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 $k$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 $P_>(l)\sim l^{-s}$ 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

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

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