Pang, Wenjun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 83-88). / Abstracts in English and Chinese. / ACKNOWLEDGEMENTS --- p.ii / ABSTRACT --- p.iv / 摘要 --- p.v / CONTENTS --- p.vi / List of Tables --- p.viii / List of Figures --- p.ix / Chapter 1. --- INTRODUCTION --- p.1 / Chapter 1.1 --- Recent research work --- p.4 / Chapter 1.2 --- Objectives --- p.8 / Chapter 1.3 --- Thesis Organization --- p.10 / Chapter 2. --- FRACTAL MODELING --- p.12 / Chapter 2.1 --- Fractal and Fractal Art --- p.12 / Chapter 2.2 --- Fractal Geometry --- p.15 / Chapter 2.3 --- Construction of Fractals --- p.21 / Chapter 2.4 --- Fractal Measurement and Aesthetics --- p.27 / Chapter 3. --- OVERVIEW OF EVOLUTIONARY DESIGN --- p.30 / Chapter 3.1 --- Initialization --- p.33 / Chapter 3.2 --- Selection --- p.33 / Chapter 3.3 --- Reproduction --- p.34 / Chapter 3.4 --- Termination --- p.36 / Chapter 4. --- EVOLUTIONARY 3D FRACTAL MODELING --- p.38 / Chapter 4.1 --- Fractal Construction --- p.38 / Chapter 4.1.1 --- Self-similar Condition of Fractal --- p.38 / Chapter 4.1.2 --- Fractal Transformation (FT) IFS Formulation --- p.39 / Chapter 4.1.3 --- IFS Genotype and Phenotype Expression --- p.41 / Chapter 4.2 --- Evolutionary Algorithm --- p.43 / Chapter 4.2.1 --- Single-point Crossover --- p.45 / Chapter 4.2.2 --- Arithmetic Gaussian mutation --- p.45 / Chapter 4.2.3 --- Inferior Elimination --- p.46 / Chapter 4.3 --- Interactive Fine-tuning using FT IFS --- p.46 / Chapter 4.4 --- Gaussian Fitness Function --- p.48 / Chapter 5. --- GAUSSIAN AESTHETIC FITNESS FUNCTION --- p.49 / Chapter 5.1 --- Fitness Considerations --- p.50 / Chapter 5.2 --- Fitness Function Formulation --- p.53 / Chapter 5.3 --- Results and Discussion on Fitness Function --- p.55 / Chapter 6. --- EXPERIMENT RESULTS and DISCUSSION --- p.59 / Chapter 6.1 --- Experiment of Evolutionary Generation --- p.59 / Chapter 6.2 --- Comparison on Different Methods --- p.60 / Chapter 7. --- 3D FRACTALS RENDERING and APPLICATION --- p.62 / Chapter 7.1 --- Transforming Property and User Modification --- p.62 / Chapter 7.2 --- Visualization and Rendering of 3D Fractals --- p.66 / Chapter 7.3 --- Applications in Design --- p.74 / Chapter 8. --- CONCLUSIONS and FUTURE WORK --- p.81 / Chapter 8.1 --- Conclusions --- p.81 / Chapter 8.2 --- Future Work --- p.81 / BIBLIOGRAPHY --- p.83 / Appendix --- p.89 / Marching Cubes Method --- p.89

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Filamentous fungal colonies show a remarkable diversity of different mycelial branching patterns. To date, the characterization of this biostructural complexity has been based on subjective descriptions. Here, computerized image analysis in conjunction with video microscopy has been used to quantify several aspects of fungal growth and differentiation. This was accomplished by applying the new branch of mathematics called Fractal Geometry to this biological system, to provide an objective description of morphological and biochemical complexity. The fractal dimension is useful for describing irregularity and shape complexity in systems that appear to display scaling correlations (between structural units) over several orders of length or size. The branching dynamics of Pycnoporus cinnabarinus have been evaluated using fractals in order to determine whether there was a correlation between branching complexity and the amount of extracellular phenol-oxidase that accumulated during growth. A non-linear branching response was observed when colonies were grown in the presence of the aminoanthraquinone dye, Remazol Brilliant Blue R. Branching complexity could be used to predict the generalized yield of phenol-oxidase that accumulated in submerged culture, or identify paramorphogens that could be used to improve yield. A method to optimize growth of discrete fungal colonies for microscopy and image analysis on microporous membranes revealed secretion sites of the phenoloxidase, laccase as well as the intracellular enzyme, acid phosphatase. This method was further improved using microwave-accelerated heating to detect tip and sheath bound enzyme. The spatial deposition of secreted laccase and acid phosphatase displayed antipersistent scaling in deposition and/or secretion pattern. To overcome inherent statistical limitations of existing methods, a new signal processing tool, called wavelets were applied to analyze both one and two-dimensional data to measure fractal scaling. Two-dimensional wavelet packet analysis (2-d WPA) measured the (i) mass fractal dimension of binary images, or the (ii) self-affine dimension of grey-scale images. Both 1- and 2-d WPA showed comparative accuracy with existing methods yet offered improvements in computational efficiency that were inherent with this multiresolution technique. The fractal dimension was shown to be a sensitive indicator of shape complexity. The discovery of power law scaling was a hallmark of fractal geometry and in many cases returned values that were indicative of a self-organized critical state. This meant that the dynamics of fungal colony branching equilibrium. Hence there was potential for biostructural changes of all sizes, which would allow the system to efficiently adapt to environmental change at both the macro and micro levels.

Ce travail présente des contributions théoriques et pratiques à la théorie des codages symboliques de systèmes dynamiques. Les applications concernent différents champs mathématiques et la modélisation en biologie moléculaire. Le but est d'illustrer comment des méthodes de discrétisation de systèmes dynamiques et une approche algorithmique permettent d'exploiter au mieux les connaissances disponibles sur le système, même partielles. Un premier objectif est d'exhiber des informations au sujet d'une dynamique que l'on connaît explicitement et les traduire en propriétés concrètes. Un deuxième objectif est de produire de la connaissance sur une dynamique ou un modèle lorsqu'on ne le connaît pas explicitement.Dans ce document, ces deux questions sont abordées sur deux grandes classes de systèmes dynamiques. <br /><br />Les premiers systèmes considérés sont des automorphismes et des translations sur un tore. Inspirés par les cas unidimensionnels (beta-numération, étude des suites sturmiennes), la question principale qui se pose est de trouver un domaine fondamental pour le tore dans lequel les trajectoires de la dynamique considérée se codent par des systèmes symboliques simples. Dans le cas où l'automorphisme du tore considéré admet une unique direction dilatante (le cas Pisot), un bon candidat pour ces partitions est donné par un domaine dont la base est fractale, introduit par G. Rauzy dans les années 1980. Nous décrivons comment une approche décidable pour décrire le bord fractal du domaine et ses propriétés de pavage, permet de s'assurer qu'il s'agit d'un domaine adéquat pour un codage du l'automorphisme. La description du bord du domaine permet de décrire ses propriétés topologiques, et de les exploiter dans les différents domaines d'informatique théorique où les automorphismes et les additions sur un tore apparaissent. Ainsi, en théorie des nombres, nous nous appuyons sur la topologie du domaine pour caractériser les propriétés des développements finis ou purement périodiques de rationnels en base non entière. En géométrie discrète, ces propriétés s'interprètent en termes de conditions pour l'engendrement de plans discrets par des méthodes itératives. <br /><br />La deuxième classe de systèmes concerne les systèmes dynamiques de grande échelle en biologie moléculaire. Il s'avère que les données et les connaissances sur les modèles de régulations transcriptionnelles dans une cellule sont souvent trop partielles pour leur appliquer les méthodes usuellement utilisées pour la modélisation de systèmes expérimentaux. Dans ce document, nous discutons d'un formalisme (inspiré par la dynamique) qui permet d'interpréter les observations en biologie moléculaire, pour aider à la correction de modèles, et, dans le futur, à la mise en place de plans expérimentaux. Au vu de la qualité des données, les aspects dynamiques sont alors remplacés par des considérations sur les déplacements d'états stationnaires, et analyser les données revient à formaliser puis résoudre des contraintes portant sur des ensembles discrets. Nous montrons ainsi comment aborder les notions de corrections de modèles et de diagnostic de réseaux grande échelle.

Nous mettons en évidence un certain nombre de liens entre la théorie des nombres et celle des automates :<br>- étude de sous-suites de la suite "somme des chiffres", étude des itérées de cette suite ;<br>- utilisation de suites automatiques particulières (baptisées q-miroirs) dans le problème de l'itération des fonctions continues unimodales réelles ;<br>- étude d'un curieux ensemble de répartition modulo 1 de nombres réels ; <br>- propriétés arithmétiques d'un automate cellulaire ;<br>- répartition modulo 1 des puissances de séries formelles à coefficients automatiques (donc algébrique sur le corps des fractions rationnelles sur un corps fini).

<p>This thesis consists of three papers, all of them on the topic of function spaces on fractals.</p><p>The papers summarised in this thesis are:</p><p>Paper I Mats Bodin, Wavelets and function spaces on Mauldin-Williams fractals, Research Report in Mathematics No. 7, Umeå University, 2005.</p><p>Paper II Mats Bodin, Harmonic functions and Lipschitz spaces on the Sierpinski gasket, Research Report in Mathematics No. 8, Umeå University, 2005.</p><p>Paper III Mats Bodin, A discrete characterisation of Lipschitz spaces on fractals, Manuscript.</p><p>The first paper deals with piecewise continuous wavelets of higher order in Besov spaces defined on fractals. A. Jonsson has constructed wavelets of higher order on fractals, and characterises Besov spaces on totally disconnected self-similar sets, by means of the magnitude of the coefficients in the wavelet expansion of the function. For a class of fractals, W. Jin shows that such wavelets can be constructed by recursively calculating moments. We extend their results to a class of graph directed self-similar fractals, introduced by R. D. Mauldin and S. C. Williams.</p><p>In the second paper we compare differently defined function spaces on the Sierpinski gasket. R. S. Strichartz proposes a discrete definition of Besov spaces of continuous functions on self-similar fractals having a regular harmonic structure. We identify some of them with Lipschitz spaces introduced by A. Jonsson, when the underlying domain is the Sierpinski gasket. We also characterise some of these spaces by means of the magnitude of the coefficients of the expansion of a function in a continuous piecewise harmonic base.</p><p>The last paper gives a discrete characterisation of certain Lipschitz spaces on a class of fractal sets. A. Kamont has discretely characterised Besov spaces on intervals. We give a discrete characterisation of Lipschitz spaces on fractals admitting a type of regular sequence of triangulations, and for a class of post critically finite self-similar sets. This shows that, on some fractals, certain discretely defined Besov spaces, introduced by R. Strichartz, coincide with Lipschitz spaces introduced by A. Jonsson and H. Wallin for low order of smoothness.</p>

This thesis consists of three papers, all of them on the topic of function spaces on fractals. The papers summarised in this thesis are: Paper I Mats Bodin, Wavelets and function spaces on Mauldin-Williams fractals, Research Report in Mathematics No. 7, Umeå University, 2005. Paper II Mats Bodin, Harmonic functions and Lipschitz spaces on the Sierpinski gasket, Research Report in Mathematics No. 8, Umeå University, 2005. Paper III Mats Bodin, A discrete characterisation of Lipschitz spaces on fractals, Manuscript. The first paper deals with piecewise continuous wavelets of higher order in Besov spaces defined on fractals. A. Jonsson has constructed wavelets of higher order on fractals, and characterises Besov spaces on totally disconnected self-similar sets, by means of the magnitude of the coefficients in the wavelet expansion of the function. For a class of fractals, W. Jin shows that such wavelets can be constructed by recursively calculating moments. We extend their results to a class of graph directed self-similar fractals, introduced by R. D. Mauldin and S. C. Williams. In the second paper we compare differently defined function spaces on the Sierpinski gasket. R. S. Strichartz proposes a discrete definition of Besov spaces of continuous functions on self-similar fractals having a regular harmonic structure. We identify some of them with Lipschitz spaces introduced by A. Jonsson, when the underlying domain is the Sierpinski gasket. We also characterise some of these spaces by means of the magnitude of the coefficients of the expansion of a function in a continuous piecewise harmonic base. The last paper gives a discrete characterisation of certain Lipschitz spaces on a class of fractal sets. A. Kamont has discretely characterised Besov spaces on intervals. We give a discrete characterisation of Lipschitz spaces on fractals admitting a type of regular sequence of triangulations, and for a class of post critically finite self-similar sets. This shows that, on some fractals, certain discretely defined Besov spaces, introduced by R. Strichartz, coincide with Lipschitz spaces introduced by A. Jonsson and H. Wallin for low order of smoothness.

This thesis explores new approaches to the analysis of functions by combining tools from the fields of complex bases, number systems, iterated function systems (IFS) and wavelet multiresolution analyses (MRA). The foundation of this work is grounded in the identification of a link between two-dimensional non-separable Haar wavelets and complex bases. The theory of complex bases and this link are generalized to higher dimensional number systems. Tilings generated by number systems are typically fractal in nature. This often yields asymmetry in the wavelet trees of functions during wavelet decomposition. To acknowledge this situation, a class of extensions of functions is developed. These are shown to be consistent with the Mallat algorithm. A formal definition of local IFS on wavelet trees (LIFSW) is constructed for MRA associated with number systems, along with an application to the inverse problem. From these investigations, a series of algorithms emerge, namely the Mallat algorithm using addressing in number systems, an algorithm for extending functions and a method for constructing LIFSW operators in higher dimensions. Applications to image coding are given and ideas for further study are also proposed. Background material is included to assist readers less familiar with the varied topics considered. In addition, an appendix provides a more detailed exposition of the fundamentals of IFS theory.

Fractal geometry is a new branch of mathematics. This report presents the tools, methods and theory required to describe this geometry. The power of Iterated Function Systems (IFS) is introduced and applied to produce fractal images or approximate complex estructures found in nature. The focus of this thesis is on how fractal geometry can be used in applications to computer graphics or to model natural objects.

In the course Discrete Deterministic Chaos, Dr. Mark Daniels introduces students to Chaos Theory and explores many topics within the field. Students prove many of the key results that are discussed in class and work through examples of each topic. Connections to the secondary mathematics curriculum are made throughout the course, and students discuss how the topics in the course could be implemented in the classroom. This paper will provide an overview of the topics covered in the course, Discrete Deterministic Chaos, and provide additional discussion on various related topics. / text