The Geometry of Fractal Sets

Author: K. J. Falconer
Publisher: Cambridge University Press
ISBN: 9780521337052
Category: Mathematics
Page: 162
View: 5595

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This book contains a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Questions of local density and the existence of tangents of such sets are studied, as well as the dimensional properties of their projections in various directions. In the case of sets of integral dimension the dramatic differences between regular 'curve-like' sets and irregular 'dust like' sets are exhibited. The theory is related by duality to Kayeka sets (sets of zero area containing lines in every direction). The final chapter includes diverse examples of sets to which the general theory is applicable: discussions of curves of fractional dimension, self-similar sets, strange attractors, and examples from number theory, convexity and so on. There is an emphasis on the basic tools of the subject such as the Vitali covering lemma, net measures and Fourier transform methods.

Fractal Geometry and Applications: Analysis, number theory, and dynamical systems

Author: Benoit B. Mandelbrot
Publisher: American Mathematical Soc.
ISBN: 0821836374
Category: Ergodic theory
Page: 517
View: 463

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This volume offers an excellent selection of cutting-edge articles about fractal geometry, covering the great breadth of mathematics and related areas touched by this subject. Included are rich survey articles and fine expository papers. The high-quality contributions to the volume by well-known researchers--including two articles by Mandelbrot--provide a solid cross-section of recent research representing the richness and variety of contemporary advances in and around fractal geometry. In demonstrating the vitality and diversity of the field, this book will motivate further investigation into the many open problems and inspire future research directions. It is suitable for graduate students and researchers interested in fractal geometry and its applications. This is a two-part volume. Part 1 covers analysis, number theory, and dynamical systems; Part 2, multifractals, probability and statistical mechanics, and applications.

Fractals in Probability and Analysis

Author: Christopher J. Bishop,Yuval Peres
Publisher: Cambridge University Press
ISBN: 1107134110
Category: Mathematics
Page: 412
View: 8102

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This is a mathematically rigorous introduction to fractals which emphasizes examples and fundamental ideas. Building up from basic techniques of geometric measure theory and probability, central topics such as Hausdorff dimension, self-similar sets and Brownian motion are introduced, as are more specialized topics, including Kakeya sets, capacity, percolation on trees and the traveling salesman theorem. The broad range of techniques presented enables key ideas to be highlighted, without the distraction of excessive technicalities. The authors incorporate some novel proofs which are simpler than those available elsewhere. Where possible, chapters are designed to be read independently so the book can be used to teach a variety of courses, with the clear structure offering students an accessible route into the topic.

Fractals And Beyond: Complexities In The Sciences

Author: Novak Miroslav M
Publisher: World Scientific
ISBN: 9814544507
Page: 372
View: 2215

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This is the second volume of 'A Course in Analysis' and it is devoted to the study of mappings between subsets of Euclidean spaces. The metric, hence the topological structure is discussed as well as the continuity of mappings. This is followed by introducing partial derivatives of real-valued functions and the differential of mappings. Many chapters deal with applications, in particular to geometry (parametric curves and surfaces, convexity), but topics such as extreme values and Lagrange multipliers, or curvilinear coordinates are considered too. On the more abstract side results such as the Stone-Weierstrass theorem or the Arzela-Ascoli theorem are proved in detail. The first part ends with a rigorous treatment of line integrals.The second part handles iterated and volume integrals for real-valued functions. Here we develop the Riemann (-Darboux-Jordan) theory. A whole chapter is devoted to boundaries and Jordan measurability of domains. We also handle in detail improper integrals and give some of their applications.The final part of this volume takes up a first discussion of vector calculus. Here we present a working mathematician's version of Green's, Gauss' and Stokes' theorem. Again some emphasis is given to applications, for example to the study of partial differential equations. At the same time we prepare the student to understand why these theorems and related objects such as surface integrals demand a much more advanced theory which we will develop in later volumes.This volume offers more than 260 problems solved in complete detail which should be of great benefit to every serious student.

Analysis on Fractals

Author: Jun Kigami
Publisher: Cambridge University Press
ISBN: 9780521793216
Category: Mathematics
Page: 226
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Self-contained introduction to analysis on fractals, a developing area of mathematics, for graduates and researchers.

Encyclopedia of mathematical physics

Author: Sheung Tsun Tsou
Publisher: Academic Pr
ISBN: 9780125126601
Category: Science
Page: 3500
View: 7014

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The Encyclopedia of Mathematical Physics provides a complete resource for researchers, students and lecturers with an interest in mathematical physics. It enables readers to access basic information on topics peripheral to their own areas, to provide a repository of the core information in the area that can be used to refresh the researcher's own memory banks, and aid teachers in directing students to entries relevant to their course-work. The Encyclopedia does contain information that has been distilled, organised and presented as a complete reference tool to the user and a landmark to the body of knowledge that has accumulated in this domain. It also is a stimulus for new researchers working in mathematical physics or in areas using the methods originated from work in mathematical physics by providing them with focused high quality background information. * First comprehensive interdisciplinary coverage * Mathematical Physics explained to stimulate new developments and foster new applications of its methods to other fields * Written by an international group of experts * Contains several undergraduate-level introductory articles to facilitate acquisition of new expertise * Thematic index and extensive cross-referencing to provide easy access and quick search functionality * Also available online with active linking.

Quantum Graphs and Their Applications

Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Quantum Graphs and Their Applications, June 19-23, 2005, Snowbird, Utah
Author: Gregory Berkolaiko
Publisher: American Mathematical Soc.
ISBN: 0821837656
Category: Mathematics
Page: 307
View: 1888

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This volume is a collection of articles dedicated to quantum graphs, a newly emerging interdisciplinary field related to various areas of mathematics and physics. The reader can find a broad overview of the theory of quantum graphs. The articles present methods coming from different areas of mathematics: number theory, combinatorics, mathematical physics, differential equations, spectral theory, global analysis, and theory of fractals. They also address various important applications, such as Anderson localization, electrical networks, quantum chaos, mesoscopic physics, superconductivity, optics, and biological modeling.

Fractal Geometry and Stochastics II

Author: Christoph Bandt,Siegfried Graf,Martina Zähle
Publisher: N.A
Category: Mathematics
Page: 292
View: 627

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A collection of contributions by outstanding mathematicians, highlighting the principal directions of research on the combination of fractal geometry and stochastic methods. Clear expositions introduce the most recent results and problems on these subjects and give an overview of their historical development.

Graph Directed Markov Systems

Geometry and Dynamics of Limit Sets
Author: R. Daniel Mauldin,Mariusz Urbanski,Mariusz Urbański
Publisher: Cambridge University Press
ISBN: 9780521825382
Category: Mathematics
Page: 281
View: 9818

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Monograph on Graph Directed Markov Systems with backgound and research level material.

Fraktalgeometrie: Selbstähnlichkeit und fraktale Dimension –

Author: Jutta Otterbein,Christina Sawatzki
Publisher: GRIN Verlag
ISBN: 3656262365
Category: Mathematics
Page: 20
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Studienarbeit aus dem Jahr 2010 im Fachbereich Mathematik - Geometrie, Note: 1,3, Universität Kassel (Institut für Mathematik), Veranstaltung: Fachwissenschaftliches Seminar, Sprache: Deutsch, Abstract: In der nachfolgenden Arbeit soll die Selbstähnlichkeit und fraktale Dimension, Teil 1 behandelt werden. Vorab wird der Begriff „Fraktale“ im Allgemeinen beschrieben und erklärt. Zur Verdeutlichung des Begriffs wird ferner auf die unterschiedlichen Eigenschaften der Fraktale, die das Grundgerüst der Fraktalgeometrie und den Schwerpunkt der Arbeit bilden, eingegangen. Des Weiteren wird die Selbstähnlichkeit dargestellt, die sich unter anderem zwischen der exakten und der statistischen Selbstähnlichkeit unterscheiden lässt. Einige Beispiele sollen diesen Unterschied deutlich machen und herauskristallisieren. Darauf aufbauend wird die Selbstähnlichkeitsdimension allgemein definiert sowie die Formel zu ihrer Berechnung abgeleitet. Anschließend wird sich den mathematischen Fraktalen zugewandt. Im Mittelpunkt stehen die Cantor-Drittelmenge und das Sierpinski-Dreieck, bei denen jeweils die Selbstähnlichkeit sowie deren Dimension beschrieben und vertiefend erklärt wird. Abschließend werden unterschiedliche Wischaktivi-täten in der Ebene und im Raum anhand zahlreicher Beispiele skizziert und diese miteinander verglichen.

Dirichlet Forms and Analysis on Wiener Space

Author: Nicolas Bouleau,Francis Hirsch
Publisher: Walter de Gruyter
ISBN: 311085838X
Category: Mathematics
Page: 335
View: 5016

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The subject of this book is analysis on Wiener space by means of Dirichlet forms and Malliavin calculus. There are already several literature on this topic, but this book has some different viewpoints. First the authors review the theory of Dirichlet forms, but they observe only functional analytic, potential theoretical and algebraic properties. They do not mention the relation with Markov processes or stochastic calculus as discussed in usual books (e.g. Fukushima’s book). Even on analytic properties, instead of mentioning the Beuring-Deny formula, they discuss “carré du champ” operators introduced by Meyer and Bakry very carefully. Although they discuss when this “carré du champ” operator exists in general situation, the conditions they gave are rather hard to verify, and so they verify them in the case of Ornstein-Uhlenbeck operator in Wiener space later. (It should be noticed that one can easily show the existence of “carré du champ” operator in this case by using Shigekawa’s H-derivative.) In the part on Malliavin calculus, the authors mainly discuss the absolute continuity of the probability law of Wiener functionals. The Dirichlet form corresponds to the first derivative only, and so it is not easy to consider higher order derivatives in this framework. This is the reason why they discuss only the first step of Malliavin calculus. On the other hand, they succeeded to deal with some delicate problems (the absolute continuity of the probability law of the solution to stochastic differential equations with Lipschitz continuous coefficients, the domain of stochastic integrals (Itô-Ramer-Skorokhod integrals), etc.). This book focuses on the abstract structure of Dirichlet forms and Malliavin calculus rather than their applications. However, the authors give a lot of exercises and references and they may help the reader to study other topics which are not discussed in this book. Zentralblatt Math, Reviewer: S.Kusuoka (Hongo)