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1 
Zeros of Jacobi polynomials and associated inequalitiesMancha, Nina 11 March 2015 (has links)
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.

2 
Rate of convergence of Hermite interpolation based on the roots of certain Jacobi polynomials /Ekong, Victor Jonathan Udo January 1972 (has links)
No description available.

3 
Applied leftdefinite theory the Jacobi polynomials, their Sobolev orthogonality, and selfadjoint operators /Bruder, Andrea S. Littlejohn, Lance L. January 2009 (has links)
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. 115119).

4 
On The Wkb Asymptotic Solutionsof Differential Equations Of The Hypergeometric TypeAksoy, Betul 01 December 2004 (has links) (PDF)
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.

5 
O teorema de comparação de Sturm e aplicações / Sturm comparison theorem and applicationsYen, Chi Lun, 1983 09 May 2013 (has links)
Orientadores: Dimitar Kolev Dimitrov, Roberto Andreani / Tese (doutorado)  Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 20180823T19:23:17Z (GMT). No. of bitstreams: 1
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Previous issue date: 2013 / 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 X1Jacobi, 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 X1Jacobi 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

6 
Free Vibrations and Static Deformations of Composite Laminates and Sandwich Plates using Ritz MethodAlanbay, Berkan 15 December 2020 (has links)
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 firstorder shear deformation theory. For the thick isotropic plates and laminates, a thirdorder 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 fiberreinforced epoxy composite) and soft core (e.g., foam) exist, multilayered structures may exhibit complex kinematic behaviors. Hence, in such case, C<sub>z</sub>⁰ 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) Nthorder shear and normal deformation theories to model sandwich structures with various facetocore 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 onestep wellknown stress recovery scheme (SRS). Furthermore, analytical solutions of threedimensional 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 facetocore stiffness ratios. In addition to natural frequency results obtained from ESL and LW theories, the solutions of the corresponding 3dimensional 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 higherorder) 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, platelike 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, platelike 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 threedimensional (3D) objects, their thickness is small as compared to the inplane 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 3D object). Hence, a plate theory can be developed to provide accurately a quantity of interest. Some issues are more challenging for lowfidelity 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 facesheets (e.g., made of carbon fiberreinforced 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.

7 
A finite family of qorthogonal polynomials and resultants of Chebyshev polynomialsGishe, Jemal Emina 01 June 2006 (has links)
Two problems related to orthogonal polynomials and special functions are considered. For q greater than 1 it is known that continuous qJacobi polynomials are orthogonal on the imaginary axis. The first problem is to find proper normalization to form a system of polynomials that are orthogonal on the real line. By introducing a degree reducing operator and a scalar product one can show that the normalized continuous qJacobi polynomials satisfies an eigenvalue equation. This implies orthogonality of the normalized continuous qJacobi polynomials. As a byproduct, different results related to the normalized system of polynomials, such as its closed form,threeterm recurrence relation, eigenvalue equation, Rodrigues formula and generating function will be computed. A discriminant related to the normalized system is also obtained. The second problem is related to recent results of Dilcher and Stolarky on resultants of Chebyshev polynomials. They used algebraic methods to evaluate the resultant of two combinations of Chebyshev polynomials of the second kind. This work provides an alternative method of computing the same resultant and also enables one to compute resultants of more general combinations of Chebyshev polynomials of the second kind. Resultants related to combinations of Chebyshev polynomials of the first kind are also considered.

8 
Núcleos positivos definidos em espaços 2homogêneos / Positive definite kernels on twopoint homogeneous spacesBarbosa, Victor Simões 26 July 2016 (has links)
Neste trabalho analisamos a positividade definida estrita de núcleos contínuos sobre um espaço compacto 2homogêneo. R. Gangolli (1967) apresentou uma caracterização completa para os núcleos que são contínuos, isotrópicos e positivos definidos sobre um espaço compacto 2homogêneo 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 2homogêneos, 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 twopoint 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 twopoint 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.

9 
Funções positivas definidas para interpolação em esferas complexas. / Positive definite functions for interpolation on complex spheres.Peron, Ana Paula 07 February 2001 (has links)
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 socalled 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.

10 
Funções de interpolação e regras de integração tensorizaveis para o metodo de elementos finitos de alta ordem / Tensorbased interpolation functions and integration rules for the high order finite elements methodsVazquez, Thais Godoy 26 February 2008 (has links)
Orientador: Marco Lucio Bittencourt / Tese (doutorado)  Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 20180810T12:57:32Z (GMT). No. of bitstreams: 1
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Previous issue date: 2008 / 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, determinamse 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, procuramse novas funçoes de base para tornar as matrizes de massa e rigidez mais esparsas possiveis. Em seguida, escolhese os pontos de integraçao que otimizam o custo do calculo dos coeficientes das matrizes de massa e rigidez usando as regras de quadratura de GaussJacobi, GaussRadauJacobi e GaussLobattoJacobi. Por fim, mostrase a construçao de uma base unidimensional nodal que permite obter uma matriz de rigidez praticamente diagonal para problemas de Poisson unidimensionais. Discutese ainda extensoes para elementos bi e tridimensionais / Abstract: The main purpose of this work is the development of tensorbased interpolation functions and integration rules for the hp Highorder 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 GaussJacobi, GaussRadauJacobi and GaussLobattoJacobi. Finally, we construct an onedimensional nodal shape function that obtains an almost diagonal stiffness matrix for the 1D Poisson problem. Extensions to two and threedimensional elements are discussed. / Doutorado / Mecanica dos Sólidos e Projeto Mecanico / Doutor em Engenharia Mecânica

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