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【书籍】《FRACTAL GEOMETRY-Mathematical Foundations...》Kenneth Falconer
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?????????é????·????ó????·ò????????·??í???ó?????????????????? ·????????? ·??????? ???§?? ???? ??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 https://d.namipan.com/d/1cfdcaf542b16319a6b4ae7e2c16e299d50188fb48483700 [ Last edited by parklyn on 2010-3-6 at 09:14 ] |
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