Related papers: Polymake and Lattice Polytopes
We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also…
We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For…
We point out that a newly introduced recursive algorithm for lattice polymers has a much wider range of applicability. In particular, we apply it to the simulation of off-lattice polymers with Lennard-Jones potentials between non-bonded…
We provide two algorithms for computing the volume of a convex polytope with half-space representation {x>=0; Ax <=b} for some (m,n) matrix A and some m-vector b. Both algorithms have a O(n^m) computational complexity which makes them…
We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software…
We introduce the convex combinatorial optimization problem, a far reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and…
A new construction of codes from old ones is considered, it is an extension of the matrix-product construction. Several linear codes that improve the parameters of the known ones are presented.
These lectures on the combinatorics and geometry of 0/1-polytopes are meant as an \emph{introduction} and \emph{invitation}. Rather than heading for an extensive survey on 0/1-polytopes I present some interesting aspects of these objects;…
Given a finite set of lattice points, we compare its sumsets and lattice points in its dilated convex hulls. Both of these are known to grow as polynomials. Generally, the former are subsets of the latter. In this paper, we will see that…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
We give a parametrization for crystal bases of Demazure modules as a set of lattice points in some convex polytope and we also describe explicitly the extremal vectors as solutions of some system of linear equations.
The reduction of covering decision systems is an important problem in data mining, and covering-based rough sets serve as an efficient technique to process the problem. Geometric lattices have been widely used in many fields, especially…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
The mixing operation for abstract polytopes gives a natural way to construct the minimal common cover of two polytopes. In this paper, we apply this construction to the regular convex polytopes, determining when the mix is again a polytope,…
We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial…
A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have…
We give a new combinatorial proof for the number of convex polyominoes whose minimum enclosing rectangle has given dimensions. We also count the subclass of these polyominoes that contain the lower left corner of the enclosing rectangle…
We describe the C program mori.x. It is part of PALP, a package for analyzing lattice polytopes. Its main purpose is the construction and analysis of three--dimensional smooth Calabi--Yau hypersurfaces in toric varieties. The ambient toric…
This is a survey article on the theory of lattice points in large planar domains and bodies of dimensions 3 and higher, with an emphasis on recent developments and new methods, including a lot of results established only during the last few…
We present a mathematical and algorithmic scheme for learning the principal geometric elements in an image or 3D object. We build on recent work that convexifies the basic problem of finding a combination of a small number shapes that…