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Fractal-Based Methods in Analysis 2012 [Kõva köide]

Franklin Mendivil, Edward R. Vrscay, Davide La Torre, Herb Kunze

Formaat: Hardback, 408 pages, kõrgus x laius: 235x155 mm, kaal: 799 g, XVI, 408 p., 1 Hardback
Ilmumisaeg: 17-Nov-2011
Kirjastus: Springer-Verlag New York Inc.
ISBN-10: 1461418909
ISBN-13: 9781461418900

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Formaat: Hardback, 408 pages, kõrgus x laius: 235x155 mm, kaal: 799 g, XVI, 408 p., 1 Hardback
Ilmumisaeg: 17-Nov-2011
Kirjastus: Springer-Verlag New York Inc.
ISBN-10: 1461418909
ISBN-13: 9781461418900

Teised raamatud teemal:

Püsilink: https://www.kriso.ee/db/9781461418900.html

Märksõnad:

Fractal-based methods are at the heart of modeling the behavior of phenomena at varying scales. This volume collates techniques for using IFS fractals, including the very latest cutting-edge methods, from more than 20 years of research in this area.

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 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 of mathematics at the University of Guelph in Ontario. Davide La Torre is an associate professor of mathematics in the Department of Economics, Management and Quantitative Methods of the University of Milan. Franklin Mendivil is a professor of mathematics at Acadia University in Nova Scotia. Edward Vrscay is a professor in the department of Applied Mathematics at the University of Waterloo in Ontario. The major focus of their research is on fractals and the applications of fractals.

Arvustused

From the reviews:

This book intends to introduce the prospective reader to a variety of fractal-based methods in analysis. The book is accessible to graduate students with a solid understanding of real analysis, functional analysis, and probability theory, but it seems to be primarily aimed at mathematicians who like to employ fractal-based methodologies in their research. (Peter R. Massopust, Mathematical Reviews, September, 2013)

1 What do we mean by "Fractal-Based Analysis"?

(20)

1.1 Fractal transforms and self-similarity

(5)

1.2 Self-similarity: A brief historical review

(4)

1.2.1 The construction of self-similar sets

(3)

1.2.2 The construction of self-similar measures

(1)

1.3 Induced fractal transforms

(4)

1.4 Inverse problems for fractal transforms and "collage coding"

(5)

1.4.1 Fractal image coding

(3)

2 Basic IFS Theory

(66)

2.1 Contraction mappings and fixed points

(6)

2.1.1 Inverse problem

(2)

2.2 Iterated Function System (IFS)

(16)

2.2.1 Motivating example: The Cantor set

(3)

2.2.2 Space of compact subsets and the Hausdorff metric

(4)

2.2.3 Definition of IFS

(5)

2.2.4 Collage theorem for IFS

(1)

2.2.5 Continuous dependence of the attractor

(3)

2.3 Code space and the address map

(5)

2.4 The chaos game

(3)

2.5 IFS with probabilities

(18)

2.5.1 IFSP and invariant measures

(11)

2.5.2 Moments of the invariant measure and M

(3)

2.5.3 The ergodic theorem for IFSP

(4)

2.6 Some classical extensions

(18)

2.6.1 IFS with condensation

(2)

2.6.2 Fractal interpolation functions

(2)

2.6.3 Graph-directed IFS

(5)

2.6.4 IFS with infinitely many maps

(8)

3 IFS on Spaces of Functions

(38)

3.1 Motivation: Fractal imaging

(5)

3.2 IFS on functions

(10)

3.2.1 Uniformly contractive IFSM

(3)

3.2.2 IFSM on Lp(X)

(3)

3.2.3 Affine IFSM

(1)

3.2.4 IFSM with infinitely many maps

(1)

3.2.5 Progression from geometric IFS to IFS on functions

100

(2)

3.3 IFS on wavelets

102

(9)

3.3.1 Brief wavelet introduction

103

(2)

3.3.2 IFS operators on wavelets (IFSW)

105

(2)

3.3.3 Correspondence between IFSW and IFSM

107

(4)

3.4 IFS and integral transforms

111

(14)

3.4.1 Fractal transforms of integral transforms

113

(1)

3.4.2 Induced fractal operators on fractal transforms

114

(2)

3.4.3 The functional equation for the kernel

116

(3)

3.4.4 Examples

119

(6)

4 IFS, Multifunctions, and Measure-Valued Functions

125

(24)

4.1 IMS and IMS with probabilities

125

(5)

4.1.1 Code space

129

(1)

4.2 Iterated function systems on multifunctions

130

(10)

4.2.1 Spaces of multifunctions

130

(2)

4.2.2 Some IFS operators on multifunctions (IFSMF)

132

(3)

4.2.3 An application to fractal image coding

135

(5)

4.3 Iterated function systems on measure-valued images

140

(9)

4.3.1 A fractal transform operator on measure-valued images

141

(4)

4.3.2 Moment relations induced by the fractal transform operator

145

(4)

5 IFS on Spaces of Measures

149

(64)

5.1 Signed measures

150

(13)

5.1.1 Complete space of signed measures

151

(2)

5.1.2 IFS operator on signed measures

153

(3)

5.1.3 "Generalized measures" as dual objects in Lip(X,R)

156

(4)

5.1.4 Noncompact case

160

(3)

5.2 Vector-valued measures

163

(27)

5.2.1 Complete space of vector measures

166

(2)

5.2.2 IFS on vector measures

168

(7)

5.2.3 Coloured fractals

175

(4)

5.2.4 Line integrals on fractal curves

179

(3)

5.2.5 Generalized vector measures

182

(1)

5.2.6 Green's theorem for planar domains with fractal boundaries

183

(3)

5.2.7 Some generalizations for vector measures

186

(4)

5.3 Set-valued measures

190

(23)

5.3.1 Complete space of multimeasures

193

(3)

5.3.2 IFS operators on multimeasures

196

(4)

5.3.3 Generalizations for spaces of multimeasures

200

(2)

5.3.4 Union-additive multimeasures

202

(4)

5.3.5 IFS on union-additive multimeasures

206

(3)

5.3.6 Generalities on union-additive multimeasures

209

(3)

5.3.7 Extension of finitely union-additive multimeasures

212

(1)

6 The Chaos Game

213

(30)

6.1 Chaos game for IFSM

214

(7)

6.1.1 Chaos game for nonoverlapping IFSM

214

(3)

6.1.2 Chaos game for overlapping IFSM

217

(4)

6.2 Chaos game for wavelets

221

(11)

6.2.1 Rendering a compactly supported scaling function

222

(2)

6.2.2 Modified chaos game algorithm for wavelet generation

224

(2)

6.2.3 Chaos game for wavelet analysis

226

(2)

6.2.4 Chaos game for wavelet synthesis

228

(2)

6.2.5 Some extensions

230

(2)

6.3 Chaos game for multifunctions and multimeasures

232

(11)

6.3.1 Chaos game for fractal measures with fractal densities

232

(2)

6.3.2 Chaos game for multifunctions

234

(5)

6.3.3 Chaos game for multimeasures

239

(4)

7 Inverse Problems and Fractal-Based Methods

243

(72)

7.1 Ordinary differential equations

244

(21)

7.1.1 Inverse problem for ODEs

248

(2)

7.1.2 Practical Considerations and examples

250

(9)

7.1.3 Multiple, partial, and noisy data sets

259

(6)

7.2 Two-point boundary value problems

265

(11)

7.2.1 Inverse problem for two-point BVPs

268

(1)

7.2.2 Practical considerations and examples

269

(7)

7.3 Quasilinear partial differential equations

276

(8)

7.3.1 Inverse problems for traffic and fluid flow

282

(2)

7.4 Urison integral equations

284

(5)

7.4.1 Inverse problem for Urison integral equations

286

(3)

7.5 Hammerstein integral equations

289

(6)

7.5.1 Inverse problem for Hammerstein integral equations

291

(4)

7.6 Random fixed-point equations

295

(7)

7.6.1 Inverse problem for random DEs

296

(6)

7.7 Stochastic differential equations

302

(1)

7.7.1 Inverse problem for SDEs

303

(1)

7.8 Applications

303

(12)

7.8.1 Population dynamics

303

(2)

7.8.2 mRNA and protein concentration

305

(2)

7.8.3 Bacteria and amoeba interaction

307

(2)

7.8.4 Tumor growth

309

(6)

8 Further Developments and Extensions

315

(42)

8.1 Generalized collage theorems for PDEs

315

(26)

8.1.1 Elliptic PDEs

318

(14)

8.1.2 Parabolic PDEs

332

(2)

8.1.3 Hyperbolic PDEs

334

(3)

8.1.4 An application: A vibrating string driven by a stochastic process

337

(4)

8.2 Self-similar objects in cone metric spaces

341

(16)

8.2.1 Cone metric space

341

(2)

8.2.2 Scalarizations of cone metrics

343

(6)

8.2.3 Cone with empty interior

349

(1)

8.2.4 Applications to image processing

350

(7)

A Topological and Metric Spaces

357

(12)

A.1 Sets

357

(1)

A.2 Topological spaces

358

(1)

A.2.1 Basic definitions

358

(1)

A.2.2 Convergence, countability, and separation axioms

359

(2)

A.2.3 Compactness

361

(1)

A.2.4 Continuity and connectedness

361

(2)

A.3 Metric spaces

363

(1)

A.3.1 Sequences in metric spaces

363

(1)

A.3.2 Bounded, totally bounded, and compact sets

364

(1)

A.3.3 Continuity

365

(1)

A.3.4 Spaces of compact subsets

366

(3)

B Basic Measure Theory

369

(16)

B.1 Introduction

369

(1)

B.2 Measurable spaces and measures

370

(1)

B.2.1 Construction of measures

371

(3)

B.2.2 Measures on metric spaces and Hausdorff measures

374

(1)

B.3 Measurable functions and integration

375

(1)

B.3.1 The integral

376

(3)

B.4 Signed measures

379

(1)

B.5 Weak convergence of measures

380

(2)

B.5.1 Monge-Kantorovich metric

382

(1)

B.6 Lp spaces

383

(2)

C Basic Notions from Set-Valued Analysis

385

(6)

C.1 Basic definitions

385

(1)

C.1.1 Contractive multifunctions, fixed points, and collage theorems

386

(2)

C.1.2 Convexity and multifunctions

388

(1)

C.2 Integral of multifunctions

388

(3)

References

391

(12)

Index

403

Franklin Mendivil is a Professor in the Department of Mathematics and Statistics at Acadia University; Herb Kunze is a Professor in the Department of Mathematics and Statistics at Guelph University; Davide La Torre is an Associate Professor in the Department of Economics, Business and Statistics at University of Milan; Edward R. Vrscay is a Professor at the University of Waterloo.

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