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8# |
$a Machine generated contents note: 1. Fractals and Fractal Distributions 1 -- 1.1 The Fractal Dimension3 -- 1.2 Determining Fractal Dimensions8 -- 1.3 Fractal Distributions13 -- 1.4 Fractals or Fractal Distributions? 17 -- 1.5 Are Fractals Useful?19 -- 1.6 Where do Fractals Come From? 21 -- 2. Recognizing Power-Law Distributions25 -- 2.1 Maximum Likelihood, Least Squares, and Linear Regression 26 -- 2.2 Do Cumulative Size Distributions Tell the Truth?28 -- 2.3 Binning31 -- 2.4 Censoring37 -- 3. Self-Affine Time Series41 -- 3.1 Brownian Motion42 -- 3.2 White Noise44 -- 3.3 Fourier Transforms46 -- 3.4 Fractional Brownian Motion48 -- 3.5 Generating FBM51 -- 3.6 Scaling Properties of FBM52 -- 3.7 Self-Affine Scale Invariance and Fractal Dimensions55 -- 3.8 Recognizing FBM56 -- 3.9 The Variogram Analysis59 -- 3.10 Predictability64 -- 4. Deterministic Chaos67 -- 4.1 The Lorenz Equations68 -- 4.2 The Physics Behind the Lorenz Equations69 -- 4.3 Phase Space, Attractors, and Bifurcations74 -- 4.4 Limit Cycles and Strange Attractors77 -- 4.5 The Lyapunov Exponent78 -- 4.6 Does it Matter whether God Plays Dice?84 -- 4.7 Deterministic Chaos and Self-Affine Fractals85 -- 5. Self-Organized Criticality87 -- 5.1 Critical-Point Phenomena88 -- 5.2 The Bak-Tang-Wiesenfeld Model90 -- 5.3 The Critical State98 -- 5.4 What is SOC?99 -- 5.5 Sandpile Dynamics and the BTW Model102 -- 6. The Forest-Fire Model - Tuning and Universality109 -- 6.1 The Forest-Fire Model109 -- 6.2 Universality119 -- 6.3 Non-Equilibrium States in SOC Systems122 -- 7. Earthquakes and Stick-Slip Motion125 -- 7.1 The Fractal Character of Earthquakes127 -- 7.2 The Burridge-Knopoff Model130 -- 7.3 Separation of Time Scales133 -- 7.4 Cellular Automata135 -- 7.5 The Olami-Feder-Christensen Model138 -- 7.6 Boundary Conditions in the OFC Model142 -- 7.7 Efficient Simulation of the OFC Model143 -- 7.8 Is the OFC Model Self-Organized Critical?145 -- 7.9 Rupture Area and Seismic Moment149 -- 7.10 The Temporal Fingerprint of the OFC Model152 -- 7.11 How Complex is the OFC Model?161 -- 8. Landslides163 -- 8.1 Fractal Properties of Landslides164 -- 8.2 Are Landslides like Sandpile Avalanches?167 -- 8.3 Data, Models, and Reality173 -- 8.4 The Role of Time-Dependent Weakening175 -- 8.5 On Predicting Slope Stability184 -- 8.6 Are SOC and Universality Important in Landform Evolution? 187 -- 9. Drainage Networks189 -- 9.1 Fractal Properties of Drainage Networks189 -- 9.2 Discharge, Drainage Areas, and Water Balance196 -- 9.3 Peano's Basin198 -- 9.4 Random-Walk Approaches200 -- 9.5 Drainage Networks and Landform Evolution201 -- 9.6 Optimal Channel Networks216 -- 9.7 Drainage Networks and Self-Organized Criticality220 -- 9.8 Optimization by Permanent Reorganization?233 -- 10. SOC and Nothing Else?235 -- 10.1 Ensembles of SOC systems236 -- 10.2 SOC in Pre-Structured Systems240 -- 10.3 Highly Optimized Tolerance245 -- 11. Where do we Stand?253 -- A. Numerics of Ordinary Differential Equations255. |
