The Geometry of Fractal Sets

Author: K. J. Falconer

Publisher: Cambridge University Press

ISBN: 9780521337052

Category: Mathematics

Page: 162

View: 9795

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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.
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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

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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.
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Caractérisation Des Roches

Author: John A. Hudson

Publisher: Thomas Telford

ISBN: 9780727716880

Category: Technology & Engineering

Page: 487

View: 8094

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The subject of rock characterization is not only about the optimal length-to-diameter ratio for a compression test specimen and other similar tactical aspects of the testing procedures, it is also about the whole strategic concept of how to characterize naturally-occurring rock masses, which have been in existence for millions of years. They have been operating as natural process-response systems for all time and are about to be perturbed by engineers in order to achieve particular objectives. By international authors, this volume is important and useful for all geotechnical engineers and related positions who need to know the latest information to succeed.
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Fractals in Probability and Analysis

Author: Christopher J. Bishop,Yuval Peres

Publisher: Cambridge University Press

ISBN: 1107134110

Category: Mathematics

Page: 412

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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.
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Fractals And Beyond: Complexities In The Sciences

Author: Novak Miroslav M

Publisher: World Scientific

ISBN: 9814544507

Category:

Page: 372

View: 6297

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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.
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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.
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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: 4472

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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.
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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: 4347

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