10,258 research outputs found
The Kauffman model on Small-World Topology
We apply Kauffman's automata on small-world networks to study the crossover
between the short-range and the infinite-range case. We perform accurate
calculations on square lattices to obtain both critical exponents and fractal
dimensions. Particularly, we find an increase of the damage propagation and a
decrease in the fractal dimensions when adding long-range connections.Comment: AMS-LaTeX v1.2, 8 pages with 8 figures Encapsulated Postscript, to be
published in Physica
Random soups, carpets and fractal dimensions
We study some properties of a class of random connected planar fractal sets
induced by a Poissonian scale-invariant and translation-invariant point
process. Using the second-moment method, we show that their Hausdorff
dimensions are deterministic and equal to their expectation dimension. We also
estimate their low-intensity limiting behavior. This applies in particular to
the "conformal loop ensembles" defined via Poissonian clouds of Brownian loops
for which the expectation dimension has been computed by Schramm, Sheffield and
Wilson.Comment: To appear in J. London Math. So
Using NMR to Measure Fractal Dimensions
A comment is made on the recent PFG NMR measurements by Stallmach, et al. on
water-saturated sands [Phys. Rev. Lett. 88, 105505 (2002)]. It is pointed out
that the usual law for the time-dependent diffusion coefficient D(t) used by
these authors is not valid for a fractal surface. It is shown that (1-D(t)/D0)
\~ t^[(3-Ds)/2] at short times for a surface of fractal dimension Ds, where D0
is the bulk diffusion coefficient. Preliminary PFG NMR data on water saturated
limestone and plastic beads are presented to illustrate this analysis.Comment: 1 page, 1 figur
Fractal fractal dimensions of deterministic transport coefficients
If a point particle moves chaotically through a periodic array of scatterers
the associated transport coefficients are typically irregular functions under
variation of control parameters. For a piecewise linear two-parameter map we
analyze the structure of the associated irregular diffusion coefficient and
current by numerically computing dimensions from box-counting and from the
autocorrelation function of these graphs. We find that both dimensions are
fractal for large parameter intervals and that both quantities are themselves
fractal functions if computed locally on a uniform grid of small but finite
subintervals. We furthermore show that there is a simple functional
relationship between the structure of fractal fractal dimensions and the
difference quotient defined on these subintervals.Comment: 16 pages (revtex) with 6 figures (postscript
Classifying Cantor Sets by their Fractal Dimensions
In this article we study Cantor sets defined by monotone sequences, in the
sense of Besicovitch and Taylor. We classify these Cantor sets in terms of
their h-Hausdorff and h-Packing measures, for the family of dimension functions
h, and characterize this classification in terms of the underlying sequences.Comment: 10 pages, revised version. To appear in Proceedings of the AMS
Scale distributions and fractal dimensions in turbulence
A new geometric framework connecting scale distributions to coverage statistics is employed to analyze level sets arising in turbulence as well as in other phenomena. A 1D formalism is described and applied to Poisson, lognormal, and power-law statistics. A d-dimensional generalization is also presented. Level sets of 2D spatial measurements of jet-fluid concentration in turbulent jets are analyzed to compute scale distributions and fractal dimensions. Lognormal statistics are used to model the level sets at inner scales. The results are in accord with data from other turbulent flows
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