Using new number-theoretic bounds on the denominators of unit fractions summing up to one, we show that in any dimension d ≥ 4 there is only one d-dimensional reflexive simplex having maximal volume. Moreover, only these reflexive simplices can admit an edge that has the maximal number of lattice points possible for an edge of a reflexive simplex. In general, these simplices are also expected t...

We study simplices whose vertices lie on a lattice and have no other lattice points. Suchèmpty lattice simplices' come up in the theory of integer programming, and in some combi-natorial problems. They have been investigated in various contexts and under varying terminology Can thèemptiness' of lattice simplices bèwell-characterized' ? Is theirìattice-width' small ? Do the integer points of the...

Let σ be a simplex of RN with vertices in the integral lattice ZN . The number of lattice points of mσ (= {mα : α ∈ σ}) is a polynomial function L(σ,m) of m ≥ 0. In this paper we present: (i) a formula for the coefficients of the polynomial L(σ, t) in terms of the elementary symmetric functions; (ii) a hyperbolic cotangent expression for the generating functions of the sequence L(σ,m), m ≥ 0; (...

Let P be a d-dimensional lattice polytope. We show that there exists a natural number cd , only depending on d, such that the multiples cP have a unimodular cover for every c ∈ N, c ≥ cd . Actually, an explicit upper bound for cd is provided, together with an analogous result for unimodular covers of rational cones. 1. STATEMENT OF RESULTS All polytopes and cones considered in this paper are as...

In this paper, (d + 1)-pencil lattices on simplicial partitions in Rd are studied. The barycentric approach naturally extends the lattice from a simplex to a simplicial partition, providing a continuous piecewise polynomial interpolant over the extended lattice. The number of degrees of freedom is equal to the number of vertices of the simplicial partition. The constructive proof of this fact l...

We give the distribution functions, the expected values, and the moments of linear combinations of lattice polynomials from the uniform distribution. Linear combinations of lattice polynomials, which include weighted sums, linear combinations of order statistics, and lattice polynomials, are actually those continuous functions that reduce to linear functions on each simplex of the standard tria...

Given a non-negative integer k, we construct a lattice 3-simplex P with the following property: The affine semigroup QP associated to P is not normal, and every element q ∈ QP \ QP has lattice distance at least k above every facet of QP .

Published descriptions of families with lattice dystrophy of the cornea reveal a much more varied clinical course and a more pleomorphic corneal picture than the rather stereotyped textbook descriptions would suggest. This report describes a family with lattice dystrophy of unusually late onset with retention of useful vision into the 7th decade. Herpes simplex keratitis was misdiagnosed in 5 m...

A theorem of Howe states that every 3-dimensional lattice polytope P whose only lattice points are its vertices, is a Cayley polytope, i.e. P is the convex hull of two lattice polygons with distance one. We want to generalize this result by classifying 3-dimensional lattice polytopes without interior lattice points. The main result will be, that they are up to finite many exceptions either Cayl...

The aim of the present study was to prepare floating matrix tablet glipizide by applying Simplex lattice design using kappa carrageenan, Hydroxypropylmethylcellulose K15M and sodium bicarbonate as independent variable. similarity factor (f2), time release 50 % drug 90 were taken dependent factors. A total 14 batches prepared given Design-Expert® Software (version- 9.0.6, Stat-Ease) response obt...