By C. H. Scholz, B. B. Mandelbrot (auth.), Christopher H. Scholz, Benoît B. Mandelbrot (eds.)
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Extra info for Fractals in Geophysics
M, which is on the order of the estimated Kolmogorov scale (that is, the smallest dynamical scale in the flow). Not resolved here are even smaller scales in the dye concentration fluctuations, expected to occur because of the large Schmidt number, Sc, of the dye. ) It is legitimate to consider the plane intersection "mathematically thin" with respect to scales of the order of the Kolmogorov scale and larger. m2 • Further processing was done on a VAX station II/GPX. The image shows a number of geometrically interesting features, one of which relates to the boundary that separates the nozzle fluid from the ambient tank fluid.
Differences between the five examples. Gross nonuniversality. All the above limits differ from one another. For example, the limits for the non-Gaussian cases differ from the limit relative to the Gaussian case. 3, this surprising result will be shown to hold very generally. It is the second most important probabilistic fact underlying the multifractals (the most important fact being the behavior of moments along low dimensional cuts; Section 8). 3. Generalization of the Multifractal Formalism by Application of the Cramer Limit Theorem.
Max) = - oo. To show that this result is intuitive, take a more general M that is continuously distributed, but can be approximated arbitrarily close by a discrete multiplier with log Mmax arbitrarily close to 0 and/or log Mmin arbitrarily close to - oo, with both extremes having correspondingly arbitrary tiny probabilities. max = 00. A new and important distinction. ) > 0 will be called manifest. 's are manifest. 's. ax) = q:,in > -00. 's on the moments. ) restricted to manifest values of a.. max fail to be infinite.
Fractals in Geophysics by C. H. Scholz, B. B. Mandelbrot (auth.), Christopher H. Scholz, Benoît B. Mandelbrot (eds.)