by Wai-Shing Luk. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 104-113). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Functional Extension of Iterative Methods --- p.2 / Chapter 1.2 --- Applications in Circuit Simulation --- p.2 / Chapter 1.3 --- Multigrid Acceleration --- p.3 / Chapter 1.4 --- Why Hilbert Space? --- p.4 / Chapter 1.5 --- Parallel Implementation --- p.5 / Chapter 1.6 --- Domain Decomposition --- p.5 / Chapter 1.7 --- Contributions of This Thesis --- p.6 / Chapter 1.8 --- Outlines of the Thesis --- p.7 / Chapter 2 --- Waveform Relaxation Methods --- p.9 / Chapter 2.1 --- Basic Idea --- p.10 / Chapter 2.2 --- Linear Operators between Banach Spaces --- p.14 / Chapter 2.3 --- Waveform Relaxation Operators for ODE's --- p.16 / Chapter 2.4 --- Convergence Analysis --- p.19 / Chapter 2.4.1 --- Continuous-time Convergence Analysis --- p.20 / Chapter 2.4.2 --- Discrete-time Convergence Analysis --- p.21 / Chapter 2.5 --- Further references --- p.24 / Chapter 3 --- Waveform Krylov Subspace Methods --- p.25 / Chapter 3.1 --- Overview of Krylov Subspace Methods --- p.26 / Chapter 3.2 --- Krylov Subspace methods in Hilbert Space --- p.30 / Chapter 3.3 --- Waveform Krylov Subspace Methods --- p.31 / Chapter 3.4 --- Adjoint Operator for WBiCG and WQMR --- p.33 / Chapter 3.5 --- Numerical Experiments --- p.35 / Chapter 3.5.1 --- Test Circuits --- p.36 / Chapter 3.5.2 --- Unstructured Grid Problem --- p.39 / Chapter 4 --- Parallel Implementation Issues --- p.50 / Chapter 4.1 --- DECmpp 12000/Sx Computer and HPF --- p.50 / Chapter 4.2 --- Data Mapping Strategy --- p.55 / Chapter 4.3 --- Sparse Matrix Format --- p.55 / Chapter 4.4 --- Graph Coloring for Unstructured Grid Problems --- p.57 / Chapter 5 --- The Use of Inexact ODE Solver in Waveform Methods --- p.61 / Chapter 5.1 --- Inexact ODE Solver for Waveform Relaxation --- p.62 / Chapter 5.1.1 --- Convergence Analysis --- p.63 / Chapter 5.2 --- Inexact ODE Solver for Waveform Krylov Subspace Methods --- p.65 / Chapter 5.3 --- Experimental Results --- p.68 / Chapter 5.4 --- Concluding Remarks --- p.72 / Chapter 6 --- Domain Decomposition Technique --- p.80 / Chapter 6.1 --- Introduction --- p.80 / Chapter 6.2 --- Overlapped Schwarz Methods --- p.81 / Chapter 6.3 --- Numerical Experiments --- p.83 / Chapter 6.3.1 --- Delay Circuit --- p.83 / Chapter 6.3.2 --- Unstructured Grid Problem --- p.86 / Chapter 7 --- Conclusions --- p.90 / Chapter 7.1 --- Summary --- p.90 / Chapter 7.2 --- Future Works --- p.92 / Chapter A --- Pseudo Codes for Waveform Krylov Subspace Methods --- p.94 / Chapter B --- Overview of Recursive Spectral Bisection Method --- p.101 / Bibliography --- p.104

Abstract We study in this thesis functional spaces involved in crack problems in unbounded domains. These spaces are defined by closing spaces of Sobolev H1 regularity functions (or vector fields) of bounded support, by the L2 norm of the gradient. In the case of linear elasticity, the closure is done under the L2 norm of the symmetric gradient. Our main result states that smooth functions are in this closure if and only if their gradient, (respectively symmetric gradient for the elasticity case), is in L2. We provide examples of functions in these newly defined spaces that are not in L2. We show however that some limited growth in dimension 2, or some decay in dimension 3 must hold for functions in those spaces: this is due to Hardy's inequalities.

Thesis (M. Sc.)--University of Canterbury, 2008. / Typescript (photocopy). Includes bibliographical references (p. [58]). Also available via the World Wide Web.

Thesis (MSc (Mathematical Sciences. Physical and Mathematical Analysis))--University of Stellenbosch, 2006. / This work studies the regularity (or smoothness) of continuous finitely supported refinable functions which are mainly encountered in multiresolution analysis, iterative interpolation processes, signal analysis, etc. Here, we present various kinds of sufficient conditions on a given mask to guarantee the regularity class of the corresponding refinable function. First, we introduce and analyze the cardinal B-splines Nm, m ∈ N. In particular, we show that these functions are refinable and belong to the smoothness class Cm−2(R). As a generalization of the cardinal B-splines, we proceed to discuss refinable functions with positive mask coefficients. A standard result on the existence of a refinable function in the case of positive masks is quoted. Following [13], we extend the regularity result in [25], and we provide an example which illustrates the fact that the associated symbol to a given positive mask need not be a Hurwitz polynomial for its corresponding refinable function to be in a specified smoothness class. Furthermore, we apply our regularity result to an integral equation. An important tool for our work is Fourier analysis, from which we state some standard results and give the proof of a non-standard result. Next, we study the H¨older regularity of refinable functions, whose associated mask coefficients are not necessarily positive, by estimating the rate of decay of their Fourier transforms. After showing the embedding of certain Sobolev spaces into a H¨older regularity space, we proceed to discuss sufficient conditions for a given refinable function to be in such a H¨older space. We specifically express the minimum H¨older regularity of refinable functions as a function of the spectral radius of an associated transfer operator acting on a finite dimensional space of trigonometric polynomials. We apply our Fourier-based regularity results to the Daubechies and Dubuc-Deslauriers refinable functions, as well as to a one-parameter family of refinable functions, and then compare our regularity estimates with those obtained by means of a subdivision-based result from [28]. Moreover, we provide graphical examples to illustrate the theory developed.

Orientadores: Daniela Mariz Silva Vieira, Jorge Tulio Ascui Mujica / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T05:01:42Z (GMT). No. of bitstreams: 1 Santos_ElisaReginados_M.pdf: 1412993 bytes, checksum: 96265b236947bac13325b8670c149359 (MD5) Previous issue date: 2009 / Resumo: Um operador linear limitado T entre espaços normados satisfaz a equação de Daugavet se II I + T II = 1+ T .Este trabalho tem como objetivo principal estudar tal equação para operadores lineares limitados no espaço das funções contínuas C(S), onde S é um espaço Hausdorff compacto. Para tanto, estudamos algumas representações de C*(S), o dual topológico de C(S), segundo as propriedades topológicas de S, e também representações de operadores definidos em C(S) ou com imagem em C(S). Fazendo uso desta teoria de representações em C(S) apresentamos então algumas classes de operadores que satisfazem a equação de Daugavet. Iniciamos apresentando a demonstração dada por H. Kamowitz em [11], de que se T é um operador linear compacto em C(S) então II I + T II = 1+ T se e somente se S não possui pontos isolados. Em seguida, apresentamos a demonstração dada por J. R. Holub em [8], provando que operadores fracamente compactos em C[0, 1] satisfazem a equação de Daugavet. Finalmente apresentamos a demonstração dada por D. Werner em [15], onde prova-se que um operador linear fracamente compacto no espaço C(S) satisfaz a equação de Daugavet se e somente se S não possui pontos isolados. / Abstract: A bounded linear operator T between normed spaces satisfies the Daugavet equation if II I + T II = 1+ T .The main purpose of this work is to study the Daugavet equation for bounded linear operators on the space C(S), where S is a compact Hausdorff space. For this, we study some representations of C_(S), the conjugate space of C(S), according the topological properties of S, and also representations of operators defined on C(S) or with range in C(S). Using this theory of representations on C(S) we present some classes of operators that satisfy the Daugavet equation. Firstly we present the proof given by H. Kamowitz in [11] that if T is a compact linear operator on C(S) then II I + T II = 1+ T if and only if S is has no isolated points. Next we present the proof given by J. R. Holub in [8], showing that weakly compact operators on C[0, 1] satisfy the Daugavet equation. Finally we present the proof given by D.Werner in [15], where it is shown that a weakly compact operator on the space C(S) satis_es the Daugavet equation if and only if S has no isolated points. / Mestrado / Analise Funcional / Mestre em Matemática

This paper investigates the compact-open topology on the set of C_k(X) of continuous real-valued functions defined on a Tychonoff space X. More precisely, we study the following problem: If P is a topological property, does there exist a topological property Q so that C_k(X) has P if and only if X has Q? Characterizations of many properties are obtained throughout the thesis, sometimes modulo some “mild” restrictions on the space X. The main properties involved are summarized in a diagram in the introduction. / Ph. D.

Neste trabalho, alguns espaços de funções que surgem naturalmente no contexto da topologia geral são estudados. Por meio da noção de bornologias, problemas de Cp-teoria e Ck-teoria são analisados simultaneamente, como a caracterização de certas funções cardinais no espaço das funções contínuas e propriedades relativas a jogos seletivos. A compactificação de Stone- Cech também é estudada, onde a existência de P-pontos no resíduo dos naturais é considerada sob a Hipótese do Contínuo. Com a adição de certas hipóteses sobre pequenos cardinais, alguns resultados obtidos ao longo do texto são utilizados em problemas relacionados com espaços de Michael e espaços de Alster / In this work, some function spaces that naturally arise in the general topology context are studied. By the notion of bornologies, Cp-theory and Ck-theory problems are simultaneously analysed, as the characterization of some cardinal functions in the space of continuous real functions and properties related to selective games. The Stone-Cech compactification is also studied, and the existence of P-points in the remainder of the set of the natural numbers is considered under the Continuum Hypothesis. With the addition of some hypotheses about small cardinals, some results obtained through the text are used in problems related to Michael spaces and Alster spaces

Neste trabalho, alguns espaços de funções que surgem naturalmente no contexto da topologia geral são estudados. Por meio da noção de bornologias, problemas de Cp-teoria e Ck-teoria são analisados simultaneamente, como a caracterização de certas funções cardinais no espaço das funções contínuas e propriedades relativas a jogos seletivos. A compactificação de Stone- Cech também é estudada, onde a existência de P-pontos no resíduo dos naturais é considerada sob a Hipótese do Contínuo. Com a adição de certas hipóteses sobre pequenos cardinais, alguns resultados obtidos ao longo do texto são utilizados em problemas relacionados com espaços de Michael e espaços de Alster / In this work, some function spaces that naturally arise in the general topology context are studied. By the notion of bornologies, Cp-theory and Ck-theory problems are simultaneously analysed, as the characterization of some cardinal functions in the space of continuous real functions and properties related to selective games. The Stone-Cech compactification is also studied, and the existence of P-points in the remainder of the set of the natural numbers is considered under the Continuum Hypothesis. With the addition of some hypotheses about small cardinals, some results obtained through the text are used in problems related to Michael spaces and Alster spaces

In this work we analyze some selection principles over some classes of hyperspaces. In the first part we consider selective variations of tightness over a class of function spaces whose topologies are determined by bornologies on the space. As results, we extend several well known translations between covering properties and closure properties of the topology of pointwise convergence. In the second part we consider artificial hyperspaces that assist the analysis of productive topological properties. We emphasize the results characterizing productively ccc preorders and the characterization of the Lindelöf property via closed projections. / Neste trabalho analisamos alguns princípios seletivos quando considerados sobre alguns tipos de hiperespaços. Na primeira parte consideramos variações seletivas do tightness sobre diversos tipos de espaços de funções, cujas topologias são determinadas por bornologias no espaço. Como resultados, estendemos diversas traduções conhecidas entre propriedades de recobrimento e propriedades de convergência na topologia da convergência pontual. Na segunda parte consideramos hiperespaços artificiais que auxiliam na análise de propriedades topológicas produtivas. Destacamos os resultados que caracterizam as pré-ordens produtivamente ccc e a caracterização da propriedade de Lindelöf em termos de projeções fechadas.

<p>This thesis consists of three papers, all of them on the topic of function spaces on fractals.</p><p>The papers summarised in this thesis are:</p><p>Paper I Mats Bodin, Wavelets and function spaces on Mauldin-Williams fractals, Research Report in Mathematics No. 7, Umeå University, 2005.</p><p>Paper II Mats Bodin, Harmonic functions and Lipschitz spaces on the Sierpinski gasket, Research Report in Mathematics No. 8, Umeå University, 2005.</p><p>Paper III Mats Bodin, A discrete characterisation of Lipschitz spaces on fractals, Manuscript.</p><p>The first paper deals with piecewise continuous wavelets of higher order in Besov spaces defined on fractals. A. Jonsson has constructed wavelets of higher order on fractals, and characterises Besov spaces on totally disconnected self-similar sets, by means of the magnitude of the coefficients in the wavelet expansion of the function. For a class of fractals, W. Jin shows that such wavelets can be constructed by recursively calculating moments. We extend their results to a class of graph directed self-similar fractals, introduced by R. D. Mauldin and S. C. Williams.</p><p>In the second paper we compare differently defined function spaces on the Sierpinski gasket. R. S. Strichartz proposes a discrete definition of Besov spaces of continuous functions on self-similar fractals having a regular harmonic structure. We identify some of them with Lipschitz spaces introduced by A. Jonsson, when the underlying domain is the Sierpinski gasket. We also characterise some of these spaces by means of the magnitude of the coefficients of the expansion of a function in a continuous piecewise harmonic base.</p><p>The last paper gives a discrete characterisation of certain Lipschitz spaces on a class of fractal sets. A. Kamont has discretely characterised Besov spaces on intervals. We give a discrete characterisation of Lipschitz spaces on fractals admitting a type of regular sequence of triangulations, and for a class of post critically finite self-similar sets. This shows that, on some fractals, certain discretely defined Besov spaces, introduced by R. Strichartz, coincide with Lipschitz spaces introduced by A. Jonsson and H. Wallin for low order of smoothness.</p>