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

La présente thèse couvre deux principaux thèmes de recherche qui seront présentés dans deux parties et précédés par un prolegomenon commun. Dans ce dernier nous introduisons les concepts essentiels et nous exploitons aussi le lien entre les deux parties.<p><p>Dans la première partie, le principal objet d’intérêt est la soi-disant fonctionnelle exponentielle de processus de Lévy. La loi de cette variable aléatoire joue un rôle primordial dans de nombreux domaines divers tant sur le plan théorique que dans des domaines appliqués. Doney dérive une factorisation de la loi arc-sinus en termes de suprema de processus stables indépendants et de même index. Une factorisation similaire de la loi arc-sinus en termes de derniers temps de passage au niveau 1 de processus de Bessel peut aussi être établie en utilisant un résultat dû à Getoor. Des factorisations semblables d’une variable de Pareto en termes des mêmes objets peut également être obtenue. Le but de cette partie est de donner une preuve unifiée et une généralisation de ces factorisations qui semblent n’avoir aucun lien à première vue. Même s’il semble n’y avoir aucune connexion entre le supremum d’un processus stable et le dernier temps de passage d’un processus de Bessel, il peut être montré que ces variables aleatoires sont liées à des fonctionnelles exponentielles de processus de Lévy spécifiques. Notre contribution principale dans cette partie et aussi au niveau de caractérisations de la loi de la fonctionnelle exponentielle sont des factorisations de la loi arc-sinus et de variables de Pareto généralisées. Notre preuve s’appuie sur une factorisation de Wiener-Hopf récente de Patie et Savov.<p>Dans la deuxième partie, motivée par le fait que la dérivée fractionnaire de Caputo et d’autres opérateurs fractionnaires classiques coïncident avec le générateur de processus de Markov auto-similaires positifs particuliers, nous introduisons des opérateurs généralisés de Caputo et nous étudions certaines propriétés. Nous nous intéressons particulièrement aux conditions sous lesquelles ces opérateurs coïncident avec les générateurs infinitésimaux de processus de Markov auto-similaires positifs généraux. Dans ce cas, nous étudions les fonctions invariantes de ces opérateurs qui admettent une représentation en termes de séries entières. Nous précisons que cette classe de fonctions contient les fonctions de Bessel modifiées, les fonctions de Mittag-Leffler ainsi que plusieurs fonctions hypergéométriques. Nous proposons une étude unifiant et en profondeur de cette classe de fonctions. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

We examine the evolution of globular star clusters, modelled as spherically symmetric stellar systems, using various techniques. Such clusters possess a central region of approximately uniform density which is referred to as the core. We concentrate our analysis on the evolution of the cluster after the core has undergone core collapse; a process where its radius decreases and its density increases. After this collapse, the system as a whole can expand in a self-similar fashion (homologous post-collapse evolution) which has long been thought to be due to gravitational interactions between different populations of single stars and binary stars in the core. We confirm this assumption by constructing a simple analytical model which combines much of the theoretical knowledge of previous research in the field. This model consists of two stellar populations, each defined by the mass of the individual stars, and a separate core. Our simple model is itself constructed from two simpler models – a twocomponent model without a core and a single mass model with a core – and takes into account the main gravitational interactions thought to drive the post-collapse evolution. To ensure that no important mechanisms have been neglected in our simple model, we will compare it with an N-body simulation. We compute our N-body models with NBODY6 (using a GPU version for large N). When we compare the N-body model with the simple model, we find qualitative agreement between them for most cases. Even though some mechanisms (e.g. escape of stars) are neglected in our simple model, we find that both models show homologous post-collapse evolution. We also review the homologous post-collapse Fokker-Planck model in the case of equal stellar masses derived by H´enon (1961) with the intention of extending this for the two-component case. We present our numerical solutions for H´enon’s model and find that our numerical solutions are in satisfactory agreement with the results shown in this paper. When we extend this work for a general two-component model (i.e. with no restriction on the number of heavier stars), we find that a homologous solution cannot be found with this approach. By contrast, we suggest that it would be possible to find a homologous two-component solution by extending the one-component solution published later by H´enon (1965), which differs from the earlier model by neglecting the external tidal field of the parent galaxy. Much of the work shown in this thesis would be relevant for such future study.

No description available.

Neste trabalho estudamos as subvariedades lagrangeanas mínimas e autossimilares do espaço paracomplexo Dn. Começamos definindo o conceito de variedade para-Kähler e, como exemplo, descrevemos o espaço projetivo paracomplexo. Em seguida, estudamos as subvariedades paracomplexas e lagrangeanas. Após mostrarmos que toda subvariedade paracomplexa não-degenerada é mínima, dedicamos a atenção ao estudo das subvariedades lagrangeanas, restringindo-nos ao ambiente Dn. Em particular, estudamos as lagrangeanas que são invariantes sob a ação canônica do grupo SO(n), e as superfícies de Castro-Chen. Em ambos os casos, analisamos a minimalidade e a autossimilaridade das mesmas. / In this work, we study minimal and self-similar Lagrangian submanifolds in the para-complex space Dn. Firstly, we define the concept of para-Kähler manifold and, to exemplify, we describe the para-complex projective space.Then, we study para-complex submanifolds and Lagrangian submanifolds. After proving that every non-degenerate para-complex submanifold is minimal, we pay attention to Lagrangian submanifolds, restricting us to the case of Dn. In particular, we study Lagrangian submanifolds which are invariant by the canonical SO(n)-action of Dn, and Castro-Chen\'s surfaces. In both cases, we analyse the minimality and self-similarity.

Neste trabalho estudamos as subvariedades lagrangeanas mínimas e autossimilares do espaço paracomplexo Dn. Começamos definindo o conceito de variedade para-Kähler e, como exemplo, descrevemos o espaço projetivo paracomplexo. Em seguida, estudamos as subvariedades paracomplexas e lagrangeanas. Após mostrarmos que toda subvariedade paracomplexa não-degenerada é mínima, dedicamos a atenção ao estudo das subvariedades lagrangeanas, restringindo-nos ao ambiente Dn. Em particular, estudamos as lagrangeanas que são invariantes sob a ação canônica do grupo SO(n), e as superfícies de Castro-Chen. Em ambos os casos, analisamos a minimalidade e a autossimilaridade das mesmas. / In this work, we study minimal and self-similar Lagrangian submanifolds in the para-complex space Dn. Firstly, we define the concept of para-Kähler manifold and, to exemplify, we describe the para-complex projective space.Then, we study para-complex submanifolds and Lagrangian submanifolds. After proving that every non-degenerate para-complex submanifold is minimal, we pay attention to Lagrangian submanifolds, restricting us to the case of Dn. In particular, we study Lagrangian submanifolds which are invariant by the canonical SO(n)-action of Dn, and Castro-Chen\'s surfaces. In both cases, we analyse the minimality and self-similarity.

No description available.

Network-on-Chip (NoC) is a developing and promising on-chip communication paradigm that improves scalability and performance of System-on-Chips. NoC design flow contains many problems from different areas, for example networking, embedded design and computer architecture. Application mapping is one of these problems, which is generally considered as a communication energy minimization problem. This dissertation approaches to this problem from a networking point of view and tries to find a mapping solution which improves the network performance in terms of the number of packets in the buffers while still minimizing the total communication energy consumption. For this purpose an on-chip network traffic model is required. Self similarity is a traffic model that is used to characterize Ethernet and/or wide area network traffic, as well as most of on-chip network traffic. In this thesis, by using an on-chip traffic characterization that contains self similarity, an application mapping problem definition that contains both energy and buffer utilization concerns is proposed. In order to solve this intractable problem a genetic algorithm based model is implemented. Execution of the algorithm on different test cases has proved that such a mapping formulation avoids high buffer utilizations while still keeping the communication energy low.

In this paper, we focus on the goodness of fit test for self-similar property of two well-known processes: the fractional Brownian motion and the fractional autoregressive integrated moving average process. The Hurst parameter of the self-similar process is estimated by the embedding branching process method proposed by Jones and Shen (2004). The goodness of fit test for self-similarity is based on the Pearson chi-square test statistic. We approximate the null distribution of the test statistic by a scaled chi-square distribution to correct the size bias problem of the conventional chi-square distribution. The scale parameter and degrees of freedom of the test statistic are determined via regression method. Simulations are performed to show the finite sample size and power of the proposed test. Empirical applications are conducted for the high frequency financial data and human heart rate data.

The central purpose of this study is the integration of modem philosophical thinking with different chaos theory principles and definitions to form relational perspectives. Relations are used in different contexts to base the causes of deterministic chaos (chaost) in the laws of nature which constitutes order. The chaost-attractor is used as subjective conception to investigate the possibilities of hidden order in a seemingly chaotic state of the objective reality. Relevant definitions of the chaos theory were analysed methodically and transcendentally with the aid of concepts of order and relations. Attention is given to the broad associations and analogies from philosophy and other disciplines which relate to the connectivity of objects to form systems. Subjective model development was done which is used to consequentially analyse some statements from published research which applied principles of chaost. It is argued that: the intrinsic properties of objects determine the causality of forces which bind objects to compose systems; a web of interactive bonds functions subjective to laws of nature which determine whether a system is in a state of order, chaost or real chaos; a dynamical transfer of many intrinsic and asymmetric properties via internal bonds constitutes non-linear connectivity which causes a sensitivity for initial conditions. It is found that the chaost of the chaos theory is not the same as real, objective chaos. The random-like evolution of a dynamic system is determined by the occurrence of irregularities and uncertainties in its internal order. A web of interactive bonds distribute small changes self-similar and scale-relevant. The difficulty in describing and explaining the complex behaviour of composed entities is simplified by the proposed web-chaost model. / Thesis (M.A. (Philosophy))--North-West University, Potchefstroom Campus, 2006.