Falconer K. — Fractal Geometry. Mathematical Foundations and applications :: Электронная библиотека попечительского совета мехмата МГУ

Главная Ex Libris Книги Журналы Статьи Серии Каталог Wanted Загрузка ХудЛит Справка Поиск по индексам Поиск Форум

Авторизация

Поиск по указателям

Falconer K. — Fractal Geometry. Mathematical Foundations and applications

Обсудите книгу на научном форуме

Нашли опечатку?
Выделите ее мышкой и нажмите Ctrl+Enter

Название: Fractal Geometry. Mathematical Foundations and applications

Автор: Falconer K.

Аннотация:

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.

Язык:

Рубрика: Математика/Геометрия и топология/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1990

Количество страниц: 288

Добавлена в каталог: 14.11.2004

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

Предметный указатель

( $\alpha$ -)well approximable numbers      141 190—191
( $\delta$ -)cover      25
( $\delta$ -)parallel body      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- $\alpha$       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

1 2

О проекте