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

Oxygen non-stoichiometry and defect structure of LaMn1-zCuzO3+δ

The quantitative model analysis of the defect structure of copper doped lanthanum manganites LaMn1-zCuzO3+δ (z = 0.05 and 0.1) was performed. In the framework of the model, the independent course of the three reactions of defect formation, including the electronic exchange between manganese and copper, completion of a lattice by the absorption of oxygen and the disproportionation of manganese was considered. It is shown that the increase in the dopant content leads to a change in the dominant electronic process. This is reflected in the lock and disproportionation and leads to the increase in the concentration of holes and decrease the concentration of electrons localized on the manganese atoms