2,938 research outputs found
Numerical investigations of discrete scale invariance in fractals and multifractal measures
Fractals and multifractals and their associated scaling laws provide a
quantification of the complexity of a variety of scale invariant complex
systems. Here, we focus on lattice multifractals which exhibit complex
exponents associated with observable log-periodicity. We perform detailed
numerical analyses of lattice multifractals and explain the origin of three
different scaling regions found in the moments. A novel numerical approach is
proposed to extract the log-frequencies. In the non-lattice case, there is no
visible log-periodicity, {\em{i.e.}}, no preferred scaling ratio since the set
of complex exponents spread irregularly within the complex plane. A non-lattice
multifractal can be approximated by a sequence of lattice multifractals so that
the sets of complex exponents of the lattice sequence converge to the set of
complex exponents of the non-lattice one. An algorithm for the construction of
the lattice sequence is proposed explicitly.Comment: 31 Elsart pages including 12 eps figure
Random walk on Sierpinski-type multifractals
A method is established which allows the calculation of the walk dimension
for Sierpinski-type multifractals. The multifractal scaling behaviour of the
average time needed to cover a distance in the mentionned multifractals is
shown. For the average-time-multifractal we calculate the Renyi dimensions and
allpy the f(alpha)-formalism.Comment: 9 pages, Postscrip
High values of disorder-generated multifractals and logarithmically correlated processes
In the introductory section of the article we give a brief account of recent
insights into statistics of high and extreme values of disorder-generated
multifractals following a recent work by the first author with P. Le Doussal
and A. Rosso (FLR) employing a close relation between multifractality and
logarithmically correlated random fields. We then substantiate some aspects of
the FLR approach analytically for multifractal eigenvectors in the
Ruijsenaars-Schneider ensemble (RSE) of random matrices introduced by E.
Bogomolny and the second author by providing an ab initio calculation that
reveals hidden logarithmic correlations at the background of the
disorder-generated multifractality. In the rest we investigate numerically a
few representative models of that class, including the study of the highest
component of multifractal eigenvectors in the Ruijsenaars-Schneider ensemble
Convolution of multifractals and the local magnetization in a random field Ising chain
The local magnetization in the one-dimensional random-field Ising model is
essentially the sum of two effective fields with multifractal probability
measure. The probability measure of the local magnetization is thus the
convolution of two multifractals. In this paper we prove relations between the
multifractal properties of two measures and the multifractal properties of
their convolution. The pointwise dimension at the boundary of the support of
the convolution is the sum of the pointwise dimensions at the boundary of the
support of the convoluted measures and the generalized box dimensions of the
convolution are bounded from above by the sum of the generalized box dimensions
of the convoluted measures. The generalized box dimensions of the convolution
of Cantor sets with weights can be calculated analytically for certain
parameter ranges and illustrate effects we also encounter in the case of the
measure of the local magnetization. Returning to the study of this measure we
apply the general inequalities and present numerical approximations of the
D_q-spectrum. For the first time we are able to obtain results on multifractal
properties of a physical quantity in the one-dimensional random-field Ising
model which in principle could be measured experimentally. The numerically
generated probability densities for the local magnetization show impressively
the gradual transition from a monomodal to a bimodal distribution for growing
random field strength h.Comment: An error in figure 1 was corrected, small additions were made to the
introduction and the conclusions, some typos were corrected, 24 pages,
LaTeX2e, 9 figure
Multifractals Competing with Solitons on Fibonacci Optical Lattice
We study the stationary states for the nonlinear Schr\"odinger equation on
the Fibonacci lattice which is expected to be realized by Bose-Einstein
condensates loaded into an optical lattice. When the model does not have a
nonlinear term, the wavefunctions and the spectrum are known to show fractal
structures. Such wavefunctions are called critical. We present a phase diagram
of the energy spectrum for varying the nonlinearity. It consists of three
portions, a forbidden region, the spectrum of critical states, and the spectrum
of stationary solitons. We show that the energy spectrum of critical states
remains intact irrespective of the nonlinearity in the sea of a large number of
stationary solitons.Comment: 5 pages, 4 figures, major revision, references adde
Modeling fractal structure of city-size distributions using correlation function
Zipf's law is one the most conspicuous empirical facts for cities, however,
there is no convincing explanation for the scaling relation between rank and
size and its scaling exponent. Based on the idea from general fractals and
scaling, this paper proposes a dual competition hypothesis of city develop to
explain the value intervals and the special value, 1, of the power exponent.
Zipf's law and Pareto's law can be mathematically transformed into one another.
Based on the Pareto distribution, a frequency correlation function can be
constructed. By scaling analysis and multifractals spectrum, the parameter
interval of Pareto exponent is derived as (0.5, 1]; Based on the Zipf
distribution, a size correlation function can be built, and it is opposite to
the first one. By the second correlation function and multifractals notion, the
Pareto exponent interval is derived as [1, 2). Thus the process of urban
evolution falls into two effects: one is Pareto effect indicating city number
increase (external complexity), and the other Zipf effect indicating city size
growth (internal complexity). Because of struggle of the two effects, the
scaling exponent varies from 0.5 to 2; but if the two effects reach equilibrium
with each other, the scaling exponent approaches 1. A series of mathematical
experiments on hierarchical correlation are employed to verify the models and a
conclusion can be drawn that if cities in a given region follow Zipf's law, the
frequency and size correlations will follow the scaling law. This theory can be
generalized to interpret the inverse power-law distributions in various fields
of physical and social sciences.Comment: 18 pages, 3 figures, 3 table
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