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Quantum walks are the quantum mechanical analogue of classical random walks. Discrete-time quantum walks have been introduced and studied mostly on the line Z or higher dimensional space Z d but rarely defined on graphs with fractal dimensions because the coin operator depends on the position and the Fourier transform on the fractals is not defined. Inspired by its nature of classical walks, different quantum walks will be defined by choosing different shift and coin operators. When the coin operator is uniform, the results of classical walks will be obtained upon measurement at each step. Moreover, with measurement at each step, our results reveal more information about the classical random walks. In this dissertation, two graphs with fractal dimensions will be considered. The first one is Sierpinski gasket, a degree-4 regular graph with Hausdorff di- mension of df = ln 3/ ln 2. The second is the Cantor graph derived like Cantor set, with Hausdorff dimension of df = ln 2/ ln 3. The definitions and amplitude functions of the quantum walks will be introduced. The main part of this dissertation is to derive a recursive formula to compute the amplitude Green function. The exiting probability will be computed and compared with the classical results. When the generation of graphs goes to infinity, the recursion of the walks will be investigated and the convergence rates will be obtained and compared with the classical counterparts. / Mathematics

In this work, we propose the application fractal compression techniques to textured images segmentation, and use the transformation coefficients as features for segmentation. The result is improved by combining fractal dimension feature and the transformation coefficients from the original and its filter versions. Feature vectors are clustered together using K-mean algorithm with features pre-smoothing. The numbers of feature are minimized to reach the compromise result. In the integrated approach, we attempt to improve segmentation of texture images using our method. Background knowledge of image segmentation and image compression will be presented. Algorithms for fractal dimension calculation, K-means clustering, and fractal compression is given. Experimental results are included, and possible future work is mentioned.

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Dans le contexte de l’acoustique on a systématisé la dérivation de modèles nonlinéaires(l’équation de Kuznetsov, l’équation KZK et la NPE). On a estimé le temps pourlequel des solutions régulières de ces modèles restent proches des solutions des systèmes deNavier-Stokes/Euler compressibles isentropiques (en précisant leur plus faible régularité) etétabli les résultats analogues entre les solutions des équations de KZK, NPE et Westerveltpar rapport à la solution de l’équation de Kuznetsov. Pour ce faire, on a étudié l’équationde Kuznetsov en commençant par le problème de Cauchy dans les cas visqueux (stabilité,unicité et existence globale des solutions régulières) et non-visqueux (caractère bien poséavec les estimations optimales du temps d’existence maximale des solutions régulières) etégalement dans un demi espace avec des conditions au limites périodiques en temps oudans un espace périodique dans une direction. On a aussi obtenu l’existence et l’unicité dessolutions faibles pour l’équation des ondes fortement amortie et l’équation deWestervelt surla plus large classe de domaines aux bords irréguliers, ainsi que la convergence asymptotiquedes solutions de l’équation de Westervelt avec conditions de Robin sur les bords préfractalsapproximant un bord fractal de type mixture de Koch. / In the framework of acoustic we systematize the derivation of nonlinear models(the Kuznetsov equation, the KZK equation and the NPE). We estimate the time for whichthe regular solutions of these models stay close of the solutions of the compressible isentropicNavier-Stokes/Euler systems (pointing out their weakest regularity) and establish similarresults between the solutions of the KZK, NPE and Westervelt equations with respectto the solutions of the Kuznetsov equation. To do so, we study the Kuznetsov equationbeginning by the Cauchy problem in the viscous case (stability, gobal well posedness ofregular solutions) and inviscid case (well posedness with optimal estimations of the maximalexistence time for regular solutions) and also in the half space with time periodic boundaryconditions or in a periodic in one direction space. We also obtain the existence and unicityof weak solutions for the strongly damped wave equation and the Westervelt equation in thelargest class of domains with irregular boundaries, along with the asymptotic convergenceof the solutions of the Westervelt equation with Robin boundary conditions on prefractalboundaries approximating a Koch mixture as fractal boundary.

Submitted by Jaqueline Silva (jtas29@gmail.com) on 2016-08-31T19:46:48Z No. of bitstreams: 2 Dissertação - Ricardo Anselmo Zanotto - 2015.pdf: 7706833 bytes, checksum: 26c6e884d0e3a03a3daebaa4ab5764a4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-08-31T19:47:01Z (GMT) No. of bitstreams: 2 Dissertação - Ricardo Anselmo Zanotto - 2015.pdf: 7706833 bytes, checksum: 26c6e884d0e3a03a3daebaa4ab5764a4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-08-31T19:47:01Z (GMT). No. of bitstreams: 2 Dissertação - Ricardo Anselmo Zanotto - 2015.pdf: 7706833 bytes, checksum: 26c6e884d0e3a03a3daebaa4ab5764a4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2015-12-12 / Outro / This is a research about a part of the non-Euclidean geometry that has recently been very studied. It was addressed initial themes of the non-Euclidean geometry and it was exposed the studies abut fractals, its history, buildings and main fractals (known as classic fractals). It was also addressed the relation among the school years contents and how to use fractals; as well as some of its applications that have helped a lot of researches to spread and show better results. / Este trabalho é uma pesquisa sobre parte da geometria não euclidiana que há pouco vem sendo muito estudada, os fractais. Abordamos temas iniciais da geometria nãoeuclidiana e no decorrer do trabalho expomos nosso estudo sobre fractais, seu histórico, construções, principais fractais (conhecidos como fractais clássicos). Também abordamos relações entre conteúdos dos anos escolares e como usar fractais nos mesmos; como também algumas de suas aplicações que vem ajudando muitas pesquisas a se difundirem e apresentarem melhores resultados.

This thesis examines the intersection of math and art by focusing on three specific branches of math: the fourth dimension, non-Euclidean geometry, and chaos and fractals. Different genres of art interact with each of these branches of math. The influence of the fourth dimension can easily be seen in Cubism and Russian Constructivism. Non-Euclidean geometry guided some of M.C. Escher’s work, and it inspired the Crochet Coral Reef project. Chaos and fractals can be found in art and architecture throughout history, but Vincent van Gogh and Jackson Pollock are notable examples of artists who used chaos in their work. Some artists incorporate math into their work in a rigorous, exacting manner, while others take inspiration from a general concept and provide a more abstract interpretation. Regardless of mathematical accuracy, mathematically inspired art can provide a greater understanding of mathematical concepts.