Elementary Introduction to Spatial and Temporal Fractals

1. Introduction.- 2. Fundamental Concepts and Definitions.- 3. Examples of Fractal Geometry.- 3.1 Cantor Set (0 < dF < 1).- 3.2 Rugged Lines (1 < dF ? 2).- 3.2.1 Koch Curve and Lake.- 3.2.2 Fractal Dimension of a Rugged Profile.- 3.2.3 Multi-fractal Dimensions.- 3.3 Irregular Surface (2 < dF ? 3).- 3.3.1 Monolayers of Different Adsorbates.- 3.3.2 Monolayers on Adsorbent Particles of Different Sizes.- 3.3.3 Pore-size Distribution.- 3.4 Growth Processes.- 3.4.1 Eden (Surface Growth) Model.- 3.4.2 Diffusion-limited Aggregation (DLA).- 4. Fractals in Time.- 4.1 Change of Commodity Prices.- 4.1.1 Stable (Levy) Distribution.- 4.2 Fractional Brownian Motion.- 4.2.1 Self-affinity.- 4.2.2 Discrete-time Fractional Noise.- 4.2.3 Rescaled Range Analysis.- 4.3 Fractal Stochastic Processes.- 4.3.1 Bernoulli's Scaling and the St. Petersburg Paradox.- 4.3.2 Fractal Random Walk.- 4.3.3 Fractal Time.- 5. Fractals in Chaos.- 5.1 Quantification of Chaos.- 5.1.1 Stochastic or Chaotic.- 5.1.2 Calculation of Attractor Dimension from a Time Series.- 5.1.3 Lyapunov Exponent and Limits of Predictability.- 6. Epilog.- A1. Appendix 1: Perimeter Fractal Dimension of Char Particles from a Downdraft Gasifier Through Image Analysis.- A1.1 Theoretical.- A1.1.1 Perimeter Fractal Dimension.- Al.1.2 Geometrical Bases for Measurement.- A1.2 Experimental.- A1.2.1 Sample Generation.- A1.2.2 Sample Preparation.- A1.2.3 Measurement of Feret Diameter.- A1.2.4 Evaluation of Perimeter Fractal Dimension.- A1.3 Results and Discussion.- A1.3.1 Comparison with Other Methods.- A1.3.2 Evaluation of Various Methods.- A2. Appendix 2: Surface Fractal Dimension of Rice Hull-Derived Charcoal from a Fluidized-Bed Reactor.- A2.1 Theoretical.- A2.2 Experimental.- A2.3 Results and Discussion.- A3. Appendix 3: Fractal Analysis of Pressure Fluctuations in Multiphase Flow Systems.- A3.1 Theoretical.- A3.1.1 Discrete-time Fractional Noise.- A3.1.2 Rescaled Range Analysis.- A3.2 Experimental.- A3.2.1 Facilities.- A3.2.2 Measurements and Computations.- A3.3 Results and Discussion.- Nomenclature.- Literature Cited.- Author Index.