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Falconer K. — Fractal geometry: mathematical foundations and applications

Falconer K. — Fractal geometry: mathematical foundations and applications

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Название: Fractal geometry: mathematical foundations and applications

Автор: Falconer K.

Аннотация:

Since its original publication in 1990, Kenneth Falconer's Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. It introduces the general mathematical theory and applications of fractals in a way that is accessible to students from a wide range of disciplines. This new edition has been extensively revised and updated. It features much new material, many additional exercises, notes and references, and an extended bibliography that reflects the development of the subject since the first edition. * Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals. * Each topic is carefully explained and illustrated by examples and figures. * Includes all necessary mathematical background material. * Includes notes and references to enable the reader to pursue individual topics. * Features a wide selection of exercises, enabling the reader to develop their understanding of the theory. * Supported by a Web site featuring solutions to exercises, and additional material for students and lecturers. Fractal Geometry: Mathematical Foundations and Applications is aimed at undergraduate and graduate students studying courses in fractal geometry. The book also provides an excellent source of reference for researchers who encounter fractals in mathematics, physics, engineering, and the applied sciences.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2

Год издания: 2003

Количество страниц: 337

Добавлена в каталог: 12.12.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

Fractally homogeneous turbulence      309
Fractional Brownian motion      173 267 269 267—271
Fractional Brownian surface(s)      273—275
Fractions, continued      153—154
Frostman's lemma      70
Full square      254
Function(s)      6 6—7
Function(s), continuous      10
Function(s), convex      181 181—182
Functional analysis      179
Gauge function      37
Gaussian distribution      23
Gaussian process      267
General construction      61—62 62
Generator      134—135
Generator, examples      132 133 134
Geometric invariance      41
Geometric measure theory      53 76
Graphs of functions      160 160—169 258 266 267
Gravitational potential      70 306—307
Group of transformations      8 110 111
Group(s) of fractional dimension      182-184
Growth      300—306
Hamilton's equations      207
Hamiltonian      207—208
Hamiltonian systems, stability of      207—208 212
Hausdorff dimension      xxiv 27 31 31—33 31 54
Hausdorff dimension and box(-counting) dimension      46 60
Hausdorff dimension and packing dimension      53
Hausdorff dimension and projections      90—93
Hausdorff dimension of a measure      209 288
Hausdorff dimension of attractor      192 193
Hausdorff dimension of self-affine set      140—144 140 144
Hausdorff dimension, Brownian motion      263 265
Hausdorff dimension, calculating      70—72 92
Hausdorff dimension, equivalent definitions      35—36
Hausdorff dimension, fractional Brownian motion      268
Hausdorff dimension, fractional Brownian surface      273
Hausdorff dimension, Levy stable process      272
Hausdorff dimension, properties      32
Hausdorff distance      124 124
Hausdorff measure      14 27—30 28
Hausdorff measure and intersections      113
Hausdorff measure and packing measure      53
Hausdorff measure and product rule      99
Hausdorff measure and quasi-circle      236
Hausdorff metric      124 145 300
Hausdorff — Besicovitch dimension      31
Heat equation      303
Hele-Shaw cell      304
Henon attractor/map      103 196—197 197 198 212 213
Heuristic calculation(s)      34—35 129
Histogram method, for multifractal spectrum      279 283
Hoelder condition      30 32 262 265 268
HoElder exponent      283 312
Hoelder function      8 10 30 161
Hoelder's inequality      297
Homeomorphism      10 33
Homogeneous turbulence      308
Horseshoe map      194—195 195 196 212
Image      7 258 275
Image encoding      145—148
In general      109 110
Independence of events      20
Independence of random variables      20
Independent increment      259 268 311
Index- $\alpha$ fractional Brownian function      273
Index- $\alpha$ fractional Brownian motion      267 267—271 269 311—312
Index- $\alpha$ fractional Brownian surface      273 274
Infimum      5
Information dimension      41
Injection      7
integral      16—17
Integral geometry      118
Interior      6
Interior of loop      223
Intermittency      308
Interpolation      169 170
Intersection formulae      110—113
Intersection(s)      4 109—118 110 265—266 275
Intersection(s), large      113—118 157
interval      4
Interval density      152
Invariance, geometric      41
Invariance, Lipschitz      41 48
Invariant curves      205 207
Invariant measure      208 208—211
Invariant set      123 123—129 187 218
Invariant tori      208
Inverse function      7
Inverse image      7
Investment calculations      186
Irregular point      78
Irregular set      78 79—80 94
Irregular set, examples      xxi 81 180
Isolated point, as attractor      201
Isometry      7 30
Isotropic      261
Isotropic turbulence      307
Iterated construction(s)      95—96 96 180—181
Iterated function system (IFS)      123 123—128
Iterated function system (IFS) and repellers      187—189
Iterated function system (IFS), advantages      128
Iterated function system (IFS), attractor for      123 123—129 146—148 194 228
Iterated function system (IFS), variations      135—139
Iterated venetian blind construction      95—96 96 180
Iteration      186—201 215—242
Jarnik's theorem      155—157 205 207
Jordan curve      53 81
Julia set      xxii 215 215—242 219 233 234
Kakeya problem      176—179
Koch curve      see von Koch curve
Kolmogorov entropy      41
Kolmogorov model of turbulence      307—309
Kolmogorov — Arnold — Moser (KAM) theorem      208
Laminar flow      307
Landscapes      273
Laplace's equation      304 305
Law of Averages      23—24
Lebesgue density theorem      77 77 93
Lebesgue measure      13 16 112 192 264
Lebesgue measure, n-dimensional      13 17 28 112 143 266
Legendre spectrum      282
Legendre transform      281 282 287
Length      13 81
Length, scaling of      29
Level set      266 275
Level-k interval      35 62 152
Level-k set      127
Levy process      267 271—273
Levy stable process      271 271—273
Liapounov exponents      208—211 209 210 212
Lim sup sets      113
LIMIT      8—9
Limit of sequence      5
Limit, lower      9
Limit, upper      9
Line segment, dimension print      56
Line segment, uniform mass distribution on      14
Line set      176 176—179
Linear transformation      8
Lipschitz equivalence      236
Lipschitz function      8 10 30 103
Lipschitz invariance      41 48 55 56
Lipschitz transformation      30 32 34
Local dimension      283
Local product      103 196 202
Local structure of fractals      76—89
Logarithmic density      41
Logarithms      10

Logistic map      189—193 191 192 212
Long range dependence      311
Loop      223 224—225
Loop, closed      201
Lorenz attractor      203—204 204
Lorenz equations      202—203
Lower box(-counting) dimension      41
Lower coarse multifractal spectrum      280
Lower density      77 84
Lower limit      9
Lubrication theory      305
Mandelbrot set      223 223—227 224 230 233 235
Mandelbrot, Benoit      xxii xxv
Mapping(s)      6 8
Markov partition      189
Martingale      248 311
Mass distribution      11 12 277
Mass distribution and distribution of digits      151—152
Mass distribution and product rule      99 101
Mass distribution principle      60 60—61 131 200
Mass distribution, construction by repeated subdivision      14—15 15
Mass distribution, uniform, on line segment      14
Maximum modulus theorem      231
Maximum range      160
Maxwell's equations      310
Mean      21
Mean-value theorem      10 137 190
Measure(s)      11 11—17
Measure(s) on a set      11
Measure(s), $\sigma$ -finite      95
Measure(s), counting      13
Measure(s), Hausdorff      14 27—30 28 53 99 113
Measure(s), Hausdorff dimension of      209 288
Measure(s), invariant      208 208—211
Measure(s), Lebesgue      13 16 28 112 192
Measure(s), multifractal      211 277—296
Measure(s), n-dimensional Lebesgue      13 17 112 143
Measure(s), net      36 68
Measure(s), packing      50—53 51 88
Measure(s), probability      19
Measure(s), restriction of      14
Measure(s), self-similar      278 279 280 286
Measure(s), tangent      89
Method of moments, for multifractal spectrum      280—281 283
Metric dimension      41
Middle $\lambda$ Cantor set      64
Middle third Cantor set      xvii xviii
Middle third Cantor set and repellers      188 189
Middle third Cantor set and self-similarity      123 124 129
Middle third Cantor set and tangents      87
Middle third Cantor set as attractor      189
Middle third Cantor set in intersections      112
Middle third Cantor set, box(-counting) dimension      47
Middle third Cantor set, construction of      xviii 127
Middle third Cantor set, features      xviii 123
Middle third Cantor set, generalization of      63—64
Middle third Cantor set, Hausdorff dimension      34—35 60 61
Middle third Cantor set, product      99 100
Minkowski content      45
Minkowski( — Bouligand) dimension      46
Modified box(-counting) dimension      49 49—50
Modified box(-counting) dimension, upper, and packing dimension      51—52
Modified von Koch curve      132 133—134
Moment sum      280 281
Monotonicity of box(-)counting dimension      48
Monotonicity of dimension print      55
Monotonicity of Hausdorff dimension      32 41
Monotonicity of packing dimension      51
Montel's theorem      218—219 221
Moser's twist theorem      207
Mountain skyline      271 298
Multifractal spectrum      277—286 283 289 292—294 315
Multifractal time      312 313 314
Multifractal(s)      211 277—296
Multifractal(s), coarse analysis      277 278—283
Multifractal(s), fine analysis      277 283—286
Multifractal(s), self-similar      286—296
Multifractional Brownian motion      271
Multiple points      265 275
Multivariate normal variable(s)      267
Natural fractals      xxvi 146 147 298—300
Natural logarithms      10
Navier — Stokes equations      203 307 309
Neighbourhood      4 5 45 see
Net measure      36 68
Neural networks      277
Newton's method      237—241
Non-linear Cantor set      136—137 154
Non-removable set      180 181
Non-singular transformation      8
Normal distribution      23 260
Normal family      218 218—219
Normal family at a point      218
Normal numbers      151
Number theory      151—158
Often      110
One-sided dimension      54
One-to-one correspondence      7
One-to-one function      7
Onto function      7
Open ball      4
Open interval      4 138
Open set      5 32 41 187
Open set condition      129 130—134 249
Orbit      186 189 216 228
Orthogonal projection      xxv 34 90—97 91 176 177
Packing dimension      50—53 51 284
Packing dimension and modified upper box dimension      51—52
Packing measure      50—53 51 88
Packing measure and Hausdorff measure      53
Parallel body      4
Parseval's theorem      73
Partial quotient      153
Percolation      251—255 252 253 254
Perfect set      220 242
Period      216
Period doubling      191—193
Period-p orbit      216 232 235
Period-p point      186 191
Periodic orbit      228 232
Periodic point      216 221
Phase transition      251
Physical applications      xxvi 298—316
Pinch point      232
Plane cross section      202
Plant growth      300
Poincare section      202
Poincare — Bendixson theorem      201 202
Point mass      13
Pointwise convergence      10
Poisson's equation      307
Polynomials, Newton's method for      237—241
Population dynamics      186 190 193
Porous medium, flow through      305
Potential theoretic methods      70—72 92 111 248—249
Power spectrum      171 270
Prandtl number      203
Pre-fractal(s)      126 127
Pre-image      7
probability      18
Probability density function      23
Probability measure      18 19
Probability space      19
Probability theory      17—24
Probability, conditional      19
PRODUCT      99
Product formula      99—107
Product, cartesian      4 87 99 100
Projection theorems      90—93 180
Projection(s)      90—97 91

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