Liu, Xin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 61-63). / Abstracts in English and Chinese. / Abstract --- p.1 / Acknowledgments --- p.4 / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- Basics of self-affine tiles --- p.8 / Chapter 1.1 --- Self-affine sets --- p.8 / Chapter 1.2 --- Self-affine tiles --- p.11 / Chapter 1.3 --- Structure of tiling sets --- p.15 / Chapter 1.4 --- Integral self-affine tiles --- p.20 / Chapter 1.4.1 --- Lebesgue measure of integral self-affine tile --- p.22 / Chapter 1.4.2 --- Classification of digit set --- p.24 / Chapter 2 --- Connectedness of self-affine tiles --- p.28 / Chapter 2.1 --- Connectedness --- p.28 / Chapter 2.2 --- Disk-likeness --- p.33 / Chapter 3 --- Tiles with rotations and reflections --- p.39 / Chapter 3.1 --- Tiles --- p.39 / Chapter 3.2 --- Integral tiles --- p.54 / Bibliography --- p.60

Li, Chung Fai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 43). / Abstracts in English and Chinese. / Acknowledgement --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Construction of the State Spaces --- p.5 / Chapter 2.1 --- The Sierpinski Gasket --- p.5 / Chapter 2.1.1 --- Neighbourhood in the Sierpinski Gasket --- p.7 / Chapter 2.2 --- The Sierpinski Carpet --- p.9 / Chapter 2.2.1 --- Neighbourhood in the Sierpinski Carpet --- p.10 / Chapter 3 --- Preliminary Random Processes on Each Level --- p.12 / Chapter 3.1 --- The Sierpinski Gasket --- p.12 / Chapter 3.1.1 --- Definitions --- p.12 / Chapter 3.1.2 --- Properties of the Random Walk --- p.13 / Chapter 3.1.3 --- Preparations for convergence and continuity --- p.16 / Chapter 3.2 --- The Sierpinski Carpet --- p.19 / Chapter 3.2.1 --- The Brownian Motion Bn on Cn --- p.19 / Chapter 3.2.2 --- Properties of Bm(t) --- p.20 / Chapter 3.2.3 --- Exit time for Bn --- p.27 / Chapter 4 --- The limiting process --- p.29 / Chapter 4.1 --- The Sierpinski Gasket --- p.29 / Chapter 4.1.1 --- Convergence and continuity --- p.29 / Chapter 4.1.2 --- Extension from to G --- p.31 / Chapter 4.1.3 --- Markov property --- p.33 / Chapter 4.2 --- The Sierpinski Carpet --- p.34 / Chapter 4.2.1 --- Continuity --- p.34 / Chapter 4.2.2 --- Existence of Markov process on C --- p.37 / Chapter 4.2.3 --- Piecing Together --- p.38 / Bibliography --- p.43

Tools to explore chaos are as far away as a personal computer or a pocket calculator. A few lines of simple equations in BASIC produce fantastic graphic displays. In the following computer experiment, the dimension of a strange attractor is found by three algorithms; Shaw's, Grassberger-Procaccia's and Guckenheimer's. The programs were tested on the Henon attractor which has a known fractal dimension. Shaw's and Guckenheimer's algorithms were tested with 1000 data points, and Grassberger's with 100 points, a data set easily handled by a PC in one hour or less using BASIC or any other language restricted to 640K RAM. Since dimension estimates are notorious for requiring many data points, the author wanted to find an algorithm to quickly estimate a low-dimensional system (around 2). Although all three programs gave results in the neighborhood of the fractal dimension for the Henon attractor, Dfracta1=1.26, none appeared to converge to the fractal dimension.

Bibliography : leaves 115-119 / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1994

The goal of this research project is to determine the fractal nature, if any, which certain surface water waves exhibit when viewed on a microscopic scale. We make use of the mathematical formulation of non-viscous fluids describing their physical properties. Using these expressions and including boundary conditions for free surfaces as well as taking the surface tension into consideration, we obtain a partial differential equation describing the dynamics of surface water waves. A brief introduction to the study of fractal geometry with several examples of well-known fractals is included. An important property of fractals is their non-integral dimension. Several methods of determining the dimension of a curve are described in this paper. Our wave equation is examined under different assumptions representing the conditions of a surface water wave near its breaking point. Solutions are developed using analytical and numerical methods. We determine the dimension of 'rough' solutions using one of the methods introduced and conclude that under certain conditions, surface water waves near their breaking point exhibit a fractal structure on a microscopic scale. / Graduation date: 1994

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Šio darbo tikslai – ištirti paprasčiausius fraktalus, išnagrinėti fraktalų savybių realizavimo ir fraktalų atvaizdavimo Logo priemonėmis galimybes, sudaryti fraktalų dėstymo mokiniams metodiką naudojant Logo. Šiems tikslams įgyvendinti iškeliami darbo uždaviniai – užrašyti matematinius fraktalų sudarymo algoritmus Logo kalba bei pateikti keletą fraktalų realizavimo Logo kalba pavyzdžių. Darbą sudaro dvi dalys. Pirmojoje pateikiamas fraktalų apibrėžimas, jų savybės, trumpa fraktalų istorija, detaliau nagrinėjami žinomi fraktalai – Julijaus ir Mandelbroto aibės. Antrojoje dalyje nagrinėjamos fraktalų vaizdavimo Logo priemonėmis galimybės, pateikiami Logo programų fraktalams kurti pavyzdžiai. Darbe ištirta, kaip modeliuojami fraktalai bei pateikta keletas programų Logo kalba fraktalams sudaryti. Programos antrojoje darbo dalyje pateikiamos nuosekliai, nuo paprasčiausių programėlių supažindinti su rekursija iki sudėtingesnių algoritmų įvairiems fraktalams sudaryti, taip pasiūlant glaustą fraktalų mokymo naudojant Logo metodiką. Nagrinėjant fraktalus susipažįstama su koordinačių sistemomis, sveikųjų skaičių aritmetika, begalybės sąvoka, lavinami skaičiavimo ir braižymo įgūdžiai. Fraktalus gali nagrinėti įvairaus amžiaus mokiniai. Perteikiant fraktalus paprasta ir visiems lengvai perprantama Logo kalba, kartu yra ugdomi algoritmavimo, programavimo ir darbo su kompiuteriu gebėjimai. Sudėtingiems fraktalams sudaryti Logo priemonės nėra racionalios dėl lėto programų darbo atliekant... [toliau žr. visą tekstą] / The goal of this master's degree thesis is to analyze properties and generation algorithms of simple fractals and to implement these algorithms into Logo programming language suggesting methodology of teaching fractals using Logo in schools and providing some examples of Logo programs for fractal generation. The thesis consists of two parts. The first part provides basic theory on fractals. It begins with a simple explanation of what a fractal is using examples of self-similarity and recursive process and going into a more mathematical definition of fractals, introduced by B. Mandelbrot. After a brief history of fractals, a more in-depth analysis of Mandelbrot and Julia sets, the two well-known fractals arising from very simple sequences of complex numbers defined by the relation z_{n+1} = z_n^2 + c is given. The last chapter of the first part points out the reasons how fractals are useful and why they should be taught at school – fractals are fun; fractals are beautiful; anyone can play with them; fractals promote curiosity; computers, when used to explain fractal theory, enhance learning. The second part focuses on using Logo to generate fractals. It provides a few Logo programs of various complexity ranging from simple recursive functions to handling operations with complex numbers. Examples of Logo programs include generation of fractal trees, Koch snowflake and Sierpinski gasket, implementation of chaos game and iterated function systems, and manipulating complex numbers... [to full text]

Thesis (M.S. in Mechanical Engineering)--Naval Postgraduate School, June 2008. / Thesis Advisor(s): Kwon, Young W. "June 2008." Description based on title screen as viewed on August 26, 2008. Includes bibliographical references (p. 51). Also available in print.

M.Sc. (Computer Science) / Please refer to full text to view abstract