Contents 8
Preface M. M. Novak 12
Selected Topics in Mathematics, Physics, and Finance Originating in Fractal Geometry B. B. Mandelbrot 14
1 Introductory comments of various kinds 14
2 Complex Brownian bridge; Brownian cluster and the dimension 4/3 of its boundary; the self-avoiding plane Brownian motion 20
3 Explosive multiplication of new fractal constructions, dimensions (including negative ones), and Holder exponents 23
4 Tools of fractal analysis other than the dimensions: ramification and lacunarity 26
5 Fractality of the major fractal clusters in statistical physics 30
6 Interrelations between fractality and smooth variability: some cases may have a common origin in the usual partial differential equations 32
7 Iterates of the complex map z2 + c. Julia and Mandelbrot sets 36
8 Limit sets of Kleinian groups 38
9 The study of power law probability distributions and the notion that variability and randomness can fall into distinct \ 41
10 The variation of financial prices 42
11 The directly useful fractal 44
12 Fractals in the college and school classroom 45
Acknowledgements 45
References 46
A Renewal Process of Mittag-Leffler Type F. Mainardi, R. Gorenflo and E. Scalas 48
1 Essentials of renewal theory 48
2 The Poisson process as a renewal process 51
3 A fractional generalization of the renewal Poisson process 52
4 The Mittag-Leffler distribution as limit for thinned renewal processes 55
5 Conclusions 56
Acknowledgements 57
References 59
On the Activity of Absorbing Irregular Interfaces J. S. Andrade Jr., H. F. Da Silva, E. A. Henrique and B. Sapoval 60
1 Introduction 60
2 Laplacian Transport and the Theorem of Makarov 61
3 The Nonequilibrium Molecular Dynamics Model 64
4 The Random-Walk Model 67
5 Summary 68
Acknowledgments 68
References 68
Fractal Deformation Using Displacement Vectors and Their Increasing Rates Based on Extended Unit Iterated Shuffle Transformation T. Fujimoto and N. Chiba 70
1 Introduction 70
2 Extended Unit Iterated Shuffle Transformation 71
3 Fractal Deformation by Applying IST to Displacement Vectors 71
4 Fractal Deformation by Applying IST to Increasing Rates of Displacement Vectors 74
5 Conclusion and Future Work 81
References 81
Multifractal and Stochastic Analysis of Electropolished Surfaces M. Haase, A. Mora and B. Lehle 82
1 Introduction 82
2 Characteristic length scales and scaling regions 84
3 Multifractal analysis based on wavelets 85
4 Stochastic Analysis 87
5 Conclusions 90
References 91
A Method for Numerical Estimation of Generalized Renyi Dimensions of Affine Recurrent IFS Invariant Measures T. Martyn 92
1 Introduction 92
2 Preliminaries 93
3 Lattice approximation of RIFS invariant measure 96
4 Computing Renyi dimensions 99
5 Discussion of results 101
Acknowledgments 103
References 103
Nonlinear Dynamics and Prediction of the Caspian Sea Level N. G. Makarenko, L. M. Karimova, Y. B. Kuandykov and M. M. Novak 104
1 Introduction 104
2 Caspian Sea Level Data Description and Methods of Analysis 105
3 CSL Forecasting by Artificial Neural Networks 110
4 Conclusions 113
5 Acknowledgments 114
References 114
Self-similarity in Plants: Integrating Mathematical and Biological Perspectives P. Prusinkiewicz 116
1 Introduction 116
2 Identity-in-parallel and iterated function systems 117
3 L-systems, recurrence systems, and self-similarity 118
4 Catenative formulas 122
5 Data-flow network representation of recurrence systems 123
6 Extension to branching structures 124
7 Relation between topological and geometric self-similarity 125
8 Conclusions 129
Acknowledgments 130
References 130
Cognitive Scale-free Networks as a Model for Intermittency in Human Natural Language P. Allegrini, P. Grigolini and L. Palatella 132
1 Introduction 132
2 DE and concepts 134
3 Scale-free networks, intermittency and the Zipf's law 139
4 Conclusions 141
References 142
The Complexity of Biological Ageing D. Stauffer 144
1 Introduction 144
2 Demography 144
3 Why do we age? 148
4 Simulations of mutation accumulation 149
5 Sex 152
6 Conclusion 153
Acknowledgements 153
References 153
Fitting Curves by Fractal Interpolation: An Application to the Quantification of Cognitive Brain Processes M. A. Navascues and M. V. Sebastian 156
1 Introduction 156
2 Fractal Interpolation of a Continuous Function in a Compact Real Interval 157
3 Fit of sampled data by fractal interpolation 160
4 Application to the quantification of cognitive brain processes 161
Acknowledgments 166
References 166
Stochastic and Regular Components in Forcing of Solar Large-scale Structures E. Tikhomolov 168
1 Introduction 168
2 Model 169
3 Results 172
4 Discussion 176
References 177
Fast, Efficient On-line Simulation of Self-similar Processes O.D.Jones 178
1 Introduction 178
2 EBP processes 178
3 Simulation algorithm 182
4 Proofs 186
References 189
Fractal Geometry in the Arts: An Overview Across The Different Cultures N.Sala 190
1 Introduction 190
2 Fractal components in the arts 191
3 Conclusions 200
4 References 201
Fractal Properties and Characterization of Road Profiles P. Legrand, J. Levy Vehel and M.-T. Do 202
1 Introduction and background 202
2 The road profiles 202
3 Fractal analysis 204
4 Results 206
References 211
Fractal Distributions of Temperature, Salinity and Fluorescence in Spring 2001-2002 in South San Francisco Bay K. Fisher and W. Kimmerer 212
1 Introduction 212
2 Data and Methods 213
3 Results 217
4 Discussion 220
5 Conclusions 222
6 Acknowledgements 223
References 223
Characterization of Fractal Structures Through a Hausdorf Measure Based Method F.Nekka and J.Li 226
1 Introduction 226
2 Hausdorff Measure Spectrum Functions of Uniform Cantor Sets. 227
3 Applications 227
4 Conclusion 233
Acknowledgments 233
References 233
Fractal Scattering Indicators for Urban Sound Diffusion P. W. Woloszyn 234
1 Introduction: Problem and purpose 234
2 Diffusion through oblique incident wave 234
3 Diffraction and structure factor of a multiscalc rough boundary surface 236
4 Application : a Facade scattering characterization 240
5 Experimental validation: In situ measurements 242
6 Conclusion 244
7 References 245
Binomial Multiplicative Model of Critical Fragmentation H. Katsuragi, D. Sugino and H. Honjo 246
1 Introduction 246
2 Experiment 247
3 Model 250
4 Discussion 252
5 Conclusions 254
Acknowledgments 254
References 255
Study on the Improved Fractal Interpolation Surface of the Attitude and Surface of Fault H. Sun and H. Xie 256
1. Introduction 256
2. The principles of fractal interpolation surface on a rectangle field 257
3. Attitude analyses of the fault surface 260
4. Improved fractal interpolation surface of the fault surface 263
5. Fractal interpolation of the fault surface 266
6. Conclusions 266
Acknowledgements 267
References 267
A Deterministic Power Domain Algorithm for Fractal Image Decompression N. Nikolaou, A. Kakos and V. Drakopoulos 268
1 Introduction 268
2 A Computational Model 269
3 The Plotkin Power Domain Algorithm 270
4 The Algorithm and its Complexity 274
5 Conclusions 275
Appendix 277
References 278
Comparative Dynamical Scaling Analysis of Quasi-2D Electrodeposited Silver Patterns under Localized and Non-localized Random Quenched Noise M. A. Pasquale, S. L. Marchiano and A. J. Arvia 280
1 Introduction 280
2 Experimental 281
3 Results and Discussion 282
A cknowledgment s 289
References 289
Epidermal Ridges: Positional Information Coded in an Orientational Field M. B. Nguyen, V. Fleury and J.-F. Gouyet 292
1 Introduction 292
2 Epidermal Ridges 294
3 The model 298
4 Results 300
5 Conclusion 302
References 303
Multiscale Principal Components A. Saucier 304
1 Introduction 304
2 Overview of our approach to the construction of multiscale principal components 305
3 Construction of the first order MFC 305
4 Construction of higher order MFCs 306
5 Diadic MPCs 308
6 Conclusions 312
References 313
Coexistence of Doublon and Dendrite Structure with Phase-Field Model S. Tokunaga and H. Sakaguchi 314
1. Introduction 314
2. Model equation 315
3. Result and discussion 317
4. Summary 318
References 321
Fractality and Fractal Dimension in Mesoamerican Pyramid Analysis G. Burkle-Elizondo, A. G. Fuentes-Larios and R. D. Valdez-Cepeda 322
References 323
Morphological Variety in Crystal Growth of Mercury (II) Chloride on Agar Slides /. A. Betancourt-Mar and E. J. Sudrez-Dominguez 324
References 325
Fractal Characteristics of Bainbridge Crater Lake Sediment Gray-scale Intensity Data Documenting the Frequency and Intensity of Holocene El Nino/Southern Oscillation Events N. A. Bryksina and W. M. Last 326
References 327
Fractals and Plant Water Use Efficiency A. Bari, G. Ayad, A. Martin, J. L. Gonzalez-Andujar, M. Naclrit, and I. Elouafi 328
References 329
Need and Feasibility of Applying L-system Models in Agricultural Crop Modeling L. Pachepsky, M. Kaul, Ch. Walthall, C. Daughtry and]. Lydon 330
Fractal Detection and Avoidance Using RS Statistics and Honeybee Navigational Skills in Dynamic Environments R.L.Walker 332
References 333
Signal and Image Processing with FracLab J. Levy Vehel and P. Legrand 334
References 335
Author Index 336