Basic properties of finite subsets of the integer lattice Z are investigated from the point of view of geometric tomography. Results obtained concern the Minkowski addition of convex lattice sets and polyominoes, discrete X-rays and the discrete and continuous covariogram, the determination of symmetric convex lattice sets from the cardinality of their projections on hyperplanes, and a discrete...

In this paper we derive an explicit formula for the HirzebruchMumford volume of an even lattice L of signature (2, n). If Γ ⊂ O(L) is an arithmetic subgroup, then an application of Hirzebruch-Mumford proportionality allows us to determine the leading term of the growth of the dimension of the spaces Sk(Γ) of cusp forms of weight k, as k goes to infinity. We compute this in a number of examples,...

We prove a number of results concerning the embedding of a Banach lattice X into an r. i. space Y. For example we show that if Y is an r. i. space on [0, oo) which is/7-convex for some/? > 2 and has nontrivial concavity then any Banach lattice X which is r-convex for some r > 2 and embeds into Y must embed as a sublattice. Similar conclusions can be drawn under a variety of hypotheses on Y; if ...

We prove that the order complex of a geometric lattice has a convex ear decomposition. As a consequence, if ∆(L) is the order complex of a rank (r+1) geometric lattice L, then the for all i ≤ r/2 the h-vector of ∆(L) satisfies, hi−1 ≤ hi and hi ≤ hr−i. We also obtain several inequalities for the flag h-vector of ∆(L) by analyzing the weak Bruhat order of the symmetric group. As an application, ...

We introduce reflexive polytopes of index l as a natural generalisation of the notion of a reflexive polytope of index 1. These l-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive polygons via a change of the underlying lattice. This allows us to efficiently classify all isomorphism classes of l-reflexive polygons up to index 200. As another ...

We introduce reflexive polytopes of index l as a natural generalisation of the notion of a reflexive polytope of index 1. These l-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive polygons via a change of the underlying lattice. This allows us to efficiently classify all isomorphism classes of l-reflexive polygons up to index 200. As another ...

Let L be a lattice. If for each a < b ∈ L there is a lattice embedding of φ : L → [a, b] then L is called a semifractal. If, in addition, 0,1 ∈ L and φ can always be chosen such that φ(0) = a and φ(1) = b then L is said to be a 0–1-semifractal. Now let L be a bounded lattice. If for each a1 < b1 ∈ L and a2 < b2 ∈ L there is a lattice embedding ψ : [a1, b1] → [a2, b2] with ψ(a1) = a2 and ψ(b1) =...

in this paper, we examine that some simple $k_4$-groups can be determined uniquely by their orders and one or two irreducible complex character degrees.

In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion of quasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the diffe...