Let Out (RG) be the set of all outer R-automorphisms of a group ring RG of arbitrary group G over a commutative ring R with 1. It is proved that there is a bijective correspondence between the set Out (RG) and a set consisting of R(G × G)-isomorphism classes of R-free R(G × G)-modules of a certain type. For the case when G is finite and R is the ring of algebraic integers of an algebraic number field the above result implies that there are only finitely many conjugacy classes of group bases in RG. A generalization of a result due to R. Sandling is also provided.

A lemma is obtained which guarantees that non-expansive mappings on contractive spaces have fixed points. An example shows that Schauders's fixed point theorem cannot be extended to contractive spaces, but a theorem for contractive spaces, analogous to a result of B. N. Sadovskii on convex spaces, is derived from the lemma. Finally, some local results for ε-chainable contractive spaces are given.

We prove a result on compactness properties of Fréchet-derivatives which implies that the Fréchet-derivative of a weakly compact map between Banach spaces is weakly compact. This result is applied to characterize certain weakly compact composition operators on Sobolev spaces which have application in the theory of nonlinear integral equations and in the calculus of variations.

The main results of this article are (I) Let B be a homogeneous Banach algebra, A a closed subalgebra of B, and I the largest closed ideal of B contained in A. We assert that for some closed subalgebra J of B. Furthermore, under suitable conditions, we show that A is an R-subalgebra if and only if J is an R-subalgebra. A number of concrete closed subalgebras of a homogeneous Banach algebra therefore are R-subalgebras. For the definition of P(A) and that of an R-subalgebra, see the introduction in Section 1. (II) We give sufficient and necessary conditions for a closed subalgebra of Lp(G), 1 ≦ p ≦ ∞, to be an R-subalgebra.

We contine our study of the following combinatorial problem: What is the largest integer N = N (t, m, p) for which there exists a set of N people satisfying the following conditions: (a) each person speaks t languages, (b) among any m people there are two who speak a common language and (c) at most p speak a common language. We obtain bounds for N(t, m, p) and evaluate N(3, m, p) for all m and infintely many values of p.

A Bhaskar Rao design is obtained from the incidence matrix of a partially balanced incomplete block design with m associate classes by negating some elements of the matrix in such a way that the inner product of rows α and β is ci if α and β are ith associates. In this paper we use nested designs constructed from unions of cyclotomic classes to give Bhaskar Rao designs.

It is shown that a cohomology theory over an admissible category, which is obtained from an additive cohomology theory over a smaller admissible category, through the Kan extension process, always admits global Adams completion.

Let Q(x, y, z, t, u) be a real indefinite 5-ary quadratic form of type (3,2) and determinant D(> 0). Then given any real numbers x0, y0, z0, t0, u0 there exist integers x, y, z, t, u such that 0 < Q(x+x0,y+y0,z+z0,t+t0,u+u0)≦(16D)1/5. All the critical forms are also determined.

We establish a method of constructing kernels of Bergman operators for second-order linear partial differential equations in two independent variables, and use the method for obtaining a new class of Bergman kernels, which we call modified class E kernels since they include certain class E kernals. They also include other kernels which are suitable for global representations of solutions (whereas Bergman operators generally yield only local representations).

Let K be an algebraically closed field of characteristic zero, and S a nonempty subset of K such that S Q = Ø and card S < card K, where Q is the field of rational numbers. By Zorn's Lemma, there exist subfields F of K which are maximal with respect to the property of being disjoint from S. This paper examines such subfields and investigates the Galois group Gal K/F along with the lattice of intermediate subfields.

A square matrix A is transposable if P(RA) = (RA)T for some permutation matrices p and R, and symmetrizable if (SA)T = SA for some permutation matrix S. In this paper we find necessary and sufficient conditions on a permutation matrix P so that A is always symmetrizable if P(RA) = (RA)T for some permutation matrix R.

In this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ: S → T between regular semigroups. This is a mapping such that (ab) θ ≦ aθ bθ in the natural partial order on T.

The problem of positive points in polar lattices, discussed by Hossain and Worley for the distance functions Ft(x1, x2) = │x1│+│tx2 │ and , is considered for a general distance function F. Best possible results are obtained.