On the Optimality of Functionals over Triangulations of Delaunay Sets

In this short paper, we consider the functional density on sets of uniformly bounded triangulations with fixed sets of vertices. We prove that if a functional attains its minimum on the Delaunay triangulation, for every finite set in the plane, then for infinite sets the density of this functional attains its minimum also on the Delaunay triangulations

Nonlinear dynamics of the interface of dielectric liquids in a strong electric field: Reduced equations of motion

The evolution of the interface between two ideal dielectric liquids in a strong vertical electric field is studied. It is found that a particular flow regime, for which the velocity potential and the electric field potential are linearly dependent functions, is possible if the ratio of the permittivities of liquids is inversely proportional to the ratio of their densities. The corresponding reduced equations for interface motion are derived. In the limit of small density ratio, these equations coincide with the well-known equations describing the Laplacian growth.Comment: 10 page

Scaling of the distribution of shortest paths in percolation

We present a scaling hypothesis for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for {\it (i)} the effect of the finite size of the system, and {\it (ii)} the dependence of this distribution on the site occupancy probability $p$. We test the hypothesis for the case of two-dimensional percolation.Comment: 7 pages, 3 figure