Subspace coverings with multiplicities

Finite Groups with p-Sylow Coverings

Exponentials with in nite multiplicities

Given a semi-ring with unit which satis es some algebraic conditions, we de ne an exponential functor on the category of sets and relations which allows to de ne a denotational model of di erential linear logic and of the lambda-calculus with resources. We show that, when the semi-ring has an element which is in nite in the sense that it is equal to its successor, this model does not validate t...

Networks with arbitrary edge multiplicities

One of the main characteristics of real-world networks is their large clustering. Clustering is one aspect of a more general but much less studied structural organization of networks, i.e. edge multiplicity, defined as the number of triangles in which edges, rather than vertices, participate. Here we show that the multiplicity distribution of real networks is in many cases scale-free, and in ge...

Indecomposable coverings with unit discs

We disprove the 1980 conjecture of János Pach about the cover-decomposability of open convex sets by showing that the unit disc is not cover-decomposable. In fact, our proof easily generalizes to any set with a smooth boundary. We also show that (the suitable variant of) the conjecture holds for unbounded sets. Let C be a collection of sets in R2. We say that C is an m-fold covering if every po...

Multiple Coverings with Closed Polygons

A planar set P is said to be cover-decomposable if there is a constant k = k(P ) such that every k-fold covering of the plane with translates of P can be decomposed into two coverings. It is known that open convex polygons are cover-decomposable. Here we show that closed, centrally symmetric convex polygons are also cover-decomposable. We also show that an infinite-fold covering of the plane wi...