Covering the Cube by Affine Hyperplanes

Maximal independent sets in the covering graph of the cube

Several familiar problems in extremal set theory can be cast as questions about the maximum possible size of an independent set defined on a suitable graph, about the total number of independent sets in such graphs, or about enumeration of the maximal independent sets. Here we find bounds on the number of maximal independent sets in the covering graph of a hypercube. © 2010 Elsevier B.V. All ri...

On-line Algorithms for the q-adic Covering of the Unit Interval and for Covering a Cube by Cubes

We present efficient algorithms for the on-line q-adic covering of the unit interval by sequences of segments. The basic method guarantees covering provided the total length of segments is at least 1 + 2 · 1 q − 1 q3 . Other algorithms improve this estimate for q ≥ 6. The unit d-dimensional cube can be on-line covered by an arbitrary sequence of cubes whose total volume is at least 2+ 53 + 5 3 ...

Normality and Covering Properties of Affine Semigroups

S̄ = {x ∈ gp(S) | mx ∈ S for some m > 0}. One calls S normal if S = S̄. For simplicity we will often assume that gp(S) = Z; this is harmless because we can replace Z by gp(S) if necessary. The rank of S is the rank of gp(S). We will only be interested in the case in which S ∩ (−S) = 0; such affine semigroups will be called positive. The positivity of S is equivalent to the pointedness of the cone...

LOWER BOUNDARY HYPERPLANES OF THE CANONICAL LEFT CELLS IN THE AFFINE WEYL GROUP Wa( Ãn−1)

Let Γ be any canonical left cell of the affine Weyl group Wa of type e An−1, n > 1. We describe the lower boundary hyperplanes for Γ, which answer two questions of Humphreys. Let Wa be an affine Weyl group with Φ the root system of the corresponding Weyl group. Fix a positive root system Φ of Φ, there is a bijection from Wa to the set of alcoves in the euclidean space E spanned by Φ. We identif...

Essential overs of the ube by hyperplanes