Universal covers of finite sets

A recurrent problem in geometry, convexity, and combinatorics is to find universal covers for various classes of sets. (See [2] pp 84–88 for example.) This paper considers that problem for finite sets in the plane. Various covers are given including one which coversall finite sets, is closed, and has empty interior. The problem of covering a finite set or a similar copy of the set is also considered.     

Minimal sets in finite rings

We describe how to calculate the (, )-minimal sets in any finite ring.     

Minimal balanced sets and finite geometries

A balanced set is a collection of subsets of a finite set that can be weighted so as to cover the whole set uniformly. Minimal balanced sets are of interest in the theory of n-person games, in particular for the existence of outcomes that cannot be improved upon by any coalition (core of the game).The object of this paper is to determine the finite geometries which are minimal balanced sets. We prove that the dual of any t-design with t ? 2 is a minimal balanced set. In particular symmetrical 2-designs (as projective spaces, biplanes, etc.) are always minimal balanced sets. For 1-designs the problem becomes much more difficult, but it is for instance easy to prove that any partial geometry which is not the dual of a 2-Steiner system is never a minimal balanced set; in particular generalized quadrangles are never minimal balanced sets. For linear graphs the problem is completely solved: the dual of a connected linear graph is a minimal balanced set if and only if this linear graph is not bichromatic.     

Minimal blocking sets in finite planes

A result on enclosed (k,n)-arcs in PG(2,q) is used to characterise all minimal blocking sets of sizes =q +n in PG(2,q) with at least twon-secants. This also permits a slight extension of the bounds on blocking sets established by Blokhuis and Brouwer.     

Minimal covers,minimal sets and canonical facets of the posynomial knapsack polytope

An irredundant representation of the 0–1 solutions to a posynomial inequality in terms of covering constraints induced by minimal covers is given. This representation is further strengthened using extended covering constraints induced by maximal extensions of minimal covers. Necessary, sufficient, and in a special case necessary and sufficient conditions for an extended covering constraint induced by a minimal set to be a facet of the posynomial knapsack polytope are given.     

Minimal generating sets of maximal size in finite monolithic groups

Denote by m(G) the largest size of a minimal generating set of a finite group G. We want to estimate the difference m(G) ? m(G/N) in the case when N is the unique minimal normal subgroup of G.     

Minimal universal covers inE n

It is shown that any plane set of constant unit width contains a semi-circle of radius 1/2, and using this a minimal univeral plane cover is explicitly constructed. It is also shown that in an n-dimensional space with n>2 there are minimal universal covers of arbitrary large diameter. This paper was written while the author was a National Science Foundation Visiting Senior Fellow at the University of Washington, Seattle, Washington, U.S.A.     

Minimal covers of the prisms and antiprisms

This paper contains a classification of the regular minimal abstract polytopes that act as covers for the convex polyhedral prisms and antiprisms. It includes a detailed discussion of their topological structure, and completes the enumeration of such covers for convex uniform polyhedra. Additionally, this paper addresses related structural questions in the theory of string C-groups.     

Maximal covers of finite groups

Abstract

Let λ(G) be the maximum number of subgroups in an irredundant covering of the finite group G. We prove that if G is a group with λ(G) ≤ 6, then G is supersolvable. We also describe the structure of groups G with λ(G) = 6. Moreover, we show that if G is a group with λ(G)?<?31, then G is solvable.     

Vertex covers and eternal dominating sets

The eternal domination problem requires a graph to be protected against an infinitely long sequence of attacks on vertices by guards located at vertices, the configuration of guards inducing a dominating set at all times. An attack at a vertex with no guard is defended by sending a guard from a neighboring vertex to the attacked vertex. We allow any number of guards to move to neighboring vertices at the same time in response to an attack. We compare the eternal domination number with the vertex cover number of a graph. One of our main results is that the eternal domination number is less than the vertex cover number of any graph of minimum degree at least two having girth at least nine.     

Minimal sets and varieties

The aim of this paper is twofold. First some machinery is established to reveal the structure of abelian congruences. Then we describe all minimal, locally finite, locally solvable varieties. For locally solvable varieties, this solves problems 9 and 10 of Hobby and McKenzie. We generalize part of this result by proving that all locally finite varieties generated by nilpotent algebras that have a trivial locally strongly solvable subvariety are congruence permutable.

     

Minimal sets of Cartan foliations

A foliation that admits a Cartan geometry as its transversal structure is called a Cartan foliation. We prove that on a manifold M with a complete Cartan foliation ?, there exists one more foliation (M, \(\mathcal{O}\)), which is generally singular and is called an aureole foliation; moreover, the foliations ? and \(\mathcal{O}\) have common minimal sets. By using an aureole foliation, we prove that for complete Cartan foliations of the type ?/? with a compactly embedded Lie subalgebra ? in ?, the closure of each leaf forms a minimal set such that the restriction of the foliation onto this set is a transversally locally homogeneous Riemannian foliation. We describe the structure of complete transversally similar foliations (M, ?). We prove that for such foliations, there exists a unique minimal set ?, and ? is contained in the closure of any leaf. If the foliation (M, ?) is proper, then ? is a unique closed leaf of this foliation.     

Cardinality bounds via covers by compact sets

Acta Mathematica Hungarica - We establish results concerning covers of spaces by compact and related sets. Several cardinality bounds follow as corollaries. Introducing the cardinal invariant...     

Small universal covers for sets of unit diameter

The current smallest convex universal cover for sets of unit diameter is described. This reduction of Sprague's cover is by 4 · 10^-11 and results in an asymmetrical cover. Another small universal cover of sets of unit diameter with an axis of symmetry reduces Sprague's cover by 0.0019. An indication is given of how to use computers in the solution of this kind of problem.     

A note on finite abelian covers

We prove that any abelian cover over a smooth variety is defined by some cyclic equations. From the defining equations, we compute explicitly the normalization, branch locus, ramification indices, global invariants, and the resolution of singularities. As an application, we construct a new algebraic surface which is the quotient of ball.     

Finite polytopes have finite regular covers

We prove that any finite, abstract n-polytope is covered by a finite, abstract regular n-polytope.     

Balanced minimum covers of a finite set

In this paper a method of enumeration for n-balanced, labelled, minimum covers of a finite set of m indistinguishable elements is developed. The method is then used to tabulate the number of such covers for small values of m and n by means of a generating function.