The Geometry of Fractal Sets

Author: K. J. Falconer

Publisher: Cambridge University Press

ISBN: 9780521337052

Category: Mathematics

Page: 162

View: 3512

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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

View: 8978

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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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Fractals in Probability and Analysis

Author: Christopher J. Bishop,Yuval Peres

Publisher: Cambridge University Press

ISBN: 1107134110

Category: Mathematics

Page: 412

View: 3270

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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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Caractérisation Des Roches

Author: John A. Hudson

Publisher: Thomas Telford

ISBN: 9780727716880

Category: Technology & Engineering

Page: 487

View: 2876

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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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Analysis on Fractals

Author: Jun Kigami

Publisher: Cambridge University Press

ISBN: 9780521793216

Category: Mathematics

Page: 226

View: 9376

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Self-contained introduction to analysis on fractals, a developing area of mathematics, for graduates and researchers.
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Encyclopedia of mathematical physics

Author: Sheung Tsun Tsou

Publisher: Academic Pr

ISBN: 9780125126601

Category: Science

Page: 3500

View: 6620

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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.
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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: 4500

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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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