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

Possibility of Geometric Description of Quasiparticles in Solids

New phenomenological approach for the description of elementary collective excitations is proposed. The crystal is considered to be an anisotropic space-time vacuum with a prescribed metric tensor in which the information on electromagnetic crystalline fields is included. The quasiparticles in this space are supposed to be described by the equations structurally similar to the relativistic wave equations for particles in empty space. The generalized Klein-Gordon-Fock equation and the generalized Dirac equation in external electromagnetic field are considered. The applicability of the proposed approach to the case of conduction electron in a crystal is discussed.Comment: 17 pages, latex; to appear in Int. Jnl. Mod. Phy

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