http://hdl.handle.net/10362/1047 DM_PhD 2013-06-19T13:26:55Z http://hdl.handle.net/10362/8677 Title: Factorization of elliptic boundary value problems by invariant embedding and application to overdetermined problems Authors: Orey, Maria de Serpa Salema Reis de Abstract: The purpose of this thesis is the factorization of elliptic boundary value problems defined in cylindrical domains, in a system of decoupled first order initial value problems. We begin with the Poisson equation with mixed boundary conditions, and use the method of invariant embedding: we embed our initial problem in a family of similar problems, defined in sub-domains of the initial domain, with a moving boundary, and an additional condition in the moving boundary. This factorization is inspired by the technique of invariant temporal embedding used in Control Theory when computing the optimal feedback, for, in fact, as we show, our initial problem may be defined as an optimal control problem. The factorization thus obtained may be regarded as a generalized block Gauss LU factorization. From this procedure emerges an operator that can be either the Dirichlet-to-Neumann or the Neumann-to-Dirichlet operator, depending on which boundary data is given on the moving boundary. In any case this operator verifies a Riccati equation that is studied directly by using an Yosida regularization. Then we extend the former results to more general strongly elliptic operators. We also obtain a QR type factorization of the initial problem, where Q is an orthogonal operator and R is an upper triangular operator. This is related to a least mean squares formulation of the boundary value problem. In addition, we obtain the factorization of overdetermined boundary value problems, when we consider an additional Neumann boundary condition: if this data is not compatible with the initial data, then the problem has no solution. In order to solve it, we introduce a perturbation in the original problem and minimize the norm of this perturbation, under the hypothesis of existence of solution. We deduce the normal equations for the overdetermined problem and, as before, we apply the method of invariant embedding to factorize the normal equations in a system of decoupled first order initial value problems. Description: Dissertação para obtenção do Grau de Doutor em Matemática 2011-01-01T00:00:00Z http://hdl.handle.net/10362/7856 Title: Spectral and homogenization problems Authors: Ferreira, Rita Alexandra Gonçalves Description: Dissertation for the Degree of Doctor of Philosophy in Mathematics 2011-01-01T00:00:00Z http://hdl.handle.net/10362/7848 Title: New strategies to detect and understand genotype-by-environment interactions and QTL-by-environment interactions Authors: Rodrigues, Paulo Jorge Canas Abstract: Genotype-by-environment interaction (GEI) is frequent in multi-environment trials, and represents differential responses of genotypes across environments. With the development of molecular markers and mapping techniques, researchers can go one step further and analyse the whole genome to detect specific locations of genes which influence a quantitative trait such as yield. These locations are called quantitative trait locus (QTL), and when these QTLs have different expression across environments we talk about QTLby-environment interactions (QEI), which is the base of GEI. Good understandings of these interactions enable researchers to select better genotypes across different environmental conditions and, consequently, to improve crops in developed and developing countries. In this thesis I intend to present new strategies to improve detection and better understanding of QTLs, especially those exhibiting QEI in the context of multi-environment trials, by using and providing open source software. The first part of this thesis presents a comparison between two of the most used methods to analyse and to structure GEI: the joint regression analysis (JRA) and the additive main effects and multiplicative interaction (AMMI) model. This comparison is made in terms of “robustness” with different incidence rates of missing values, and in terms of dominant/winner genotypes. In the following chapters two- and threestages approaches are presented in which the AMMI model is used to gain accuracy in the phenotypic data, and their scores used to order the environments to find ecological or biological patterns. The first approach (two stages) is appropriated when the error variance is constant across environments, whereas the second (three stages) is more general and accounts for differences in the error variances by using the proposed weighted AMMI model (WAMMI). The final part of the thesis illustrates a strategy to simulate and to model GEI and QEI in complex traits, with the example of yield, based on a number of physiological parameters purely genotype dependent. This is done by using an eco-physiological genotype-to-phenotype model with seven parameters defined with a simple QTL basis. Description: Dissertação para obtenção do Grau de Doutor em Estatística e Gestão do Risco, especialidade em Estatística 2012-01-01T00:00:00Z http://hdl.handle.net/10362/6647 Title: Monóides de transformações Authors: Quinteiro, Teresa Maria de Araújo Melo Abstract: Na Teoria dos Semigrupos é extremamente importante o papel dos semigrupos de transformações. De facto, estes desempenham o papel, na Teoria dos Semigrupos, correspondente ao dos grupos de permutações, na Teoria dos Grupos. Estão ainda presentes de modo crucial na Teoria dos Autómatos e Linguagens Formais, tendo assim aplicabilidade na Computação Teórica e na Linguística, bem como em muitas outras áreas do conhecimento. As cardinalidades e as características de diversas classes de semigrupos de transformações(totais, parciais, parciais injectivas, que preservam a ordem, a orientação ou uma relação de equivalência) têm sido objecto de pesquisa de um número considerável de autores. Na primeira parte desta dissertação apresentamos a nossa contribuição para este estudo calculando as cardinalidades e as características de alguns monóides de transformações sobre uma cadeia nita que preservam uma participação uniforme. A segunda parte deste trabalho é dedicada a uma construção de semigrupos, o produto semidirecto bilateral, introduzida para grupos por Zappa e estudada para semigrupos por Kunze. Usando várias estratégias, decompomos certos monóides de transformações como quocientes de um produto semidirecto bilateral de dois dos seus submonóides. Um dos procedimentos que utilizamos resulta de um processo geral para obter produtos semidirectos bilaterais, o qual consiste na construção de um produto semidirecto bilateral de dois monóides livres que, sob determinadas condições, induz um produto semidirecto bilateral de dois monóides de nidos por apresentações associadas a esses monóides livres. Como aplicação, deduzimos decomposições de alguns monóides de transformações sobre uma cadeia nita, entre os quais salientamos o monóide das transformações crescentes. Os resultados obtidos têm aplicabilidade imediata às pseudovariedades geradas pelos monóides em questão permitindo-nos em particular concluir que a pseudovariedade O, gerada pela família dos monóides de transformações totais e crescentes sobre uma cadeia com n elementos, está propriamente contida no produto semidirecto bilateral da pseudovariedade J, dos monóides J -triviais, por ela própria. Description: Dissertação para obtenção do Grau de Doutor em Matemática 2011-01-01T00:00:00Z