Hyperspaces And Fractals

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Fractals and Hyperspaces

Author : Keith R. Wicks
Publisher : Springer
ISBN 13 : 354046610X
Total Pages : 175 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Fractals and Hyperspaces by : Keith R. Wicks

Download or read book Fractals and Hyperspaces written by Keith R. Wicks and published by Springer. This book was released on 2006-11-14 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: Addressed to all readers with an interest in fractals, hyperspaces, fixed-point theory, tilings and nonstandard analysis, this book presents its subject in an original and accessible way complete with many figures. The first part of the book develops certain hyperspace theory concerning the Hausdorff metric and the Vietoris topology, as a foundation for what follows on self-similarity and fractality. A major feature is that nonstandard analysis is used to obtain new proofs of some known results much more slickly than before. The theory of J.E. Hutchinson's invariant sets (sets composed of smaller images of themselves) is developed, with a study of when such a set is tiled by its images and a classification of many invariant sets as either regular or residual. The last and most original part of the book introduces the notion of a "view" as part of a framework for studying the structure of sets within a given space. This leads to new, elegant concepts (defined purely topologically) of self-similarity and fractality: in particular, the author shows that many invariant sets are "visually fractal", i.e. have infinite detail in a certain sense. These ideas have considerable scope for further development, and a list of problems and lines of research is included.

Hyperspaces and Fractals

Author : Christophe Segers
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (98 download)

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Book Synopsis Hyperspaces and Fractals by : Christophe Segers

Download or read book Hyperspaces and Fractals written by Christophe Segers and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Clifford Wavelets, Singular Integrals, and Hardy Spaces

Author : Marius Mitrea
Publisher : Springer
ISBN 13 : 9783540578840
Total Pages : 136 pages
Book Rating : 4.5/5 (788 download)

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Book Synopsis Clifford Wavelets, Singular Integrals, and Hardy Spaces by : Marius Mitrea

Download or read book Clifford Wavelets, Singular Integrals, and Hardy Spaces written by Marius Mitrea and published by Springer. This book was released on 1994-04-28 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book discusses the extensions of basic Fourier Analysis techniques to the Clifford algebra framework. Topics covered: construction of Clifford-valued wavelets, Calderon-Zygmund theory for Clifford valued singular integral operators on Lipschitz hyper-surfaces, Hardy spaces of Clifford monogenic functions on Lipschitz domains. Results are applied to potential theory and elliptic boundary value problems on non-smooth domains. The book is self-contained to a large extent and well-suited for graduate students and researchers in the areas of wavelet theory, Harmonic and Clifford Analysis. It will also interest the specialists concerned with the applications of the Clifford algebra machinery to Mathematical Physics.

Fractals and Spectra

Author : Hans Triebel
Publisher : Springer Science & Business Media
ISBN 13 : 3034800339
Total Pages : 278 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Fractals and Spectra by : Hans Triebel

Download or read book Fractals and Spectra written by Hans Triebel and published by Springer Science & Business Media. This book was released on 2010-10-28 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with the symbiotic relationship between the theory of function spaces, fractal geometry, and spectral theory of (fractal) pseudodifferential operators as it has emerged quite recently. Most of the presented material is published here for the first time.

Classics On Fractals

Author : Gerald A. Edgar
Publisher : CRC Press
ISBN 13 : 042969122X
Total Pages : 384 pages
Book Rating : 4.4/5 (296 download)

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Book Synopsis Classics On Fractals by : Gerald A. Edgar

Download or read book Classics On Fractals written by Gerald A. Edgar and published by CRC Press. This book was released on 2019-03-08 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: Read the masters! Experience has shown that this is good advice for the serious mathematics student. This book contains a selection of the classical mathematical papers related to fractal geometry. For the convenience of the student or scholar wishing to learn about fractal geometry, nineteen of these papers are collected here in one place. Twelve of the nineteen have been translated into English from German, French, or Russian. In many branches of science, the work of previous generations is of interest only for historical reasons. This is much less so in mathematics.1 Modern-day mathematicians can learn (and even find good ideas) by reading the best of the papers of bygone years. In preparing this volume, I was surprised by many of the ideas that come up.

A Concise Introduction to Hypercomplex Fractals

Author : Andrzej Katunin
Publisher : CRC Press
ISBN 13 : 135180121X
Total Pages : 91 pages
Book Rating : 4.3/5 (518 download)

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Book Synopsis A Concise Introduction to Hypercomplex Fractals by : Andrzej Katunin

Download or read book A Concise Introduction to Hypercomplex Fractals written by Andrzej Katunin and published by CRC Press. This book was released on 2017-10-05 with total page 91 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents concisely the full story on complex and hypercomplex fractals, starting from the very first steps in complex dynamics and resulting complex fractal sets, through the generalizations of Julia and Mandelbrot sets on a complex plane and the Holy Grail of the fractal geometry – a 3D Mandelbrot set, and ending with hypercomplex, multicomplex and multihypercomplex fractal sets which are still under consideration of scientists. I tried to write this book in a possibly simple way in order to make it understandable to most people whose math knowledge covers the fundamentals of complex numbers only. Moreover, the book is full of illustrations of generated fractals and stories concerned with great mathematicians, number spaces and related fractals. In the most cases only information required for proper understanding of a nature of a given vector space or a construction of a given fractal set is provided, nevertheless a more advanced reader may treat this book as a fundamental compendium on hypercomplex fractals with references to purely scientific issues like dynamics and stability of hypercomplex systems.

Fractals and Universal Spaces in Dimension Theory

Author : Stephen Lipscomb
Publisher : Springer
ISBN 13 : 9780387854939
Total Pages : 242 pages
Book Rating : 4.8/5 (549 download)

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Book Synopsis Fractals and Universal Spaces in Dimension Theory by : Stephen Lipscomb

Download or read book Fractals and Universal Spaces in Dimension Theory written by Stephen Lipscomb and published by Springer. This book was released on 2008-12-03 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: Historically, for metric spaces the quest for universal spaces in dimension theory spanned approximately a century of mathematical research. The history breaks naturally into two periods - the classical (separable metric) and the modern (not-necessarily separable metric). The classical theory is now well documented in several books. This monograph is the first book to unify the modern theory from 1960-2007. Like the classical theory, the modern theory fundamentally involves the unit interval. Unique features include: * The use of graphics to illustrate the fractal view of these spaces; * Lucid coverage of a range of topics including point-set topology and mapping theory, fractal geometry, and algebraic topology; * A final chapter contains surveys and provides historical context for related research that includes other imbedding theorems, graph theory, and closed imbeddings; * Each chapter contains a comment section that provides historical context with references that serve as a bridge to the literature. This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. Being the first monograph to focus on the connection between generalized fractals and universal spaces in dimension theory, it will be a natural text for graduate seminars or self-study - the interested reader will find many relevant open problems which will create further research into these topics.

Fractals and Universal Spaces in Dimension Theory

Author : Stephen Lipscomb
Publisher : Springer Science & Business Media
ISBN 13 : 0387854940
Total Pages : 259 pages
Book Rating : 4.3/5 (878 download)

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Book Synopsis Fractals and Universal Spaces in Dimension Theory by : Stephen Lipscomb

Download or read book Fractals and Universal Spaces in Dimension Theory written by Stephen Lipscomb and published by Springer Science & Business Media. This book was released on 2008-10-28 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: Historically, for metric spaces the quest for universal spaces in dimension theory spanned approximately a century of mathematical research. The history breaks naturally into two periods - the classical (separable metric) and the modern (not-necessarily separable metric). The classical theory is now well documented in several books. This monograph is the first book to unify the modern theory from 1960-2007. Like the classical theory, the modern theory fundamentally involves the unit interval. Unique features include: * The use of graphics to illustrate the fractal view of these spaces; * Lucid coverage of a range of topics including point-set topology and mapping theory, fractal geometry, and algebraic topology; * A final chapter contains surveys and provides historical context for related research that includes other imbedding theorems, graph theory, and closed imbeddings; * Each chapter contains a comment section that provides historical context with references that serve as a bridge to the literature. This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. Being the first monograph to focus on the connection between generalized fractals and universal spaces in dimension theory, it will be a natural text for graduate seminars or self-study - the interested reader will find many relevant open problems which will create further research into these topics.

Some Novel Types of Fractal Geometry

Author : Stephen Semmes
Publisher : Oxford University Press
ISBN 13 : 9780198508069
Total Pages : 180 pages
Book Rating : 4.5/5 (8 download)

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Book Synopsis Some Novel Types of Fractal Geometry by : Stephen Semmes

Download or read book Some Novel Types of Fractal Geometry written by Stephen Semmes and published by Oxford University Press. This book was released on 2001 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with fractal geometries that have features similar to ones of ordinary Euclidean spaces, while at the same time being quite different from Euclidean spaces.. A basic example of this feature considered is the presence of Sobolev or Poincaré inequalities, concerning the relationship between the average behavior of a function and the average behavior of its small-scale oscillations. Remarkable results in the last few years through Bourdon-Pajot and Laakso have shown that there is much more in the way of geometries like this than have been realized, only examples related to nilpotent Lie groups and Carnot metrics were known previously. On the other had, 'typical' fractals that might be seen in pictures do not have these same kinds of features. This text examines these topics in detail and will interest graduate students as well as researchers in mathematics and various aspects of geometry and analysis.

Cantor Set Approximations and Dimension Computations in Hyperspaces

Author : Matthew G. Zapf
Publisher :
ISBN 13 :
Total Pages : 244 pages
Book Rating : 4.:/5 (785 download)

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Book Synopsis Cantor Set Approximations and Dimension Computations in Hyperspaces by : Matthew G. Zapf

Download or read book Cantor Set Approximations and Dimension Computations in Hyperspaces written by Matthew G. Zapf and published by . This book was released on 2011 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: Given a metric space (K, d), the hyperspace of K is defined by H(K) = {F c K: F is compact, F =/ 0}. H(K) is itself a metric space under the Hausdorff metric dH. Hyperspaces have been extensively studied by topologists since the 1970's, but the measure theoretical study of hyperspaces has lagged, Boardman and Goodey concurrently provided a characterization of a one-parameter family of Hausdorff gauge functions that determine the dimension of H([0, 1]), and this result was extended by McClure to H(X) where X is a self-similar fractal satisfying the Open Set Condition. This dissertation further generalizes these results to include graph-self-similar and self-conformal fractals satisfying the Open Set Condition in Rd. In Chapter 2 it is shown that the dimensions of the underlying fractals may be approximated by the dimensions of sets invariant under particularly constructed subiterated function systems that satisfy the Strong Separation Condition. In Chapter 3, a one-parameter family of gauge functions is constructed which computes the dimensions of the hyperspaces of graph-self-similar sets that satisfy the Strong Separation Condition, after which the approximations of Chapter 2 are applied to extend the result to graph-self-similar sets which satisfy the Open Set Condition. The analogous results for self-conformal sets that satisfy the Open Set Condition are developed in Chapter 4.

Secrets of the Extra Dimensions

Author :
Publisher :
ISBN 13 :
Total Pages : 64 pages
Book Rating : 4.:/5 (837 download)

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Book Synopsis Secrets of the Extra Dimensions by :

Download or read book Secrets of the Extra Dimensions written by and published by . This book was released on 2009 with total page 64 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Analysis on Fractals

Author : Jun Kigami
Publisher : Cambridge University Press
ISBN 13 : 0521793211
Total Pages : 238 pages
Book Rating : 4.5/5 (217 download)

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Book Synopsis Analysis on Fractals by : Jun Kigami

Download or read book Analysis on Fractals written by Jun Kigami and published by Cambridge University Press. This book was released on 2001-06-07 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers analysis on fractals, a developing area of mathematics which focuses on the dynamical aspects of fractals, such as heat diffusion on fractals and the vibration of a material with fractal structure. The book provides a self-contained introduction to the subject, starting from the basic geometry of self-similar sets and going on to discuss recent results, including the properties of eigenvalues and eigenfunctions of the Laplacians, and the asymptotical behaviors of heat kernels on self-similar sets. Requiring only a basic knowledge of advanced analysis, general topology and measure theory, this book will be of value to graduate students and researchers in analysis and probability theory. It will also be useful as a supplementary text for graduate courses covering fractals.

Lectures On Fractal Geometry

Author : Martina Zaehle
Publisher : World Scientific
ISBN 13 : 9811283354
Total Pages : 141 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Lectures On Fractal Geometry by : Martina Zaehle

Download or read book Lectures On Fractal Geometry written by Martina Zaehle and published by World Scientific. This book was released on 2023-12-27 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on a series of lectures at the Mathematics Department of the University of Jena, developed in the period from 1995 up to 2015. It is completed by additional material and extensions of some basic results from the literature to more general metric spaces.This book provides a clear introduction to classical fields of fractal geometry, which provide some background for modern topics of research and applications. Some basic knowledge on general measure theory and on topological notions in metric spaces is presumed.

Fractals Everywhere

Author : Michael F. Barnsley
Publisher : Academic Press
ISBN 13 : 148325769X
Total Pages : 565 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Fractals Everywhere by : Michael F. Barnsley

Download or read book Fractals Everywhere written by Michael F. Barnsley and published by Academic Press. This book was released on 2014-05-10 with total page 565 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fractals Everywhere, Second Edition covers the fundamental approach to fractal geometry through iterated function systems. This 10-chapter text is based on a course called "Fractal Geometry", which has been taught in the School of Mathematics at the Georgia Institute of Technology. After a brief introduction to the subject, this book goes on dealing with the concepts and principles of spaces, contraction mappings, fractal construction, and the chaotic dynamics on fractals. Other chapters discuss fractal dimension and interpolation, the Julia sets, parameter spaces, and the Mandelbrot sets. The remaining chapters examine the measures on fractals and the practical application of recurrent iterated function systems. This book will prove useful to both undergraduate and graduate students from many disciplines, including mathematics, biology, chemistry, physics, psychology, mechanical, electrical, and aerospace engineering, computer science, and geophysical science.

Fractal Functions, Fractal Surfaces, and Wavelets

Author : Peter R. Massopust
Publisher : Academic Press
ISBN 13 : 0128044705
Total Pages : 428 pages
Book Rating : 4.1/5 (28 download)

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Book Synopsis Fractal Functions, Fractal Surfaces, and Wavelets by : Peter R. Massopust

Download or read book Fractal Functions, Fractal Surfaces, and Wavelets written by Peter R. Massopust and published by Academic Press. This book was released on 2016-09-02 with total page 428 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fractal Functions, Fractal Surfaces, and Wavelets, Second Edition, is the first systematic exposition of the theory of local iterated function systems, local fractal functions and fractal surfaces, and their connections to wavelets and wavelet sets. The book is based on Massopust’s work on and contributions to the theory of fractal interpolation, and the author uses a number of tools—including analysis, topology, algebra, and probability theory—to introduce readers to this exciting subject. Though much of the material presented in this book is relatively current (developed in the past decades by the author and his colleagues) and fairly specialized, an informative background is provided for those entering the field. With its coherent and comprehensive presentation of the theory of univariate and multivariate fractal interpolation, this book will appeal to mathematicians as well as to applied scientists in the fields of physics, engineering, biomathematics, and computer science. In this second edition, Massopust includes pertinent application examples, further discusses local IFS and new fractal interpolation or fractal data, further develops the connections to wavelets and wavelet sets, and deepens and extends the pedagogical content. Offers a comprehensive presentation of fractal functions and fractal surfaces Includes latest developments in fractal interpolation Connects fractal geometry with wavelet theory Includes pertinent application examples, further discusses local IFS and new fractal interpolation or fractal data, and further develops the connections to wavelets and wavelet sets Deepens and extends the pedagogical content

Measure, Topology, and Fractal Geometry

Author : Gerald Edgar
Publisher : Springer Science & Business Media
ISBN 13 : 0387747494
Total Pages : 293 pages
Book Rating : 4.3/5 (877 download)

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Book Synopsis Measure, Topology, and Fractal Geometry by : Gerald Edgar

Download or read book Measure, Topology, and Fractal Geometry written by Gerald Edgar and published by Springer Science & Business Media. This book was released on 2007-10-23 with total page 293 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on a course given to talented high-school students at Ohio University in 1988, this book is essentially an advanced undergraduate textbook about the mathematics of fractal geometry. It nicely bridges the gap between traditional books on topology/analysis and more specialized treatises on fractal geometry. The book treats such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology. It takes into account developments in the subject matter since 1990. Sections are clear and focused. The book contains plenty of examples, exercises, and good illustrations of fractals, including 16 color plates.

Besov Spaces on Fractals

Author : Per Bylund
Publisher : LAP Lambert Academic Publishing
ISBN 13 : 9783843369633
Total Pages : 124 pages
Book Rating : 4.3/5 (696 download)

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Book Synopsis Besov Spaces on Fractals by : Per Bylund

Download or read book Besov Spaces on Fractals written by Per Bylund and published by LAP Lambert Academic Publishing. This book was released on 2010-10 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: A physical state in a domain is often described by a model containing a linear partial differential equation. As an example of this, consider the steady state temperature distribution in a homogenous isotropic body. The problem, called Dirichlet's problem, is to find a function u, given that u=f in the interior of the body and u=g on the surface (where u denotes the laplacian of u). The solution depends on f and g, but also on the geometry of the surface S. If the given functions f and g, as well as the subset S of 3-space, are smooth enough, then there exists a unique solution. However, since there are numerous non-smooth structures in nature, it is clear that the study of Dirichlet's problem in the case when f, g and S are less smooth becomes an important task. Function spaces defined on subsets of n-space originates from the study of Dirichlet's problem in the non-smooth case of f, g and S. An important class of functions in this respect are Besov spaces, defined in n-space in the 60's. In the 80's Besov spaces were extended to d-sets, typically fractal sets with non-integer local dimension d. In this book we extend Besov space theory to sets with varying local dimension."