Fractal Geometry - Kenneth Falconer

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Auteur(s) : Kenneth Falconer

Editeur : John Wiley & Sons

Langue : Anglais

Parution : 01/01/2014

Format : Moyen, de 350g à 1kg

Nombre de pages : 400

Expédition : 629

Dimensions : 23.6 x 15.4 x 2.5

ISBN : 111994239X

Résumé :

The seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions.

Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals.? The book introduces and develops the general theory and applications of fractals in a way that is accessible to students and researchers from a wide range of disciplines.

Fractal Geometry: Mathematical Foundations and Applications is an excellent course book for undergraduate and graduate students studying fractal geometry, with suggestions for material appropriate for a first course indicated. The book also provides an invaluable foundation and reference for researchers who encounter fractals not only in mathematics but also in other areas across physics, engineering and the applied sciences.

• Provides a comprehensive and accessible introduction to the mathematical theory and???? applications of fractals
• Carefully explains each topic using illustrative examples and diagrams
• Includes the necessary mathematical background material, along with notes and references to enable the reader to pursue individual topics
• Features a wide range of exercises, enabling readers to consolidate their understanding
• Supported by a website with solutions to exercises and additional material www.wileyeurope.com/fractal

Leads onto the more advanced sequel Techniques in Fractal Geometry (also by Kenneth Falconer and available from Wiley)

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Biographie:
Kenneth Falconer, University of St Andrews, UK....

Sommaire:

Preface to the first edition ix

Preface to the second edition xiii

Preface to the third edition xv

Course suggestions xvii

Introduction xix

Part I Foundations 1

1 Mathematical background 3

1.1 Basic set theory 3

1.2 Functions and limits 7

1.3 Measures and mass distributions 11

1.4 Notes on probability theory 17

1.5 Notes and references 24

Exercises 24

2 Box-counting dimension 27

2.1 Box-counting dimensions 27

2.2 Properties and problems of box-counting dimension 34

2.3 Modified box-counting dimensions 38

2.4 Some other definitions of dimension 40

2.5 Notes and references 41

Exercises 42

3 Hausdorff and packing measures and dimensions 44

3.1 Hausdorff measure 44

3.2 Hausdorff dimension 47

3.3 Calculation of Hausdorff dimension - simple examples 51

3.4 Equivalent definitions of Hausdorff dimension 53

3.5 Packing measure and dimensions 54

3.6 Finer definitions of dimension 57

3.7 Dimension prints 58

3.8 Porosity 60

3.9 Notes and references 63

Exercises 64

4 Techniques for calculating dimensions 66

4.1 Basic methods 66

4.2 Subsets of finite measure 75

4.3 Potential theoretic methods 77

4.4 Fourier transform methods 80

4.5 Notes and references 81

Exercises 81

5 Local structure of fractals 83

5.1 Densities 84

5.2 Structure of 1-sets 87

5.3 Tangents to s-sets 92

5.4 Notes and references 96

Exercises 96

6 Projections of fractals 98

6.1 Projections of arbitrary sets 98

6.2 Projections of s-sets of integral dimension 101

6.3 Projections of arbitrary sets of integral dimension 103

6.4 Notes and references 105

Exercises 106

7 Products of fractals 108

7.1 Product formulae 108

7.2 Notes and references 116

Exercises 116

8 Intersections of fractals 118

8.1 Intersection formulae for fractals 119

8.2 Sets with large intersection 122

8.3 Notes and references 128

Exercises 128

Part II Applications and Examples 131

9 Iterated function systems - self-similar and self-affine sets 133

9.1 Iterated function systems 133

9.2 Dimensions of self-similar sets 139

9.3 Some variations 143

9.4 Self-affine sets 149

9.5 Applications to encoding images 155

9.6 Zeta functions and complex dimensions 158

9.7 Notes and references 167

Exercises 167

10 Examples from number theory 169

10.1 Distribution of digits of numbers 169

10.2 Continued fractions 171

10.3 Diophantine approximation 172

10.4 Notes and references 176

Exercises 176

11 Graphs of functions 178

11.1 Dimensions of graphs 178

11.2 Autocorrelation of fractal functions 188

11.3 Notes and references 192

Exercises 192

12 Examples from pure mathematics 195

12.1 Duality and the Kakeya problem 195

12.2 Vitushkin's conjecture 198

12.3 Convex functions 200

12.4 Fractal groups and rings 201

12.5 Notes and references 204

Exercises 204

13 Dynamical systems 206

13.1 Repellers and iterated function systems 208

13.2 The logistic map 209

13.3 Stretching and folding transformations 213

13.4 The solenoid 217

13.5 Continuous dynamical systems 220

13.6 Small divisor theory 225

13.7 Lyapunov exponents and entropies 228

13.8 Notes and references 231

Exercises 232