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Author: Alain Le Mehaute
Publisher: CRC Press
ISBN: 9780849377228
Category : Mathematics
Languages : en
Pages : 214
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Book Description
Fractal geometry, based on recursive mathematical schemas, provides a means for modeling a great number of natural phenomena. For this reason, it is of increasing interest to physicists, chemists, biologists, and geographers, among others. A major quality of fractality is that it not only unifies phenomena previously thought to be anomalous or disparate in a single theoretical framework, but it also promotes a return to graphical treatment, which had been almost completely banished from scientific thought in favor of analysis. This book casts a new, lively light on scientific territories still not fully explored. It is designed for research workers, engineers, and experimentalists faced with problems of measure and action in heterogenous materials and environments. Several color plates illustrate the implications and consequences of this theory for most of the questions raised by the taking into consideration of time in a fractal space.
Author: Alain Le Mehaute
Publisher: CRC Press
ISBN: 9780849377228
Category : Mathematics
Languages : en
Pages : 214
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Book Description
Fractal geometry, based on recursive mathematical schemas, provides a means for modeling a great number of natural phenomena. For this reason, it is of increasing interest to physicists, chemists, biologists, and geographers, among others. A major quality of fractality is that it not only unifies phenomena previously thought to be anomalous or disparate in a single theoretical framework, but it also promotes a return to graphical treatment, which had been almost completely banished from scientific thought in favor of analysis. This book casts a new, lively light on scientific territories still not fully explored. It is designed for research workers, engineers, and experimentalists faced with problems of measure and action in heterogenous materials and environments. Several color plates illustrate the implications and consequences of this theory for most of the questions raised by the taking into consideration of time in a fractal space.
Author: Hillel Furstenberg
Publisher: American Mathematical Society
ISBN: 1470410346
Category : Mathematics
Languages : en
Pages : 82
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Book Description
Fractal geometry represents a radical departure from classical geometry, which focuses on smooth objects that "straighten out" under magnification. Fractals, which take their name from the shape of fractured objects, can be characterized as retaining their lack of smoothness under magnification. The properties of fractals come to light under repeated magnification, which we refer to informally as "zooming in". This zooming-in process has its parallels in dynamics, and the varying "scenery" corresponds to the evolution of dynamical variables. The present monograph focuses on applications of one branch of dynamics--ergodic theory--to the geometry of fractals. Much attention is given to the all-important notion of fractal dimension, which is shown to be intimately related to the study of ergodic averages. It has been long known that dynamical systems serve as a rich source of fractal examples. The primary goal in this monograph is to demonstrate how the minute structure of fractals is unfolded when seen in the light of related dynamics. A co-publication of the AMS and CBMS.
Author: Kenneth Falconer
Publisher: John Wiley & Sons
ISBN: 0470299452
Category : Mathematics
Languages : en
Pages : 367
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Book Description
Since its original publication in 1990, Kenneth Falconer's Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. It introduces the general mathematical theory and applications of fractals in a way that is accessible to students from a wide range of disciplines. This new edition has been extensively revised and updated. It features much new material, many additional exercises, notes and references, and an extended bibliography that reflects the development of the subject since the first edition. * Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals. * Each topic is carefully explained and illustrated by examples and figures. * Includes all necessary mathematical background material. * Includes notes and references to enable the reader to pursue individual topics. * Features a wide selection of exercises, enabling the reader to develop their understanding of the theory. * Supported by a Web site featuring solutions to exercises, and additional material for students and lecturers. Fractal Geometry: Mathematical Foundations and Applications is aimed at undergraduate and graduate students studying courses in fractal geometry. The book also provides an excellent source of reference for researchers who encounter fractals in mathematics, physics, engineering, and the applied sciences. Also by Kenneth Falconer and available from Wiley: Techniques in Fractal Geometry ISBN 0-471-95724-0 Please click here to download solutions to exercises found within this title: http://www.wileyeurope.com/fractal
Author: Michel L. Lapidus
Publisher: Springer Science & Business Media
ISBN: 1461421764
Category : Mathematics
Languages : en
Pages : 583
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Book Description
Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. Throughout Geometry, Complex Dimensions and Zeta Functions, Second Edition, new results are examined and a new definition of fractality as the presence of nonreal complex dimensions with positive real parts is presented. The new final chapter discusses several new topics and results obtained since the publication of the first edition.
Author: Michel L. Lapidus
Publisher: Springer Science & Business Media
ISBN: 1461253144
Category : Mathematics
Languages : en
Pages : 277
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Book Description
A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem is to describe the relationship between the shape (geo metry) of the drum and its sound (its spectrum). In this book, we restrict ourselves to the one-dimensional case of fractal strings, and their higher dimensional analogues, fractal sprays. We develop a theory of complex di mensions of a fractal string, and we study how these complex dimensions relate the geometry with the spectrum of the fractal string. We refer the reader to [Berrl-2, Lapl-4, LapPol-3, LapMal-2, HeLapl-2] and the ref erences therein for further physical and mathematical motivations of this work. (Also see, in particular, Sections 7. 1, 10. 3 and 10. 4, along with Ap pendix B.) In Chapter 1, we introduce the basic object of our research, fractal strings (see [Lapl-3, LapPol-3, LapMal-2, HeLapl-2]). A 'standard fractal string' is a bounded open subset of the real line. Such a set is a disjoint union of open intervals, the lengths of which form a sequence which we assume to be infinite. Important information about the geometry of . c is contained in its geometric zeta function (c(8) = L lj. j=l 2 Introduction We assume throughout that this function has a suitable meromorphic ex tension. The central notion of this book, the complex dimensions of a fractal string . c, is defined as the poles of the meromorphic extension of (c.
Author: Kenneth Falconer
Publisher: Wiley
ISBN: 9780471957249
Category : Mathematics
Languages : en
Pages : 0
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Book Description
Following on from the success of Fractal Geometry: Mathematical Foundations and Applications, this new sequel presents a variety of techniques in current use for studying the mathematics of fractals. Much of the material presented in this book has come to the fore in recent years. This includes methods for studying dimensions and other parameters of fractal sets and measures, as well as more sophisticated techniques such as thermodynamic formalism and tangent measures. In addition to general theory, many examples and applications are described, in areas such as differential equations and harmonic analysis. This book is mathematically precise, but aims to give an intuitive feel for the subject, with underlying concepts described in a clear and accessible manner. The reader is assumed to be familiar with material from Fractal Geometry, but the main ideas and notation are reviewed in the first two chapters. Each chapter ends with brief notes on the development and current state of the subject. Exercises are included to reinforce the concepts. The author's clear style and up-to-date coverage of the subject make this book essential reading for all those who with to develop their understanding of fractal geometry.
Author: Benoit Mandelbrot
Publisher: Echo Point Books & Media, LLC
ISBN: 9781648370410
Category :
Languages : en
Pages : 0
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Book Description
Written in a style that is accessible to a wide audience, The Fractal Geometry of Nature inspired popular interest in this emerging field. Mandelbrot's unique style, and rich illustrations will inspire readers of all backgrounds.
Author: K. J. Falconer
Publisher: Cambridge University Press
ISBN: 9780521337052
Category : Mathematics
Languages : en
Pages : 184
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Book Description
A mathematical study of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Considers questions of local density, the existence of tangents of such sets as well as the dimensional properties of their projections in various directions.
Author: Gerald A. Edgar
Publisher: Springer Science & Business Media
ISBN: 1475741340
Category : Mathematics
Languages : en
Pages : 252
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Book Description
From the reviews: "In the world of mathematics, the 1980's might well be described as the "decade of the fractal". Starting with Benoit Mandelbrot's remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology. However, the book also contains many good illustrations of fractals (including 16 color plates), together with Logo programs which were used to generate them. ... Here then, at last, is an answer to the question on the lips of so many: 'What exactly is a fractal?' I do not expect many of this book's readers to achieve a mature understanding of this answer to the question, but anyone interested in finding out about the mathematics of fractal geometry could not choose a better place to start looking." #Mathematics Teaching#1
Author: Martin J. Turner
Publisher: Academic Press
ISBN: 9780127039701
Category : Computers
Languages : en
Pages : 352
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Book Description
This book is concerned with the theory and application of fractal geometry in digital imaging. Throughout the book, a series of new approaches to defining fractals are illustrated, such as the analysis of the fractal power spectrum and the use of fractional differentials. Several new algorithms and applications are also discussed and applied to real life images. Fractal Geometry in Digital imaging will appeal to postgraduates, researchers and practitioners in image processing, mathematics and computing, information technology and engineering.
