The properties of graphene-like BC$_6$N semiconductor are studied using density functional theory taking into account the attractive interaction between B and N atoms. In presence a strong dopant atoms, electron charge distribution is highly localized along B-N bonds, while for weaker electrons delocalized entire hexagonal ring BC$_6$N. Furthermore, when both atoms doped at same site hexagon, b...

The properties of graphene-like BC$_6$N semiconductor are studied using density functional theory taking into account the attractive interaction between B and N atoms. In presence a strong dopant atoms, electron charge distribution is highly localized along B-N bonds, while for weaker electrons delocalized entire hexagonal ring BC$_6$N. Furthermore, when both atoms doped at same site hexagon, b...

Consider critical percolation on a rhombus in the hexagonal lattice, and let A the event of a left to right crossing (by 1’s). Suppose we sample configurations from A by starting from some fixed configuration in A, and then at each step a uniformly random hexagon is resampled as long as the resulting configuration is in A. It is easy to see that this sampling procedure converges to the uniform ...

We generalize a theorem of W. Jockusch and J. Propp on quartered Aztec diamonds by enumerating the tilings of quartered Aztec rectangles. We use subgraph replacement method to transform the dual graph of a quartered Aztec rectangle to the dual graph of a quartered lozenge hexagon, and then use Lindström-GesselViennot methodology to find the number of tilings of a quartered lozenge hexagon.

The well-known dual pair of Napoleon equilateral triangles intrinsic to each triangle is extended to infinite sequences of them, shown to be special cases of infinite regular hexagon sequences on each triangle. A set of hexagon-to-hexagon transformations, the hex operators, is defined for this purpose, a set forming an abelian monoid under function composition. The sequences result from arbitra...

By investigating the construction of the split Cayley generalized hexagon, H(2), we get that there do not exist five distinct hexagon lines each a distance two apart from each other. From this we prove that the smallest Paige loop has a covering number of seven and that its automorphism group permutes these coverings transitively.

Abstract Fully localised patterns involving cellular hexagons or squares have been found experimentally and numerically in various continuum models. However, there is currently no mathematical theory for the emergence of these from a quiescent state. A key issue that standard techniques one-dimensional proven insufficient understanding localisation higher dimensions. In this work, we present co...