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[资源] 【书籍】《FRACTAL GEOMETRY-Mathematical Foundations...》Kenneth Falconer

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??FRACTAL GEOMETRY-Mathematical Foundations and Applications(Second Edition) ?·
Kenneth Falconer University of St Andrews,UK
2003 John Wiley&Sons,Ltd ISBNs:0-470-84861-8(HB);0-470-84862-6(PB)

PART I FOUNDATIONS 1
Chapter 1 Mathematical background...............................3
1.1 Basic set theory...................................3
1.2 Functions and limits.................................6
1.3 Measures and mass distributions........................11
1.4 Notes on probability theory............................17
1.5 Notes and references................................24
Exercises........................................25
Chapter 2 Hausdorff measure and dimension.........................27
2.1 Hausdorff measure.................................27
2.2 Hausdorff dimension................................31
2.3 Calculation of Hausdorff dimension??simple examples...........34
*2.4 Equivalent definitions of Hausdorff dimension.................35
*2.5 Finer definitions of dimension...........................36
2.6 Notes and references................................37
Exercises........................................37
Chapter 3 Alternative definitions of dimension.......................39
3.1 Box-counting dimensions.............................41
3.2 Properties and problems of box-counting dimension............47
vvi Contents
*3.3 Modified box-counting dimensions.......................49
*3.4 Packing measures and dimensions.......................50
3.5 Some other definitions of dimension.......................53
3.6 Notes and references................................57
Exercises........................................57
Chapter 4 Techniques for calculating dimensions.....................59
4.1 Basic methods....................................59
4.2 Subsets of finite measure.............................68
4.3 Potential theoretic methods............................70
*4.4 Fourier transform methods.............................73
4.5 Notes and references................................74
Exercises........................................74
Chapter 5 Local structure of fractals...............................76
5.1 Densities........................................76
5.2 Structure of 1-sets..................................80
5.3 Tangents to s-sets..................................84
5.4 Notes and references................................89
Exercises........................................89
Chapter 6 Projections of fractals..................................90
6.1 Projections of arbitrary sets............................90
6.2 Projections of s-sets of integral dimension...................93
6.3 Projections of arbitrary sets of integral dimension..............95
6.4 Notes and references................................97
Exercises........................................97
Chapter 7 Products of fractals....................................99
7.1 Product formulae...................................99
7.2 Notes and references................................107
Exercises........................................107
Chapter 8 Intersections of fractals.................................109
8.1 Intersection formulae for fractals........................110
*8.2 Sets with large intersection............................113
8.3 Notes and references................................118
Exercises........................................119
PART II APPLICATIONS AND EXAMPLES 121
Chapter 9 Iterated function systems??self-similar and self-affine sets....123
9.1 Iterated function systems.............................123
9.2 Dimensions of self-similar sets..........................128vii
9.3 Some variations...................................135
9.4 Self-affine sets....................................139
9.5 Applications to encoding images.........................145
9.6 Notes and references................................148
Exercises........................................149
Chapter 10 Examples from number theory............................151
10.1 Distribution of digits of numbers.........................151
10.2 Continued fractions.................................153
10.3 Diophantine approximation............................154
10.4 Notes and references................................158
Exercises........................................158
Chapter 11 Graphs of functions.....................................160
11.1 Dimensions of graphs................................160
*11.2 Autocorrelation of fractal functions.......................169
11.3 Notes and references................................173
Exercises........................................173
Chapter 12 Examples from pure mathematics.........................176
12.1 Duality and the Kakeya problem.........................176
12.2 Vitushkin??s conjecture...............................179
12.3 Convex functions...................................181
12.4 Groups and rings of fractional dimension....................182
12.5 Notes and references................................184
Exercises........................................185
Chapter 13 Dynamical systems.....................................186
13.1 Repellers and iterated function systems....................187
13.2 The logistic map...................................189
13.3 Stretching and folding transformations.....................193
13.4 The solenoid......................................198
13.5 Continuous dynamical systems..........................201
*13.6 Small divisor theory.................................205
*13.7 Liapounov exponents and entropies.......................208
13.8 Notes and references................................211
Exercises........................................212
Chapter 14 Iteration of complex functions??Julia sets..................215
14.1 General theory of Julia sets............................215
14.2 Quadratic functions??the Mandelbrot set...................223
14.3 Julia sets of quadratic functions.........................227
14.4 Characterization of quasi-circles by dimension................235
14.5 Newton??s method for solving polynomial equations.............237
14.6 Notes and references................................241
Exercises........................................242viii Contents
Chapter 15 Random fractals.......................................244
15.1 A random Cantor set.................................246
15.2 Fractal percolation..................................251
15.3 Notes and references................................255
Exercises........................................256
Chapter 16 Brownian motion and Brownian surfaces...................258
16.1 Brownian motion...................................258
16.2 Fractional Brownian motion............................267
16.3 Le??vy stable processes...............................271
16.4 Fractional Brownian surfaces...........................273
16.5 Notes and references................................275
Exercises........................................276
Chapter 17 Multifractal measures..................................277
17.1 Coarse multifractal analysis............................278
17.2 Fine multifractal analysis..............................283
17.3 Self-similar multifractals..............................286
17.4 Notes and references................................296
Exercises........................................296
Chapter 18 Physical applications...................................298
18.1 Fractal growth....................................300
18.2 Singularities of electrostatic and gravitational potentials..........306
18.3 Fluid dynamics and turbulence..........................307
18.4 Fractal antennas...................................309
18.5 Fractals in finance..................................311
18.6 Notes and references................................315
Exercises........................................316
References...........................................317
Index................................................329
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[ Last edited by parklyn on 2010-3-6 at 09:14 ]

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