Aggregate and fractal tessellations

Consider a sequence of stationary tessellations {‹n}, n=0,1,..., of  d consisting of cells {Cn(xin)}with the nuclei {xin}. An aggregate cell of level one, C01(xi0), is the result of merging the cells of ‹1 whose nuclei lie in C0(xi0). An aggregate tessellation ‹0n consists of the aggregate cells of level n, C0n(xi0), defined recursively by merging those cells of ‹n whose nuclei lie in Cnm1(xi0). We find an expression for the probability for a point to belong to atypical aggregate cell, and obtain bounds for the rate of itsexpansion. We give necessary conditions for the limittessellation to exist as nMX and provide upperbounds for the Hausdorff dimension of its fractal boundary and forthe spherical contact distribution function in the case ofPoisson-Voronoi tessellations {‹n}

On the number of two-dimensional threshold functions

A two-dimensional threshold function of k-valued logic can be viewed as coloring of the points of a k x k square lattice into two colors such that there exists a straight line separating points of different colors. For the number of such functions only asymptotic bounds are known. We give an exact formula for the number of two-dimensional threshold functions and derive more accurate asymptotics.Comment: 17 pages, 2 figure