Format:
Online Ressource (408 S.)
Edition:
Online-Ausg. Springer eBook Collection. Mathematics and Statistics Electronic reproduction; Available via World Wide Web
ISBN:
1283446413
,
1461418917
,
9781283446419
,
9781461418917
Content:
The idea of modeling the behavior of phenomena at multiple scales has become a useful tool in both pure and applied mathematics. Fractal-based techniques lie at the heart of this area, as fractals are inherently multiscale objects; they very often describe nonlinear phenomena better than traditional mathematical models. In many cases they have been used for solving inverse problems arising in models described by systems of differential equations and dynamical systems. ¡ "Fractal-Based Methods in Analysis" draws together, for the first time in book form, methods and results from almost twenty years of research in this topic, including new viewpoints and results in many of the chapters.¡ For each topic the theoretical framework is carefully explained using examples and applications. ¡ The second chapter on basic iterated function systems theory is designed to be used as the basis for a course and includes many exercises.¡ This chapter, along with the three background appendices on topological and metric spaces, measure theory, and basic results from set-valued analysis, make the book suitable for self-study or as a source book for a graduate course. The other chapters illustrate many extensions and applications of fractal-based methods to different areas. This book is intended for graduate students and researchers in applied mathematics, engineering and social sciences. ¡ Herb Kunze is a Professor in the Department of Mathematics and Statistics, University of Guelph.¡ Davide La Torre is an Associate Professor in the Department of Economics, Business and Statistics, University of Milan.¡¡ Franklin Mendivil is a Professor in the Department of¡Mathematics and Statistics, Acadia University.¡Edward R. Vrscay is a Professor in the Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo.¡ A major focus of their research is fractals and their applications
Content:
The idea of modeling the behaviour of phenomena at multiple scales has become a useful tool in both pure and applied mathematics. Fractal-based techniques lie at the heart of this area, as fractals are inherently multiscale objects; they very often describe nonlinear phenomena better than traditional mathematical models. In many cases they have been used for solving inverse problems arising in models described by systems of differential equations and dynamical systems. "Fractal-Based Methods in Analysis" draws together, for the first time in book form, methods and results from almost
Note:
Includes bibliographical references and index
,
Fractal-Based Methodsin Analysis; Preface; Contents; Chapter 1 What do we mean by "Fractal-Based Analysis"?; 1.1 Fractal transforms and self-similarity; 1.2 Self-similarity: A brief historical review; 1.2.1 The construction of self-similar sets; 1.2.2 The construction of self-similar measures; 1.3 Induced fractal transforms; 1.4 Inverse problems for fractal transforms and "collage coding"; 1.4.1 Fractal image coding; Chapter 2 Basic IFS Theory; 2.1 Contraction mappings and fixed points; 2.1.1 Inverse problem; 2.2 Iterated Function System (IFS); 2.2.1 Motivating example: The Cantor set
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2.2.2 Space of compact subsets and the Hausdorff metric2.2.3 Definition of IFS; 2.2.4 Collage theorem for IFS; 2.2.5 Continuous dependence of the attractor; 2.3 Code space and the address map; 2.4 The chaos game; 2.5 IFS with probabilities, self-similar measures, and the ergodic theorem; 2.5.1 IFSP and invariant measures; 2.5.1.1 Continuous dependence; 2.5.2 Moments of the invariant measure and M*; 2.5.3 The ergodic theorem for IFSP; 2.6 Some classical extensions; 2.6.1 IFS with condensation; 2.6.2 Fractal interpolation functions; 2.6.3 Graph-directed IFS; 2.6.4 IFS with infinitely many maps
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2.6.4.1 IFS indexed by a compact set2.6.4.2 IFSP indexed by a compact set; Chapter 3 IFS on Spaces of Functions; 3.1 Motivation: Fractal imaging; 3.2 IFS on functions; 3.2.1 Uniformly contractive IFSM; 3.2.2 IFSM on Lp(X); 3.2.2.1 The case of fractal block-coding and local IFSM; 3.2.3 Affine IFSM; 3.2.4 IFSM with infinitely many maps; 3.2.4.1 Uniformly contractive f?; 3.2.4.2 IFSM in Lp(X); 3.2.5 Progression from geometric IFS to IFS on functions; 3.3 IFS on wavelets; 3.3.1 Brief wavelet introduction; 3.3.2 IFS operators on wavelets (IFSW); 3.3.3 Correspondence between IFSW and IFSM
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3.3.3.1 IFSM for IFSW of Example 3.73.3.3.2 IFSM for IFSW of Example 3.8; 3.4 IFS and integral transforms; 3.4.1 Fractal transforms of integral transforms; 3.4.1.1 Derivation of a functional equation for the kernel; 3.4.2 Induced fractal operators on fractal transforms; 3.4.3 The functional equation for the kernel; 3.4.4 Examples; 3.4.4.1 Fourier transform; 3.4.4.2 Wavelet transform; 3.4.4.3 Lebesgue transform; 3.4.4.4 Moments of measures; Chapter 4 Iterated Function Systems, Multifunctions, and Measure-Valued Functions
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4.1 Iterated multifunction systems and iterated multifunction systems with probabilities4.1.1 Code space; 4.2 Iterated function systems on multifunctions; 4.2.1 Spaces of multifunctions; 4.2.2 Some IFS operators on multifunctions (IFSMF); 4.2.3 An application to fractal image coding; 4.3 Iterated function systems on measure-valued images; 4.3.1 A fractal transform operator on measure-valued images; 4.3.2 Moment relations induced by the fractal transform operator; Chapter 5 IFS on Spaces of Measures; 5.1 Signed measures; 5.1.1 Complete space of signed measures
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5.1.2 IFS operator on signed measures
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Electronic reproduction; Available via World Wide Web
Additional Edition:
ISBN 1461418909
Additional Edition:
ISBN 1283446367
Additional Edition:
ISBN 1461418909
Additional Edition:
ISBN 9781461418900
Additional Edition:
ISBN 9781461418900
Additional Edition:
Druckausg. Fractal-based methods in analysis New York : Springer, 2012 ISBN 1461418909
Additional Edition:
ISBN 9781461418900
Language:
English
Subjects:
Mathematics
Keywords:
Analysis
;
Ableitung gebrochener Ordnung
DOI:
10.1007/978-1-4614-1891-7
URL:
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