Algorithms for many geometric queries rely on representations that are comprised of combinatorial (logical, incidence) information, usually in a form of a graph or a cell complex, and geometric data that represents embeddings of the cells in the Euclidean space E. Whenever geometric embeddings are imprecise, their incidence relationships may become inconsistent with the associated combinatorial...

We will present a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes. But cubical sets can also express topo...

1. Preliminaries. A functor from a category of combinatorial geometries, or equivalently a category of geometric lattices, to a category of commutative algebras will be described, and some properties of this functor will be investigated. In particular, a cohomology will be associated to each point of a geometry and will be derived from the associated algebra. If (G, S) is a geometry on a set S ...

This issue of the journal is devoted to the themes of Geometry, Combinatorial Designs and Cryptology. The guest editors selected sixteen contributions covering various areas within these themes, ranging from public-key cryptography to matters related to key distribution and authentication, from problems in graph theory to resolvability issues in designs, from finite projective planes to higher-...

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding maps and by taking infinite subsequences. That embedding can be realized by ...

A significant group of problems coming from the realm of Combinatorial Geometry can only be approached through the use of Algebraic Topology. From the first such application to Kneser’s problem in 1978 by Lovász [13] through the solution of the Lovász conjecture [1], [8], many methods from Algebraic Topology have been used. Specifically, it appears that the understanding of equivariant theories...

We discuss certain extremal problems in combinatorial geometry, including Sylvester’s problem and its generalizations.