This thesis discusses several topics related to interpolation and how it is used in numerical analysis. It begins with an overview of the aspects of interpolation theory that are relevant to the discussion at hand before presenting three new contributions to the field. The first new result is a detailed error analysis of the barycentric formula for trigonometric interpolation in equally-spaced points. We show that, unlike the barycentric formula for polynomial interpolation in Chebyshev points (and contrary to the main view in the literature), this formula is not always stable. We demonstrate how to correct this instability via a rewriting of the formula and establish the forward stability of the resulting algorithm. Second, we consider the problem of trigonometric interpolation in grids that are perturbations of equally-spaced grids in which each point is allowed to move by at most a fixed fraction of the grid spacing. We prove that the Lebesgue constant for these grids grows at a rate that is at most algebraic in the number of points, thus answering questions put forth by Trefethen and Weideman about the robustness of numerical methods based on trigonometric interpolation in points that are uniformly distributed but not equally-spaced. We use this bound to derive theorems about the convergence rate of trigonometric interpolation in these grids and also discuss the related question of quadrature. Specifically, we prove that if a function has V ≥ 1 derivatives, the Vth of which is Hölder continuous (with a Hölder exponent that depends on the size of the maximum allowable perturbation), then the interpolants converge uniformly to the function at an algebraic rate; larger values of V lead to more rapid convergence. A similar statement holds for the corresponding quadrature rule. We also consider what analogue, if any, there is for trigonometric interpolation of the famous 1/4 theorem of Kadec from sampling theory that restricts the size of the perturbations one can make to the integers and still be guaranteed to have a set of stable sampling for the Paley-Wiener space. We present numerical evidence suggesting that in the discrete case, the 1/4 threshold takes the form of a threshold for the boundedness of a "2-norm Lebesgue constant" and does not appear to have much significance in practice. We believe that these are the first results regarding this problem to appear in the literature. While we do not believe the results we establish are the best possible quantitatively, they do (rigorously) capture the main features of trigonometric interpolation in perturbations of equally-spaced grids. We make several conjectures as to what the optimal results may be, backed by extensive numerical results. Finally, we consider a new application of interpolation to numerical linear algebra. We show that recently developed methods for computing the eigenvalues of a matrix by dis- cretizing contour integrals of its resolvent are equivalent to computing a rational interpolant to the resolvent and finding its poles. Using this observation as the foundation, we develop a method for computing the eigenvalues of real symmetric matrices that enjoys the same advantages as contour integral methods with respect to parallelism but employs only real arithmetic, thereby cutting the computational cost and storage requirements in half.

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Adviser: Peter B. Gilkey. ix, 128 leaves / A print copy of this title is available through the UO Libraries under the call number: MATH QA613 .Z43 2000 / Relative to a non-degenerate metric of signature (p, q), an algebraic curvature tensor is said to be IP if the associated skew-symmetric curvature operator R(π) has constant eigenvalues and if the kernel of R(π) has constant dimension on the Grassmanian of non-degenerate oriented 2-planes. A pseudo-Riemannian manifold with a non-degenerate indefinite metric of signature (p, q) is said to be IP if the curvature tensor of the Levi-Civita connection is IP at every point; the eigenvalues are permitted to vary with the point. In the Riemannian setting (p, q) = (0, m), the work of Gilkey, Leahy, and Sadofsky and the work of Ivanov and Petrova have classified the IP metrics and IP algebraic curvature tensors if the dimension is at least 4 and if the dimension is not 7. We use techniques from algebraic topology and from differential geometry to extend some of their results to the Lorentzian setting (p, q) = (1, m – 1) and to the setting of metrics of signature (p, q) = (2, m – 2).

Orientadores: Sueli Irene Rodrigues Costa, Simão Nicolau Stelmastchuk / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-10T23:29:23Z (GMT). No. of bitstreams: 1 Tavares_FabianoPinto_M.pdf: 877532 bytes, checksum: 0081c7182db3aab71edcbe274822bea5 (MD5) Previous issue date: 2008 / Resumo: Neste trabalho focalizamos a deformação de cônicas e quádricas por transformações lineares. Deduzimos de forma explícita os autovalores e autovetores ortonormais de matrizes reais 2 x 2 e 3 x 3, para os quais não há quase referências na literatura e nem incorporação nos programas computacionais de cálculo simbólico usuais. Esta determinação levou -nos a estudar um pouco da história da resolução das equações de terceiro grau e das condições e formulações das raízes reais destas. Os resultados foram utilizados na determinação explícita das deformações por transformações lineares de cônicas e quádricas, sendo estas discutidas em termos de características das matrizes associadas / Abstract: We discuss here the deformations of conics and quadrics under linear mappings. We set explicitly the eingenvalues and the orthonormal eigenvectors of real symmetric 2 X 2 and 3 X 3 matrices. These expressions are scarce in the literature and not incorporated in symbolic calculus software. The determination of those eigenvalues leaded us to the study of the solution of third degree equations and some of related historical aspects with focus on conditions and expressions for their real solutions Those results are used in the exact determination of the linear deformation of conics and quadrics in terms of the characteristics of their associated matrices / Mestrado / Geometria Topologia / Mestre em Matemática

Ph.D. / In this thesis, the ranks of many types of regular and strongly regular graphs are determined. Also determined are ranks of regular graphs under unary operations: the line graph, the complement, the subdivision graph, the connected cycle, the complete subdivision graph, and the total graph. The binary operations considered are the Cartesian product and the complete product. The ranks of the Cartesian product of regular graphs have been investigated previously in [BBD1]; here, we summarise and extend those results to include more regular graphs. We also examine a special nonregular graph, the path. Ranks of paths and products of graphs involving paths are presented as well

Orientador: Alagacone Sri Ranga / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T19:37:15Z (GMT). No. of bitstreams: 1 Felix_HeronMartins_D.pdf: 783541 bytes, checksum: cea4459f391a5da7e61d9cff02244ec0 (MD5) Previous issue date: 2015 / Resumo: O objetivo do presente trabalho se divide em duas partes: na primeira, estudaremos uma regra de quadratura interpolatória sobre os zeros de polinômios núcleo obtidos a partir de uma sequência de polinômios L-ortogonais, oferecendo técnicas numéricas para a obtenção dos nós e pesos dessa regra de quadratura. Na segunda parte, forneceremos uma caracterização dos polinômios de Szegö em termos de duas sequências reais, dentre as quais uma é sequência encadeada. Tal caracterização afeta a relação entre os polinômios núcleo e os polinômios ortogonais no círculo unitário aos quais estes estão associados / Abstract: The main goal of the present work falls under two parts: firstly, we'll study a quadrature rule over the zeros of the kernel polynomials obtained from a sequence of L-orthogonal polynomials, offering numerical techniques for evaluating the nodes and weights of such quadrature rule. Secondly, we'll give a characterization for Szegö polynomials in terms of two real sequences, in which one is a chained sequence. Such characterization influences the connection between the kernel polynomials and the related orthogonal polynomials over the unit circle / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

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abstract: Presented is a design approach and test of a novel compact waveguide that demonstrated the outer dimensions of a rectangular waveguide through the introduction of parallel raised strips, or flanges, which run the length of the rectangular waveguide along the direction of wave propagation. A 10GHz waveguide was created with outer dimensions of a=9.0mm and b=3.6mm compared to a WR-90 rectangular waveguide with outer dimensions of a=22.86mm and b=10.16mm which the area is over 7 times the area. The first operating bandwidth for a hollow waveguide of dimensions a=9.0mm and b=3.6mm starts at 16.6GHz a 40% reduction in cutoff frequency. The prototyped and tested compact waveguide demonstrated an operating close to the predicted 2GHz with predicted vs measured injection loss generally within 0.25dB and an overall measured injection loss of approximately 4.67dB/m within the operating bandwidth. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2019

In 1980 Feigin and Fuchs classified the length 2 bounded representations of Vec(R), the Lie algebra of polynomial vector fields on the line, as a result of their work on the cohomology of Vec(R). This dissertation is concerned mainly with the uniserial (completely indecomposable) representations of Vec(R) with a single Casimir eigenvalue and weights bounded below. Such representations are composed of irreducible representations with semisimple Euler operator action, bounded weight space dimensions, and weights bounded below. These are known to be the tensor density modules with lowest weight λ, for any non-zero complex number λ, and the trivial module C, with Vec(R) actions π_λ and π_C, respectively. Our proofs are cohomology arguments involving the first cohomology groups of Vec(R) with values in the space of homomorphisms between two irreducible representations. These results classify the finite length uniserial extensions, with a single Casimir eigenvalue, of admissible irreducible Vec(R) representations with weights bounded below. In almost every case there is at most one uniserial representation with a given composition series. However, in the case of an odd length extension with composition series {π_1,π_C,π_1,…,π_C,π_1}, there is a one-parameter family of extensions. We also give preliminary results on uniserial representations of the Virasoro Lie algebra.

Equations of the form 𝑎𝑥³ + 𝑏𝑦³ = 1, where the constants 𝑎 and 𝑏 are integers of some number field such that 𝑎𝑥³ + 𝑏𝑦³ is irreducible, are a particularly significant class of cubic Thue equations that notably includes the cubic Pell equation. For a positive cubefree rational integer 𝑑, we consider the family of equations of the form 𝑚𝑥³ − 𝑑𝑛𝑦³ = 1 where 𝑚 and 𝑛 are squarefree. We define an 𝐿-function associated to 𝑑 whose nonvanishing coefficients correspond to the nontrivial solutions of those equations. That definition uses expressions related to the cubic theta function Q (􏰇√ 􏰈-), and we study that 𝐿-function’s analytic properties by using a method generalizing the approach used by Takhtajan and Vinogradov to derive a trace formula using the quadratic theta function for Q. We construct its meromorphic continuation and determine the locations and orders of its poles. Specifically, the poles occur at the eigenvalues of the Laplacian for the Maass forms 𝑢_𝑗 , 𝑗 = 1, 2, 3, · · · in the discrete spectrum, with a double pole at 𝑠 = ½ and possible simple poles at 𝑠=𝑠_𝑗,1−𝑠_𝑗,where𝜆𝑗 =2𝑠_𝑗(2−2𝑠_𝑗)istheLaplaceeigenvalueof𝑢𝑗 and𝜆𝑗 ≠1.