Let n ≥ 3 be an integer. A convex lattice n-gon is a polygon whose n vertices are points on the integer lattice Z 2 and whose interior angles are strictly less than π. Let a n denote the least possible area enclosed by a convex lattice n-gon, then [1, 2, 3] {a n } ∞ n=3 = n 1 2

In this paper, persents the definitions of strongly prime ideal, strongly prime N-subgroup, Pseudo-valuation near ring and Pseudo-valuation N-group. Some of their properties have also been proven by theorems. Then it is shown that, if N be near ring with quotient near-field K and P be a strongly prime ideal of near ring N, then is a strongly prime ideal of ‎‎, for any multiplication subset S of...

Suppose that P is a convex polyhedron in the hyperbolic 3-space with finite volume and P has integer ( > 1) submultiples of it as dihedral angles. We prove that if the rank of the abelianization of a normal torsion-free finite index subgroup of the polyhedral group G associated to P is one, then P has exactly one ideal vertex of type (2,2,2,2) and G has an index two subgroup which does not cont...

We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmüller space. Given a subgroup G of MCG de ning an extension 1 ! 1(S) ! ΓG ! G ! 1, we prove that if ΓG is a word hyperbolic group then G is a convex cocompact subgroup of MCG. When G is free and convex cocompact, called a Schottky ...

We provide a Rademacher theorem for intrinsically Lipschitz functions \(\phi\colon U\subseteq\mathbf{W}\to\mathbf{L}\), where \(U\) is Borel set, \(\mathbf{W}\) and \(\mathbf{L}\) are complementary subgroups of Carnot group, we require that normal subgroup. Our hypotheses satisfied example when horizontal Moreover, an area formula this class functions.

Minkowski’s second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski’s bound by replacing the volume by the lattice point enumerator of a convex body. To this we are interested in bounds on the coefficients of Ehrhart polynomials of lattice polytopes via the successive minima. Our resu...

Given a lattice Λ, a lattice polyhedra is a convex polyhedra P , such that all vertices of P are lattice points and no other other point in P is a lattice point. By lattice-free convex set we mean a convex set with no lattice point from Λ in its strict interior. However lattice points are allowed on the boundary. We will usually work with the standard lattice in R3, i.e. points which have all t...