyperspectral unmixing problem. The alternating direction method of multipliers (ADMM) is investigated to solve the related optimization problems

Abstract in English / The research will analyze and compare the current research on the spectral factorization of non-singular and singular matrices. We show that a nonsingular non-scalar matrix A can be written as a product A = BC where the eigenvalues of B and C are arbitrarily prescribed subject to the condition that the product of the eigenvalues of B and C must be equal to the determinant of A. Further, B and C can be simultaneously triangularised as a lower and upper triangular matrix respectively. Singular matrices will be factorized in terms of nilpotent matrices and otherwise over an arbitrary or complex field in order to present an integrated and detailed report on the current state of research in this area. Applications related to unipotent, positive-definite, commutator, involutory and Hermitian factorization are studied for non-singular matrices, while applications related to positive-semidefinite matrices are investigated for singular matrices. We will consider the theorems found in Sourour [24] and Laffey [17] to show that a non-singular non-scalar matrix can be factorized spectrally. The same two articles will be used to show applications to unipotent, positive-definite and commutator factorization. Applications related to Hermitian factorization will be considered in [26]. Laffey [18] shows that a non-singular matrix A with det A = ±1 is a product of four involutions with certain conditions on the arbitrary field. To aid with this conclusion a thorough study is made of Hoffman [13], who shows that an invertible linear transformation T of a finite dimensional vector space over a field is a product of two involutions if and only if T is similar to T−1. Sourour shows in [24] that if A is an n × n matrix over an arbitrary field containing at least n + 2 elements and if det A = ±1, then A is the product of at most four involutions. We will review the work of Wu [29] and show that a singular matrix A of order n ≥ 2 over the complex field can be expressed as a product of two nilpotent matrices, where the rank of each of the factors is the same as A, except when A is a 2 × 2 nilpotent matrix of rank one. Nilpotent factorization of singular matrices over an arbitrary field will also be investigated. Laffey [17] shows that the result of Wu, which he established over the complex field, is also valid over an arbitrary field by making use of a special matrix factorization involving similarity to an LU factorization. His proof is based on an application of Fitting's Lemma to express, up to similarity, a singular matrix as a direct sum of a non-singular and nilpotent matrix, and then to write the non-singular component as a product of a lower and upper triangular matrix using a matrix factorization theorem of Sourour [24]. The main theorem by Sourour and Tang [26] will be investigated to highlight the necessary and sufficient conditions for a singular matrix to be written as a product of two matrices with prescribed eigenvalues. This result is used to prove applications related to positive-semidefinite matrices for singular matrices. / National Research Foundation of South Africa / Mathematical Sciences / M Sc. (Mathematics)

This study investigates the possibilities and implications for the teaching of the solving of polynomial equations. It is historically directed and also focusses on the working procedures in algebra which target the cognitive and affective domains. The teaching implications of the development of representational styles of equations and their solving procedures are noted. Since concepts in algebra can be conceived as processes or objects this leads to cognitive obstacles, for example: a limited view of the equal sign, which result in learning and reasoning problems. The roles of sense-making, visual imagery, mental schemata and networks in promoting meaningful understanding are scrutinised. Questions and problems to solve are formulated to promote the processes associated with the solving of polynomial equations, and the solving procedures used by a group of college students are analysed. A teaching model/method, which targets the cognitive and affective domains, is presented. / Mathematics Education / M.A. (Mathematics Education)

This study investigates the possibilities and implications for the teaching of the solving of polynomial equations. It is historically directed and also focusses on the working procedures in algebra which target the cognitive and affective domains. The teaching implications of the development of representational styles of equations and their solving procedures are noted. Since concepts in algebra can be conceived as processes or objects this leads to cognitive obstacles, for example: a limited view of the equal sign, which result in learning and reasoning problems. The roles of sense-making, visual imagery, mental schemata and networks in promoting meaningful understanding are scrutinised. Questions and problems to solve are formulated to promote the processes associated with the solving of polynomial equations, and the solving procedures used by a group of college students are analysed. A teaching model/method, which targets the cognitive and affective domains, is presented. / Mathematics Education / M.A. (Mathematics Education)