Discrete convexity and unimodularity. I

In this paper we develop a theory of convexity for a free Abelian group M (the lattice of integer points), which we call theory of discrete convexity. We characterize those subsets X of the group M that could be call "convex". One property seems indisputable: X should coincide with the set of all integer points of its convex hull co(X) (in the ambient vector space V). However, this is a first approximation to a proper discrete convexity, because such non-intersecting sets need not be separated by a hyperplane. This issue is closely related to the question when the intersection of two integer polyhedra is an integer polyhedron. We show that unimodular systems (or more generally, pure systems) are in one-to-one correspondence with the classes of discrete convexity. For example, the well-known class of g-polymatroids corresponds to the class of discrete convexity associated to the unimodular system A_n:={\pm e_i, e_i-ej} in Z^n.Comment: 26 pages, Late

Effect of rhythmic photostimulation on monkeys with hyperkinesis of post-encephalitic genesis

In hyperkinetic monkeys a response opposite to that of healthy monkeys was observed during rhythmic photostimulation (frequency 3, 9, 18, 20, and 25/sec), i.e., the hyperkinesis disappeared. The significance of rhythmic excitatory cycles for interconnections between different brain structures is discussed

Condorcet domains of tiling type

A Condorcet domain (CD) is a collection of linear orders on a set of candidates satisfying the following property: for any choice of preferences of voters from this collection, a simple majority rule does not yield cycles. We propose a method of constructing "large" CDs by use of rhombus tiling diagrams and explain that this method unifies several constructions of CDs known earlier. Finally, we show that three conjectures on the maximal sizes of those CDs are, in fact, equivalent and provide a counterexample to them.Comment: 16 pages. To appear in Discrete Applied Mathematic