4,868 research outputs found
Numerics and Fractals
Local iterated function systems are an important generalisation of the
standard (global) iterated function systems (IFSs). For a particular class of
mappings, their fixed points are the graphs of local fractal functions and
these functions themselves are known to be the fixed points of an associated
Read-Bajactarevi\'c operator. This paper establishes existence and properties
of local fractal functions and discusses how they are computed. In particular,
it is shown that piecewise polynomials are a special case of local fractal
functions. Finally, we develop a method to compute the components of a local
IFS from data or (partial differential) equations.Comment: version 2: minor updates and section 6.1 rewritten, arXiv admin note:
substantial text overlap with arXiv:1309.0243. text overlap with
arXiv:1309.024
Are galaxy distributions scale invariant? A perspective from dynamical systems theory
Unless there is evidence for fractal scaling with a single exponent over
distances .1 <= r <= 100 h^-1 Mpc then the widely accepted notion of scale
invariance of the correlation integral for .1 <= r <= 10 h^-1 Mpc must be
questioned. The attempt to extract a scaling exponent \nu from the correlation
integral n(r) by plotting log(n(r)) vs. log(r) is unreliable unless the
underlying point set is approximately monofractal. The extraction of a spectrum
of generalized dimensions \nu_q from a plot of the correlation integral
generating function G_n(q) by a similar procedure is probably an indication
that G_n(q) does not scale at all. We explain these assertions after defining
the term multifractal, mutually--inconsistent definitions having been confused
together in the cosmology literature. Part of this confusion is traced to a
misleading speculation made earlier in the dynamical systems theory literature,
while other errors follow from confusing together entirely different
definitions of ``multifractal'' from two different schools of thought. Most
important are serious errors in data analysis that follow from taking for
granted a largest term approximation that is inevitably advertised in the
literature on both fractals and dynamical systems theory.Comment: 39 pages, Latex with 17 eps-files, using epsf.sty and a4wide.sty
(included) <[email protected]
Hopping Conductivity of a Nearly-1d Fractal: a Model for Conducting Polymers
We suggest treating a conducting network of oriented polymer chains as an
anisotropic fractal whose dimensionality D=1+\epsilon is close to one.
Percolation on such a fractal is studied within the real space renormalization
group of Migdal and Kadanoff. We find that the threshold value and all the
critical exponents are strongly nonanalytic functions of \epsilon as \epsilon
tends to zero, e.g., the critical exponent of conductivity is \epsilon^{-2}\exp
(-1-1/\epsilon). The distribution function for conductivity of finite samples
at the percolation threshold is established. It is shown that the central body
of the distribution is given by a universal scaling function and only the
low-conductivity tail of distribution remains -dependent. Variable
range hopping conductivity in the polymer network is studied: both DC
conductivity and AC conductivity in the multiple hopping regime are found to
obey a quasi-1d Mott law. The present results are consistent with electrical
properties of poorly conducting polymers.Comment: 27 pages, RevTeX, epsf, 5 .eps figures, to be published in Phys. Rev.
Random walk through fractal environments
We analyze random walk through fractal environments, embedded in
3-dimensional, permeable space. Particles travel freely and are scattered off
into random directions when they hit the fractal. The statistical distribution
of the flight increments (i.e. of the displacements between two consecutive
hittings) is analytically derived from a common, practical definition of
fractal dimension, and it turns out to approximate quite well a power-law in
the case where the dimension D of the fractal is less than 2, there is though
always a finite rate of unaffected escape. Random walks through fractal sets
with D less or equal 2 can thus be considered as defective Levy walks. The
distribution of jump increments for D > 2 is decaying exponentially. The
diffusive behavior of the random walk is analyzed in the frame of continuous
time random walk, which we generalize to include the case of defective
distributions of walk-increments. It is shown that the particles undergo
anomalous, enhanced diffusion for D_F < 2, the diffusion is dominated by the
finite escape rate. Diffusion for D_F > 2 is normal for large times, enhanced
though for small and intermediate times. In particular, it follows that
fractals generated by a particular class of self-organized criticality (SOC)
models give rise to enhanced diffusion. The analytical results are illustrated
by Monte-Carlo simulations.Comment: 22 pages, 16 figures; in press at Phys. Rev. E, 200
One-dimensional wave equations defined by fractal Laplacians
We study one-dimensional wave equations defined by a class of fractal
Laplacians. These Laplacians are defined by fractal measures generated by
iterated function systems with overlaps, such as the well-known infinite
Bernoulli convolution associated with the golden ratio and the 3-fold
convolution of the Cantor measure. The iterated function systems defining these
measures do not satisfy the post-critically finite condition or the open set
condition. By using second-order self-similar identities introduced by
Strichartz et al., we discretize the equations and use the finite element and
central difference methods to obtain numerical approximations to the weak
solutions. We prove that the numerical solutions converge to the weak solution,
and obtain estimates for the rate of convergence
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