Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references (leaves 31). Also available in electronic version. Access restricted to campus users.

Thesis (doctoral)--Rheinischen Friedrich-Wilhelms-Universität, Bonn, 1980. / Includes bibliographical references (p. 252-258).

The aim of this thesis is derive a set of polynomials defined on simply connected domains, the Faberpolynomials, in which all analytic function on the domain can be uniformly approximated. Importantconcepts and theorems such as isomorphisms, automorphisms and the Riemann mapping theorem areintroduced. Examples and applications are also included. Furthermore, the thesis will aim to introducean important consequence of the Faber polynomials, the method of the Grunsky inequalities. The first section introduces important properties of analytic functions and the concept of isomor-phisms, in particular the form of all automorphisms of the unit disc will be derived. The second sectionconsiders the Riemann mapping theorem, a theorem that relates any simply connected region that is notall of ℂ to the unit disc. A proof of the theorem beginning with the Arzelá-Ascoli theorem is provided.An application in constructing harmonic functions on arbitrary simply connected regions will be pre-sented. In the third section, definitions and properties of the Faber polynomials are developed; followedby simple examples. The section concludes with a proof and example of the statement that analyticfunctions can be approximated by Faber polynomials. In the fourth and last section of the thesis, themethod of Grunsky inequalities is presented. Starting off, the Grunsky coefficients are defined using theFaber polynomials. Properties of Grunsky coefficients such as the symmetry property and the Grunskyinequalities are then derived. To conclude it will be shown that the Grunsky inequalities provide aunivalence criterion for analytic functions defined on the unit disc.

The object of my thesis is the realization of grammatical evolution in the Java programming language for solving problems of approximation of functions and synthesis of logical circuits. The application is practical used for testing and gathering data in context of using different purpose function and parallel grammatical evolution. The data are analyzed and evaluated.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2015-03-24T21:17:09Z No. of bitstreams: 3 Dissertação - Luiz Mario Lustosa Pascoal - 2014.pdf: 7280181 bytes, checksum: 68a6ac0602e3e51f6e6952bbd6916150 (MD5) FunctionApproximator.zip: 2288624 bytes, checksum: 178c2e6a0b080b3d0548836974016236 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Erika Demachki (erikademachki@gmail.com) on 2015-03-24T21:19:16Z (GMT) No. of bitstreams: 3 Dissertação - Luiz Mario Lustosa Pascoal - 2014.pdf: 7280181 bytes, checksum: 68a6ac0602e3e51f6e6952bbd6916150 (MD5) FunctionApproximator.zip: 2288624 bytes, checksum: 178c2e6a0b080b3d0548836974016236 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-03-24T21:19:16Z (GMT). No. of bitstreams: 3 Dissertação - Luiz Mario Lustosa Pascoal - 2014.pdf: 7280181 bytes, checksum: 68a6ac0602e3e51f6e6952bbd6916150 (MD5) FunctionApproximator.zip: 2288624 bytes, checksum: 178c2e6a0b080b3d0548836974016236 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2014-08-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / With the development of web 2.0, social networks have achieved great space on the internet, with that many users provide information and interests about themselves. There are expert systems that make use of the user’s interests to recommend different products, these systems are known as Recommender Systems. One of the main techniques of a Recommender Systems is the Collaborative Filtering (User-based) which recommends products to users based on what other similar people liked in the past. Therefore, this work presents model approximation of functions that generates rankings, that through a Genetic Algorithm, is able to learn an approximation function composed by different social variables, customized for each Facebook user. The learned function must be able to reproduce a ranking of people (friends) originally created with user’s information, that apply some influence in the user’s decision. As a case study, this work discusses the context of events through information regarding the frequency of participation of some users at several distinct events. Two different approaches on learning and applying the approximation function have been developed. The first approach provides a general model that learns a function in advance and then applies it in a set of test data and the second approach presents an specialist model that learns a specific function for each test scenario. Two proposals for evaluating the ordering created by the learned function, called objective functions A and B, where the results for both objective functions show that it is possible to obtain good solutions with the generalist and the specialist approaches of the proposed method. / Com o desenvolvimento da Web 2.0, as redes sociais têm conquistado grande espaço na internet, com isso muitos usuários acabam fornecendo diversas informações e interesses sobre si mesmos. Existem sistemas especialistas que fazem uso dos interesses do usuário para recomendar diferentes produtos, esses sistemas são conhecidos como Sistemas de Recomendação. Uma das principais técnicas de um Sistema de Recomendação é a Filtragem Colaborativa (User-based) que recomenda produtos para seus usuários baseados no que outras pessoas similares à ele tenham gostado no passado. Portanto, este trabalho apresenta um modelo de aproximação de funções geradora de rankings que, através de um Algoritmo Genético, é capaz de aprender uma função de aproximação composta por diferentes atributos sociais, personalizada para cada usuário do Facebook. A função aprendida deve ser capaz de reproduzir um ranking de pessoas (amigos) criado originalmente com informações do usuário, que exercem certa influência na decisão do usuário. Como estudo de caso, esse trabalho aborda o contexto de eventos através de informações com relação a frequência de participação de alguns usuários em vários eventos distintos. Foram desenvolvidas duas abordagens distintas para aprendizagem e aplicação da função de aproximação. A primeira abordagem apresenta um modelo generalista, que previamente aprende uma função e em seguida a aplica em um conjunto de dados de testes e a segunda abordagem apresenta um modelo especialista, que aprende uma função específica para cada cenário de teste. Também foram apresentadas duas propostas para avaliação da ordenação criada pela função aprendida, denominadas funções objetivo A e B, onde os resultados para ambas as funções objetivo A e B mostram que é possível obter boas soluções com as abordagens generalista e especialista do método proposto.

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A..=2b, where A 2 Rnn is a large, sparse symmetric positive definite matrix and b 2 Rn is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LLT is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L..T z, with x = A..1=2z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form n = A..=2b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t..=2 and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A..=2b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.