Let M be a planar centrally symmetric convex body. If H is an a‰ne regular hexagon of the smallest (the largest) possible area inscribed in M, then M contains (respectively, the interior of M does not contain) an additional pair of symmetric vertices of the a‰neregular 12-gon TH whose every second vertex is a vertex of H. Moreover, we can inscribe in M an octagon whose three pairs of opposite v...

Let G = (V,E) be a graph and T be a spanning tree of G. We consider the following strategy in advancing in G from a vertex x towards a target vertex y: from a current vertex z (initially, z = x), unless z = y, go to a neighbor of z in G that is closest to y in T (breaking ties arbitrarily). In this strategy, each vertex has full knowledge of its neighborhood in G and can use the distances in T ...

Intrinsic complete characterizations of constructive, context-free and regular languages have been formulated by means of configurations of languages. The definition of a semiconfiguration is given here by generalizing the definition of a configuration. By means of semiconfigurations, an intrinsic complete characterization of context-sensitive languages is formulated. 1. Languages and generaliz...

The compatible expansion in series of solutions of both the equations of P–Q pair at neighborhood of the singular point is obtained in closed form for regular and irregular singularities. The conservation laws of the system of ordinary differential equations to arise from the compatibility condition of the P–Q pair are derived.

The compatible expansion in series of solutions of both the equations of P–Q pair at neighborhood of the singular point is obtained in closed form for regular and irregular singularities. The conservation laws of the system of ordinary differential equations to arise from the compatibility condition of the P–Q pair are derived.

Let G = (V, E) be a graph and T be a spanning tree of G. We consider the following strategy in advancing in G from a vertex x towards a target vertex y: from a current vertex z (initially, z = x), unless z = y, go to a neighbor of z in G that is closest to y in T (breaking ties arbitrarily). In this strategy, each vertex has full knowledge of its neighborhood in G and can use the distances in T...

This paper provides a bridge between the classical tiling theory and cellular automata on one side, and the complex neighborhood self-assembling situations that exist in practice, on the other side. A neighborhood N is a finite set of pairs (i, j) ∈ Z, indicating that the neighbors of a position (x, y) are the positions (x + i, y + j) for (i, j) ∈ N . This includes classical neighborhoods of si...

The question of when a given graph can be the underlying graph of a regular map has roots a hundred years old and is currently the object of several threads of research. This paper outlines this topic briefly and proves that a product of graphs which have regular embeddings also has such an embedding. We then present constructions for members of three families: (1) circulant graphs, (2) wreath ...

An oriented graph dominates pairs if for every pair of vertices u, v, there exists a vertex w such that the edges ~ wu and ~ wv both lie in G. We construct regular oriented triangle free graphs with this property, and thereby we disprove a conjecture of Myers. We also construct oriented graphs for which each pair of vertices is dominated by a unique vertex.