612 research outputs found
Superdiffusion in a class of networks with marginal long-range connections
A class of cubic networks composed of a regular one-dimensional lattice and a
set of long-range links is introduced. Networks parametrized by a positive
integer k are constructed by starting from a one-dimensional lattice and
iteratively connecting each site of degree 2 with a th neighboring site of
degree 2. Specifying the way pairs of sites to be connected are selected,
various random and regular networks are defined, all of which have a power-law
edge-length distribution of the form with the marginal
exponent s=1. In all these networks, lengths of shortest paths grow as a power
of the distance and random walk is super-diffusive. Applying a renormalization
group method, the corresponding shortest-path dimensions and random-walk
dimensions are calculated exactly for k=1 networks and for k=2 regular
networks; in other cases, they are estimated by numerical methods. Although,
s=1 holds for all representatives of this class, the above quantities are found
to depend on the details of the structure of networks controlled by k and other
parameters.Comment: 10 pages, 9 figure
What makes a space have large weight?
We formulate several conditions (two of them are necessary and sufficient)
which imply that a space of small character has large weight. In section 3 we
construct a ZFC example of a first countable 0-dimensional space X of size
2^omega with w(X)=2^omega and nw(X)=omega, we show that CH implies the
existence of a 0-dimensional space Y of size omega_1 with w(Y)=nw(Y)=omega_1
and chi(Y)=R(Y)=omega, and we prove that it is consistent that 2^omega is as
large as you wish and there is a 0-dimensional space Z of size 2^omega such
that w(Z)=nw(Z)=2^omega but chi(Z)=R(Z^omega)=omega
Long-range epidemic spreading in a random environment
Modeling long-range epidemic spreading in a random environment, we consider a
quenched disordered, -dimensional contact process with infection rates
decaying with the distance as . We study the dynamical behavior
of the model at and below the epidemic threshold by a variant of the
strong-disorder renormalization group method and by Monte Carlo simulations in
one and two spatial dimensions. Starting from a single infected site, the
average survival probability is found to decay as up to
multiplicative logarithmic corrections. Below the epidemic threshold, a
Griffiths phase emerges, where the dynamical exponent varies continuously
with the control parameter and tends to as the threshold is
approached. At the threshold, the spatial extension of the infected cluster (in
surviving trials) is found to grow as with a
multiplicative logarithmic correction, and the average number of infected sites
in surviving trials is found to increase as with
in one dimension.Comment: 12 pages, 6 figure
On the family of B-spline surfaces obtained by knot modification
B-spline surfaces are piecewisely defined surfaces where the section points of the domain of definition are called knots. In [2] the authors proved some theorems in terms of knot modification of B-spline curves. Here we generalize these results for one- and two-parameter family of surfaces. An additional result concerning a higher order contact of these surfaces and an envelope is also proved
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