## Gaussian Self-Affinity and Fractals: Globality, The Earth, 1/f Noise, and R/SBenoit Mandelbrot¿s pioneering research in fractal geometry has affected many areas of mathematics, physics, finance and other disciplines. The papers reprinted in this third volume of his Selected Works center on a detailed study of fractional Brownian functions, best known as the mathematical tools behind the celebrated fractal landscapes. Extensive introductory material preceding the reprints incorporates striking new observations and conjectures. This book explores the fractal themes of ¿self-affinity¿ and ¿globality.¿ The ubiquity of ¿wild¿ temporal and spatial variability led Mandelbrot, in the early 1960¿s, to conclude that those phenomena lie beyond the usual statistical techniques and represent a new state of indeterminism. New mathematical tools are needed, and this book contributes to their development. |

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### Contents

Preface | 1 |

Overview of fractals and multifractals | 9 |

ADVANCES IN OLD BUT OPEN TOPICS | 49 |

Selfaffinity closeup on a versatile family | 50 |

Toward a definition of selfaffine functions | 83 |

Fractal dimensions of Wiener Brownian motion random walk and their clusters 2 32 431 and the new transient 53 | 111 |

Still growing Weierstrass family of functions | 142 |

Iso and heterodiffusion and statistics using the bridge range mesodiffusion | 155 |

Poisson approximation of the multitemporal Brownian functions and generalizations | 363 |

Geometry of homogeneous scalar turbulence isosurface fractal dimensions 52 and 83 | 368 |

Earths relief shape and fractal dimension of coastlines and numberarea rule for islands | 390 |

Midpoint displacement cartoon surfaces | 402 |

SELFAFFINE CARTOONS IN GRIDS THEIR MULTIPLE FRACTAL DIMENSIONS | 423 |

Selfaffinity and fractal dimension | 425 |

Diagonally selfaffine fractal cartoons Part 1 mass box and gap fractal dimensions local or global | 437 |

Length and area anomalies | 453 |

Diffusion selfaffine fractal functions their stationarity in logarithmic time | 172 |

BROAD CONTINUING ISSUES | 187 |

Recorded history and personal recollections | 204 |

I | 231 |

41968 909918 HlO Noah Joseph and operational hydrology M Wallis 1968 | 236 |

FRACTIONAL BROWNIAN MOTIONS | 253 |

Fractional Brownian motions fractional noises and applications M Van Ness 1968 | 254 |

Computer experiments with fractional Gaussian noises Part 1 Sample graphs averages and variances M Wallis 1969a | 283 |

Computer experiments with fractional Gaussian noises Part 2 Rescaled bridge range and pox diagrams M Wallis 1969a | 306 |

Computer experiments with fractional Gaussian noises Part 3 Mathematical appendix M Wallis 1969a | 327 |

Fast fractional Gaussian noise generator | 339 |

Brokenline approximation to fractional noise | 357 |

III | 361 |

Diagonally selfaffine fractal cartoons Part 3 anomalous Hausdorff dimension and multifractal localization | 463 |

IV | 481 |

Robustness of RIS in measuring noncyclic global statistical dependence M Wallis 1969c | 483 |

Limit theorems on the selfnormalized bridge range | 517 |

Global dependence in geophysical records M Wallis 1969b | 538 |

Secular pole motion and Chandler wobble M McCamy 1970J | 562 |

Clustering in a point process intertoken histograms and RS pox diagrams Damerau M 1973 | 584 |

Global longterm dependence in economics and finance long foreword and excerpts from M 1969e M 1971n and M 1972c | 601 |

Fractal aspects of computer memories foreword and excerpts from Voldman M Hoevel Knight Rosenfeld 1983 | 611 |

613 | |

637 | |

### Common terms and phrases

affine approximation asymptotic average behavior BH(t bridge range called Cantor dust cartoons Chandler wobble Chapter cluster conjecture construction continuous converges correlation corresponding covariance curve defined definition described diffusion distribution example Fickian Figure finite follows Fourier fractal dimension fractal geometry fractional Brownian motion fractional Gaussian noise fractional noise frequency Gaussian noise Gaussian processes global dependence graph Hausdorff-Besicovitch dimension hence Hurst hydrology increments infinite integer interpolation interval intuitive limit linear MANDELBROT mathematical measure mesofractal midpoint displacement multifractal noise with H paper physics plane Poisson pox diagram R/S analysis random function random process random variables ratio records recursive replaced result sample scaling Section self-affine self-similar sequence slope spectral density spectrum square stationary statistical statistically independent term theorem transform turbulence unifractal value of H variance Wallis Weierstrass Weierstrass function white noise Wiener Brownian yields zero