Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1993. / Vita. Abstract. Includes bibliographical references (leaves 132-134). Also available via the Internet.

Thesis (doctoral)--University of Stockholm, 1984. / Abstract (1 leaf) inserted. Bibliography: p. 79-80.

We study the problem of classifying all Poisson-Lie structures on the group Gy of local diffeomorphisms of the real line R¹ which leave the origin fixed, as well as the extended group of diffeomorphisms G₀_∞ ⊃ G_∞ whose action on R¹ does not necessarily fix the origin. A complete classification of all Poisson-Lie structures on the group G_∞ is given. All Poisson-Lie structures of coboundary type on the group G₀_∞ are classified. This includes a classification of all Lie-bialgebra structures on the Lie algebra G_∞ of G_∞, which we prove to be all of coboundary type, and a classification of all Lie-bialgebra structures of coboundary type on the Lie algebra Go_∞ of Go_∞ which is the Witt algebra. A large class of Poisson structures on the space V_λ of λ-densities on the real line is found such that V_λ becomes a homogeneous Poisson space under the action of the Poisson-Lie group G_∞. We construct a series of finite-dimensional quantum groups whose quasiclassical limits are finite-dimensional Poisson-Lie factor groups of G_∞ and G₀_∞. / Ph. D.

Neste trabalho, estabelecemos uma equação que relaciona a curvatura de Ricci de uma variedade riemanniana M e as segundas formas fundamentais de duas folheações ortogonais de dimensões complementares, F e F, definidas em M. Usando essa equação, encontramos uma estimativa da curvatura média da folheação F e uma condição necessária e suficiente para que tal folheação seja totalmente geodésica. Mostramos também uma condição suficiente para que M seja localmente um produto riemanniano das folhas de F e F, se uma das folheações for totalmente umbílica. Por fim, provamos ainda uma fórmula integral válida para tais folheações. / In this work, we and an equation that relates the Ricci curvature of a riemannian manifold M and the second fundamental forms of two orthogonal foliations of complementary dimensions, F and F, defined on M. Using this equation, we and an estimate of the mean curvature of the foliation F and a necessary and suficient condition for the foliation F to be totally geodesic. We also show a suficient condition for the manifold M to be locally a riemannian product of the leaves of F and F, if one of the foliations is totally umbilical. Finally, we also prove an integral formula for such foliations.

This thesis presents new numerical algorithms for approximating functions by trigonometric polynomials and trigonometric rational functions. We begin by reviewing trigonometric polynomial interpolation and the barycentric formula for trigonometric polynomial interpolation in Chapter 1. Another feature of this chapter is the use of the complex plane, contour integrals and phase portraits for visualising various properties and relationships between periodic functions and their Laurent and trigonometric series. We also derive a periodic analogue of the Hermite integral formula which enables us to analyze interpolation error using contour integrals. We have not been able to find such a formula in the literature. Chapter 2 discusses trigonometric rational interpolation and trigonometric linearized rational least-squares approximations. To our knowledge, this is the first attempt to numerically solve these problems. The contribution of this chapter is presented in the form of a robust algorithm for computing trigonometric rational interpolants of prescribed numerator and denominator degrees at an arbitrary grid of interpolation points. The algorithm can also be used to compute trigonometric linearized rational least-squares and trigonometric polynomial least-squares approximations. Chapter 3 deals with the problem of trigonometric minimax approximation of functions, first in a space of trigonometric polynomials and then in a set of trigonometric rational functions. The contribution of this chapter is presented in the form of an algorithm, which to our knowledge, is the first description of a Remez-like algorithm to numerically compute trigonometric minimax polynomial and rational approximations. Our algorithm also uses trigonometric barycentric interpolation and Chebyshev-eigenvalue based root finding. Chapter 4 discusses the Fourier-Padé (called trigonometric Padé) approximation of a function. We review two existing approaches to the problem, both of which are based on rational approximations of a Laurent series. We present a numerical algorithm with examples and compute various type (m, n) trigonometric Padé approximants.

Neste trabalho, estabelecemos uma equação que relaciona a curvatura de Ricci de uma variedade riemanniana M e as segundas formas fundamentais de duas folheações ortogonais de dimensões complementares, F e F, definidas em M. Usando essa equação, encontramos uma estimativa da curvatura média da folheação F e uma condição necessária e suficiente para que tal folheação seja totalmente geodésica. Mostramos também uma condição suficiente para que M seja localmente um produto riemanniano das folhas de F e F, se uma das folheações for totalmente umbílica. Por fim, provamos ainda uma fórmula integral válida para tais folheações. / In this work, we and an equation that relates the Ricci curvature of a riemannian manifold M and the second fundamental forms of two orthogonal foliations of complementary dimensions, F and F, defined on M. Using this equation, we and an estimate of the mean curvature of the foliation F and a necessary and suficient condition for the foliation F to be totally geodesic. We also show a suficient condition for the manifold M to be locally a riemannian product of the leaves of F and F, if one of the foliations is totally umbilical. Finally, we also prove an integral formula for such foliations.

This paper is an examination of the theory of discrete complex analysis that arises from the framework of a planar cell complex. Construction of this theory is largely integration-based. A combination of two cell complexes, the double and its associated diamond complex, allows for the development of a discrete Cauchy Integral Formula. / Master of Science

Orientador: Manoel Ferreira Borges Neto / Banca: Antônio Luís Venezuela / Banca: Sandra Regina Monteiro Masalshiene Roveda / Resumo: O objetivo deste trabalho ée estabelecer similaridades entre a análise complexa e os quatérnios. Nele é feito um estudo da regularidade de funções quaterniônicas e são estabelecidas as funções exponencial e logarítmica para os quatérnios sendo feito um estudo da hiperpe- riodicidade dessas funções. Outro resultado apresentado é a generalização quaterniônica da fórmula integral de Cauchy um dos principais teoremas da análise complexa. / Abstract: The objective of this work is to establish similarities between the complex analysis and the quaternions. In it is made a study of the regularity of quaternionic functions and are established the exponential and logarithmic functions for the quaternions being made a study of the hiperperiodicity of these functions. Another presented result is the quater- nionic generalization of the Cauchy's integral formula one of the main theorems of the complex analysis. / Mestre

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-02-17Bitstream added on 2014-06-13T18:30:54Z : No. of bitstreams: 1 barreiro_rc_me_sjrp.pdf: 585027 bytes, checksum: 039155145a6c7b9e6e1fc03a02180b55 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo deste trabalho ée estabelecer similaridades entre a análise complexa e os quatérnios. Nele é feito um estudo da regularidade de funções quaterniônicas e são estabelecidas as funções exponencial e logarítmica para os quatérnios sendo feito um estudo da hiperpe- riodicidade dessas funções. Outro resultado apresentado é a generalização quaterniônica da fórmula integral de Cauchy um dos principais teoremas da análise complexa. / The objective of this work is to establish similarities between the complex analysis and the quaternions. In it is made a study of the regularity of quaternionic functions and are established the exponential and logarithmic functions for the quaternions being made a study of the hiperperiodicity of these functions. Another presented result is the quater- nionic generalization of the Cauchy's integral formula one of the main theorems of the complex analysis.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-12-03T08:37:01Z No. of bitstreams: 2 Dissertação - Saulo Henrique de Oliveira - 2015.pdf: 1917786 bytes, checksum: 72281ae1c7a550ab53f962bb0da58d07 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-12-03T08:39:30Z (GMT) No. of bitstreams: 2 Dissertação - Saulo Henrique de Oliveira - 2015.pdf: 1917786 bytes, checksum: 72281ae1c7a550ab53f962bb0da58d07 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-12-03T08:39:30Z (GMT). No. of bitstreams: 2 Dissertação - Saulo Henrique de Oliveira - 2015.pdf: 1917786 bytes, checksum: 72281ae1c7a550ab53f962bb0da58d07 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-04-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work ... / Este trabalho ...