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

The use of fractal-based methods in imaging was first popularized with fractal image compression in the early 1990s. In this application, one seeks to approximate a given target image by the fixed point of a contractive operator called the fractal transform. Typically, one uses Local Iterated Function Systems with Grey-Level Maps (LIFSM), where the involved functions map a parent (domain) block in an image to a smaller child (range) block and the grey-level maps adjust the shading of the shrunken block. The fractal transform is defined by the collection of optimal parent-child pairings and parameters defining the grey-level maps. Iteration of the fractal transform on any initial image produces an approximation of the fixed point and, hence, an approximation of the target image. Since the parameters defining the LIFSM take less space to store than the target image does, image compression is achieved.This thesis extends the theoretical and practical frameworks of fractal imaging to one involving a particular type of multifunction that captures the idea that there are typically many near-optimal parent-child pairings. Using this extended machinery, we treat three application areas. After discussing established edge detection methods, we present a fractal-based approach to edge detection with results that compare favourably to the Sobel edge detector. Next, we discuss two methods of information hiding: first, we explore compositions of fractal transforms and cycles of images and apply these concepts to image-hiding; second, we propose and demonstrate an algorithm that allows us to securely embed with redundancy a binary string within an image. Finally, we discuss some theory of certain random fractal transforms with potential applications to texturing. / The Natural Sciences and Engineering Research Council and the University of Guelph helped to provide financial support for this research.

Previous studies on Leafy reduced-stature (LRS) maize found that it had extremely early maturity and a higher harvest index (HI), leading to high yields for its maturity rating. Whether this apparent high HI is relaxed to its earliness, or can also exist among the medium or late maturity LRS maize has not been previously investigated. It was also of interest to know if the traits that produced the LRS canopy structure have pleiotropic effects on root architecture. Finally, field observations indicated that LRS maize had a lower incidence of common smut. It is not known whether this apparent resistance is specific to smut or includes other diseases. / Using a wide range of the most recently developed LRS hybrids and some conventional hybrids, a two-year field experiment was conducted to examine the HI and disease resistance of LRS maize. HI, yield, and yield components were compared between the two genotype groups (LRS and conventional) under different population densities. The resistance to the natural incidence of common smut and artificially inoculated Gibberella ear rot was also tested. Morphology and fractal dimension analyses of roots at an early development stage were conducted in indoor experiments. These analyses were performed with WinRHIZO (version 3.9), an interactive scanner-based image analysis system. / This work showed that: (1) There was no relationship between the HI and maturity; higher HIs can also exist among the medium and late maturity LRS hybrids. (2) While LRS maize hybrids have the potential for high yield this was not realized in the LRS hybrids used in this work. Further breeding and development of optimum management practices are needed to fully exploit this potential. (3) During early development LRS hybrids generally had more branching and more complex root systems than conventional hybrids. (4) Fractal dimension, as a comprehensive estimation of root complexity, was highly related to major root morphological variables, such as root total length, surface area, branching frequency and dry mass. (5) Of the hybrids tested the greatest resistance to both common smut and Gibberella ear rot, two major ear diseases, occurred in some of the LRS types.