Cette thèse est divisée en cinq parties portant sur les thèmes suivants: l’interprétation physique et algébrique de familles de fonctions orthogonales multivariées et leurs applications, les systèmes quantiques superintégrables en deux et trois dimensions faisant intervenir des opérateurs de réflexion, la caractérisation de familles de polynômes orthogonaux appartenant au tableau de Bannai-Ito et l’examen des structures algébriques qui leurs sont associées, l’étude de la relation entre le recouplage de représentations irréductibles d’algèbres et de superalgèbres et les systèmes superintégrables, ainsi que l’interprétation algébrique de familles de polynômes multi-orthogonaux matriciels. Dans la première partie, on développe l’interprétation physico-algébrique des familles de polynômes orthogonaux multivariés de Krawtchouk, de Meixner et de Charlier en tant qu’éléments de matrice des représentations unitaires des groupes SO(d+1), SO(d,1) et E(d) sur les états d’oscillateurs. On détermine les amplitudes de transition entre les états de l’oscillateur singulier associés aux bases cartésienne et polysphérique en termes des polynômes multivariés de Hahn. On examine les coefficients 9j de su(1,1) par le biais du système superintégrable générique sur la 3-sphère. On caractérise les polynômes de q-Krawtchouk comme éléments de matrices des «q-rotations» de U_q(sl_2). On conçoit un réseau de spin bidimensionnel qui permet le transfert parfait d’états quantiques à l’aide des polynômes de Krawtchouk à deux variables et on construit un modèle discret de l’oscillateur quantique dans le plan à l’aide des polynômes de Meixner bivariés. Dans la seconde partie, on étudie les systèmes superintégrables de type Dunkl, qui font intervenir des opérateurs de réflexion. On examine l’oscillateur de Dunkl en deux et trois dimensions, l’oscillateur singulier de Dunkl dans le plan et le système générique sur la 2-sphère avec réflexions. On démontre la superintégrabilité de chacun de ces systèmes. On obtient leurs constantes du mouvement, on détermine leurs algèbres de symétrie et leurs représentations, on donne leurs solutions exactes et on détaille leurs liens avec les polynômes orthogonaux du tableau de Bannai-Ito. Dans la troisième partie, on caractérise deux familles de polynômes du tableau de Bannai-Ito: les polynômes de Bannai-Ito complémentaires et les polynômes de Chihara. On montre également que les polynômes de Bannai-Ito sont les coefficients de Racah de la superalgèbre osp(1,2). On détermine l’algèbre de symétrie des polynômes duaux -1 de Hahn dans le cadre du problème de Clebsch-Gordan de osp(1,2). On propose une q - généralisation des polynômes de Bannai-Ito en examinant le problème de Racah pour la superalgèbre quantique osp_q(1,2). Finalement, on montre que la q -algèbre de Bannai-Ito sert d’algèbre de covariance à osp_q(1,2). Dans la quatrième partie, on détermine le lien entre le recouplage de représentations des algèbres su(1,1) et osp(1,2) et les systèmes superintégrables du deuxième ordre avec ou sans réflexions. On étudie également les représentations des algèbres de Racah-Wilson et de Bannai-Ito. On montre aussi que l’algèbre de Racah-Wilson sert d’algèbre de covariance quadratique à l’algèbre de Lie sl(2). Dans la cinquième partie, on construit deux familles explicites de polynômes d-orthogonaux basées sur su(2). On étudie les états cohérents et comprimés de l’oscillateur fini et on caractérise une famille de polynômes multi-orthogonaux matriciels. / This thesis is divided into five parts concerned with the following topics: the physical and algebraic interpretation of families of multivariate orthogonal functions and their applications, the study of superintegrable quantum systems in two and three dimensions involving reflection operators, the characterization of families of orthogonal polynomials of the Bannai-Ito scheme and the study of the algebraic structures associated to them, the investigation of the relationship between the recoupling of irreducible representations of algebras and superalgebras and superintegrable systems, as well as the algebraic interpretation of families of matrix multi-orthogonal polynomials. In the first part, we develop the physical and algebraic interpretation of the Krawtchouk, Meixner and Charlier families of multivariate orthogonal polynomials as matrix elements of unitary representations of the SO(d + 1), SO(d, 1) and E(d) groups on oscillator states. We determine the transition amplitudes between the states of the singular oscillator associated to the Cartesian and polyspherical bases in terms of the multivariate Hahn polynomials. We examine the 9j coefficients of su(1,1) through the generic superintegrable system on the 3-sphere. We characterize the q-Krawtchouk polynomials as matrix elements of "q-rotations" of U_q(sl_2). We show how to design a two-dimensional spin network that allows perfect state transfer using the two-variable Krawtchouk polynomials and we construct a discrete model of the two-dimensional quantum oscillator using the two-variable Meixner polynomials. In the second part, we study superintegrable systems of Dunkl type, which involve reflections. We examine the Dunkl oscillator in two and three dimensions, the singular Dunkl oscillator in the plane and the generic system on the 2-sphere with reflections. We show that each of these systems is superintegrable. We obtain their constants of motion, we find their symmetry algebras as well as their representations, we give their exact solutions and we exhibit their relationship with the orthogonal polynomials of the Bannai-Ito scheme. In the third part, we characterize two families of polynomials belonging to the Bannai-Ito scheme: the complementary Bannai-Ito polynomials and the Chihara polynomials. We also show that the Bannai–Ito polynomials arise as Racah coefficients for the osp(1,2) superalgebra. We determine the symmetry algebra associated with the dual − 1 Hahn polynomials in the context of the Clebsch-Gordan problem for osp(1,2). We introduce a q -generalization of the Bannai-Ito polynomials by examining the Racah problem for the quantum superalgebra osp_q(1,2). Finally, we show that the q-deformed Bannai-Ito algebra serves as a covariance algebra for osp_q(1,2). In the fourth part, we determine the relationship between the recoupling of representations of the su(1,1) and osp(1,2) algebras and second-order superintegrable systems with or without reflections. We also study representations of Racah–Wilson and Bannai-Ito algebras. Moreover, we show that the Racah Wilson algebra serves as a quadratic covariance algebra for sl(2). In the fifth part, we explicitly construct two families of d-orthogonal polynomials based on su(2). We investigate the squeezed/coherent states of the finite oscillator and we characterize a family of matrix multi-orthogonal polynomials.

Cette thèse se situe dans le domaine de la combinatoire algébrique et porte sur l'étude et les applications de trois ordres sur les permutations : les deux ordres faibles (gauche et droit) et l'ordre fort ou de Bruhat. Dans un premier temps, nous étudions l'action du groupe symétrique sur les polynômes multivariés. En particulier, les opérateurs de emph{différences divisées} permettent de définir des bases de l'anneau des polynômes qui généralisent les fonctions de Schur aussi bien du point de vue de leur construction que de leur interprétation géométrique. Nous étudions plus particulièrement la base des polynômes de Grothendieck introduite par Lascoux et Schützenberger. Lascoux a montré qu'un certain produit de polynômes peut s'interpréter comme un produit d'opérateurs de différences divisées. En développant ce produit, nous ré-obtenons un résultat de Lenart et Postnikov et prouvons de plus que le produit s'interprète comme une somme sur un intervalle de l'ordre de Bruhat. Nous présentons aussi l'implantation que nous avons réalisée sur Sage des polynômes multivariés. Cette implantation permet de travailler formellement dans différentes bases et d'effecteur des changements de bases. Elle utilise l'action des différences divisées sur les vecteurs d'exposants des polynômes multivariés. Les bases implantées contiennent en particulier les polynômes de Schubert, les polynômes de Grothendieck et les polynômes clés (ou caractères de Demazure).Dans un second temps, nous étudions le emph{treillis de Tamari} sur les arbres binaires. Celui-ci s'obtient comme un quotient de l'ordre faible sur les permutations : à chaque arbre est associé un intervalle de l'ordre faible formé par ses extensions linéaires. Nous montrons qu'un objet plus général, les intervalles-posets, permet de représenter l'ensemble des intervalles du treillis de Tamari. Grâce à ces objets, nous obtenons une formule récursive donnant pour chaque arbre binaire le nombre d'arbres plus petits ou égaux dans le treillis de Tamari. Nous donnons aussi une nouvelle preuve que la fonction génératrice des intervalles de Tamari vérifie une certaine équation fonctionnelle décrite par Chapoton. Enfin, nous généralisons ces résultats aux treillis de $m$-Tamari. Cette famille de treillis introduite par Bergeron et Préville-Ratelle était décrite uniquement sur les chemins. Nous en donnons une interprétation sur une famille d'arbres binaires en bijection avec les arbres $m+1$-aires. Nous utilisons cette description pour généraliser les résultats obtenus dans le cas du treillis de Tamari classique. Ainsi, nous obtenons une formule comptant le nombre d'éléments plus petits ou égaux qu'un élément donné ainsi qu'une nouvelle preuve de l'équation fonctionnelle des intervalles de $m$-Tamari. Pour finir, nous décrivons des structures algébriques $m$ qui généralisent les algèbres de Hopf $FQSym$ et $PBT$ sur les permutations et les arbres binaires / This thesis comes within the scope of algebraic combinatorics and studies problems related to three orders on permutations: the two said weak orders (right and left) and the strong order or Bruhat order.We first look at the action of the symmetric group on multivariate polynomials. By using the emph{divided differences} operators, one can obtain some generalisations of the Schur function and form bases of non symmetric multivariate polynomials. This construction is similar to the one of Schur functions and also allows for geometric interpretations. We study more specifically the Grothendieck polynomials which were introduced by Lascoux and Schützenberger. Lascoux proved that a product of these polynomials can be interpreted in terms of a product of divided differences. By developing this product, we reobtain a result of Lenart and Postnikov and also prove that it can be interpreted as a sum over an interval of the Bruhat order. We also present our implementation of multivariate polynomials in Sage. This program allows for formal computation on different bases and also implements many changes of bases. It is based on the action of the divided differences operators. The bases include Schubert polynomials, Grothendieck polynomials and Key polynomials. In a second part, we study the emph{Tamari lattice} on binary trees. This lattice can be obtained as a quotient of the weak order. Each tree is associated with the interval of its linear extensions. We introduce a new object called, emph{interval-posets} of Tamari and show that they are in bijection with the intervals of the Tamari lattice. Using these objects, we give the recursive formula counting the number of elements smaller than or equal to a given tree. We also give a new proof that the generating function of the intervals of the Tamari lattice satisfies some functional equation given by Chapoton. Our final contributions deals with the $m$-Tamari lattices. This family of lattices is a generalization of the classical Tamari lattice. It was introduced by Bergeron and Préville-Ratelle and was only known in terms of paths. We give the description of this order in terms of some family of binary trees, in bijection with $m+1$-ary trees. Thus, we generalize our previous results and obtain a recursive formula counting the number of elements smaller than or equal to a given one and a new proof of the functional equation. We finish with the description of some new $"m"$ Hopf algebras which are generalizations of the known $FQSym$ on permutations and $PBT$ on binary trees

O método de elementos analíticos simula escoamentos subterrâneos por meio da superposição de soluções conceituais. No contexto do método, inomogeneidade é uma região bem definida de condutividade hidráulica constante. A diferença de condutividade hidráulica entre a inomogeneidade e o meio em que está inserida causa uma descontinuidade (salto) no potencial de descarga. Tradicionalmente este salto é simulado usando polinômios de primeiro ou segundo grau. O presente trabalho usa polinômios splines quadráticos para interpolar os saltos ocorridos no potencial de descarga ao longo das bordas de inomogeneidades. Paralelamente, a formulação tradicional de interpolação dos saltos no potencial de descarga é estendida para qualquer grau. Os principais elementos que compõe o método são descritos e implementados. O programa computacional resultante (AEM) foi desenvolvido integrado a um sistema de informações geográficas de código-aberto (JUMP). O programa permite a integração com outros sistemas de informações geográficas baseados em JAVA, guardando independência do SIG residente. O projeto do programa AEM/JUMP é baseado na programação orientada a objetos e apresentou grande afinidade com o método de elementos analíticos, havendo identificação entre os conceitos de elemento (usado pelo método) e de objeto (usado pela programação). Conceitos de padrões de projeto são utilizados objetivando ampliar as facilidades de leitura, entendimento, otimização e modificação do código fonte, já disponibilizadas pela programação orientada a objetos. Problemas conceituais são abordados usando as formulações propostas. A interpolação por splines quadráticas mostrou-se eficiente e precisa. Considerando as soluções exatas, o erro médio sobre a área de estudo foi inferior a 0,12%. O AEM/JUMP foi aplicado à região da Lagoa do Bonfim - RN com o objetivo de determinar as isolinhas de cargas hidráulicas. Os resultados foram comparados com estudo anterior, onde obteve resultados compatíveis, comprovando a aplicação do método e de sua implementação. Foram incorporadas ao problema da Lagoa do Bonfim características geométricas do contorno do oceano e de aluviões existentes no entorno da lagoa, demonstrando a utilidade do programa para gerar diferentes cenários de simulação. / The analytical elements method simulates underground draining through the superposition of conceptual solutions. In the method\'s context, inhomogeneity in defined as a clearly set region of constant hydraulic conductivity. Inhomogeneity hydraulic conductivity differences and the environment in which they are inserted cause a discontinuity (jump) in the discharge potential. Traditionally, this jump is simulated using first or second degree polynomials.The present work presents a formulation that uses quadratic spline polynomials to interpolate jumps occurred in the discharge potential through inhomogeneity borders. At the same time, the traditional formulation of discharge potential jump interpolation is extended to any degree. The main elements that compose the method are described and implemented. The resulting computational program (AEM) was developed integrated to an open code geographic information system (JUMP). The program permits the integration with other geographic information systems based on JAVA, keeping its independence from resident SIG. The architecture project program AEM/JUMP is based on object-oriented programming and presented great affinity with the analytical element method, showing identification among element concepts (used by the method) and the object (used by the program). Standard project concepts are used, seeking to widen source code reading possibilities, understanding, optimization and modifications already available through the object-oriented programming. Conceptual problems are approached with proposed formulations. Quadratic spline interpolation proved to be efficient and precise. Considering exact solutions, average mistake on study area was lower than 0.12%. AEM/JUMP was applied to the Lagoa do Bonfim (RN) lake region with the aim of establishing hydraulic charge isolines. Results were compared with the previous study, where compatible results had been obtained, thus proving method feasibility and implementation. Geometric features of surrounding areas and alluvion regions present around the lake area were incorporated to the original problem, demonstrating the usefulness of the program to generate different simulation scenarios.

A tese apresenta um método para a solução dos problemas do valor inicial (PVIs) com margens de erro comparáveis às de métodos numéricos consagrados (MN), tanto para a função quanto para suas derivadas. O método é aplicável a equações diferenciais (EDs) lineares ou não, sendo o ferramental desenvolvido até a quarta ordem, que pode ser expandido para ordens superiores. A solução é uma expressão polinomial de alto grau com coeficientes expressos pela razão entre dois inteiros. O método se mostra eficaz mesmo em alguns casos em que os MN não conseguiram dar a partida. As resoluções são obtidas considerando que o espaço de soluções é um espaço de Hilbert, equipado com a base completa dos polinômios de Legendre. Em decorrência do método aqui desenvolvido, os majorantes de erros para a função e derivadas são determinados analiticamente por um cálculo matricial também deduzido nesta tese. Paralelamente a toda fundamentação analítica, foi desenvolvido o software SAM, que automatiza todas as tarefas na busca de soluções dos PVIs. A tese propõe e verifica a validade de um novo critério de erro no qual pesam tanto os erros locais quanto os erros globais, simultaneamente. Como subprodutos dos resultados já descritos, igualmente integrados ao SAM, obtiveram-se também: (1) Um critério objetivo para analisar a qualidade de um MN, sem necessidade do conhecimento de seu algoritmo; (2) Uma ferramenta para aproximações polinomiais de alta precisão para funções de quadrado integrável em determinado intervalo limitado, com um majorante de erro; (3) Um ferramental analítico para transposição genérica (linear ou não) dos PVIs até 4ª ordem, nas mudanças de domínio; (4) As matrizes de integração e diferenciação genéricas para todas as bases polinomiais do espaço de Hilbert. / This thesis shows a new method to get polynomial solutions to the initial value problems (IVP), with an error margin comparable to the consecrate numerical methods (NM), for both the function and its derivatives. The method works with differential equations (DEs) linear or not, beeing the developed tolls available until 4th order, whose can be expanded to higher orders. The solution is a polynomial high degree expression with coefficients expressed by the ratio between two integers. The method behaves efficiently even in some cases that NM cannot get started. The resolutions are gotten considering that, the solution space is a Hilbert space, equipped with a complete set basis of Legendre Polynomials. Due the method here developed, the errors majoratives for the function and its derivatives are found analytically by a matrix calculus, also derived in this thesis. Beside all analytical foundation, a software (SAM) was developed to automate the whole process, joining all the tasks involved in the search for solutions to the IVP. This thesis proposes, verifies and validates a new error criterion, which takes in account simultaneously the local and global errors. As sub-products of the results described before, also integrated to the SAM, the following achievements should be highlighted: (1) An objective criterion to analyze the quality of any NM, despite of the knowledge of its algorithm; (2) A tool for a polynomial approximation, of high precision, for functions whose square is integrable in a given limited domain, with an errors majorative; (3) A tool-kit for a generically transpose (linear or not) of the IVPs domain and form, taking into account its derivatives, until the 4th order; (4) The generic matrices for integration and differentiation for all the polynomial basis of the Hilbert space.

A avicultura é uma das principais atividades agropecuárias do Brasil, em 2011 o país produziu 12.230 toneladas de carne de frango, sendo o 3º maior produtor de frango do mundo, apenas atrás dos EUA e China. Parte deste êxito se deve principalmente ao melhoramento genético animal implantado nas últimas décadas. Os objetivos nesse estudo foram: 1º - Comparar as funções das curvas de crescimento: von Bertalanffy, Gompertz, Logística, Richards e Brody, pelo método dos Mínimos Quadrados Ordinários (QMO) e Quadrados Mínimos Ponderados (QMP) a dados de peso vivo para as linhagens experimentais de frango caipira (7P, Caipirão da ESALQ, Caipirinha da ESALQ e Carijó Barbado) e selecionar uma curva de crescimento que melhor descreva o padrão de crescimento para cada linhagem, e estimar os componentes genéticos (herdabilidades e correlações genéticas) dos parâmetros destas funções sob análise bicaracterística; 2º - Comparar modelos com diferentes ordens de ajuste por meio de funções polinomiais de Legendre, sob modelos de regressão aleatória, com variância residual heterogênea, para a estimação dos componentes de (co) variância e avaliação genética de linhagens experimentais de frango caipira (7P, Caipirão da ESALQ, Caipirinha da ESALQ e Carijó Barbado). O modelo que melhor se adequou a curva de crescimento para as linhagens estudadas foi à função de Gompertz ajustado pelo método dos Quadrados Mínimos Ponderados (QMP). Os parâmetros genéticos estimados para as medidas e dos modelos Gompertz ponderado podem ser utilizados como critérios de seleção, pois parecem ter efeito genético considerável para estas características. No entanto, deve-se haver cautela na utilização do parâmetro como critério de seleção para a linhagem Carijó Barbado devido a baixa herdabilidade. As correlações genéticas e fenotípicas entre as características e foram negativas e altas. Indicando que quanto maior o peso assintótico menor a taxa de crescimento. Dentre os modelos de Regressão Aleatória estudados o polinômio de 3ª ordem foi o que melhor se adequou para descrever as curvas de crescimento das linhagens estudadas. As estimativas de variâncias, herdabilidades foram afetadas pela modelagem da variância residual. Em geral as herdabilidades estimadas para as idades de 1 a 84 dias variaram de moderadas a altas para as linhagens estudadas, indicando que qualquer idade pode ser utilizada como critério de seleção. A seleção aos 42 dias de idade pode ser mantida como critério de seleção. / Aviculture is one of the main agribusiness activities in Brazil, in 2011 the country produced 12,230 tons of broiler meat, was the 3rd largest producer of broiler in the world, only behind the USA and China. Part of this success is mainly due to animal genetic improvement implemented in recent decades. In this study, our objectives were: 1 - to compare the functions of the Von Bertalanffy, Gompertz, Logistic, Richards and Brody growth curves by the Ordinary Least Squares and Weighted Least Squares method, from data for body weight from experimental free-range chicken lines (7P, Caipirão da ESALQ, Caipirinha da ESALQ and Carijó Barbado) and select a growth curve that best describes their growth. From this, estimates of the genetic components (heritability and genetic correlations) of the parameters of these functions under bivariate analysis; 2 - Comparing models with different orders of adjustment by means of Legendre polynomial functions under random regression models with heterogeneous residual variance for the estimation of (co) variance and genetic evaluation of experimental free-range chicken lines (7P, Caipirão da ESALQ, Caipirinha da ESALQ and Carijó Barbado). The model that best adapted the growth curve for all lines studied was the Gompertz function adjusted using weighted least squares. Genetic parameters for measurements and can be used as selection criteria because they seem to have considerable genetic effects for these characteristics. There should be caution in using the parameter as a selection criterion for the Carijó Barbado line due to low heritability. The genetic and phenotypic correlations between traits and were negative and high, indicating that the higher the asymptotic weight, the lower the growth rate. Among the Random Regression models studied the 3rd order polynomial was best adapted to describe the growth curves of the lines studied. Estimates of variances and heritabilities were affected by residual variance modeling. Overall heritability estimates between 1 to 84 days of age ranged from moderate to high for all lines, indicating that any age can be used as a selection criterion, including maintaining the current selection at 42 days of age.

Com o objetivo de avaliar o desempenho dos modelos mistos quando se assumem bases de funções ortonormais de Legendre, Jacobi modificadas e trigonométricas como covariáveis dos coeficientes aleatórios, os dados referentes à pesagem corporal de animais da raça Nelore do nascimento aos 800 dias, foram analisados com modelos que assumiram inicialmente coeficientes aleatórios de efeito genético direto e efeito permanente animal (dois fatores aleatórios), em seguida foi acrescentado o efeito genético materno (três fatores aleatórios) e finalmente assumiram-se também os coeficientes aleatórios de efeito permanente materno (quatro fatores aleatórios). Foram considerados como efeitos fixos, as idades da mãe ao parto, os grupos contemporâneos e uma regressão linear por polinômios de Legendre. Os dados oriundos da fazenda Mundo Novo fornecidos pelo Grupo de Melhoramento Animal da FZEA/USP continham 61.975 pesagens corporais de 20.543 animais e informações de 26.275 animais da raça Nelore no "pedigree". O número de pesagem por animal não ultrapassou a seis e cada animal forneceu apenas uma medida em cada um dos seguintes intervalos de idade (em dias): 1 – 69, 70 – 159, 160 – 284, 285 – 454, 455 – 589 e 590 – 800. O propósito desse estudo foi comparar o ajuste da curva média de crescimento dos animais por intermédio de modelos mistos sob influência das funções ortonormais com dois, três e quatro fatores aleatórios. Um segundo propósito do trabalho foi investigar o comportamento das curvas dos componentes aleatórios estimados por meio dos modelos selecionados em cada base de funções nos três grupos distintos de efeitos aleatórios e examinar o comportamento das curvas dos coeficientes de herdabilidade obtidas a partir das curvas dos componentes aleatórios. Por meio do aplicativo WOMBAT, as análises foram realizadas usando-se o algoritmo PX-AI. Em função da parcimônia, o critério de informação bayesiano de Schwarz (BIC) foi adotado para selecionar os modelos que melhor se adequaram aos dados, que em ordem crescente de seus valores foram: com dois fatores aleatórios, os modelos de Legendre com seis covariáveis (ML26), de Jacobi Modificado com cinco covariáveis (MJ25) e o trigonométrico com seis covariáveis (MT26); com três fatores aleatórios, os modelos com seis covariáveis (MJ36, ML36, MT36); e com quatro fatores aleatórios, os modelos de Jacobi Modificado com cinco covariáveis (MJ45), de Legendre com cinco covariáveis (ML45), e o trigonométrico com seis covariáveis (MT46). Dentre os nove modelos selecionados, o modelo com o menor BIC foi o modelo MJ36, porém o modelo MJ45 apresentou estimativas de componentes de variância muito próximas do modelo MJ36. As estimativas dos componentes de variância e dos coeficientes de herdabilidade obtidas pelos modelos com funções de Jacobi modificadas, nos extremos do intervalo, ficaram abaixo das obtidas pelos modelos com funções de Legendre e no interior do intervalo elas foram concordantes, ficando entre 0,2 e 0,3. As estimativas obtidas dos modelos com funções trigonométricas se diferenciaram dos demais e foram muito baixas no extremo do intervalo para modelos com mais de dois fatores aleatórios. A média das curvas de crescimento que mais se aproximou da tendência média dos dados em cada ponto do intervalo foi obtida pelo modelo MJ26. / This work's statistical objective is to assess the performance of random coef- ficient regression models when Legendre, modified Jacobi and trigonometric functions are used as the covariate basis. This was studied with an application to a genetic analysis of the body weight of cattle from the Nellore breed. In the period 1981 to 2002 body weight data of animals were collected from the birth to the 800th day of life. An initial two random factor model used random coefficients for the direct genetic and environment animal effects. A second three random factors model introduced an additional random term for coefficients maternal genetic effects. Our final model, with four random factors, included environment maternal effects. Average growth curve was modeled by a fixed linear regression on days of age nested within contemporary group and ages of dams at calving. The data come from the Mundo Novo farm, and were provided by the Animal Breeding Genetic Group of the FZEA/USP. There were 61,975 body weights measured on 20,543 animals. In addition, information from 26,275 pedigree Nellore animals was included. No animal was weighed more than six times, and each animal supplied at most one measure within each of the following age intervals (in days): 1-69, 0-159, 160-284, 285-454, 455-589 and 590-800. This study aimed to compare the animal's mean growth curve using mixed models with the orthonormal function bases, in the case of two, three and four random factors. A second aim was to investigate the estimated random components curve behaviour using the selected models with each base of functions in the three distinct random effect groups and to examine the behaviour of the heritability coefficient curves obtained through the random component curves. The analysis was done using the PX-AI and the WOMBAT device. For parsimony, the Schwartz Bayesian information criterion (BIC) was adopted to select the best models. This criterion suggested two random factors, the for Legendre model, six covariates (ML26), for the Modified Jacobi model, five covariates (MJ25) and for the trigonometric model, six covariates (MT26). With three random factors, the models all required six covariates (MJ36, ML36, MT36). Finally, with four random factors, the Modified Jacobi model required five covariates (MJ45), the Legendre model required five covariates (ML45), and the trigonometric model required six covariates (MT46). Within the nine selected models, the MJ36 model was the one with the smaller BIC, however the MJ45 model presented variance components estimates very similar to the MJ36 model. The variance components and heritability coefficient estimates from the models with modified Jacobi functions were bellow the ones obtained with Legendre functions even at the extreme end of the intervals. In the interior of the interval, however, they were in agreement, staying between 0.2 and 0.3. The estimates obtained with trigonometric functions differed from the others and were much lower at the interval extremes for models with more than two random factors.

L’objectif de cette thèse est de construire des estimateurs non-paramétriques d’une fonction de distribution, d’une densité de probabilité et d’une fonction de régression en utilisant les méthodes d’approximation stochastiques afin de corriger l’effet du bord créé par les estimateurs à noyaux continus classiques. Dans le premier chapitre, on donne quelques propriétés asymptotiques des estimateurs continus à noyaux. Puis, on présente l’algorithme stochastique de Robbins-Monro qui permet d’introduire les estimateurs récursifs. Enfin, on rappelle les méthodes utilisées par Vitale, Leblanc et Kakizawa pour définir des estimateurs d’une fonction de distribution et d’une densité de probabilité en se basant sur les polynômes de Bernstein.Dans le deuxième chapitre, on a introduit un estimateur récursif d’une fonction de distribution en se basant sur l’approche de Vitale. On a étudié les propriétés de cet estimateur : biais, variance, erreur quadratique intégré (MISE) et on a établi sa convergence ponctuelle faible. On a comparé la performance de notre estimateur avec celle de Vitale et on a montré qu’avec le bon choix du pas et de l’ordre qui lui correspond notre estimateur domine en terme de MISE. On a confirmé ces résultatsthéoriques à l’aide des simulations. Pour la recherche pratique de l’ordre optimal, on a utilisé la méthode de validation croisée. Enfin, on a confirmé les meilleures qualités de notre estimateur à l’aide des données réelles. Dans le troisième chapitre, on a estimé une densité de probabilité d’une manière récursive en utilisant toujours les polynômes de Bernstein. On a donné les caractéristiques de cet estimateur et on les a comparées avec celles de l’estimateur de Vitale, de Leblanc et l’estimateur donné par Kakizawa en utilisant la méthode multiplicative de correction du biais. On a appliqué notre estimateur sur des données réelles. Dans le quatrième chapitre, on a introduit un estimateur récursif et non récursif d’une fonction de régression en utilisant les polynômes de Bernstein. On a donné les caractéristiques de cet estimateur et on les a comparées avec celles de l’estimateur à noyau classique. Ensuite, on a utilisé notre estimateur pour interpréter des données réelles. / The aim of this thesis is to construct nonparametric estimators of distribution, density and regression functions using stochastic approximation methods in order to correct the edge effect created by kernels estimators. In the first chapter, we givesome asymptotic properties of kernel estimators. Then, we introduce the Robbins-Monro stochastic algorithm which creates the recursive estimators. Finally, we recall the methods used by Vitale, Leblanc and Kakizawa to define estimators of distribution and density functions based on Bernstein polynomials. In the second chapter, we introduced a recursive estimator of a distribution function based on Vitale’s approach. We studied the properties of this estimator : bias, variance, mean integratedsquared error (MISE) and we established a weak pointwise convergence. We compared the performance of our estimator with that of Vitale and we showed that, with the right choice of the stepsize and its corresponding order, our estimator dominatesin terms of MISE. These theoretical results were confirmed using simulations. We used the cross-validation method to search the optimal order. Finally, we applied our estimator to interpret real dataset. In the third chapter, we introduced a recursive estimator of a density function using Bernstein polynomials. We established the characteristics of this estimator and we compared them with those of the estimators of Vitale, Leblanc and Kakizawa. To highlight our proposed estimator, we used real dataset. In the fourth chapter, we introduced a recursive and non-recursive estimator of a regression function using Bernstein polynomials. We studied the characteristics of this estimator. Then, we compared our proposed estimator with the classical kernel estimator using real dataset.

Une Fonction Latticielle Polynômiale (FLP) sur un treillis L est une fonction p : Ln → L, qui peut être exprimée à partir de variables, de constantes et des opérateurs de treillis ∧ et ∨ . Dans les cas où L est distributif et borné, les FLP incluent les intégrales de Sugeno. Celles-ci sont des fonctions d'agrégation qui permettent de fusionner des valeurs sur des échelles ordinales non numériques, et qui sont utilisées notamment dans l'approche qualitative de l'Aide à la Décision Multi Critères en tant qu'alternatives ordinales aux intégrales de Choquet. Dans une première partie, nous traitons la tâche d'interpolation par des FLP, c'est à dire : pour un treillis L, un sous-ensemble fini D de Ln et une fonction f : D → L, retourner une FLP p : Ln → L telle que p(x) = f(x) pour tout x ∊ D (si une telle FLP existe). Nous traitons successivement le cas où L est un treillis fini et le cas où L est une treillis distributif borné. Dans les deux cas, nous donnons des algorithmes qui résolvent ce problème en temps polynomial. Dans une seconde partie, nous abordons les généralisations des intégrales de Sugeno appelées Fonctions d'Utilité de Sugeno (FUS), qui permettent la fusion de valeurs appartenant à des échelles ordinales différentes, ainsi que leur application à la tâche de classification monotone. Nous introduisons un modèle composé de plusieurs FUS, ainsi qu'un algorithme d'apprentissage d'un tel modèle. Nous comparons ce modèle aux ensembles de règles de décision appris par VC-DomLEM, et étudions le nombre de FUS nécessaires afin de modéliser des données empiriques / A Lattice Polynomial Function (LPF) over a lattice L is a map p : Ln → L that can be defined by an expression involving variables, constants and the lattice operators ∧ and ∨. If L is a distributive lattice, these maps include the so-called Sugeno integrals that are aggregation functions capable of merging ordinal values, not necessarily numerical. They are widely used in the qualitative approach to Multiple Criteria Decision Aiding (MCDA), and they can be thought of as the ordinal counterparts of Choquet integrals. In the first part of this thesis, we tackle the task of interpolating a partial function by an LPF, stated as follows: for a lattice L, a finite subset D of Ln, and a function f : D → L, return an LPF p : Ln → L such that p(x) = f(x) for all x ∊ D (if such an LPF exists). We treat the cases where L is a finite lattice, and then the cases where L is a bounded distributive lattice. In both cases, we provide algorithms that solve this problem in polynomial time. In the second part, we consider generalizations of Sugeno integrals in the multi-attribute setting, in particular, the Sugeno Utility Functions (SUFs), that are able to merge values coming from different ordinal scales. We consider the their use in monotonic classification tasks. We present a model based on a set of SUFs and an algorithm for learning such model from data. We compare this model to the sets of monotonic decision rules learned by VC-DomLEM, and study the number of SUFs that are required in order to model empirical data

Neste trabalho analisamos a positividade definida estrita de núcleos contínuos sobre um espaço compacto 2-homogêneo. R. Gangolli (1967) apresentou uma caracterização completa para os núcleos que são contínuos, isotrópicos e positivos definidos sobre um espaço compacto 2-homogêneo Md: a parte isotrópica do núcleo é uma série de Fourier uniformemente convergente, com coeficientes não negativos, em relação a certos polinômios de Jacobi atrelados a Md. Uma das contribuições de nosso trabalho é uma caracterização para a positividade definida estrita de tais núcleos, complementando a caracterização apresentada por Chen et al. (2003) no caso em que Md é uma esfera unitária de dimensão maior ou igual a 2. Outra contribuição do trabalho é uma extensão do resultado de Gangolli para núcleos sobre produtos cartesianos de espaços compactos 2-homogêneos, e a consequente caracterização para núcleos estritamente positivos definidos neste mesmo contexto. Por fim, a última contribuição do trabalho envolve a análise do grau de diferenciabilidade da parte isotrópica de um núcleo contínuo, isotrópico e positivo definido sobre Md e a aplicabilidade de tal análise em resultados envolvendo a positividade definida estrita. / In this work we analyze the strict positive definiteness of continuous kernels on compact two-point homogeneous spaces Md. R. Gangolli (1967) presented a complete characterization for continuous, isotropic and positive definite kernels on Md: the isotropic part of the kernel is a uniformly convergent Fourier series of certain Jacobi polynomials associated to Md, with nonnegative coefficients. One of the contributions of our work is a characterization for the strict positive definiteness of such kernels, completing that one presented by Chen et al. (2003) in the case Md is the unit sphere of dimension at least 2. Another contribuition of this work is an extension of Gangolli\'s result for kernels on a product of compact two-point homogeneous spaces, and the subsequent characterization of strict positive definiteness in this same context. Finally, the last contribution in this work involves the analysis of the differentiability of the isotropic part of a continuous, isotropic and positive definite kernel on Md and the applicability of such analysis in results involving the strict positive definiteness.

A Teoria das equações diferenciais faz parte de uma área da Matemática muito rica em aplicações. Os métodos numéricos para a solução de equações diferenciais ordinárias são, da mesma forma que as próprias equações, fontes importantes de problemas a serem pesquisados. Como destaque tem-se os métodos multiderivadas de passo múltiplo, que são importantes na solução de problemas stiff. Os métodos numéricos mais conhecidos para a solução desses problemas são os BDF, que compõem, para L = 1, a família dos métodos (K, L) de Brown. Algumas questões relacionadas à estabilidade dos métodos (K, L) ainda não foram solucionadas como, por exemplo, uma conjectura de Jeltsch. Para analisá-la, é necessário estudar o comportamento dos zeros dos polinômios característicos associados aos métodos (K, L). Neste trabalho é apresentado um estudo sobre zeros de polinômios com o objetivo de demonstrar a validade da conjectura de Jeltsch para K \'< OU =\' \'K IND; L\' . As regiões de estabilidade para alguns valores de K e L fixos são apresentadas e também é utilizada a teoria das order stars para mostrar algumas propriedades dos métodos (K, L). Portanto, este trabalho apresenta um estudo sobre os métodos (K, L) de Brown e usa uma ferramenta pouco utilizada na literatura, que são as order stars, para demonstrar alguns resultados / THe theory of differential equations is part of one area of Mathematics very rich in applications. The numerical methods for the solutions of ordinary differential equations are, in the same way as the equations themselves, important sources of problems to be studied. As prominence one has the multiderivative multistep methods which are important for the solution of stiff problems. The best known numerical methods for the solutions of these kind of problems are the BDF methods, which is part of the family of the Brown (K,L) methods with L = 1. Some questions about stability of the (K, L) methods has not been solved yet as, for example, a conjecture by Jeltsch. In order to tackle this open problem, it becomes necessary to study the behavior of the zeros of the characteristic polynomials associated to the (K, L) methods. In this work a study of the zeros of the characteristic polynomial is carried out aiming at proving Jeltsch conjecture for K < OR = \'K IND.L\'. Regions of stability is shown for some fixed values of K and L, as well as the use of order stars techniques are applied to show some properties of (K, L) methods. Therefore, this work presents a study of Brown\'s (K, L) methods, that makes use of a tool that seems not to have been used very often in the literature, the order stars, in order to prove the main results