A subset S of an additive group G is called a maximal sum-free set in G if (S+S) ∩ S = ø and ∣S∣ ≥ ∣T∣ for every sum-free set T in G. It is shown that if G is an elementary abelian p–group of order pn, where p = 3k ± 1, then a maximal sum-free set in G has kpn-1 elements. The maximal sum-free sets in Zp are characterized to within automorphism.

Maximal sum-free sets in groups Zn, where n is any positive integer such, that every prime divisor of n is congruent to 1 modulo 3, are completely characterized.

In this paper we calculate the number of congugacy classes in the following finite classical groups: GLn(Fq); PGLn(Fq), SLn(Fq), and more generally G(Fq), where G is any algebraic group isogenous to SLn; PSLn(Fq); ; , , and more generally where G is any group isogenous to SUn over Fq; and .

We provide bounds on the average distance between two points uniformly and independently chosen from a compact convex subset of the s-dimensional Euclidean space.

Linckelmann and Murphy have classified the Morita equivalence classes of p-blocks of finite groups whose basic algebra has dimension at most $12$. We extend their classification to dimension $13$ and $14$. As predicted by Donovan’s conjecture, we obtain only finitely many such Morita equivalence classes.

Trace inequalities for sums and products of matrices are presented. Relations between the given inequalities and earlier results are discussed. Among other inequalities, it is shown that if A and B are positive semidefinite matrices then for any positive integer k.

A convex lattice polygon is a polygon whose vertices are points on the integer lattice and whose interior angles are strictly less than π radians. We define a(2n) to be the least possible area of a convex lattice polygon with 2n vertices. A method for constructing convex lattice polygons with area a(2n) is described, and values of a(2n) for low n are obtained.

The conjugacy classes in the finite-dimensional projective full linear, special linear and projective special linear groups over an arbitrary commutative field are determined. The results over a finite field are applied to certain enumerative problems.

Suppose the edges of the complete graph on 17 vertices are coloured in three colours. It is shown that at least five monochromatic triangles must aris.

We consider two problems concerning Kolmogorov widths of compacts in Banach spaces. The first problem is devoted to relations between the asymptotic behavior of the sequence of n-widths of a compact and of its projections onto a subspace of codimension one. The second problem is devoted to comparison of the sequence of n-widths of a compact in a Banach space 𝒴 and of the sequence of n-widths of the same compact in other Banach spaces containing 𝒴 as a subspace.