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Produktbild: Fractal Geometry

Fractal Geometry Mathematical Foundations and Applications

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Beschreibung

Produktdetails

Einband

Gebundene Ausgabe

Erscheinungsdatum

24.01.2014

Verlag

John Wiley & Sons Inc

Seitenzahl

400

Maße (L/B/H)

23,6/15,4/2,5 cm

Gewicht

629 g

Auflage

3. Auflage

Sprache

Englisch

ISBN

978-1-119-94239-9

Beschreibung

Rezension

"Falconer's book is excellent in many respects and the reviewer strongly recommends it. May every university library own a copy, or three! And if you're a student reading this, go check it out today!." ( Mathematical Association of America , 11 June 2014)

Produktdetails

Einband

Gebundene Ausgabe

Erscheinungsdatum

24.01.2014

Verlag

John Wiley & Sons Inc

Seitenzahl

400

Maße (L/B/H)

23,6/15,4/2,5 cm

Gewicht

629 g

Auflage

3. Auflage

Sprache

Englisch

ISBN

978-1-119-94239-9

Herstelleradresse

Libri GmbH
Europaallee 1
36244 Bad Hersfeld
DE

Email: gpsr@libri.de

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  • Produktbild: Fractal Geometry

  • 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

    14 Iteration of complex functions - Julia sets and the Mandelbrot set 235

    14.1 General theory of Julia sets 235

    14.2 Quadratic functions - the Mandelbrot set 243

    14.3 Julia sets of quadratic functions 248

    14.4 Characterisation of quasi-circles by dimension 256

    14.5 Newton's method for solving polynomial equations 258

    14.6 Notes and references 262

    Exercises 262

    15 Random fractals 265

    15.1 A random Cantor set 266

    15.2 Fractal percolation 272

    15.3 Notes and references 277

    Exercises 277

    16 Brownian motion and Brownian surfaces 279

    16.1 Brownian motion in R 279

    16.2 Brownian motion in R n 285

    16.3 Fractional Brownian motion 289

    16.4 Fractional Brownian surfaces 294

    16.5 Lévy stable processes 296

    16.6 Notes and references 299

    Exercises 299

    17 Multifractal measures 301

    17.1 Coarse multifractal analysis 302

    17.2 Fine multifractal analysis 307

    17.3 Self-similar multifractals 310

    17.4 Notes and references 320

    Exercises 320

    18 Physical applications 323

    18.1 Fractal fingering 325

    18.2 Singularities of electrostatic and gravitational potentials 330

    18.3 Fluid dynamics and turbulence 332

    18.4 Fractal antennas 334

    18.5 Fractals in finance 336

    18.6 Notes and references 340

    Exercises 341

    References 342

    Index 357