Dissertations / Theses on the topic 'Jacobi polynomials'

Orientador: Eliana Xavier Linhares de Andrade
Banca: Ana Paula Peron
Banca: Alagacone Sri Ranga
Resumo: Sejam Sn(x); n ¸ 0; os polinômios de Sobolev, ortogonais com relação ao produto interno hf; giS = ZR f(x)g(x)dÃ0(x) + ¸ ZR f0(x)g0(x)dÃ1(x); ¸ > 0; onde fdÃ0; dÃ1g forma um par coerente de medidas relacionadas às medidas de Jacobi ou de Laguerre. Denotemos por PÃ0 n (x) e PÃ1 n (x); n ¸ 0; os polinômios ortogonais com respeito a dÃ0 e dÃ1; respectivamente. Neste trabalho, estudamos o comportamento assintótico, quando n ! 1; das razões entre os polinômios de Sobolev, Sn(x); e os polinômios ortogonais PÃ0 n (x) e PÃ1 n (x); além do comportamento limite da razão entre esses dois últimos polinômios. Propriedades assintóticas para os coeficientes da relação de recorrência satisfeita pelos polinômios de Sobolev também foram estudadas.
Abstract: Let Sn(x); n ¸ 0; be the Sobolev polynomials, orthogonal with respect to the inner product hf; giS = ZR f(x)g(x)dÃ0(x) + ¸ ZR f0(x)g0(x)dÃ1(x); ¸ > 0; where fdÃ0; dÃ1g forms a coherent pair of measures related to the Jacobi measure or Laguerre measure. Let PÃ0 n (x) and PÃ1 n (x); n ¸ 0; denote the orthogonal polynomials with respect to dÃ0 and dÃ1; respectively. In this work we study the asymptotic behaviour, as n ! 1; of the ratio between the Sobolev polynomials, Sn(x); and the ortogonal polynomials PÃ0 n (x) and PÃ1 n (x); as well as the limit behaviour of the ratio between the last two polynomials. Furthermore, we also give asymptotic results for the coefficients of the recurrence relation satisfied by the Sobolev polynomials.
Mestre

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Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
Sejam Sn(x); n ¸ 0; os polinômios de Sobolev, ortogonais com relação ao produto interno hf; giS = ZR f(x)g(x)dÃ0(x) + ¸ ZR f0(x)g0(x)dÃ1(x); ¸ > 0; onde fdÃ0; dÃ1g forma um par coerente de medidas relacionadas às medidas de Jacobi ou de Laguerre. Denotemos por PÃ0 n (x) e PÃ1 n (x); n ¸ 0; os polinômios ortogonais com respeito a dÃ0 e dÃ1; respectivamente. Neste trabalho, estudamos o comportamento assintótico, quando n ! 1; das razões entre os polinômios de Sobolev, Sn(x); e os polinômios ortogonais PÃ0 n (x) e PÃ1 n (x); além do comportamento limite da razão entre esses dois últimos polinômios. Propriedades assintóticas para os coeficientes da relação de recorrência satisfeita pelos polinômios de Sobolev também foram estudadas.
Let Sn(x); n ¸ 0; be the Sobolev polynomials, orthogonal with respect to the inner product hf; giS = ZR f(x)g(x)dÃ0(x) + ¸ ZR f0(x)g0(x)dÃ1(x); ¸ > 0; where fdÃ0; dÃ1g forms a coherent pair of measures related to the Jacobi measure or Laguerre measure. Let PÃ0 n (x) and PÃ1 n (x); n ¸ 0; denote the orthogonal polynomials with respect to dÃ0 and dÃ1; respectively. In this work we study the asymptotic behaviour, as n ! 1; of the ratio between the Sobolev polynomials, Sn(x); and the ortogonal polynomials PÃ0 n (x) and PÃ1 n (x); as well as the limit behaviour of the ratio between the last two polynomials. Furthermore, we also give asymptotic results for the coefficients of the recurrence relation satisfied by the Sobolev polynomials.

Dans ce travail, nous construisons et étudions des familles de polynômes orthogonaux généralisés définis dans l'espace des matrices hermitiennes qui sont associées à une famille de polynômes orthogonaux sur R. Nous considérons plusieurs normalisations pour ces polynômes, et obtenons des formules classiques à partir des formules correspondantes pour des polynômes définis sur R. Nous construisons également des semi-groupes d'opérateurs associés aux polynômes orthogonaux généralisés, et donnons l'expression du générateur infinitésimal de ce semi-groupe ; nous prouvons que ce semi-groupe est markovien dans les cas classiques. En ce qui concerne les expansions d-dimensionnelles de Jacobi nous étudions les notions d'intégrale fractionnelle (potentiel de Riesz), de potentiel de Bessel et de dérivées fractionnelles. Nous donnons une nouvelle décomposition de l'espace L2 associé à la mesure de Jacobi d-dimensionnelle, et obtenons un analogue du théorème du multiplicateur de Meyer dans ce cadre. Nous étudions aussi les espaces de Jacobi-Sobolev
In this work we construct and study families of generalized orthogonal polynomials with hermitian matrix argument associated to a family of orthogonal polynomials on R. Different normalizations for these polynomials are considered and we obtain some classical formulas for orthogonal polynomials from the corresponding formulas for the one–dimensional polynomials. We also construct semigroups of operators associated to the generalized orthogonal polynomials and we give an expression of the infinitesimal generator of this semigroup and, in the classical cases, we prove that this semigroup is also Markov. For d–dimensional Jacobi expansions we study the notions of fractional integral (Riesz potentials), Bessel potentials and fractional derivatives. We present a novel decomposition of the L2 space associated with the d–dimensional Jacobi measure and obtain an analogous of Meyer's multiplier theorem in this setting. Sobolev Jacobi spaces are also studied

Thesis (Ph.D.)--Baylor University, 2009.
Subscript in abstract: n and n=0 in {Pn([alpha],[beta])(x)} [infinity] n=0, [mu] in (f,g)[mu], and R in [integral]Rfgd[mu]. Superscript in abstract: ([alpha],[beta]) and [infinity] in {Pn([alpha],[beta])(x)} [infinity] n=0. Includes bibliographical references (p. 115-119).

An isospectral algorithm is one which manipulates a matrix without changing its spectrum. In this thesis we study three interrelated examples of isospectral algorithms, all pertaining to Toeplitz matrices in some fashion, and one directly involving orthogonal polynomials. The first set of algorithms we study come from discretising a continuous isospectral flow designed to converge to a symmetric Toeplitz matrix with prescribed eigenvalues. We analyse constrained, isospectral gradient flow approaches and an isospectral flow studied by Chu in 1993. The second set of algorithms compute the spectral measure of a Jacobi operator, which is the weight function for the associated orthogonal polynomials and can include a singular part. The connection coefficients matrix, which converts between different bases of orthogonal polynomials, is shown to be a useful new tool in the spectral theory of Jacobi operators. When the Jacobi operator is a finite rank perturbation of Toeplitz, here called pert-Toeplitz, the connection coefficients matrix produces an explicit, computable formula for the spectral measure. Generalisation to trace class perturbations is also considered. The third algorithm is the infinite dimensional QL algorithm. In contrast to the finite dimensional case in which the QL and QR algorithms are equivalent, we find that the QL factorisations do not always exist, but that it is possible, at least in the case of pert-Toeplitz Jacobi operators, to implement shifts to generate rapid convergence of the top left entry to an eigenvalue. A fascinating novelty here is that the infinite dimensional matrices are computed in their entirety and stored in tailor made data structures. Lastly, the connection coefficients matrix and the orthogonal transformations computed in the QL iterations can be combined to transform these pert-Toeplitz Jacobi operators isospectrally to a canonical form. This allows us to implement a functional calculus for pert-Toeplitz Jacobi operators.

Orientadores: Dimitar Kolev Dimitrov, Roberto Andreani
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: O objetivo deste trabalho é apresentar uma nova formulação do Teorema de comparação de Sturm e suas aplicações na teoria dos zeros de polinômios ortogonais, que são: monotonicidade dos zeros dos polinômios ortogonais X1-Jacobi, desigualdades de Gautschi sobre os zeros dos polinômios ortogonais de Jacobi e o comportamento assintótico dos zeros dos polinômios ultrasféricos
Abstract: In this thesis we state a new formulation of the Sturm comparison Theorem and its applications to the zeros of orthogonal polynomials. Specifically, these applications deal with the monotonicity of zeros of X1-Jacobi orthogonal polynomials, Gautschi's conjectures about inequalities of zeros of Jacobi polynomials and the asymptotic of zeros of ultrasphricals polynomials
Doutorado
Matematica Aplicada
Doutor em Matemática Aplicada

WKB procedure is used in the study of asymptotic solutions of differential equations of the hypergeometric type. Hence asymptotic forms of classical orthogonal polynomials associated with the names Jacobi, Laguerre and Hermite have been derived. In particular, the asymptotic expansion of the Jacobi polynomials $P^{(alpha, beta)}_n(x)$ as $n$ tends to infinity is emphasized.

Apresentamos uma caracterização das funções positivas definidas em esferas complexas, generalizando assim, um resultado de Schoenberg ([41]). Como no caso real, uma classe importante dessas funções é aquela composta pelas funções estritamente positivas definidas de uma certa ordem; estas podem ser utilizadas para resolver certos problemas de interpolação de dados arbitrários associados a pontos distintos distribuídos nas esferas. Com esse objetivo, obtivemos algumas condições necessárias e suficientes (separadamente) para que funções positivas definidas sejam estritamente positivas definidas. Os resultados apresentados fornecem uma caracterização final elementar para funções estritamente positivas definidas de todas as ordens em quase todas as esferas complexas. Funções estritamente positivas definidas de ordem 2 são caracterizadas em todas as esferas complexas. Analisamos também a relação entre funções estritamente positivas definidas em esferas complexas e funções estritamente positivas definidas em esferas reais.
We characterize positive definite functions on complex spheres, generalizing a famous result due to I. J. Schoenberg ([41]). As in the real case, we study the so-called strictly positive definite functions. They can be used to perform interpolation of scattered data on those spheres. We present (separated) necessary and sufficient conditions for a positive definite function to be strictly positive definite of a certain order. These conditions produce a final characterization for those positive definite functions which are strictly positive definite of all orders, on almost all spheres. Strictly positive definite functions of order 2 are identified. Finally, we study a connection between strictly positive definite functions on real spheres and strictly positive definite functions on complex spheres.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
Nesta dissertação estudamos sequências de polinômios ortogonais que surgem como auto-funções polinomiais do problema de Sturm-Liouville, sob a condição de que, nem todos os graus das auto-funções polinomiais estejam presentes na sequência de graus dos polinômios que formam o conjunto ortogonal completo. Estas sequências são chamadas de sequências de polinômios ortogonais excepcionais. Emparticular,realizamosumestudodospolinômiosortogonaisexcepcionais X1-Jacobi e X1-Laguerre.
In this dissertation we study sequences of orthogonal polynomials that arise as polynomial eigenfunctions of the Sturm-Liouville problem, with the condition that not all degrees of polynomial eigenfunctions are present in the sequence of degrees of the polynomials that form a complete orthogonal set. These sequences are called exceptional orthogonal polynomial sequences. In particular, we study the exceptional orthogonal polynomials X1-Jacobi and X1-Laguerre.

In this study, Ritz method has been employed to analyze the following problems: free vibrations of plates with curvilinear stiffeners, the lowest 100 frequencies of thick isotropic plates, free vibrations of thick quadrilateral laminates and free vibrations and static deformations of rectangular laminates, and sandwich structures. Admissible functions in the Ritz method are chosen as a product of the classical Jacobi orthogonal polynomials and weight functions that exactly satisfy the prescribed essential boundary conditions while maintaining orthogonality of the admissible functions. For free vibrations of plates with curvilinear stiffeners, made possible by additive manufacturing, both plate and stiffeners are modeled using a first-order shear deformation theory. For the thick isotropic plates and laminates, a third-order shear and normal deformation theory is used. The accuracy and computational efficiency of formulations are shown through a range of numerical examples involving different boundary conditions and plate thicknesses. The above formulations assume the whole plate as an equivalent single layer. When the material properties of individual layers are close to each other or thickness of the plate is small compared to other dimensions, the equivalent single layer plate (ESL) theories provide accurate solutions for vibrations and static deformations of multilayered structures. If, however, sufficiently large differences in material properties of individual layers such as those in sandwich structure that consists of stiff outer face sheets (e.g., carbon fiber-reinforced epoxy composite) and soft core (e.g., foam) exist, multilayered structures may exhibit complex kinematic behaviors. Hence, in such case, Cz0 conditions, namely, piecewise continuity of displacements and the interlaminar continuity of transverse stresses must be taken into account. Here, Ritz formulations are extended for ESL and layerwise (LW) Nth-order shear and normal deformation theories to model sandwich structures with various face-to-core stiffness ratios. In the LW theory, the C0 continuity of displacements is satisfied. However, the continuity of transverse stresses is not satisfied in both ESL and LW theories leading to inaccurate transverse stresses. This shortcoming is remedied by using a one-step well-known stress recovery scheme (SRS). Furthermore, analytical solutions of three-dimensional linear elasticity theory for vibrations and static deformations of simply supported sandwich plates are developed and used to investigate the limitations and applicability of ESL and LW plate theories for various face-to-core stiffness ratios. In addition to natural frequency results obtained from ESL and LW theories, the solutions of the corresponding 3-dimensional linearly elastic problems obtained with the commercial finite element method (FEM) software, ABAQUS, are provided. It is found that LW and ESL (even though its higher-order) theories can produce accurate natural frequency results compared to FEM with a considerably lesser number of degrees of freedom.
Doctor of Philosophy
In everyday life, plate-like structures find applications such as boards displaying advertisements, signs on shops and panels on automobiles. These structures are typically nailed, welded, or glued to supports at one or more edges. When subjected to disturbances such as wind gusts, plate-like structures vibrate. The frequency (number of cycles per second) of a structure in the absence of an applied external load is called its natural frequency that depends upon plate's geometric dimensions, its material and how it is supported at the edges. If the frequency of an applied disturbance matches one of the natural frequencies of the plate, then it will vibrate violently. To avoid such situations in structural designs, it is important to know the natural frequencies of a plate under different support conditions. One would also expect the plate to be able to support the designed structural load without breaking; hence knowledge of plate's deformations and stresses developed in it is equally important. These require mathematical models that adequately characterize their static and dynamic behavior. Most mathematical models are based on plate theories. Although plates are three-dimensional (3D) objects, their thickness is small as compared to the in-plane dimensions. Thus, they are analyzed as 2D objects using assumptions on the displacement fields and using quantities averaged over the plate thickness. These provide many plate theories, each with its own computational efficiency and fidelity (the degree to which it reproduces behavior of the 3-D object). Hence, a plate theory can be developed to provide accurately a quantity of interest. Some issues are more challenging for low-fidelity plate theories than others. For example, the greater the plate thickness, the higher the fidelity of plate theories required for obtaining accurate natural frequencies and deformations. Another challenging issue arises when a sandwich structure consists of strong face-sheets (e.g., made of carbon fiber-reinforced epoxy composite) and a soft core (e.g., made of foam) embedded between them. Sandwich structures exhibit more complex behavior than monolithic plates. Thus, many widely used plate theories may not provide accurate results for them. Here, we have used different plate theories to solve problems including those for sandwich structures. The governing equations of the plate theories are solved numerically (i.e., they are approximately satisfied) using the Ritz method named after Walter Ritz and weighted Jacobi polynomials. It is shown that these provide accurate solutions and the corresponding numerical algorithms are computationally more economical than the commonly used finite element method. To evaluate the accuracy of a plate theory, we have analytically solved (i.e., the governing equations are satisfied at every point in the problem domain) equations of the 3D theory of linear elasticity. The results presented in this research should help structural designers.

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.

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.

This thesis considers symmetric functions and algebraic combinatorics via the polynomial representation theory of GL(N,C). In particular, we utilise the theory of Jacobi-Trudi determinants to prove some new results pertaining to the Littlewood-Richardson coefficients. Our results imply, under some hypotheses on the strictness of the partition an equality between Littlewood-Richardson coefficients and Kostka numbers. For the case that a suitable partition has two rows, an explicit formula is then obtained for the Littlewood-Richardson coefficient using the Hook Length formula. All these results are then applied to compute branching laws for GL(m+n,C) restricting to GL(m,C) x GL(n,C). The technique also implies the well-known Racah formula.

Este trabalho é composto de duas partes distintas, ambas dentro de um mesmo tema: núcleos positivos definidos sobre variedades. Na primeira delas fornecemos uma caracterização para os núcleos contínuos, isotrópicos e positivos definidos a valores matriciais sobre um espaço compacto 2-homogêneo. Utilizando-a, investigamos a positividade definida estrita destes núcleos, apresentando inicialmente algumas condições suficientes para garantir tal propriedade. No caso em que o espaço 2-homogêneo não é uma esfera, descrevemos uma caracterização definitiva para a positividade definida estrita do núcleo. Neste mesmo caso, para núcleos a valores no espaço das matrizes de ordem 2, apresentamos uma caraterização alternativa para a positividade definida estrita do núcleo via os dois elementos na diagonal principal da representação matricial do núcleo. Na segunda parte, nos restringimos a núcleos positivos definidos escalares sobre os mesmos espaços e determinamos condições necessárias e suficientes para a positividade definida estrita de um produto de núcleos positivos definidos sobre um mesmo espaço compacto 2-homogêneo. Apresentamos ainda uma extensão deste resultado para núcleos positivos definidos sobre o produto cartesiano de um grupo localmente compacto com uma esfera de dimensão alta, mantendo-se a isotropia na componente esférica.
In this work we present a characterization for the continuous, isotropic and positive definite matrix-valued kernels on a compact two-point homogeneous space. After that, we consider the strict positive definiteness of the kernels, describing some independent sufficient conditions for that property to hold. In the case the space is not a sphere, one of the conditions becomes necessary and sufficient for the strict positive definiteness of the kernel. Further, for 22- matrix-valued kernels on a compact two-point homogeneous space which is not a sphere, we present a characterization for the strict positive definiteness of the kernels based upon the main diagonal elements in its matrix representation. In the last part of this work, we restrict ourselves to scalar kernels and determine necessary and sufficient conditions in order that the product of two continuous, isotropic and positive definite kernels on a compact two-point homogeneous space be strictly positive definite. We also discuss the extension of this result for kernels defined on a product of a locally compact group and a high dimensional sphere.

Orientador: Marco Lucio Bittencourt
Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica
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Resumo: Este trabalho tem por objetivo principal o desenvolvimento de funções de interpolaçao e regras de integraçao tensorizaveis para o Metodo dos Elementos Finitos (MEF) de alta ordem hp, considerando os sistemas de referencias locais dos elementos. Para isso, primeiramente, determinam-se ponderaçoes especficas para as bases de funçoes de triangulos e tetraedros, formada pelo produto tensorial de polinomios de Jacobi, de forma a se obter melhor esparsidade e condicionamento das matrizes de massa e rigidez dos elementos. Alem disso, procuram-se novas funçoes de base para tornar as matrizes de massa e rigidez mais esparsas possiveis. Em seguida, escolhe-se os pontos de integraçao que otimizam o custo do calculo dos coeficientes das matrizes de massa e rigidez usando as regras de quadratura de Gauss-Jacobi, Gauss-Radau-Jacobi e Gauss-Lobatto-Jacobi. Por fim, mostra-se a construçao de uma base unidimensional nodal que permite obter uma matriz de rigidez praticamente diagonal para problemas de Poisson unidimensionais. Discute-se ainda extensoes para elementos bi e tridimensionais
Abstract: The main purpose of this work is the development of tensor-based interpolation functions and integration rules for the hp High-order Finite Element Method (FEM), considering the local reference systems of the elements. We first determine specific weights for the shape functions of triangles and tetrahedra, constructed by the tensorial product of Jacobi polynomials, aiming to obtain better sparsity and numerical conditioning for the mass and stiffness matrices of the elements. Moreover, new shape functions are proposed to obtain more sparse mass and stiffness matrices. After that, integration points are chosen that optimize the cost for the calculation of the coefficients of the mass and stiffness matrices using the rules of quadrature of Gauss-Jacobi, Gauss-Radau-Jacobi and Gauss-Lobatto-Jacobi. Finally, we construct an one-dimensional nodal shape function that obtains an almost diagonal stiffness matrix for the 1D Poisson problem. Extensions to two and three-dimensional elements are discussed.
Doutorado
Mecanica dos Sólidos e Projeto Mecanico
Doutor em Engenharia Mecânica

Ce travail comporte deux parties dont la première concerne l'ensemble asymptotique $S_F$ d'une application polynomiale $F: C^n to C^n$. Dans les année 90s, Jelonek a montré que cet ensemble est une variété algébrique complexe singulière de dimension (complexe) $n-1$. Nous donnons une méthode, appelée {it méthode des fa{c c}ons}, pour stratifier cet ensemble. Nous obtenons une stratification de Thom-Mather. Par ailleurs, il existe une stratification de Whitney de $S_F$ telle que l'ensemble des fa{c c}ons possibles soit constant sur chaque strate. En utilisant les fa{c c}ons, nous donnons un algorithme pour expliciter l'ensemble asymptotique d'une application quadratique dominante en trois variables. Nous obtenons aussi une liste des ensembles asymptotiques possibles dans ce cas. La deuxième partie concerne l'ensemble $V_F$ : En 2010, Anna et Guillaume Valette ont construit une pseudo-variété réelle $V_F subset R^{2n + p}$, où $p > 0$, associée à une application polynomiale $F: C^n to C^n$. Dans le cas $n = 2$, ils ont prouvé que si $F$ est une application polynomiale de déterminant jacobien partout non nul, alors $F$ n'est pas propre si et seulement si l'homologie d'intersection de $V_F$ n'est pas triviale en dimension 2. Nous donnons une généralisation de ce résultat, dans le cas d'une application polynomiale $F : C^n to C^n$ de jacobien partout non nul. Nous donnons aussi une méthode pour stratifier l'ensemble $V_F$. Comme applications, nous obtenons des stratifications de l'ensemble des valeurs critiques asymptotiques de $F$ et de l'ensemble des points de bifurcation de $F$
There are two parts in the present work. The first part concerns the asymptotic set of a polynomial mapping $F: C^n to C^n$. In the 90s, Zbigniew Jelonek showed that this set is a $(n-1)$ - (complex) dimensional singular variety. We give a method, called {it m'ethode des fa{c c}ons}, for stratifying this set. We obtain a Thom-Mather stratification. Moreover, there exists a Whitney stratification such that the set of possible fa{c c}ons is constant on every stratum. By using the fa{c c}ons, we give an algorithm for expliciting the asymptotic sets of a dominant quadratic polynomial mapping in three variables. As a result, we have a complete list of the asymptotic sets in this case. The second part concerns the set called Valette set $V_F$. In 2010, Anna and Guillaume Valette constructed a real pseudomanifold $V_F subset R^{2n + p}$, where $p > 0$, associated to a polynomial mapping $F: C^n to C^n$. In the case $n = 2$, they proved that if $F$ is a polynomial mapping with nowhere vanishing Jacobian, then $F$ is not proper if and only if the homology (or intersection homology) of $V_F$ is not trivial in dimension 2. We give a generalization of this result, in the case of a polynomial mapping $F : C^n to C^n$ with nowhere vanishing Jacobian. We give also a method for stratifying the set $V_F$. As applications, we have the stratifications of the set of asymptotic critical values of $F$ and the set of bifurcation points of $F$

Ce travail est dédié à l'étude des inégalités de type Landau-Kolmogorov en normes L2. Les mesures utilisées sont celles d'Hermite, de Laguerre-Sonin et de Jacobi. Ces inégalités sont obtenues en utilisant une méthode variationnelle. Elles font intervenir la norme d'un polynômes p et celles de ces dérivées. Dans un premier temps, on s'intéresse aux inégalités en une variable réelle qui font intervenir un nombre quelconque de normes. Les constantes correspondantes sont prises dans le domaine où une certaine forme bilinéaire est définie positive. Ensuite, on généralise ces résultats aux polynômes à plusieurs variables réelles en utilisant le produit tensoriel dans L2 et en faisant intervenir au plus les dérivées partielles secondes. Pour les mesures d'Hermite et de Laguerre-Sonin, ces inégalités sont étendues à toutes les fonctions d'un espace de Sobolev. Pour la mesure de Jacobi on donne des inégalités uniquement pour les polynômes d'un degré fixé par rapport à chaque variable.

We analyze a possible minimal counterexample to theJacobian Conjecture P;Q with gcd(deg(P); deg(Q)) = 16 and show that its existence depends only on the existence of solutions for a certain Abel dierential equation of the second kind.
Analizamos un posible contraejemplo P;Q a la conjetura del jacobiano con gcd(deg(P); deg(Q)) = 16 y mostramos que su existencia depende exclusivamente de la existencia de soluciones de una cierta ecuacion diferencial de Abel de segundo tipo.

We consider the affine varieties which arise by considering invertible polynomial maps from C2 to itself of less than or equal to a given-degree. These varieties arise naturally in the investigation of the long-standing Jacobian Conjecture. We start with some calculations in the lower degree cases. These calculations provide a proof of the Jacobian conjecture in these cases and suggest how the investigation in the higher degree cases should proceed. We then show how invertible polynomial maps can be decomposed as products of what we call triangular maps and we are able to prove a uniqueness result which gives a stronger version of Jung's theorem [j] which is one of the most important results in this area. Our proof also gives a new derivation of Jung's theorem from Segre's lemma. We give a different decomposition of an invertible polynomial map as a composition of "irreducible maps" and we are able to write down standard forms for these irreducibles. We use these standard forms to give a description of the structure of the varieties of invertible maps. We consider some interesting group actions on our varieties and show how these fit in with the structure we describe. Finally, we look at the problem of identifying polynomial maps of finite order. Our description of the structure of the above varieties allows us to solve this problem completely and we are able to show that the only elements of finite order are those which arise from conjugating linear elements of finite order.

We compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture using a recursive formula for the Catalan numbers.
En este artículo calculamos la base de Groebner de un sistema polinomial de ecuaciones relacionada con la conjetura del jacobiano utilizando una fórmula recursiva para los numeros de Catalan.

Les travaux de recherche de cette thèse concernent l'encadrement de la constante de Markov Bernstein pour la norme L2 associée aux mesures de Jacobi et Gegenbauer généralisée. Ce travail est composé de deux parties : dans la première partie, nous avons développé une généralisation de l'algorithme qd pour les matrices symétriques définies positives à largeur de bande $\ell$ et nous avons construit l'algorithme qd pour les matrices de Jacobi par blocs. Ensuite, nous l'avons généralisé aux cas des matrices par bloc à largeur de bande $\ell$. Ces algorithmes nous permettent de trouver un majorant de la constante. Enfin, nous avons développé le déterminant caractéristique d'une matrice symétrique définie positive pentadiagonale, ce qui nous permet d'obtenir un minorant de la constante en utilisant la méthode de Newton. La deuxième partie est consacrée à l'application de tous les outils développés à l'encadrement de la constante de Markov Bernstein pour la norme L2 associée à la mesure de Gegenbauer généralisée.

De plus en plus de constructeurs proposent des accélérateurs de calculs à base de circuits reconfigurables FPGA, cette technologie présentant bien plus de souplesse que le microprocesseur. Valoriser cette flexibilité dans le domaine de l'accélération de calcul flottant en utilisant les langages de description de circuits classiques (VHDL ou Verilog) reste toutefois très difficile, voire impossible parfois. Cette thèse a contribué au développement du logiciel FloPoCo, qui offre aux utilisateurs familiers avec VHDL un cadre C++ de description d'opérateurs arithmétiques génériques adapté au calcul reconfigurable. Ce cadre distingue explicitement la fonctionnalité combinatoire d'un opérateur, et la problématique de son pipeline pour une précision, une fréquence et un FPGA cible donnés. Afin de pouvoir utiliser FloPoCo pour concevoir des opérateurs haute performance en virgule flottante, il a fallu d'abord concevoir des blocs de bases optimisés. Nous avons d'abord développé des additionneurs pipelinés autour des lignes de propagation de retenue rapides, puis, à l'aide de techniques de pavages, nous avons conçu de gros multiplieurs, possiblement tronqués, utilisant des petits multiplieurs. L'évaluation de fonctions élémentaires en flottant implique souvent l'évaluation en virgule fixe d'une fonction. Nous présentons un opérateur générique de FloPoCo qui prend en entrée l'expression de la fonction à évaluer, avec ses précisions d'entrée et de sortie, et construit un évaluateur polynomial optimisé de cette fonction. Ce bloc de base a permis de développer des opérateurs en virgule flottante pour la racine carrée et l'exponentielle qui améliorent considérablement l'état de l'art. Nous avons aussi travaillé sur des techniques de compilation avancée pour adapter l'exécution d'un code C aux pipelines flexibles de nos opérateurs. FloPoCo a pu ainsi être utilisé pour implanter sur FPGA des applications complètes.

Parallel robots with configurable platform are a class of parallel robots in which the end-effector is a closed-loop flexible chain of rigid links. We have developed a 5-RRR planar mechanism that features a flexible 5-bar chain as end-effector. The angles between adjacent sides of this chain can be controlled through the actuated revolute joints attached to the base of the mechanism. This thesis consists in the geometrical design of n-RRR planar parallel robots and in the study of the Direct Kinematics for 4-, 5- and 6-RRR mechanisms using Bilateration, a method that greatly reduces the computational time for the kinematic analysis. The next step is the singularity analysis for the n-RRR robot architectures; finally, in the last part of this thesis we present the results from experimental tests that have been performed on a 5-RRR robot prototype.

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the Degree of Master of Science. Johannesburg 2015.
This Dissertation focuses on the Jacobi polynomial. Specifically, it discusses certain aspects of the zeros of the Jacobi polynomial such as the interlacing property and quasiorthogonality. Also found in the Dissertation is a chapter on the inequalities of the zeros of the Jacobi polynomial, mainly those developed by Walter Gautschi.

碩士
國立中央大學
數學研究所
98
The acoustic problems usually happens around us in our daily life when we drive a car, take a bus or take a plane. From the problems of acoustic vibrations with damping, a polynomial eigenvalue problem is obtained by applying the Galerkin finite element method. For particular applications, we are interested in finding some selected low frequency eigenvalues which are located within the interior of the spectrum. The size of the resulting eigenproblem is typically large especially for very fined mesh case so that the parallel polynomial eigensolver is need to deal with such problem. The Jacobi-Davidson method provides a fast and efficient manner for solving the interior eigenvalues for the large sparse polynomial eigenvalue problems. We proposed an Jacobi-Davidson method based on an additive Schwarz framework in parallel implementation and used it to solve the polynomial eigenvalue problem arising from the acoustic. And we showed some parallel performance of the additive Schwarz preconditioned Jacobi-Davidson method by numerical experiments. With help of Krylov-Schwarz algorithm for the correction equation, the efficiency of JD algorithm is greatly improved.

Simulation of flow in porous media requires the numerical approximation of elliptic partial differential equations. Mixed finite element methods are frequently employed, because of local mass conservation and accurate approximation of both pressure and velocity. Mixed methods give rise to "elliptic" saddle-point (ESP) matrices, which are difficult to solve numerically. In addition, the problems to be modelled in ground water flow require that the hydraulic conductivity or absolute permeability be a tensor, which adds additional complexity to the resulting saddle-point matrices. This research develops several preconditioners for restarted GMRES solution of the ESP linear systems. These preconditioners are based on a new class of polynomial matrices, which we refer to as Multi-band Jacobi Polynomial (JMP) matrices. Applications of these preconditioners to the numerical solution of two and three spatial dimensional flow equations with tensor coefficients using rectangular lowest order Raviart-Thomas spaces are presented.

碩士
國立中央大學
數學研究所
100
Many scientific and engineering applications require accurate, fast, robust, and scalable numerical solution of large sparse algebraic polynomial eigenvalue problems (PEVPs) arising from some appropriate discretization of partial differential equations. The polynomial Jacobi-Davidson (PJD) algorithm has been numerically shown as a promising approach for the PEVPs and has gained its popularity for finding their interior spectrum of the PEVPs. The PJD algorithm is a subspace method, which extracts the candidate approximate eigenpair from a search space and the space undated by embedding the solution of the correction equation at the JD iteration. In this research, we propose the two-level PJD algorithm for PEVPs with emphasis on the application of the dissipative acoustic cubic eigenvalue problem. The proposed two-level PJD algorithm is based on the Schwarz framework. The initial basis for the search space is constructed on the current level by using the solution of the same eigenvalue problem, but defined on the previous coarser grid. On the other hand, a low-cost and efficient preconditioner based on Schwarz framework, coarse restricted additive Schwarz (RAS_c) preconditioner for the correction equation, which plays a crucial role in parallel computing for large-scale problems by using a large number of processors. Some numerical examples obtained on a parallel cluster of computers are given to demonstrate the robustness and scalability of our PJD algorithm.