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...

A well-known theorem of Fong states that over large enough fields of any characteristic, the principal indecomposable modules of a soluble finite group are induced from subgroups of order prime to the characteristic. It is shown that this property in fact characterises soluble finite groups.

Combinatorial functions are used to replace indecomposable cardinals in certain types of set theoretic arguments. This allows us to extend decidability results from nonchoice set theories to set theories with a principle of linear ordering.

In this note, we describe an extension of Hubert's Theorem 90 for K2 of fields to the relative K<i of semilocal PID's containing a field. Most of the results for K2 of fields proven in [M-S and S] then carry over to the relative K% of semilocal PID's containing a field, e.g. computation of the torsion subgroup, and the norm residue isomorphism. Applying this to the semilocal ring of {0,1} in A^...

Let (R,m) be a local ring (commutative and Noetherian). If R is complete (or, more generally, Henselian), one has the Krull-Schmidt uniqueness theorem for direct sums of indecomposable finitely generated R-modules. By passing to the m-adic completion b R, we can get a measure of how badly the Krull-Schmidt theorem can fail for a more general local ring. We assign to each finitely generated R-mo...

The recent increase in the notifications of tuberculosis infections has focused the attention of healthcare professionals on reasons for the disease's resurgence. This article describes how directly observed therapy and an enlightened approach to health promotion for groups of people perceived as being at risk, could prevent further increases in incidence.