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Falconer K. Ч Fractal Geometry. Mathematical Foundations and applications
Falconer K. Ч Fractal Geometry. Mathematical Foundations and applications

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Ќазвание: Fractal Geometry. Mathematical Foundations and applications

јвтор: Falconer K.

јннотаци€:

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: en

–убрика: ћатематика/√еометри€ и топологи€/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1990

 оличество страниц: 288

ƒобавлена в каталог: 14.11.2004

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
($\alpha$-)well approximable numbers      141 190Ч191
($\delta$-)cover      25
($\delta$-)parallel body      4 41 113
(s-)capacity      66
(s-)energy      64
(s-)potential      64
1-set      73Ч77 79Ч80 86Ч87
Affine transformation      8 126
Affinity      8
Almost all      15
Almost everywhere      15
Almost surely      18
Analytic function      126 164 197Ч223
Area      13 26
Attractive orbit      209 223
Attractive point      197
Attractor      170 173Ч188
Autocorrelation      155Ч158 247
Autocorrelation function      156
Average      20
Avogadro's number      237
Baker's transformation      177 177Ч178 193Ч195
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 38Ч49 41 255 259 267
Branching process      230
Brownian motion      33Ч34 237Ч253 238 239 265 268Ч270
Brownian motion, index-$\alpha$      245Ч248 246
Brownian motion, multiple points      243
Brownian surface      250 250Ч252
Cantor dust      xvi 31
Cantor product      94 96
Cantor set, middle third      xiii xiv 31Ч32 43 54 55 104 113 116 173
Cantor set, non-linear      124 140
Cantor set, random      225Ч230 235Ч236
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 $C^S$      104Ч109 105 143Ч144
Closed ball      3
Closed set      5
Coastline      50 265
Codomain      6
Collage theorem      134Ч135
Compact set      5
Complement of a set      4
Composition of functions      7
Computer drawings      116Ч117 135 208 215Ч216 239 247Ч248 252
Conditional probability      18
Conformal mapping      126 222
Congruence      7 103Ч104
Conjugacy      204
Connected component      6
Connected set      6 233
Content, Minkowski      42
Content, s-dimensional      42
Continued fractions      139Ч141 140 144Ч145
Continuous function      9
contours      252
Contraction      113 113Ч136
Convergence      5 8
Convergence pointwise      10
Convergence uniform      10 16
Convex surface      165Ч167
Convolution theorem      67 157
Copper sulphate      267Ч272
Correlation      155Ч158 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 74Ч78 166
Curve-free set      75 75Ч77
Curve-like set      75 75Ч77
Darcy's law      272
Data compression      132Ч137
Decomposition of 1-sets      73
Dendrite      215
Dense set      5
density      60Ч62 69 69Ч73 77 81Ч82 139
Density lower      69Ч77 70
Density upper      69Ч77 70
Derivative      9
Diameter      5 25
Difference of sets      4
Differentiability      9 77Ч79 146 159 165Ч167
Differentiability, continuous      9
Diffusion equation      270
Diffusion Limited Aggregation      267Ч272 277
Digital sundial      89Ч90
DIMENSION      xixЧxxi 36Ч53
Dimension box(-counting)      38 38Ч49 41 255 259 267
Dimension capacity      38
Dimension divider      36 49 49Ч50
Dimension entropy      38
Dimension function      33
Dimension of attractors and repellers      170Ч188 191Ч196
Dimension of graphs of functions      146Ч155
Dimension of intersections      101Ч110
Dimension of products      92Ч100
Dimension of projections      83Ч91
Dimension of random sets      224Ч231 237Ч253
Dimension of self-affine sets      99 126Ч133
Dimension of self-similar sets      xix 117Ч123
Dimension print      50Ч52
Dimension, approximations to      267
Dimension, calculation of      54Ч68
Dimension, experimental estimation of      36Ч37 265Ч267
Dimension, finer definitions      33Ч34
Dimension, Fourier      67
Dimension, Hausdorff      28Ч33 29 37
Dimension, Hausdorff Ч Besicovitch      29
Dimension, information      38 260
Dimension, lower box(-counting)      38 38Ч49 41
Dimension, metric      38
Dimension, Minkowski      42
Dimension, modified box-counting      45Ч49 46
Dimension, one-sided      50
Dimension, packing      47 47Ч49
Dimension, similarity      xix
Dimension, upper box(-counting)      38 38Ч49 41
Dimensionally homogeneous set      46
Diophantine approximation      108Ч109 141Ч14
Diophantine equations      145
Direct congruence      7
Distance set      168
Distribution of digits      34 138Ч139
Distribution, Gaussian      22
Distribution, multidimensional normal      241
Distribution, normal      22
Distribution, uniform      22
Domain      6
Duality      161Ч164
Dynamical systems      170Ч196 254 263
Dynamical systems, continuous      184Ч188
Dynamical systems, discrete      170 170Ч184 188Ч196
Egoroff's theorem      16
Electrical discharge      272
Electrolysis      267Ч272
Electrostatic potential      64 272Ч273
entropy      38 192 192Ч194 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      36Ч37 263 265Ч267
Fatou set      198 198Ч223
Figure of eight      205
First return map      185
Fixed point      170 197
Fluid dynamics      273Ч276
Fourier series      188Ч190
Fourier transform      66 66Ч67 157Ч158
Fractal growth      267Ч272
Fractal, definition of      xviiiЧxxii
Fractional Brownian motion      245Ч248
Full square      233
Function      6Ч10
Functional analysis      163Ч164
Gaussian distribution      22
General construction      56 56Ч60
Generator      122
Geometric measure theory      49 69Ч82
Graphs of functions      146 146Ч160 237
Gravitational potential      64 272Ч273
Group      167Ч168
Group of transformations      8 101Ч104
Growth      267Ч272
Hamilton's equations      190
Hamiltonian      190Ч191
Hausdorff dimension      28Ч33 29 37
Hausdorff dimension of a measure      192
Hausdorff measure      25 25Ч28
Hausdorff metric      114
Henon map      179Ч181 195
Heuristic calculation      31Ч32 117Ч118
Hoelder function      8 27Ч29 28 147 241 246 250
Homeomorphism      9 30
Horseshoe      178Ч179
Image      6
Image, encoding      132Ч137
Independence of events      18
Independence of random variables      19
Independent increments      238 248
Indifferent point      197
Infimum      4
Injection      6
integral      15Ч16
Integral geometry      109Ч110
Interior      5
Intermittency      274Ч276
Interpolation      154
Intersection      4 101Ч110 243
Intersection, large      104Ч109 143Ч144
interval      4
Invariance, geometric      37
Invariance, Lipschitz      37 217
Invariant measure      191 191Ч194
Invariant set      113 131Ч137 171Ч174 200 209
Invariant tori      190Ч191
Inverse function      7
Inverse image      6
Irregular point      70 73
Irregular set      70 70Ч82 86Ч87 164
Isometry      7
Isotropic      239
Iterated function scheme      113 131Ч137 171Ч174 209
Iterated venetian blind construction      88Ч90 165
Iteration      170Ч184 191Ч223
J-set      29 63 69Ч73 77 80
Jarnik's theorem      142Ч143 190
Jordan curve      49 74
Julia set      xvii 197 197Ч223 204
Kakeya problem      161Ч164
Kam theorem      191
Koch curve      see Уvon Koch curveФ
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      191Ч194 192
LIMIT      5 6Ч10 8
Limit lower      8
Limit upper      8
Line set      161 161Ч164
Linear transformation      7
Lipschitz function      8 28 30
Lipschitz invariance      37 217
Local product      96 180 185
Local structure      69Ч82
Logarithmic density      38
Logarithms      10
Logistic map      173Ч176 195
Loop      205
Lorenz attractor      186
Lorenz equations      186
Mandelbrot set      204 204Ч217
Mapping      6
Martingale      228
Mass distribution      10Ч16 11
Mass distribution, construction by repeated subdivision      13Ч15
Mass distribution, principle      55
Mean      20
Mean value theorem      9
Measure      10 10Ч16
Measure net      33 62
Measure on a set      11
Measure, counting      12
Measure, Hausdorff      25 25Ч28
Measure, Lebesgue      12 15 26
Measure, Lebesgue n-dimensional      12
Measure, packing      47 47Ч49 81
Measure, restriction of      13
Minkowski content      42
Monotonicity      37
Montel's theorem      199
Moser's twist theorem      190
Multifractal measures      254Ч264
Multifractal spectrum      259 261
Multiple points      243
Natural fractals      xxi 135 265Ч267
Navier Ч Stokes equation      186 273 276
Neighbourhood      5
Net measure      33 62
Newton's method      219Ч222
Normal distribution      22
Normal family      198 198Ч204
Normal family at a point      199
Normal numbers      138
Number theory      138Ч145
One-to-one correspondence      6
One-to-one function      6
Onto function      6
Open ball      3
1 2
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