Modern semiconductor fabrication techniques allow for the fabrication of semiconductor heterostructures which host electron transport with a minimum of scattering sites. In such devices, electrons populate a two-dimensional electron gas (2DEG) in which electrons propagate in exactly two dimensions, and may be further confined by potential barriers to form electron billiards. At sub-Kelvin temperatures, electron trajectories are determined largely by reflections from the billiard walls, while net conduction through the device depends on quantum mechanical wave interference. Measurements of magnetoconductance fluctuations (MCF) serve as a probe of dynamics within the electron billiard. Many prior studies have utilized heterostructures employing the modulation doping architecture, in which the 2DEG is spatially removed from the donor atoms to minimize electron scattering. Theoretical studies have claimed that MCF will be fractal when the confinement potential defining the billiard is soft-walled, regardless of the presence of smooth potentials within the billiard such as those introduced by remote ionized donors. The small-angle scattering sites resulting from these potentials are often disregarded as negligible; we use MCF measurements to investigate such claims. To probe the effect of remote ionized donor scattering on the phase space in electron billiards, we compare MCF measured on billiards in a modulation-doped heterostructure to those measured on billiards in an undoped heterostructure, in which this potential landscape is believed to be absent. Fractal studies are performed on these MCF traces, and we find that MCF measured on the undoped billiards do not exhibit measurably different fractal characteristics than those measured on the modulation-doped billiards. Having confirmed that the potential landscapes in modulation-doped heterostructures do not affect the electron phase space, we then investigate the effect of these impurities on the distribution of electron trajectories through the billiards. By employing thermal cycling experiments, we demonstrate that this distribution is highly sensitive to the precise potential landscape within the billiard, suggesting that modulation-doped heterostructures do not support fully ballistic electron transport. We compare our MCF correlation data with the dynamics of charge transfer within heterostructure systems to make qualitative conclusions regarding these dynamics.

The aesthetic response is a multifaceted and subtle behavior that ranges in magnitude from sublime to mundane. Few studies have investigated the more subtle, weak aesthetic responses to mundane scenes. But all aesthetic responses rely upon sensory-perceptual processes, which serve as a crucial first step in contemporary models of the aesthetic response. As such, understanding the roles of perceptual processes in aesthetic responses to the mundane provides insights into all aesthetic responses. Variation in the physical properties of aesthetic objects must cause such responses, but to understand the relationship, such physical properties must be quantified. Then, the mechanism can be determined. Here, I present the theoretical basis and reason for interest in such a test of mundane aesthetic responses in Chapter I. In Chapter II, I present metrics that quantify the physical properties of natural scenes, using computer-generated images that model the complexity of natural scenes to validate these measurement techniques. The methods presented in Chapter II are adapted to analyze the physical properties of natural scenes in Chapter III, extending the analysis to photographs and clarifying the relationship between the properties fractal dimension and spectral scaling decay rate. A behavioral study is presented in Chapter IV that investigates the extent that perceptual responses about complexity serve as an intermediary between aesthetic ratings and the physical properties of the images described in Chapters II and III. Chapter V summarizes the results of these studies and explores future directions. This dissertation includes previously published and unpublished coauthored material.

Estudos empíricos [1],[2] demonstraram que o trafego das redes Internet Protocol (IP) possui propriedades fractais tais como impulsividade, auto-similaridade e dependência de longa duração em diversas escalas de agregação temporal, na faixa de milissegundos a minutos. Essas características tem motivado o desenvolvimento de novos modelos fractais de teletráfego e de novos algoritmos de controle de trafego em redes convergentes. Este trabalho propõe um novo modelo de trafego no espaço de estados baseado numa aproximação finito-dimensional do processo AutoRegressive Fractionally Integrated Moving Average (ARFIMA). A modelagem por meio de processos auto-regressivos (AR) também é investigada. A analise estatística de series simuladas e de series reais de trafego mostra que a aplicação de modelos AR de ordem alta em esquemas de previsão de teletráfego é fortemente prejudicada pelo problema da identificação da ordem do modelo. Também demonstra-se que a modelagem da memória longa pode ser obtida as custas do posicionamento de um ou mais pólos nas proximidades do circulo de raio unitário. Portanto, a implementação do modelo AR ajustado pode ser instável devido a efeitos de quantização dos coeficientes do filtro digital. O modelo de memória longa proposto oferece as seguintes vantagens: a) possibilidade de implementação pratica, pois não requer memória infinita, b) modelagem (explícita) da região das baixas freqüências do espectro e c) viabilização da utilização do filtro de Kalman. O estudo de caso apresentado demonstra que é possível aplicar o modelo de memória longa proposto em trechos estacionários de sinais de teletráfego fractal. Os resultados obtidos mostram que a dinâmica do parâmetro de Hurst de sinais de teletráfego pode ser bastante lenta na pratica. Sendo assim, o novo modelo proposto é adequado para esquemas de previsão de trafego, tais como Controle de Admissão de Conexões (CAC) e alocação dinâmica de banda, dado que o parâmetro de Hurst pode ser estimado em tempo real por meio da aplicação da transformada wavelet discreta (Discrete Wavelet Transform (DWT)). / Empirical studies [1],[2] demonstrated that heterogeneous IP traffic has fractal properties such as impulsiveness, self-similarity, and long-range dependence over several time scales, from miliseconds to minutes. These features have motivated the development of new traffic models and traffic control algorithms. This work presents a new state-space model for teletraffic which is based on a finite-dimensional representation of the ARFIMA random process. The modeling via AutoRegressive (AR) processes is also investigated. The statistical analysis of simulated time series and real traffic traces show that the application of high-order AR models in schemes of teletraffic prediction can be highly impaired by the model identification problem. It is also demonstrated that the modeling of the long memory can be obtained at the cost of positioning one or more poles near the unit circle. Therefore, the implementation of the adjusted AR model can be unstable due to the quantization of the digital filter coefficients. The proposed long memory model has the following advantages: a) possibility of practical implementation, inasmuch it does not require infinite memory, b) explicit modeling of the low frequency region of the power spectrum, and c) forecasts can be performed via the Kalman predictor. The presented case study suggests one can apply the proposed model in periods where stationarity can be safely assumed. The results indicate that the dynamics of the Hurst parameter can be very slow in practice. Hence, the new proposed model is suitable for teletraffic prediction schemes, such as CAC and dynamic bandwidth allocation, given that the Hurst parameter can be estimated on-line via DWT.

Os principais resultados originais relatados ao longo desse texto provêm de observações em experimentos numéricos, entretanto, na maioria dos casos, os resultados são fundamentados com instrumentos teóricos ou com modelos heurísticos. Inicialmente, introduzimos, nas equações que descrevem osciladores caóticos, uma pequena perturbação periódica a fim de observar no espaço de parâmetros a porção de parâmetros cujo comportamento caótico é extinto. Assim, constatamos que o conjunto de parâmetros correspondentes às orbitas caóticas extintas correspondem à replicas de janelas periódicas complexas previamente existentes no sistema não-perturbado. Posteriormente, utilizando as propriedades de torsão do espaço de estados dos osciladores caóticos, visualizamos transições existentes no interior das janelas periódicas complexas. Quando consideramos sequências dessas janelas sob a ótica da torsão do espaço de estados, observamos a existência de regras que relacionam janelas consecutivas ao longo dessa sequência. Adicionalmente, no espaço de parâmetros de osciladores caóticos e sistemas dinâmicos adicionais, fizemos uma estimativa da dimensão da fronteira entre o conjunto de parâmetros que leva às soluções periódicas e o conjunto que leva aos atratores caóticos. Para os sistemas investigados, os valores obtidos para essa dimensão estão no mesmo intervalo de confiança, indicando que essa dimensão é universal. / The main results reported along this text come from observations in numerical experiments, however, in most cases, results are explained by theoretical instruments or heuristic models. Initially we introduced in the equations that describe chaotic oscillators, a small periodic perturbation to observe, in the parameter space, the portion of parameters whose chaotic behavior is extinguished. Thus, we find that the set of parameters corresponding to the extinct chaotic orbits correspond to replicas of previously complex periodic windows existing in the unperturbed system. Subsequently, using the torsion properties of state spaces of chaotic oscillators, we visualize transitions within the complex periodic windows. When we consider sequences of these windows from the perspective of torsion properties of the state space, we observe the existence of rules that relate consecutive windows along these sequences. Additionally, in the parameter space of chaotic oscillators and additional dynamical systems, we estimate the dimension of the boundary between the set of parameters that leads to periodic solutions and the set that leads to chaotic attractors. For the systems considered here, the values for this dimension are in the same confidence interval, indicating that this dimension is universal.

Estudamos as propriedades aritméticas, geométricas e topológicas de uma classe dos chamados Fractais de Rauzy. Estudamos partucularmente o azulejamento periódico do plano complexo C induzido por eles, assim como a dimensão de Hausdorff de suas fronteiras. Tal trabalho exige um estudo detalhado da fronteira destes conjuntos, que está associada às propriedades aritméticas da \'alpha\' -representação dos números complexos com respeito a um certo número algébrico \'alfa\' / We study the arithmetic, geometric and topological properties of a class of the so-called Rauzy\'s fractals. In particular we study the periodic tiling of the complex plane C induced by them and the Hausdorff dimension of its boundary. Such work is connected to a detailed study of the boundary of such sets and the arithmetic properties of the \'alpha\' representation of complex numbers with respect to a certain algebraic number \'alpha\'

Survey is time-consuming and expensive. Therefore, it needs to be both effective and efficient. Some archaeologists have argued that current survey techniques are not effective (Shott 1985, 1989), but most archaeologists continue to employ these methods and therefore must believe they are effective. If our survey techniques are effective, why do simulations suggest otherwise? If they are ineffective, can we improve them? The answers to these practical questions depend on the topological characteristics of archaeological site distributions. In this study I analyze archaeological site distributions in the Valley of Oaxaca, Mexico, using lacunarity and fractal dimension. Fractal dimension is a parameter of fractal patterns, which are complex, space-filling designs exhibiting self-similarity and power-law scaling. Lacunarity is a statistical measure that describes the texture of a spatial dispersion. It is useful in understanding how archaeological tests should be spaced during surveys. Between these two measures, I accurately describe the regional topology and suggest new considerations for archaeological survey design. / Includes bibliography. / Thesis (M.A.)--Florida Atlantic University, 2014. / FAU Electronic Theses and Dissertations Collection

Redes neurais caracterizadas por cadeias de osciladores acoplados são um dentre vários tipos de redes que possuem propriedades peculiares relacionadas com a sua estrutura topológica. A dinâmica que descreve o comportamento dessas redes é modelada por sistemas de equações diferenciais, nos quais cada neurônio (nó) é considerado como um oscilador. Estudos realizados em redes desse tipo, em tarefas de reconhecimento de padrões estáveis gerados aleatoriamente, têm apresentado resultados computacionais satisfatórios. Esta tese propôs um desenvolvimento teórico e computacional que forneceu um algoritmo, para o estudo do desempenho de redes neurais em forma de osciladores de Ciclo-Limite de Stuart-Landau, no reconhecimento de figuras fractais. Neste trabalho apresentaremos contextos reais em que podemos encontrar características deste tipo de redes e motivações. Em seguida, serão expostos conceitos de redes de Hopfield, reconhecimento de padrões, teorias dos fractais e dos osciladores de Ciclo-Limite de Stuart-Landau; tais conceitos, por sua vez, serviram como ferramentas principais para o algoritmo construído que será explicado posteriormente. Antes de apresentá-lo, será exposta a maneira como a dinâmica desses osciladores pode se tornar caótica, por meio de simulações computacionais alterando numericamente variáveis intrínsecas, como tempos de disparos entre neurônios, ou quantidades destes no sistema. Estas descobertas serviram como confirmações para elaborar e compor do algoritmo, bem como orientaram as simulações de reconhecimento de figuras fractais. Por fim, será apresentada a conclusão dos resultados encontrados. / Neural networks characterized by chains of coupled oscillators are one of several types of networks which have peculiar properties related with their topological structure. The dynamics that describes the behavior of these networks is modeled by systems of differential equations, of which each neuron (node) is considered as an oscillator. Studies on such networks, in tasks of recognizing randomly generated stable patterns, have presented satisfactory computational results. This thesis proposed a theoretical and computational development that provided an algorithm for the study of the performance of neural networks in the form of Cycle-Limit oscillators of Stuart-Landau, in the recognition of fractals. In this work we will present real contexts in which we can find characteristics of this type of networks and motivations. Next, concepts of Hopfield networks, pattern recognition, fractals theories and the Stuart-Landau Cycle-Limit oscillators will be presented; these concepts, in turn, served as the main tools for the algorithm constructed that will be explained later. Before presenting it, it will be exposed how the dynamics of these oscillators can become chaotic, through computer simulations numerically altering intrinsic variables, such as firing times between neurons, or quantities of these in the system. These findings served as confirmations for elaborating and composing the algorithm, as well as guiding the simulations of the recognition of fractals. Finally, the results will be presented.

Stochastic analysis on fractals is, as one might expect, a subfield of analysis on fractals. An intuitive starting point is to observe that on many fractals, one can define diffusion processes whose law is in some sense invariant with respect to the symmetries and self-similarities of the fractal. These can be interpreted as fractal-valued counterparts of standard Brownian motion on Rd. One can study these diffusions directly, for example by computing heat kernel and hitting time estimates. On the other hand, by associating the infinitesimal generator of the fractal-valued diffusion with the Laplacian on Rd, it is possible to pose stochastic partial differential equations on the fractal such as the stochastic heat equation and stochastic wave equation. In this thesis we investigate a variety of questions concerning the properties of diffusions on fractals and the parabolic and hyperbolic SPDEs associated with them. Key results include an extension of Kolmogorov's continuity theorem to stochastic processes indexed by fractals, and existence and uniqueness of solutions to parabolic SPDEs on fractals with Lipschitz data.

分形分析的主題是研究分形上的Dirichlet形式和Laplacian. 壓縮的自相似群有一個與之關聯的極限空間,此空間通常具備分形結構，因而引發了分形分析和自相似群兩個分支的結合. / 我們回顧了自相似群和它們的極限空間極限空間可以用Schreier 圖來逼近,事實上其可以看成由Schreier圖構造出來的雙曲圖的雙曲邊界.我們探究了迭代單值群. 通過增加專門的條件我們可以得到迭代單值群的極限空間同胚於某個Julia集. / 通過運用[31] 中的想法和[47] 中自相似隨機游動的方法，我們闡明了極限空間上Laplacian和Dirichlet形式的構造步驟我們介紹了加法器， Basilica群以及Hanoi塔群的極限空間(在第三種情況下是Sierpiríski墊片)上的Laplacian 這裡得到的Dirichlet形式是局部且正則的. / 通過採用[53] 的設置, 我們描述了加法器的極限空間上的誘發型Dirichlet形式在構造了加法器的自相似圖上的嚴格可逆隨機游動後，我們可以得到一個非局部的Dirichlet形式. / The major theme of fractal analysis is studying Dirichlet forms and Laplacians on fractals. For a contracting self-similar group there is an associated limit space, which usually exhibits a fractal structure, thereby triggering the combination of fractal analysis and self-similar groups. / We give reviews of self-similar groups and their limit spaces. Limit space can be approximated by Schreier graphs, and it is in fact identied as a hyperbolic boundary of a hyperbolic graph constructed from Schreier graphs. We explore the iterated monodromy groups. By adding technical conditions, we have that the limit space of an iterated monodromy group is homeomorphic to a Julia set. / We show the construction process of Laplacians and Dirichlet forms on limit spaces using the idea of [31] and the method of self-similar random walks from [47]. We present examples of Laplacians of the limit spaces of adding machine, the Basilica group and the Hanoi Tower group (it is Sierpi´nski gasket in this case). In this context these forms are local and regular. / We describe the induced Dirichlet forms on limit space of the adding machine by adopting the settings of [53] . By constructing strictly reversible random walks on self-similarity graph of the adding machine, we can obtain a non-local Dirichlet form. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Lin, Dateng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 71-76). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Review of fractal analysis --- p.6 / Chapter 1.2 --- Applications to self-similar groups --- p.7 / Chapter 1.3 --- Boundary theory method --- p.8 / Chapter 1.4 --- Summary of the thesis --- p.9 / Chapter 2 --- Self-similar groups --- p.11 / Chapter 2.1 --- Basic definitions --- p.11 / Chapter 2.2 --- Limit spaces of self-similar groups --- p.18 / Chapter 2.3 --- Schreier graphs approximations --- p.24 / Chapter 2.4 --- Iterated monodromy groups --- p.28 / Chapter 3 --- Construction of Laplacians on limit spaces --- p.35 / Chapter 3.1 --- Dirichlet forms, Laplacians and resistance forms --- p.35 / Chapter 3.2 --- Representations of groups and functions --- p.42 / Chapter 3.3 --- Laplacians on limit spaces --- p.45 / Chapter 4 --- Induced Dirichlet form on limit space of the adding machine --- p.53 / Chapter 4.1 --- Martin boundary and hyperbolic boundary --- p.53 / Chapter 4.2 --- Graph energy and the induced form --- p.62 / Chapter 4.3 --- Induced Dirichlet form of the adding machine --- p.65 / Bibliography --- p.71

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