Hadwiger showed by computing the intrinsic volumes of a regular simplex that a rectangular simplex is a counterexample to Wills’s conjecture for the relation between the lattice point enumerator and the intrinsic volumes in dimensions not less than 441. Here we give formulae for the volumes of spherical polytopes related to the intrinsic volumes of the regular crosspolytope and of the rectangul...

We study the asymptotic behaviour of resistance scaling and fluctuation of resistance that give rise to flicker noise in an n-simplex lattice. We propose a simple method to calculate the resistance scaling and give a closed-form formula to calculate the exponent, βL, associated with resistance scaling, for any n. Using current cumulant method we calculate the exact noise exponent for n-simplex ...

We propose the lattice version of BF gravity action whose partition function leads to the product of a particular form of 15-j symbol which corresponds to a 4-simplex. The action is explicitly constructed by lattice B field defined on triangles and link variables defined on dual links and is shown to be invariant under lattice local Lorentz transformation and Kalb-Ramond gauge transformation. W...

We develop an interesting relationship between /nite sets in a lattice and the minimal density of simplex coverings of n-space. c © 2002 Elsevier Science B.V. All rights reserved.

In [1], the author generalized Ehrhart’s idea ([2]) of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its description as the intersection of halfspaces, which determine the facets of the polytope. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We proved ...

In [1], the author generalized Ehrhart’s idea ([2]) of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its description as the intersection of halfspaces, which determine the facets of the polytope. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We proved ...

In [1], the author generalized Ehrhart’s idea ([2]) of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its description as the intersection of halfspaces, which determine the facets of the polytope. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We proved ...

In [1], the author generalized Ehrhart’s idea ([2]) of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its description as the intersection of halfspaces, which determine the facets of the polytope. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We proved ...

Integral polytopes (see [Z ]for the basic definitions) are of interest in combinatorics, linear programming, algebraic geometry-toric varieties [D,O], number theory [K-L.]. We study here lattice-free simplices, that is simplices intersecting the lattice only at their vertices. A natural question is to measure the " flatness " of these polytopes, with respect to integral dual vectors. This (arit...

In this paper, we consider integral maximal lattice-free simplices. Such simplices have integer vertices and contain integer points in the relative interior of each of their facets, but no integer point is allowed in the full interior. In dimension three, we show that any integral maximal latticefree simplex is equivalent to one of seven simplices up to unimodular transformation. For higher dim...