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Falconer K. — Fractal Geometry. Mathematical Foundations and applications
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Íàçâàíèå: Fractal Geometry. Mathematical Foundations and applicationsÀâòîð: Falconer K. Àííîòàöèÿ: An accessible introduction to fractals, useful as a text or reference. Part I is concerned with the general theory of fractals and their geometry, covering dimensions and their methods of calculation, plus the local form of fractals and their projections and intersections. Part II contains examples of fractals drawn from a wide variety of areas of mathematics and physics, including self-similar and self-affine sets, graphs of functions, examples from number theory and pure mathematics, dynamical systems, Julia sets, random fractals and some physical applications. Also contains many diagrams and illustrative examples, includes computer drawings of fractals, and shows how to produce further drawings for themselves.
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /Ãåîìåòðèÿ è òîïîëîãèÿ /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Ãîä èçäàíèÿ: 1990Êîëè÷åñòâî ñòðàíèö: 288Äîáàâëåíà â êàòàëîã: 14.11.2004Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
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Ïðåäìåòíûé óêàçàòåëü
( 141 190—191 ( 25 ( 4 41 113 (s-)capacity 66 (s-)energy 64 (s-)potential 64 1-set 73—77 79—80 86—87 Affine transformation 8 126 Affinity 8 Almost all 15 Almost everywhere 15 Almost surely 18 Analytic function 126 164 197—223 Area 13 26 Attractive orbit 209 223 Attractive point 197 Attractor 170 173—188 Autocorrelation 155—158 247 Autocorrelation function 156 Average 20 Avogadro's number 237 Baker's transformation 177 177—178 193—195 Ball 3 Basic interval 31 56 Basic set 116 Basin of attraction 171 203 Bi-lipschitz function 8 30 217 Bifurcation diagram 176 Bijection 6 Binary interval 15 33 Borel set 6 Boundary 5 Bounded variation 79 Box(-counting) dimension 38 38—49 41 255 259 267 Branching process 230 Brownian motion 33—34 237—253 238 239 265 268—270 Brownian motion, index- 245—248 246 Brownian motion, multiple points 243 Brownian surface 250 250—252 Cantor dust xvi 31 Cantor product 94 96 Cantor set, middle third xiii xiv 31—32 43 54 55 104 113 116 173 Cantor set, non-linear 124 140 Cantor set, random 225—230 235—236 Cantor set, uniform 58 95 Cantor target 96 Cat map 196 Central limit theorem 22 238 Chaos 171 173 Chaotic attractor 171 Characteristic function 248 Chernoff's theorem 261 Class 104—109 105 143—144 Closed ball 3 Closed set 5 Coastline 50 265 Codomain 6 Collage theorem 134—135 Compact set 5 Complement of a set 4 Composition of functions 7 Computer drawings 116—117 135 208 215—216 239 247—248 252 Conditional probability 18 Conformal mapping 126 222 Congruence 7 103—104 Conjugacy 204 Connected component 6 Connected set 6 233 Content, Minkowski 42 Content, s-dimensional 42 Continued fractions 139—141 140 144—145 Continuous function 9 contours 252 Contraction 113 113—136 Convergence 5 8 Convergence pointwise 10 Convergence uniform 10 16 Convex surface 165—167 Convolution theorem 67 157 Copper sulphate 267—272 Correlation 155—158 247 274 Covering lemma 60 Critical point 208 209 223 Cross section 185 cube 4 Curve 49 74 Curve, fractal 117 122 Curve, Jordan 49 74 Curve, rectifiable 74 74—78 166 Curve-free set 75 75—77 Curve-like set 75 75—77 Darcy's law 272 Data compression 132—137 Decomposition of 1-sets 73 Dendrite 215 Dense set 5 density 60—62 69 69—73 77 81—82 139 Density lower 69—77 70 Density upper 69—77 70 Derivative 9 Diameter 5 25 Difference of sets 4 Differentiability 9 77—79 146 159 165—167 Differentiability, continuous 9 Diffusion equation 270 Diffusion Limited Aggregation 267—272 277 Digital sundial 89—90 DIMENSION xix—xxi 36—53 Dimension box(-counting) 38 38—49 41 255 259 267 Dimension capacity 38 Dimension divider 36 49 49—50 Dimension entropy 38 Dimension function 33 Dimension of attractors and repellers 170—188 191—196 Dimension of graphs of functions 146—155 Dimension of intersections 101—110 Dimension of products 92—100 Dimension of projections 83—91 Dimension of random sets 224—231 237—253 Dimension of self-affine sets 99 126—133 Dimension of self-similar sets xix 117—123 Dimension print 50—52 Dimension, approximations to 267 Dimension, calculation of 54—68 Dimension, experimental estimation of 36—37 265—267 Dimension, finer definitions 33—34 Dimension, Fourier 67 Dimension, Hausdorff 28—33 29 37 Dimension, Hausdorff — Besicovitch 29 Dimension, information 38 260 Dimension, lower box(-counting) 38 38—49 41 Dimension, metric 38 Dimension, Minkowski 42 Dimension, modified box-counting 45—49 46 Dimension, one-sided 50 Dimension, packing 47 47—49 Dimension, similarity xix Dimension, upper box(-counting) 38 38—49 41 Dimensionally homogeneous set 46 Diophantine approximation 108—109 141—14 Diophantine equations 145 Direct congruence 7 Distance set 168 Distribution of digits 34 138—139 Distribution, Gaussian 22 Distribution, multidimensional normal 241 Distribution, normal 22 Distribution, uniform 22 Domain 6 Duality 161—164 Dynamical systems 170—196 254 263 Dynamical systems, continuous 184—188 Dynamical systems, discrete 170 170—184 188—196 Egoroff's theorem 16 Electrical discharge 272 Electrolysis 267—272 Electrostatic potential 64 272—273 entropy 38 192 192—194 260 Euclidean distance 3 Euclidean space 3 Event space 17 Expectation 20 Expectation equation 227 230 Expectation, conditional 21 Experiment (probabilistic) 17 Experimental approach to fractals 36—37 263 265—267 Fatou set 198 198—223 Figure of eight 205 First return map 185 Fixed point 170 197 Fluid dynamics 273—276 Fourier series 188—190 Fourier transform 66 66—67 157—158 Fractal growth 267—272 Fractal, definition of xviii—xxii Fractional Brownian motion 245—248 Full square 233 Function 6—10 Functional analysis 163—164 Gaussian distribution 22 General construction 56 56—60 Generator 122 Geometric measure theory 49 69—82 Graphs of functions 146 146—160 237 Gravitational potential 64 272—273 Group 167—168 Group of transformations 8 101—104 Growth 267—272 Hamilton's equations 190 Hamiltonian 190—191 Hausdorff dimension 28—33 29 37 Hausdorff dimension of a measure 192 Hausdorff measure 25 25—28 Hausdorff metric 114 Henon map 179—181 195 Heuristic calculation 31—32 117—118 Hoelder function 8 27—29 28 147 241 246 250 Homeomorphism 9 30 Horseshoe 178—179 Image 6 Image, encoding 132—137 Independence of events 18 Independence of random variables 19 Independent increments 238 248 Indifferent point 197 Infimum 4 Injection 6 integral 15—16 Integral geometry 109—110 Interior 5 Intermittency 274—276 Interpolation 154 Intersection 4 101—110 243 Intersection, large 104—109 143—144 interval 4 Invariance, geometric 37 Invariance, Lipschitz 37 217 Invariant measure 191 191—194 Invariant set 113 131—137 171—174 200 209 Invariant tori 190—191 Inverse function 7 Inverse image 6 Irregular point 70 73 Irregular set 70 70—82 86—87 164 Isometry 7 Isotropic 239 Iterated function scheme 113 131—137 171—174 209 Iterated venetian blind construction 88—90 165 Iteration 170—184 191—223 J-set 29 63 69—73 77 80 Jarnik's theorem 142—143 190 Jordan curve 49 74 Julia set xvii 197 197—223 204 Kakeya problem 161—164 Kam theorem 191 Koch curve see “von Koch curve” Kolmogorov entropy 38 Kolmogorov model of turbulence 274 Laminar flow 274 Laplace's equation 271 Lebesgue density theorem 69 Lebesgue measure 12 15 26 Lebesgue measure, n-dimensional 12 Legendre transform pair 259 Length 12 26 74 Level sets 245 248 252 Liapounov exponents 191—194 192 LIMIT 5 6—10 8 Limit lower 8 Limit upper 8 Line set 161 161—164 Linear transformation 7 Lipschitz function 8 28 30 Lipschitz invariance 37 217 Local product 96 180 185 Local structure 69—82 Logarithmic density 38 Logarithms 10 Logistic map 173—176 195 Loop 205 Lorenz attractor 186 Lorenz equations 186 Mandelbrot set 204 204—217 Mapping 6 Martingale 228 Mass distribution 10—16 11 Mass distribution, construction by repeated subdivision 13—15 Mass distribution, principle 55 Mean 20 Mean value theorem 9 Measure 10 10—16 Measure net 33 62 Measure on a set 11 Measure, counting 12 Measure, Hausdorff 25 25—28 Measure, Lebesgue 12 15 26 Measure, Lebesgue n-dimensional 12 Measure, packing 47 47—49 81 Measure, restriction of 13 Minkowski content 42 Monotonicity 37 Montel's theorem 199 Moser's twist theorem 190 Multifractal measures 254—264 Multifractal spectrum 259 261 Multiple points 243 Natural fractals xxi 135 265—267 Navier — Stokes equation 186 273 276 Neighbourhood 5 Net measure 33 62 Newton's method 219—222 Normal distribution 22 Normal family 198 198—204 Normal family at a point 199 Normal numbers 138 Number theory 138—145 One-to-one correspondence 6 One-to-one function 6 Onto function 6 Open ball 3
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