Considerable study has been made concerning the properties of the iterations of a homeomorphism on a metric space. Much of this material is scattered throughout the literature and understood solely by a specialist. The main object of this paper is to put into readable form proofs of theorems found in G.T. Whyburn's "Analytic Topology" pertaining to this topic in topology. Properties of the decomposition space of point-orbits induced by the iterations of a homeomorphism will compose a major part of the study. Some theorems will be established through series of lemmas required to fill in much of the detail lacking in proofs found the book. Although an elementary knowledge of topology is assumed throughout the paper, references are given for basic definitions and theorems used in developing some of the proofs. The following symbols and notation will be used throughout the paper. X will denote a metric space with metric p, S a topological space, I the set of positive integers, A, B, C... sets of points or elements. Small letters, such as a, b, c, x, y, z... will designate elements or points of sets. U and V will denote open sets Sr(x) a spherical neighborhood of x with radius r. A' denotes the set of limit points of A. A- the set of closure points of A/ U, N, C will denote union, intersection, and set inclusion respectively. The symbol E will mean "is an element of". 0 denotes the empty set. S - A is the set of points in S which are not in A.

This thesis deals mainly with introduction to iterations and with implementation possibilities to high-school education. Extends the overview of elementary functions and depth of understanding by students. There are prepared working sheets and propositions of motivation to ease making sense of this area of mathematics.

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Submitted by Erika Demachki (erikademachki@gmail.com) on 2014-08-28T19:49:36Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Revisão Dilermano V Regis.PDF: 739794 bytes, checksum: d08c982161f7e5a51c6e06ede504f527 (MD5) / Made available in DSpace on 2014-08-28T19:49:36Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Revisão Dilermano V Regis.PDF: 739794 bytes, checksum: d08c982161f7e5a51c6e06ede504f527 (MD5) Previous issue date: 2013-04-11 / O objetivo deste estudo é apresentar uma ferramenta que todos possuem, que é fácil de carregar e que portanto tem alcance popular. Nas primeiras seções apresenta-se algumas operações matemáticas elementares executadas com uma calculadora simples. Começa-se por estudar a operação de divisão de inteiros e como encontrar o resto na calculadora, como operar com frações em uma calculadora e responder na forma de fração, como encontrar a fração geratriz de uma dízima periódica ou ainda como extrair a raíz "n-ésima"de um número real. Para cada operação destas há a preocupação de se expôr também suas devidas aplicações e, quando necessário, de se realizar estudos matemáticos mais profundos como, por exemplo, no caso das congruências módulo m nas dízimas ou derivadas para raízes n-ésima. Em iterações com a adição, por exemplo, trabalha-se o uso adequado de algumas teclas para um procedimento rápido no estudo das progressões aritméticas, juro simples e funções polinomiais do 1o grau. Já na seção de iterações com o produto, propõe-se um estudo das progressões geométricas, funções exponenciais e juro composto. Iterando com as fórmulas de Maclaurin, veremos como é possível utilizar a calculadora simples para encontrar valores das funções trigonométricas circulares e hiperbólicas, bem como os valores dos logaritmos decimais e neperianos. Esse trabalho mostra ao leitor que pelo conhecimento matemático pode se otimizar a calculadora simples, responder questões que antes os cálculos ficariam apenas indicados, ganhar tempo nos cálculos e utilizar o tempo restante para melhor elaborar a estratégia de resolução, e demais vantagens que o leitor descobrirá. / Quero agradecer aos amigos, àqueles que o tempo separou mas as lembranças não, e de forma especial aos 29 novos amigos que o mestrado me proporcionou: Amanda, Viviane, Leozão, Hélio, Flávio, Carlos André, Fabão, Rogério, Marquinhos, Junior César, Paulo César, Edmaldo, Haniel, Simão, Leandro, Kadu, Frederico, Kariton, Marcão, Welington. Àqueles que foram além da amizade e se fizeram irmãos: Túlio, Robison, Alan, Hugo César, César Pereira, Leo Alcântara, Maradona, Eduardo Vasconcelos e Mateus. A esses devo meus reconhecimentos. A esses que leram ou ouviram com cuidado as primeiras versões, que opniram, corrigiram e criticaram, meus sinceros agradecimentos. Desejo agradecer de forma mais pessoal à minha família, pelo apoio incondicional, por me ouvirem mesmo quando nada entendiam. À Dagma e à Dalma pelas lições diárias de força e coragem; a minha esposa Juliana que conseguiu suportar a construção de um livro dentro de casa e por acreditar que eu conseguiria escrevê-lo. Meu maior agradecimento vai aos meus filhos: Priscila pelas horas ao telefone me ouvindo; Pedro, Mariana e Gabriel, pela correção ou reescrita de meu português, pelas brincadeiras, pelas partidas de futebol no vídeo game buscando aliviar meu cansaço, por me obrigarem a ir ao cinema ou por fazermos nada ao final dos meus dias de trabalho. Sem isto, e sem eles, eu não sobreviveria à escrita deste.

Objects such as documents, people, words or utilities, that are related in some way, for instance by citations, friendship, appearance in definitions or physical connections, may be conveniently represented using graphs or networks. An increasing number of such relational databases, as for instance the World Wide Web, digital libraries, social networking web sites or phone calls logs, are available. Relevant information may be hidden in these networks. A user may for instance need to get authority web pages on a particular topic or a list of similar documents from a digital library, or to determine communities of friends from a social networking site or a phone calls log. Unfortunately, extracting this information may not be easy. This thesis is devoted to the study of problems related to information extraction in large graphs with the help of dominant vectors of nonnegative matrices. The graph structure is indeed very useful to retrieve information from a relational database. The correspondence between nonnegative matrices and graphs makes Perron--Frobenius methods a powerful tool for the analysis of networks. In a first part, we analyze the fixed points of a normalized affine iteration used by a database matching algorithm. Then, we consider questions related to PageRank, a ranking method of the web pages based on a random surfer model and used by the well known web search engine Google. In a second part, we study optimal linkage strategies for a web master who wants to maximize the average PageRank score of a web site. Finally, the third part is devoted to the study of a nonlinear variant of PageRank. The simple model that we propose takes into account the mutual influence between web ranking and web surfing.

The solution of initial value problems (IVPs) provides the evolution of dynamic system state history for given initial conditions. Solving boundary value problems (BVPs) requires finding the system behavior where elements of the states are defined at different times. This dissertation presents a unified framework that applies modified Chebyshev-Picard iteration (MCPI) methods for solving both IVPs and BVPs. Existing methods for solving IVPs and BVPs have not been very successful in exploiting parallel computation architectures. One important reason is that most of the integration methods implemented on parallel machines are only modified versions of forward integration approaches, which are typically poorly suited for parallel computation. The proposed MCPI methods are inherently parallel algorithms. Using Chebyshev polynomials, it is straightforward to distribute the computation of force functions and polynomial coefficients to different processors. Combining Chebyshev polynomials with Picard iteration, MCPI methods iteratively refine estimates of the solutions until the iteration converges. The developed vector-matrix form makes MCPI methods computationally efficient. The power of MCPI methods for solving IVPs is illustrated through a small perturbation from the sinusoid motion problem and satellite motion propagation problems. Compared with a Runge-Kutta 4-5 forward integration method implemented in MATLAB, MCPI methods generate solutions with better accuracy as well as orders of magnitude speedups, prior to parallel implementation. Modifying the algorithm to do double integration for second order systems, and using orthogonal polynomials to approximate position states lead to additional speedups. Finally, introducing perturbation motions relative to a reference motion results in further speedups. The advantages of using MCPI methods to solve BVPs are demonstrated by addressing the classical Lambert’s problem and an optimal trajectory design problem. MCPI methods generate solutions that satisfy both dynamic equation constraints and boundary conditions with high accuracy. Although the convergence of MCPI methods in solving BVPs is not guaranteed, using the proposed nonlinear transformations, linearization approach, or correction control methods enlarge the convergence domain. Parallel realization of MCPI methods is implemented using a graphics card that provides a parallel computation architecture. The benefit from the parallel implementation is demonstrated using several example problems. Larger speedups are achieved when either force functions become more complicated or higher order polynomials are used to approximate the solutions.

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2016-05-05T20:25:53Z No. of bitstreams: 2 Dissertação - Carlos Eduardo Rosado de Barros - 2015.pdf: 1334566 bytes, checksum: 373183ed73dd83bfac5d91d2670c2e36 (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-05-06T11:43:27Z (GMT) No. of bitstreams: 2 Dissertação - Carlos Eduardo Rosado de Barros - 2015.pdf: 1334566 bytes, checksum: 373183ed73dd83bfac5d91d2670c2e36 (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Made available in DSpace on 2016-05-06T11:43:27Z (GMT). No. of bitstreams: 2 Dissertação - Carlos Eduardo Rosado de Barros - 2015.pdf: 1334566 bytes, checksum: 373183ed73dd83bfac5d91d2670c2e36 (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) Previous issue date: 2015-08-07 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this work, mainly based on the articles, [1], [2] and [7], one studies a reactiondiffusion- convection system, related the propagation of a combustion front through a porous medium, giving origin a Cauchy problem. Such a problem has been approached by the methode of the monotone iterations, which leads to an unique time-global solution. / Nesse trabalho, baseado principalmente nos artigos [1], [2] e [7], estuda-se um sistema reação-difusão-convecção, relacionado à propagação de uma frente de combustão em um meio poroso, recaindo sobre um problema de Cauchy. Tal problema é abordado através do método de iterações monótonas, o qual conduz a uma única solução global no tempo.

Asynchronous iterations often converge under different conditions than their syn- chronous counterparts. In this paper we will study the global convergence of Jacobi- Newton-like methods for nonlinear equationsF x = 0. It is a known fact, that the synchronous algorithm converges monotonically, ifF is a convex M-function and the starting valuesx0 andy0 meet the conditionF x04 04F y0 . In the paper it will be shown, which modifications are necessary to guarantee a similar convergence behavior for an asynchronous computation.

Asynchronous iterations often converge under different conditions than their syn- chronous counterparts. In this paper we will study the global convergence of Jacobi- Newton-like methods for nonlinear equationsF x = 0. It is a known fact, that the synchronous algorithm converges monotonically, ifF is a convex M-function and the starting valuesx0 andy0 meet the conditionF x04 04F y0 . In the paper it will be shown, which modifications are necessary to guarantee a similar convergence behavior for an asynchronous computation.

No description available.