This paper continues the second author's investigation of the normal structure of the automorphism group г of a free abelian group of countably infinite rank. It is shown firstly that, in contrast with the case of finite degree, for each prime p every linear transformation of the vector space of countably infinite dimension over Zp, the field of p elements, is induced by an element of г Since by a result of Alex Rosenberg GL(אo, Zp ) has a (unique) maximal normal subgroup, it then follows that г has maximal normal subgroups, one for each prime.

A theorem on the existence of a solution under feasibility assumptions to a convex minimization problem over polyhedral cones in complex space is given by using the fact that the problem of solving a convex minimization program naturally leads to the consideration of the following nonlinear complementarity problem: given g: Cn → Cn, find z such that g(z) ∈ S*, z ∈ S, and Re〈g(z), z〉 = 0, where S is a polyhedral cone and S* its polar.

We prove that most group algebras of free products have left zero divisors that are not right zero divisors.

To each associative (but not necessarily commutative) ring R we assign the complete distributive lattice R-tors of (hereditary) torsion theories over R-mod. We consider two ways of making this process functorial – once contravariantly and once covariantly – by selecting appropriate subcategories of the category of associative rings. Combined with a functor due to Rota, this gives us functors from these subcategories to the category of commutative rings.

This paper first assigns specific uniform convergence structures to the products and function spaces of pairs of uniform convergence spaces, and then establishes a bijection between corresponding sets of morphisms which yields cartesian closedness.

Let P*(α, β) denote the class of functions

analytic and univalent in |z| < 1 for which

where α є [0, 1), β є (0, 1].

Sharp results concerning coefficients, distortion theorem and radius of convexity for the class P*(α, β) are determined. A comparable theorem for the classes C*(α, β) and P*(α, β) is also obtained. Furthermore, it is shown that the class P*(α, ß) is closed under ‘arithmetic mean’ and ‘convex linear combinations’.

The fact that the group studied in the original paper, M.F. Newman, Bull. Austral. Math. Soc. 14 (1976), 293–297, is infinite follows immediately from a result in a 1940 paper of Coxeter. The computer aided methods give more detailed information about the group and some related groups.

The author has noticed two small errors in his paper [1]: In Figure 1 on p. 290, the vertices labelled 4n, 4n + 1, 4n + 2 should be labelled 4n − 1, 4n, 4n + 1, respectively. In the 14th line on p. 291, “at a distance greater than 2n − 1” should read “at a distance greater than 2n − 2”.