In this thesis examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained. Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S. Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}} {{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}} {{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.

This thesis concerns an active research area within fractal geometry. In the first part, in Chapters 2 and 3, for directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing moment of μ[subscript(u)], the self-similar measure at a vertex u, for the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any q ∈ ℝ if the strong separation condition holds, (ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μ[subscript(u)] to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μ[subscript(u)], of the set consisting of all the intersections of the components of F[subscript(u)], is strictly less than the multifractal q Hausdorff dimension with respect to μ[subscript(u)] of F[subscript(u)].

L'analyse multifractale est l'étude des propriétés locales des ensembles de mesures ou de fonctions. Son importance est apparue dans le cadre de la turbulence pleinement développée. Dans ce cadre, l'expérimentateur n'a pas accès à la vitesse en tout point d'un fluide mais il peut mesurer sa valeur en un point en fonction du temps. On ne mesure donc pas directement la fonction vitesse du fluide, mais sa trace. Cette thèse sera essentiellement consacrée à l'étude du comportement local de traces de fonctions d'espaces de Besov : nous déterminerons la dimension de Hausdorff des ensembles de points ayant un exposant de Hölder donné (spectre multifractal). Afin de caractériser facilement l'exposant de Hölder et l'appartenance à un espace de Besov, on utilisera la décomposition de fonctions sur les bases d'ondelettes.Nous n'obtiendrons pas la valeur du spectre de la trace de toute fonction d'un espace de Besov mais sa valeur pour un ensemble générique de fonctions. On fera alors appel à deux notions de généricité différentes : la prévalence et la généricité au sens de Baire. Ces notions ne coïncident pas toujours, mais, ici on obtiendra les mêmes résultats. Dans la dernière partie, afin de déterminer la forme que peut prend un spectre multifractal, on construira une fonction qui est son propre spectre / Multifractal analysis consists in the study of local properties of set of measures or functions. Its importance appeared in the frame of fully developed turbulence. In this area, physicists do not know the velocity of a fluid at all points but they can measure its value in one point in function of time. Hence, they do not measure the velocity function of the fluid but its trace.This thesis will be mainly dedicated to the study of local behavior of traces of Besov functions: we will determine the Hausdorff dimension of sets of points with a given Hölder exponent (the so-called multifractal spectrum). In order to easily characterize Hölder exponent and Besov spaces, we will use wavelet decomposition. We will not get the value of the multifractal spectrum of the trace of all functions of a Besov space, but its value for a generic set of functions. Then, we will use two notions of genericity : prevalence and Baire's genericity. Even if generic and prevalent properties can be different, here they will be the same.In the last part, in order to establish what a multifractal spectrum shape can be, we will construct a function which is its own spectrum

The study of ancient Egyptian monumental art is based on subjective and qualitative analyses by art historians and Egyptologists who use the change in stylistic trends as Dynastic chronological markers. The art of the ancient Egyptians is recognized the world over due to its specific and consistent style that lasted the whole of Dynastic Egypt. This artwork exhibits fractal qualities that support the applicability of applying fractal analysis as a quantitative and statistical tool to be used in this field. In this thesis, I show the fractality of ancient Egyptian monumental art by analyzing black and white line drawings of twenty-eight spearate bas-reliefs with three separate programs : Benoit 1.3, ImageJ, and Fractal3e. After preparing the images with GIMP2 software - used to remove non-original lines - I analyzed each image using the fractal box-counting analysis function in the above programs and calculated their fractal dimension, D. The resulting fractal dimension supported the consistency visually identified in the artwork from ancient Egypt, both chronologically and geographically. / by Jessica Robkin. / Thesis (M.A.)--Florida Atlantic University, 2012. / Includes bibliography. / Mode of access: World Wide Web. / System requirements: Adobe Reader.

Esse trabalho é um estudo sobre circuitos resistivos que apresentam a característica da autossimilaridade em sua configuração. A construção desses circuitos é feita de uma maneira recursiva, de forma análoga a um fractal autossimilar. Os circuitos são analisados pelas suas resistências equivalentes, sendo obtida uma condição para convergência desse valor. Os conceitos auxiliares necessários ao tema desta dissertação abordam a representação de um circuito resistivo como um grafo, além de conceitos envolvendo fractais autossimilares. São propostas ao final de cada capítulo atividades interdisciplinares acessíveis a alunos de ensino médio, com conteúdos envolvendo resistência equivalente, sequências, conjuntos, e noções de área e perímetro. / This work is a study of resistive circuits which present a characteristic of self similarity in their configuration. The construction of these circuits is made in a self recursive way, analogously to a self similar fractal. The circuits are analyzed by their equivalent resistance, and a condition for convergence of this quantity is obtained. Auxiliary concepts that are necessary to this dissertation theme treat the resistive circuit as a graph, and concepts involving self similar fractals. It is proposed at the end of each chapter interdisciplinary activities that are accessible to high school students, with topics involving equivalent resistence, sequences, sets, and notions of area and perimeter.

Este estudo tem como problema de pesquisa a previsão de retorno de ativos financeiros. Buscou verificar a existência de relação entre memória ou dependência de longo prazo em séries temporais fractais e erro de previsão de retornos de ativos financeiros obtida por meio de Redes Neurais Artificiais (RNA). Espera-se que séries temporais fractais com maior memória de longo prazo permitam obter previsões com menor nível de erro, na medida em que a correlação entre os elementos da série favoreça a qualidade de previsão de RNA. Como medida de memória de longo prazo, foi calculado o expoente de Hurst de cada série temporal, o qual sofreu uma transformação para atuar como um índice de previsibilidade. Para medir o erro de previsão, foi utilizada a Raiz do Erro Quadrado Médio (REQM) produzida pela RNA em cada série temporal. O cálculo do expoente de Hurst foi realizado por meio do algoritmo da análise Rescaled Range (R/S). A arquitetura de RNA utilizada foi a de Rede Neural com Atraso Alimentada Adiante (TLFN), tendo como processo de aprendizagem supervisionada o modelo de retropropagação com gradiente descendente para minimização do erro. A amostra foi composta por ativos financeiros brasileiros negociados na Bolsa de Valores, Mercadorias e Futuros de São Paulo (BM&FBovespa), especificamente ações de companhias abertas e fundos de investimentos imobiliários em um período de 10 anos. Os resultados mostraram que a relação entre as variáveis foi significativa para previsões de retornos médios diários de 126 e 252 dias úteis e não significativa para previsão de retorno de 1 dia útil. Quando a análise foi realizada em somente ativos financeiros com expoentes de Hurst persistentes, a relação foi significativa para previsão de 1 dia útil e ainda mais significativa para previsão de 126 e 252 dias úteis, não sendo significativa quando realizada a análise em somente os ativos financeiros antipersistentes. A amostra foi também particionada entre os ativos que participaram e os que não participaram do índice Bovespa (IBOVESPA) no terceiro quadrimestre de 2013. Quando analisados somente os ativos que participaram do IBOVESPA, não houve relação significativa entre as variáveis estudadas, havendo relação significativa somente quando analisados os ativos não participantes. A participação no IBOVESPA apresentou relação significativa com memória de longo prazo e não foi encontrada relação significativa dessa participação com o erro de previsão de RNA. Os resultados encontrados sugerem que o expoente de Hurst pode ser utilizado previamente para selecionar séries temporais de retornos de ativos financeiros que são mais viáveis de serem previstos, particularmente escolhendo aqueles ativos com retornos mais persistentes e que não participem do IBOVESPA. Um gestor que deseje imprimir uma administração mais ativa de seus investimentos poderia utilizá-lo para selecionar uma carteira de ativos com essas características e realizar previsões com qualidade superior ao utilizar RNA. Um investidor que execute uma administração passiva de investimentos deveria compô-la com ativos com expoentes de Hurst característicos de processos em passeio aleatório, a fim de que não seja prejudicado por movimentos não aleatórios do mercado contra os quais não esteja se protegendo. / This study has the research problem of forecasting financial assets return. It aimed to verify the existence of relationship between long-term memory or dependence in fractal time series and prediction error of financial assets returns obtained by Artificial Neural Networks (ANN). It is expected that fractal time series with larger memory could achieve predictions with lower error, since the correlation between the elements of the series favors the quality of ANN prediction. As a long-term memory measure, the Hurst exponent of each time series was calculated, which has undergone a transformation to act as an index of predictability. To measure the prediction error, the Root Mean Square Error (RMSE) produced by ANN in each time series was used. The Hurst exponent computation was conducted through the rescaled range analysis (R/S) algorithm. The ANN architecture was Time Lagged Feedforward Neural Network (TLFN), with backpropagation supervised learning process and gradient descent for error minimization. The sample was composed of Brazilian financial assets traded in the Securities, Commodities & Futures Exchange of Sao Paulo (BM&FBovespa), more specifically public companies shares and real estate investment funds. The results showed that the relationship between the variables was significant for forecasting daily average returns of 126 and 252 business days, and not significant for predicting returns of 1 business day. When the analysis was performed only in financial assets with persistent Hurst exponents, the relationship was significant for predicting returns of 1 business day and even more significant for prediction returns of 126 and 252 business days. The relationship was not significant when the analysis was performed in only antipersistent financial assets. The sample was also partitioned among the assets participating and not participating in the Bovespa Index (IBOVESPA) of the third quarter of 2013. When only assets that participated in the IBOVESPA are considered, there was no significant relationship between the variables studied, existing significant correlation only when no participants are considered. Participation in IBOVESPA showed a significant relationship with long-term memory and no significant relationship of such participation with ANN prediction error was found. The results suggest that the Hurst exponent can be used to previously select time series of financial assets returns that are most feasible to predict, particularly choosing those assets with more persistent returns and not participating in the IBOVESPA. A manager who wishes to make a more active investment management could use it to select a portfolio with these characteristics and make predictions with superior quality when using artificial neural networks. An investor who accomplishes a passive investment management should compound his portfolio with assets that follows Hurst exponents characteristic of random walk processes, so that his is not impaired by no random market movement that he is not protected.

Made available in DSpace on 2016-04-27T17:13:00Z (GMT). No. of bitstreams: 1 Andrea Gomes Nazuto Goncalves.pdf: 11389774 bytes, checksum: af30b407d8ab80be3ce67b30707b85ed (MD5) Previous issue date: 2007-05-29 / Made available in DSpace on 2016-08-25T17:25:36Z (GMT). No. of bitstreams: 2 Andrea Gomes Nazuto Goncalves.pdf.jpg: 2104 bytes, checksum: c4715912a635b5fbde63d2a9b070733f (MD5) Andrea Gomes Nazuto Goncalves.pdf: 11389774 bytes, checksum: af30b407d8ab80be3ce67b30707b85ed (MD5) Previous issue date: 2007-05-29 / Secretaria da Educação do Estado de São Paulo / The objective of this research is to investigate the learning of Geometric Progressions by fractals and their influences on the construction of the knowledge of this subject. Starting from this objective our research questions emerge: How the use of the fractals motivate can be in the perception of the solemnity-similarity? How can the solemnity-similarity contribute in the process of generalization of the formulas of the geometric progression to High School students? So, we developed a teaching sequence, using some elements of the methodology of research denominated engineering didacticism. The conceived sequence is constituted by three blocks, and in the first, we worked the fractals construction; in the second we used the Dynamic Geometry to represent them; and in the third party we focused the generalizations. We used in our research the theoretical presuppositions of Parzysz for the geometry teaching, in what it concerns at their four levels of development of the geometric thought; Machado's ideas that suggest in the construction of a geometric object an articulation among four processes: perception, physical construction, representation and conceptual organization; the situations of resolutions of problems for development of significant concepts proposed by Vergnaud; and also the Dynamic Geometry to motivate the student to investigate. The analysis of the results obtained in the application of the didactic sequence showed that the construction, the manipulation and the observation take to the perception of the solemnity-similarity this, has the aim to facilitate the process of generalization of the mathematical elements that compound the study of Geometric Progressions. In spite of, the number of students used in the sequence (22 couples) brought us great difficulties in the application of the activities, however, it reflected an atmosphere similar to the found at classroom / O objetivo desta pesquisa é investigar o aprendizado de Progressões Geométricas via fractais e as suas influências sobre a construção do conhecimento deste assunto. A partir deste objetivo emergem as nossas questões de pesquisa: Como a utilização dos fractais pode ser motivadora na percepção da autosemelhança? Como a auto-semelhança pode contribuir no processo de generalização das fórmulas da progressão geométrica para alunos do Ensino Médio? Para isto, desenvolvemos uma seqüência de ensino, utilizando alguns elementos da metodologia de pesquisa denominada engenharia didática. A seqüência concebida é constituída por três blocos, sendo que no primeiro, trabalhamos a construção de fractais; no segundo utilizamos a Geometria Dinâmica para representá-los; e no terceiro enfocamos as generalizações. Empregamos em nossa pesquisa os pressupostos teóricos de Parzysz para o ensino de geometria, no que concerne aos seus quatro níveis de desenvolvimento do pensamento geométrico; as idéias de Machado que sugere na construção de um objeto geométrico uma articulação entre quatro processos: percepção, construção física, representação e organização conceitual; as situações de resoluções de problemas para desenvolvimento de conceitos significativos propostas por Vergnaud; e também a Geometria Dinâmica para incentivar o espírito investigativo do aluno. A análise dos resultados obtidos na aplicação da seqüência didática mostrou que a construção, a manipulação e a observação levam à percepção da auto-semelhança, esta, por sua vez, facilita o processo de generalização dos elementos matemáticos que compõem o estudo de Progressões Geométricas. Não obstante, o número de alunos utilizado na seqüência (22 duplas) nos trouxe grandes dificuldades na aplicação das atividades, porém, refletiu um ambiente semelhante ao encontrado em sala de aula

Les modèles géométriques de calcul permettent d’effectuer des calculs à l’aide de primitives géométriques. Parmi eux, le modèle des machines à signaux se distingue par sa simplicité, ainsi que par sa puissance à réaliser efficacement de nombreux calculs. Nous nous proposons ici d’illustrer et de démontrer cette aptitude, en particulier dans le cas de processus massivement parallèles. Nous montrons d’abord à travers l’étude de fractales que les machines à signaux sont capables d’une utilisation massive et parallèle de l’espace. Une méthode de programmation géométrique modulaire est ensuite proposée pour construire des machines à partir de composants géométriques de base les modules munis de certaines fonctionnalités. Cette méthode est particulièrement adaptée pour la conception de calculs géométriques parallèles. Enfin, l’application de cette méthode et l’utilisation de certaines des structures fractales résultent en une résolution géométrique de problèmes difficiles comme les problèmes de satisfaisabilité booléenne SAT et Q-SAT. Ceux-ci, ainsi que plusieurs de leurs variantes, sont résolus par machines à signaux avec une complexité en temps intrinsèque au modèle, appelée profondeur de collisions, qui est polynomiale, illustrant ainsi l’efficacité et le pouvoir de calcul parallèle des machines a signaux. / Geometrical models of computation allow to compute by using geometrical elementary operations. Among them, the signal machines model distinguishes itself by its simplicity, along with its power to realize efficiently various computations. We propose here an illustration and a study of this ability, especially in the case of massively parallel processes. We show first, through a study of fractals, that signal machines are able to make a massive and parallel use of space. Then, a framework of geometrical modular programmation is proposed for designing machines from basic geometrical components —called modules— supplied with given functionnalities. This method fits particulary with the conception of geometrical parallel computations. Finally, the joint use of this method and of fractal structures provides a geometrical resolution of difficult problems such as the boolean satisfiability problems SAT and Q-SAT. These ones, as well as several variants, are solved by signal machines with a model-specific time complexity, called collisions depth, which is polynomial, illustrating thus the efficiency and the parallel computational abilities of signal machines.

本研究資料取自教育部ＥＰＳ資料庫，研究期間為民國７６年一月到８４年一月之分類股價指數，共二千二百九十四筆資料。結果發現台灣證券交易所之分類股價指每日報酬率的行為，顯著拒絕iid之虛無假設，顯示台灣股票市埸有強烈的非線性現象，拒絕原因不是來自不穩定性、也非市場為一混沌系統。本研究利用自我相關函數圖形觀察分類股價指數每日收盤價，發現每筆資料皆呈現緩慢下降的情形，因此將資料取自然對數及一階差分作資料轉換，將符合穩定性的要求。實驗結果可歸納出以下的結論：(1)國內分類股價指數每日報酬率配適AR(3)模型，利用Ljung-Box Q統計量檢定除了金融、食品及加權三筆資料不甚理想外，其它資料均可除去自我相關性。(2)以BDS統計量的結果顯示，股價報酬率均拒絕iid的假設，亦即市埸報酬不具有隨機的形態，其中以水泥類最為強烈。(3)雖然原始資與經亂數編排後的相關維度，已有所不同，但所有原始資料的相關維度均不呈現收斂的現象，顯示市場不具有碎形結構。(4)close returns檢定方法檢定結果顯示，股票市埸不具有混沌現象。

L'objectif principal de cette thèse est la description des propriétés de taille et de grande intersection des ensembles apparaissant lors de l'analyse multifractale de certains processus stochastiques. Dans ce but, nous introduisons de nouvelles classes d'ensembles à grande intersection associées à des fonctions de jauge générales et nous prouvons, à l'aide de techniques d'ubiquité, des résultats d'appartenance à ces classes pour certains ensembles limsup. Cela nous permet en particulier de décrire exhaustivement les propriétés de taille et de grande intersection des ensembles issus de la théorie classique de l'approximation diophantienne comme l'ensemble des points bien approchables par des rationnels ou l'ensemble des nombres de Liouville. Nous fournissons aussi des résultats du même type lorsque les rationnels intervenant dans l'approximation doivent vérifier certaines conditions, comme les conditions de Besicovitch. Nos techniques d'ubiquité nous permettent en outre de décrire complètement les propriétés de taille et de grande intersection des ensembles intervenant dans l'analyse multifractale des processus de Lévy et d'un modèle de séries lacunaires d'ondelettes. Nous obtenons des résultats similaires pour un nouveau modèle de séries aléatoires d'ondelettes dont les coefficients sont corrélés via une chaîne de Markov indexée par un arbre. Nous déterminons en particulier la loi du spectre de singularités de ce modèle. Pour mener cette étude, nous nous intéressons à une large classe de fractals aléatoires généralisant les constructions récursives aléatoires précédemment introduites par de nombreux auteurs.