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The purpose of this paper is to explore some of the properties of the Cantor set and to extend the idea of this set to metric spaces, in general, and to other sets of real numbers and sets in N-dimensional Euclidean space, in particular.

Thesis (doctoral)--University of Göteborg, 1992. / Extra t.p. with thesis statement inserted. Includes bibliographical references (p. [83]-88) and indexes.

The objective of my thesis is to find optimal points and the quantization error for a probability measure defined on a Cantor set. The Cantor set, we have considered in this work, is generated by two self-similar contraction mappings on the real line with distinct similarity ratios. Then we have defined a nonhomogeneous probability measure, the support of which lies on the Cantor set. For such a probability measure first we have determined the n-optimal points and the nth quantization error for n = 2 and n = 3. Then by some other lemmas and propositions we have proved a theorem which gives all the n-optimal points and the nth quantization error for all positive integers n. In addition, we have given some properties of the optimal points and the quantization error for the probability measure. In the end, we have also given a list of n-optimal points and error for some positive integers n. The result in this thesis is a nonhomogeneous extension of a similar result of Graf and Luschgy in 1997. The techniques in my thesis could be extended to discretise any continuous random variable with another random variable with finite range.

Thesis (doctoral)--University of Göteborg, 1992. / Extra t.p. with thesis statement inserted. Includes bibliographical references (p. [83]-88) and indexes.

AGUIAR, Francisco Fagner Portela. Um background na teoria dos conjuntos. 2015. 50 f. Dissertação (Mestrado em Matemática em Rede Nacional) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2015. / Submitted by Erivan Almeida (eneiro@bol.com.br) on 2015-11-13T15:41:09Z No. of bitstreams: 1 2015_dis_ffpaguiar.pdf: 1566390 bytes, checksum: 114ad96172cfa622234e88e05d73ffff (MD5) / Approved for entry into archive by Rocilda Sales(rocilda@ufc.br) on 2015-11-18T13:38:59Z (GMT) No. of bitstreams: 1 2015_dis_ffpaguiar.pdf: 1566390 bytes, checksum: 114ad96172cfa622234e88e05d73ffff (MD5) / Made available in DSpace on 2015-11-18T13:38:59Z (GMT). No. of bitstreams: 1 2015_dis_ffpaguiar.pdf: 1566390 bytes, checksum: 114ad96172cfa622234e88e05d73ffff (MD5) Previous issue date: 2015 / The set theory sometimes left out in some high schools, is in a key element for understanding the functions in particular. Failure to address this issue or its superficial approach leaves the student a difficult gap to be filled in later studies. Incidentally, the left gap may hinder student performance in higher education. If this is so, is the main objective of this work to a reinterpretation of the main topics linked to the high school set theory, while making a bridge between these and other equally important points dealing with sets in a more academic language. Will be covered from the properties and theorems related to finite sets up its generalization to infinite sets, culminating in the Cantor-Schroeder-Bernstein theorem, the Axiom of Choice and Zorn’s Lemma. To this end, there were literature searches in various sources. / A teoria de conjuntos por vezes deixada de lado em algumas escolas de ensino médio, constitui-se em um elemento primordial para o entendimento das funções, em especial. A não abordagem, ou a sua abordagem superficial, deixa no estudante uma lacuna difícil de ser suprida em estudos posteriores. Aliás, a lacuna deixada pode dificultar o desempenho do estudante no ensino superior. Diante desta constatação, é objetivo principal desta dissertação fazer uma leitura dos principais tópicos ligados à Teoria de Conjuntos do ensino médio, ao mesmo tempo em que faz uma ponte entre estes e outros pontos não menos importantes, tratando conjuntos em uma linguagem mais acadêmica. Serão abordados desde as propriedades e teoremas relacionados a conjuntos finitos, até a sua generalização para conjuntos infinitos, culminando com o teorema de Cantor-Schroeder-Bernstein, o Axioma da Escolha, e o Lema de Zorn. Para tantos, realizaram-se pesquisas bibliográficas em fontes variadas.

TEODOSIO, Elaine de Sousa. Números complexos e suas aplicações à geometria no ensino médio. 2016. 102 f. Dissertação (Mestrado em Matemática em Rede Nacional) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2016. / Submitted by Rocilda Sales (rocilda@ufc.br) on 2016-08-16T16:27:26Z No. of bitstreams: 1 2016_dis_esteodosio.pdf: 1393071 bytes, checksum: d25cd4da29b9031386959bad85406cf4 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2016-08-16T16:55:56Z (GMT) No. of bitstreams: 1 2016_dis_esteodosio.pdf: 1393071 bytes, checksum: d25cd4da29b9031386959bad85406cf4 (MD5) / Made available in DSpace on 2016-08-16T16:55:56Z (GMT). No. of bitstreams: 1 2016_dis_esteodosio.pdf: 1393071 bytes, checksum: d25cd4da29b9031386959bad85406cf4 (MD5) Previous issue date: 2016 / The set of complex numbers is often subject forgotten high school. When approached, there is a restriction on the content in their own curriculum. This is perhaps the ignorance of their applicability, making their study is specifically addressed in the graduation of some university courses. This study aims to use these numbers in the statement of some geometry theorems, showing its application in the area of mathematics. For this, initially we discuss the algebraic concepts: operations of addition, multiplication, powers of i, conjugate and modulus of a complex number. Then simple geometric concepts such as distance between two points, angle measurements, linearity condition, orthogonality and cocircular, similar triangles and analytic geometry, which until then, it uses only real numbers, now extends to the complex numbers . With the inclusion of complex numbers iin geometry it is possible to prove theorems as: The circle of nine points, the straight of Simson, Cantor's theorem, Feuerbach's theorem and the theorem Morley. Finally, we will develop the project: short course complex numbers and geometry with a group of students of the 3rd year of high school at the State School Professor Eudes Veras, located in Maracanau, Ceará. To collect the data were utizados, a socioeconomic questionnaire and a test on powers of i, conjugate module, equation of the line, Napoleon's theorem and circle of nine points. These data were organized and analyzed. / O conjunto dos números complexos muitas vezes é assunto esquecido no ensino médio. Quando abordado, há uma restrição em relação aos conteúdos na própria matriz curricular. Isso ocorre, talvez, pela ignorância quanto a sua aplicabilidade, fazendo com que seu estudo seja tratado especificamente na graduação de alguns cursos superiores. O presente trabalho tem como objetivo utilizar esses números na demonstração de alguns teoremas de geometria, mostrando assim sua aplicação nessa área da matemática. Para isso, inicialmente abordaremos os conceitos algébricos: operações de adição, multiplicação, potências de i, conjugado e módulo de um número complexo. Em seguida, conceitos geométricos simples como distância entre dois pontos, medidas de ângulo, condição de linearidade, ortogonalidade e cocircular, semelhança de triângulos e geometria analítica, que até então, utiliza-se apenas números reais, agora amplia-se para os números complexos. Com a iinserção de números complexos na geometria é possível demonstrar teoremas como: O círculo de nove pontos, a reta de Simson, o teorema de Cantor, o teorema de Feuerbach e o teorema de Morley. Por fim, desenvolveremos o projeto: minicurso números complexos e geometria com um grupo de alunos do 3o ano do Ensino Médio da Escola Estadual Professora Eudes Veras, localizada em Maracanau, Ceará. Para a coleta dos dados foram utilizados, um questionário socioeconômico e um teste sobre potências de i, conjugado, módulo, equação da reta, teorema de Napoleão e círculo de nove pontos. Esses dados foram organizados e analisados.

We study finite generation, 2-generation and simplicity of subgroups of H[sub]c, the group of homeomorphisms of Cantor space. In Chapter 1 and Chapter 2 we run through foundational concepts and notation. In Chapter 3 we study vigorous subgroups of H[sub]c. A subgroup G of H[sub]c is vigorous if for any non-empty clopen set A with proper non-empty clopen subsets B and C there exists g ∈ G with supp(g) ⊑ A and Bg ⊆ C. It is a corollary of the main theorem of Chapter 3 that all finitely generated simple vigorous subgroups of H[sub]c are in fact 2-generated. We show the family of finitely generated, simple, vigorous subgroups of H[sub]c is closed under several natural constructions. In Chapter 4 we use a generalised notion of word and the tight completion construction from [13] to construct a family of subgroups of H[sub]c which generalise Thompson's group V . We give necessary conditions for these groups to be finitely generated and simple. Of these we show which are vigorous. Finally we give some examples.

A Cantor set is a topological space which admits a hierarchy of clopen covers. A minimal Cantor set is a Cantor set together with a map such that every orbit is dense in the Cantor set. In this thesis we us inverse limits to study minimal Cantor sets and their properties. In particular, under certain hypothesis we find an upper bound for the number of ergodic measures for minimal Cantor set. / En Cantor mängd är ett topologiskt rum med en hierarki av slöppna täcken. En minimal Cantor mängd är en Cantor mängd tillsammans med en avblidning så att varje omlppsbana är tät. I den här uppsatsen använder vi omvända gränser för att studera minimala Cantor mängder och deras egenskaper. Särskilt under vissa omständigheter hittar vi en övre begränsning på antalet ergodiska mått på minimala Cantor mängder.

This paper examines self-similar sets and some of their properties, including the natural equivalence relation found in bilipschitz equivalence. Both dimension and preservation of paths are determined to be invariant under this equivalence. Also, sophisticated techniques, one involving the use of directed graphs, show the equivalence of two spaces. / Master of Science