636 research outputs found
Diffusion on middle- Cantor sets
In this paper, we study -calculus on generalized Cantor sets,
which have self-similar properties and fractional dimensions that exceed their
topological dimensions. Functions with fractal support are not differentiable
or integrable in terms of standard calculus, so we must involve local
fractional derivatives. We have generalized the -calculus on the
generalized Cantor sets known as middle- Cantor sets. We have suggested a
calculus on the middle- Cantor sets for different values of with
. Differential equations on the middle- Cantor sets have been
solved, and we have presented the results using illustrative examples. The
conditions for super-, normal, and sub-diffusion on fractal sets are given.Comment: 15 pages, 11 figure
Systems of Navier-Stokes equations on cantor sets
We present systems of Navier-Stokes equations on Cantor sets, which are described by the local fractional vector calculus. It is shown that the results for Navier-Stokes equations in a fractal bounded domain are efficient and accurate for describing fluid flow in fractal media
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
Fracton pairing mechanism for "strange" superconductors: Self-assembling organic polymers and copper-oxide compounds
Self-assembling organic polymers and copper-oxide compounds are two classes
of "strange" superconductors, whose challenging behavior does not comply with
the traditional picture of Bardeen, Cooper, and Schrieffer (BCS)
superconductivity in regular crystals. In this paper, we propose a theoretical
model that accounts for the strange superconducting properties of either class
of the materials. These properties are considered as interconnected
manifestations of the same phenomenon: We argue that superconductivity occurs
in the both cases because the charge carriers (i.e., electrons or holes)
exchange {\it fracton excitations}, quantum oscillations of fractal lattices
that mimic the complex microscopic organization of the strange superconductors.
For the copper oxides, the superconducting transition temperature as
predicted by the fracton mechanism is of the order of K. We suggest
that the marginal ingredient of the high-temperature superconducting phase is
provided by fracton coupled holes that condensate in the conducting
copper-oxygen planes owing to the intrinsic field-effect-transistor
configuration of the cuprate compounds. For the gate-induced superconducting
phase in the electron-doped polymers, we simultaneously find a rather modest
transition temperature of K owing to the limitations imposed by
the electron tunneling processes on a fractal geometry. We speculate that
hole-type superconductivity observes larger onset temperatures when compared to
its electron-type counterpart. This promises an intriguing possibility of the
high-temperature superconducting states in hole-doped complex materials. A
specific prediction of the present study is universality of ac conduction for
.Comment: 12 pages (including separate abstract page), no figure
A topological approximation of the nonlinear Anderson model
We study the phenomena of Anderson localization in the presence of nonlinear
interaction on a lattice. A class of nonlinear Schrodinger models with
arbitrary power nonlinearity is analyzed. We conceive the various regimes of
behavior, depending on the topology of resonance-overlap in phase space,
ranging from a fully developed chaos involving Levy flights to pseudochaotic
dynamics at the onset of delocalization. It is demonstrated that quadratic
nonlinearity plays a dynamically very distinguished role in that it is the only
type of power nonlinearity permitting an abrupt localization-delocalization
transition with unlimited spreading already at the delocalization border. We
describe this localization-delocalization transition as a percolation
transition on a Cayley tree. It is found in vicinity of the criticality that
the spreading of the wave field is subdiffusive in the limit
t\rightarrow+\infty. The second moment grows with time as a powerlaw t^\alpha,
with \alpha = 1/3. Also we find for superquadratic nonlinearity that the analog
pseudochaotic regime at the edge of chaos is self-controlling in that it has
feedback on the topology of the structure on which the transport processes
concentrate. Then the system automatically (without tuning of parameters)
develops its percolation point. We classify this type of behavior in terms of
self-organized criticality dynamics in Hilbert space. For subquadratic
nonlinearities, the behavior is shown to be sensitive to details of definition
of the nonlinear term. A transport model is proposed based on modified
nonlinearity, using the idea of stripes propagating the wave process to large
distances. Theoretical investigations, presented here, are the basis for
consistency analysis of the different localization-delocalization patterns in
systems with many coupled degrees of freedom in association with the asymptotic
properties of the transport.Comment: 20 pages, 2 figures; improved text with revisions; accepted for
publication in Physical Review
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