<p>This paper is about basic set theory and cardinalities for infinite sets. One of the results are that the line R and the plane R2 contains exactly the same number of points. Because of that the set theory is described with a formal language this the paper has an appendix about formal languages.</p> / <p>Denna uppsats behandlar grundläggande mängdlära och inriktar sig sedan på kardinaliteter för oändliga mängder. Bland de resultat som redovisas finns bland annat resultatet som säger att linjen R och planet R2 innehåller precis lika många punkter. Då mängdläran beskrivs av ett formellt språk så innehåller uppsatsen en bilaga om formella språk.</p>

Nous nous intéressons à l'étude de la cyclicité et la bicyclicité dans les espaces $ell^p(Z)$ à poids et à l'étude de la cyclicité dans les espaces de Dirichlet. Alors que Wiener a caractérisé la bicyclicité des vecteurs de $ell^1(Z)$ et $ell^2(Z)$ grâce à l'ensemble des zéros de la transformée de Fourier, Lev et Olevski ont démontré que cet ensemble ne peut caractériser la bicyclicité dans $ell^p(Z)$ lorsque $1<p<2$ pour des suites $u in ell^1(Z)$. Beurling, Salem et Newman se sont aussi intéressés à la bicyclicité de vecteurs de $ell^p(Z)$ pour $1<p<2$. Dans ce travail, nous étendons tout d'abord les résultats de Beurling, Salem et Newman aux espaces $ell^p(Z)$ à poids, en étudiant la dimension de Hausdorff et la capacité de l'ensemble des zéros de la transformée de Fourier. Ensuite nous démontrons que le résultat de Lev-Olevskii reste valide pour la cyclicité dans $ell^p(Z)$, $1<p<2$. De plus, nous donnons des conditions suffisantes à la cyclicité dans les espaces $ell^p(Z)$ à poids. Enfin nous démontrons que, pour une fonction $f$ appartenant à l'algèbre du disque et à un espace de type Dirichlet, si $f$ est extérieure et si l'ensemble des zéros de $f$ est réduit à un point alors $f$ est cyclique. Ceci généralise le résultat de Hedenmalm et Shields qui ont traité le cas du Dirichlet classique. / We are interested in the study of cyclicity and bicyclicity in weighted $ell^p(Z)$ spaces and the study of cyclicity in Dirichlet spaces. While Wiener characterized the bicyclicity in $ell^1(Z)$ and $ell^2(Z)$, thanks to the zero set of the Fourier transform, Lev and Olevski have shown that this set cannot characterize bicyclicity in $ell^p(Z)$ when $1 < p < 2$ for sequences in $ell^1(Z)$. Also Beurling, Salem and Newman were interested in the bicyclicity in $ell^p(Z)$ when $1 < p < 2$. In this work, we first extend the results of Beurling, Salem and Newman to the weighted $ell^p(Z)$ spaces, by studying the Hausdorff dimension and the capacity of the zero set of the Fourier transform. Then we prove that the Lev-Olevskii result remains valid for cyclicity in $ell^p(Z)$, $1 < p < 2$. In addition, we give sufficient conditions for the cyclicity in the weighted $ell^p(Z)$ spaces. Finally, we prove that, for a function $f$ in the disk algebra and in a generalized Dirichlet space, if $f$ is outer and the zero set of $f$ is reduced to a point then $f$ is cyclic. This generalizes the result of Hedenmalm and Shields who have treated the case of the classical Dirichlet space.

Submitted by Lara Oliveira (lara@ufersa.edu.br) on 2017-07-18T21:43:56Z No. of bitstreams: 1 ThiagoELM_DISSERT.pdf: 3255456 bytes, checksum: 6ce3964b5242ccf305c4e0f3d94ee65e (MD5) / Approved for entry into archive by Vanessa Christiane (referencia@ufersa.edu.br) on 2017-07-25T14:46:31Z (GMT) No. of bitstreams: 1 ThiagoELM_DISSERT.pdf: 3255456 bytes, checksum: 6ce3964b5242ccf305c4e0f3d94ee65e (MD5) / Approved for entry into archive by Vanessa Christiane (referencia@ufersa.edu.br) on 2017-07-25T14:46:56Z (GMT) No. of bitstreams: 1 ThiagoELM_DISSERT.pdf: 3255456 bytes, checksum: 6ce3964b5242ccf305c4e0f3d94ee65e (MD5) / Made available in DSpace on 2017-07-25T14:47:21Z (GMT). No. of bitstreams: 1 ThiagoELM_DISSERT.pdf: 3255456 bytes, checksum: 6ce3964b5242ccf305c4e0f3d94ee65e (MD5) Previous issue date: 2013-11-22 / The microstrip antennas are in constant evidence in current research due to its numerous advantages. Fractal geometry proposed by Mandelbrot(1975 ) combined with the performance and convenience of planar structures are an excellent combination used in the design of antennas in order to reduce the dimensions and enhance its bandwidth, and allows the emergence of best bands frequency by virtue of ownership of high -similarity. Compared with the conventional microstrip antennas, patch antennas with fractal type substrates have lower resonance frequency, enabling the manufacture of even more compact antennas. The aim of this work consists of the design of patch antennas with dielectric substrates organized almost periodic basis through the use of fractal geometry sequence Cantor applied to a circular patch antenna fed by microstrip line, designed for a resonant frequency of 10 GHz. Analysis of this microstrip antenna is made in various types of dielectric substrates by simulation through software commercial Ansoft HFSS - Designer, used for accurate analysis of the electromagnetic behavior of the antennas by the finite element method by presenting results from resonant frequency and radiation pattern, making comparisons with other results in the literature. This dissertation also presents a bibliographic study on theories of antennas while also addressing about fractal geometry, emphasizing its characteristics and properties as well as its applicability. This paper also presents a study of almost periodic structures and their mathematical formalism considered throughout this work / As antenas de microfita estão em constante evidência nas pesquisas atuais, devido às suas inúmeras vantagens. A geometria fractal proposta por Mandelbrot (1975) aliada ao bom desempenho e comodidade das estruturas planares são uma excelente combinação utilizada no projeto de antenas com o intuito de reduzir suas dimensões e realçar sua largura de banda, além de permitir o surgimento de melhores bandas de frequência em consequência da propriedade da alto-similaridade. Em comparação com as antenas em microfita convencionais, as antenas tipo patch com substratos fractais apresentam frequência de ressonância inferiores, possibilitando a fabricação de antenas ainda mais compactas. O objetivo desse trabalho consiste no projeto de antenas patch com substrato dielétrico organizado de forma quase periódica por meio da utilização da geometria fractal da sequência de Cantor aplicada a uma antena de patch circular alimentada por linha de microfita, projetada para uma frequência ressonância de 10 GHz. É feita análise dessa antena de microfita em vários tipos de substratos dielétricos por simulação através do software comercial Ansoft Designer-HFSS, usado para análise precisa do comportamento eletromagnético das antenas através do método dos elementos finitos apresentando resultados de frequência de ressonância, diagrama de radiação, carta de Smith e de campos elétricos e magnéticos fazendo-se comparações com outros resultados obtidos na literatura. Esta dissertação ainda apresenta um estudo bibliográfico em teorias de antenas, abordando também a respeito da geometria fractal, dando ênfase a suas características e propriedades como também a sua aplicabilidade. Este trabalho ainda apresenta um estudo sobre as estruturas quase periódicas e seu formalismo matemático / 2017-07-18

Cette thèse est consacrée à l’étude du phénomène de Newhouse et des bifurcations en dynamique holomorphe à plusieurs variables. Elle comporte trois Théorèmes principaux. Le premier de ces trois résultats est un Gap Lemma complexe. En dynamique réelle, le Gap Lemma de Newhouse donne un critère sur le produit des épaisseurs de deux ensembles de Cantor dynamiques pour prouver que leur intersection est non vide. On en donne une généralisation partielle au cas des ensembles de Cantor dynamiques dans C. Plus précisément, on introduit une notion d’épaisseur pour un ensemble de Cantor dynamique planaire et on fournit un critère sur le produit de deux épaisseurs afin d’obtenir une intersection entre deux ensembles de Cantor dynamiques. On montre également que l’épaisseur est une quantité qui varie continûment, ce qui permet d’obtenir des intersections persistantes d’ensembles de Cantor dynamiques. Le second Théorème de cette thèse démontre l’existence du phénomène de Newhouse dans l’espace des automorphismes polynomiaux de degré d pour n’importe quel degré d ≥ 2 dans C^{3}. Au contraire de la situation dans C^{2}, le degré est ici connu et optimal. Le point clef de la preuve est l’introduction dans le domaine complexe d’un outil issu de la dynamique réelle : le blender de Bonatti et Diaz. On formalise le concept de blender complexe et on donne un automorphisme polynomial de C^{3} de degré 2 possédant un blender. Puis, on l’utilise afin de construire successivement des tangences persistantes et des sous-ensembles résiduels d’automorphismes ayant une infinité de puits. Enfin, le dernier résultat porte sur les bifurcations d’endomorphismes holomorphes de P^{2}(C) très particuliers, appelés exemples de Lattès, semi-conjugués à une application affine sur un tore. Dujardin a conjecturé que ces derniers étaient accumulés par des ouverts de bifurcations. On montre que tout exemple de Lattès de degré suffisamment élevé est accumulé par de telles bifurcations robustes. Ceci implique en particulier que tout exemple de Lattès possède un itéré dans l’adhérence de l’intérieur du lieu de bifurcation. La démonstration est basée sur l’obtention d’intersections persistantes entre l’ensemble postcritique et un ensemble hyperbolique répulsif contenu dans l’ensemble de Julia. La preuve est divisée en deux parties : on donne tout d’abord un toy-model qui permet d’obtenir des intersections persistantes entre l’ensemble limite d’un certain type d’IFS, appelé IFS correcteur, et une courbe. Ensuite, dans un second temps, on perturbe l’exemple de Lattès pour créer simultanément un IFS correcteur dans l’ensemble de Julia et une courbe bien orientée dans l’ensemble postcritique / In this PhD thesis, we study Newhouse’s phenomenon and bifurcations in the context of dynamics in several complex variables. We prove three main Theorems. The first one is a complex Gap Lemma. In real dynamics, Newhouse’s Gap Lemma gives a criterion on the product of the thicknesses of two dynamical Cantor sets K and L to show that K ∩ L is not empty. We show a partial generalization of this result for dynamical Cantor sets in C. A relevant notion of thickness in this case is defined and we give some criterion on the product of two thicknesses to show that two dynamical Cantor sets in C must intersect. We also show that the thickness varies continuously, which generates persistent intersections of dynamical Cantor sets. In the second Theorem, we show that there exists a polynomial automorphism f of C^{3} of degree 2 such that for every automorphism g sufficiently close to f, g admits a tangency between the stable and unstable laminations of some hyperbolic set. As a consequence, for each d ≥ 2, there exists an open set of polynomial automorphisms of degree at most d in which the automorphisms having infinitely many sinks are dense. In contrary to the case of C^{2}, the degree is known. To prove these results, we give a complex analogous to the notion of blender introduced by Bonatti and Diaz. In particular, we use a blender to produce robust tangencies. In the third and last result, we study the phenomenon of robust bifurcations in the space of holomorphic maps of P^{2}(C). We prove that any Lattès example of sufficiently high degree belongs to the closure of the interior of the bifurcation locus. This gives a partial answer to a conjecture of Dujardin. In particular, every Lattès map has an iterate with this property. To show this, we design a method creating robust intersections between the limit set of a particular type of iterated functions system in C^{2} with a well-oriented complex curve. Then we show that any Lattès map of sufficiently high degree can be perturbed so that the perturbed map exhibits this geometry

Submitted by Cristine Bello Guse null (tinebelgus@yahoo.com.br) on 2018-02-09T14:19:19Z No. of bitstreams: 1 TESE Cristine Bello Guse completa.pdf: 4289840 bytes, checksum: 3d303902d2cef06e40552603c94d9599 (MD5) / Approved for entry into archive by Laura Mariane de Andrade null (laura.andrade@ia.unesp.br) on 2018-02-09T15:54:29Z (GMT) No. of bitstreams: 1 guse_cb_dr_ia.pdf: 4289840 bytes, checksum: 3d303902d2cef06e40552603c94d9599 (MD5) / Made available in DSpace on 2018-02-09T15:54:29Z (GMT). No. of bitstreams: 1 guse_cb_dr_ia.pdf: 4289840 bytes, checksum: 3d303902d2cef06e40552603c94d9599 (MD5) Previous issue date: 2018-01-30 / O termo cantor-ator refere-se a cantores líricos que levem em conta qualquer grau de responsabilidade cênica existente em qualquer tipo de performance de seu repertório, considerando que o ato de cantar em público já é em si um ato cênico. Dessa forma, existe a necessidade do cantor agregar outras habilidades artísticas que vão além da produção técnico-musical, pois sua atitude cênica não é apenas complemento, mas parte fundamental de seu fazer artístico. Junto a isso, identifica-se uma lacuna na formação profissional dos cantores líricos referente a seu desenvolvimento cênico. O objetivo dessa pesquisa é propor atividades direcionadas ao desenvolvimento de habilidades relacionadas às necessidades cênicas do cantor-ator. A pesquisa fundamenta-se na revisão da literatura específica sobre a atuação cênica do cantor lírico como referencial teórico, tendo os princípios da metodologia do diretor norte-americano Wesley Balk (1981, 1989 e 1991) como fundamentação principal. A proposta de atividades é composta por exercícios e jogos cujo foco de desenvolvimento cênico contemple pelo menos um dos objetivos a seguir: 1) A construção de uma inteligência relativa ao uso cooperativo e assertivo dos recursos expressivos do cantor lírico – corpo, voz e rosto, proporcionando-lhe a integração dos processos de cantar e atuar; 2) o estímulo da sua sensibilidade e criatividade como forma de lhe desenvolver autonomia criativa que lhe permita o senso de verdade cênica e adaptabilidade na performance. Parte-se de exercícios presentes na metodologia de Wesley Balk em que são criados detalhes sobre a sua aplicação, bem como novos exercícios que se fundamentem dentro dos mesmos princípios. Por fim, sugere-se a organização de um laboratório orientado para o desenvolvimento cênico do cantor-ator cujas atividades apresentadas sirvam como procedimentos viáveis. / The singer - actor term refers to classical singers that consider any degree of scenic responsibility existed in any kind of his/her repertory’s performance, keeping in mind that the act of singing in public is a scenic act by itself . In this way, there is a demand for the singer to aggregate other abilities beyond the technical - musical production, because his/her scenic attitude is not just a complement, but a fundamental part of his/her artistic practice. In addition, there is a gap in singers professional qualification regarding his/her acting development. The objective of this research is to propose activities aimed at the development of abilities related to the scenic needs of the singer - actor. The research is based on the review of the specific literature on the classi cal singer's scenic performance as a theoretical reference, with the principles of Wesley Balk's methodology (1981, 1989 and 1991) as the main foundation. The proposal of activities is composed of exercises and games whose focus of scenic development contemplates at least one of the following objectives: 1) The construction of an intelligence related to the cooperative and assertive use of the expressive resources of the classical singer - body, voice and face, provid ing to him/her the integration of the singing and acting processes; 2) the stimulation of his/her sensibility and creativity as a way to develop creative autonomy that allows him/her the sense of scenic truth and adaptability in performance. It starts with the exercises in the methodology of Wesley Balk, and it creates details about its application, as well as new exercises that are based on the same principles. Finally, it is suggested the organization of a laboratory oriented to the scenic development of the singer - actor in which these activities presented could serve as viable procedures.

[pt] Um diagrama de Bratteli B é um objeto combinatório representado por um grafo dividido em infinitos níveis, cada um com número finito de vértices e de arestas entre vértices de níveis consecutivos. Além disso, todo vértice possui ligação com vértices dos níveis precedente e sucessor. Estudamos, do ponto de vista topológico, o espaço dos caminhos infinitos formados pelas arestas de um diagrama de Bratteli, denotado por XB. Estabelecemos uma relação de equivalência neste espaço, denominada AF. Quando é possível definir uma ordem parcial em XB o diagrama é dito ordenado; neste caso, definimos um homeomorfismo em XB denominado de função de Bratteli-Vershik. Consideramos sistemas dinâmicos minimais definidos em conjuntos de Cantor e associamos a estes diagramas de Bratteli ordenados. Um exemplo paradigmático de um conjunto de Cantor é o espaço das sequências infinitas formadas por 00s e 10s, munido de uma métrica apropriada. Neste espaço são definidas as funções odômetro. Definimos a relação de equivalência orbital, na qual duas sequências são equivalentes se estão na mesma órbita do odômetro, e a relação de equivalência de caudas, onde duas sequências são equivalentes se a partir de alguma entrada elas são iguais. Estudamos como estas duas relações estão relacionadas. Provamos que o odômetro diádico é um homeomorfismo minimal definido em um conjunto de Cantor e, portanto, pode ser associado a um diagrama de Bratteli ordenado. Uma relação de equivalência é dita étale quando admite uma topologia gerada por uma ação local. Dois exemplos são as relações AF e orbital. Dada uma relação de equivalência étale R em um espaço X, definimos um invariante algébrico D(X,R). Construímos o grupo de dimensão de um diagrama de Bratteli. Provamos, então, que dado um diagrama de Bratteli B, seu grupo de dimensão é isomorfo a D(XB,RB), onde RB é relação AF de B. Finalmente, estudamos sob quais condições um grupo abeliano ordenado é o grupo de dimensão de um diagrama de Bratteli. Esta dissertação é baseada no livro de Ian F. Putnam Cantor minimal systems, publicado em University Lecture Series, 70. American Mathematical Society, Providence, RI, 2018. [6]. / [en] A Bratteli diagram B is a combinatorial object represented by a graph divided into infinite levels, each level with a finite number of vertices and edges between vertices of consecutive levels. Moreover, every vertex is connected to vertices of the preceding and successor levels. We study, from a topological point of view, the space of infinite paths formed by the edges of a Bratteli diagram, denoted by XB. We establish an equivalence relation on this space, called the AF relation. When it is possible to define a partial order in XB the Bratteli diagram is called ordered; in this case, we define a homeomorphism on XB called the Bratteli-Vershik function. We consider minimal dynamic systems defined on Cantor sets and associate to these systems ordered Bratteli diagrams. A paradigmatic example of a Cantor set is the space of the infinite sequences formed by 00s and 10s, equipped with an appropriate metric. In this space, are defined the odometer functions. We define the orbital equivalence relation, in which two elements of the Cantor set are equivalent if they are in the same orbit of the odometer, and the tail equivalence relation, where two sequences are equivalents if they differ in only finitely many entries. We study how these equivalence relations are related. We prove that the dyadic odometer is a minimal homeomorphism and, therefore, it can be associated to a ordered Bratteli diagram. An equivalence relation is called étale if it admits a topology generated by a local action. Two examples are the AF equivalence relation and the orbital equivalence relation above. Given an étale equivalence relation R on a space X, we define an algebraic invariant D(X,R). We construct the dimension group of a Bratteli diagram. Then, we prove that given a Bratteli diagram B, its dimension group is isomorphic to D(XB,RB), where RB is the AF equivalence relation of B. Finally, we study under which conditions an ordered abelian group is the dimension group for some Bratteli diagram. This master thesis is based on the book by Ian F. Putnam Cantor minimal systems, published in University Lecture Series, 70. American Mathematical Society, Providence, RI, 2018. [6].

Holomorphic correspondences are multivalued functions from the Riemann sphere to itself. This thesis is concerned with a certain type of holomorphic correspondence known as a covering correspondence. In particular we are concerned with a one complexdimensional family of correspondences constructed by post-composing a covering correspondence with a conformal involution. Correspondences constructed in this manner have varied and intricate dynamics. We introduce and analyze two subfamilies of this parameter space. The first family consists of correspondences for which the limit set is a Cantor set, the second family consists of correspondences for which the limit set is connected and for which the action of the correspondence on the complement of this limit set exhibits certain group like behaviour.

This work details an O(n^2) algorithm for computing the spectra of discrete Schroedinger operators with periodic potentials. Spectra of these objects enhance our understanding of fundamental aperiodic physical systems and contain rich theoretical structure of interest to the mathematical community. Previous work on the Harper model led to an O(n^2) algorithm relying on properties not satisfied by other aperiodic operators. Physicists working with the Fibonacci Hamiltonian, a popular quasicrystal model, have instead used a problematic dynamical map approach or a sluggish O(n^3) procedure for their calculations. The algorithm presented in this work, a blend of well-established eigenvalue/vector algorithms, provides researchers with a more robust computational tool of general utility. Application to the Fibonacci Hamiltonian in the sparsely studied intermediate coupling regime reveals structure in canonical coverings of the spectrum that will prove useful in motivating conjectures regarding band combinatorics and fractal dimensions.

An important part of mathematics is the construction of good definitions. Some things, like planar graphs, are trivial to define, and other concepts, like compact sets, arise from putting a name on often used requirements (although the notion of compactness has changed over time to be more general). In other cases, such as in set theory, the natural definitions may yield undesired and even contradictory results, and it can be necessary to use a more complicated formalization.    The notion of a curve falls in the latter category. While it is intuitively clear what a curve is – line segments, empty geometric shapes, and squiggles like this: – it is not immediately clear how to make a general definition of curves. Their most obvious characteristic is that they have no width, so one idea may be to view curves as what can be drawn with a thin pen. This definition, however, has the weakness that even such a line has the ability to completely fill a square, making it a bad definition of curves. Today curves are generally defined by the condition of having no width, that is, being one-dimensional, together with the conditions of being compact and connected, to avoid strange cases.    In this thesis we investigate this definition and a few examples of curves.

University, Diss., 2004--Heidelberg.