Gaussian Self-Affinity and Fractals: Globality, The Earth, 1/f Noise, and R/S

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Springer Science & Business Media, 2002 - Mathematics - 654 pages
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Benoit 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
Cumulative bibliography including copyright credits
613
Index
637
Copyright

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About the author (2002)

Benoit Mandelbrot is the Abraham Robinson Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus at the IBM T.J. Watson Research Center.