A general definition for a measure of nonprecompactness for bounded subsets of a locally convex linear topological space is given. Fixed point theorems for condensing multivalued mappings have been proved. These fixed point theorems are further generalizations of Kakutani's fixed point theorems.

In this paper the right global divisible dimension and the right global codivisible dimension of a ring R are studied relative to a torsion theory of modR. The main result shows that if (A, B) is a central splitting torsion theory on modR. then the right global divisible dimension of R with respect to (B, A) is equal to the right global codivisible dimension of R with respect to (A, B).

Infinite families of amicable orthogonal designs are constructed with

(i) both of type (1, q) in order q + 1 when q ≡ 3 (mod 4) is a prime power,

(ii) both of type (1, q) in order 2(q + 1) where q ≡ 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix,

(iii) both of type (2, 2q) in order 2(q + 1) when q ≡ 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix.

The theory of F-finite linear operators developed by Robert T. Moore is used to construct a differential calculus in locally convex spaces. This note contains the fundamental theory up to the implicit function theorem.

The maximal left ideal, the maximal right ideal and the maximal two-sided ideal are all characterized for quite general semigroups of endomorphisms. These results are then applied to such semigroups as the semigroup of all continuous selfmaps of a topological space, the semigroup of all closed selfmaps of a topological space, and the semigroup of all linear endomorphisms of a vector space.

The ring C(X) is the ring of all continuous real-valued functions on a completely regular Hausdorff space X, and βX is the Stone-Čech compactification of X.

The author proves a theorem which leads to a characterization of those zero-sets in X whose closures (in βX) are zero-sets in βX, and relates this characterization to the ideals in the ring C(X).

A general method is described for constructing examples of soluble groups whose normal subgroups form a well-ordered chain under the ordering of inclusion. This method is a variant of one introduced in a recent paper by Heineken and Wilson. Each of the resulting groups is obtained by an embedding procedure from a pair of iterated wreath products A1 wr A2 wr … wr An, B1 wr B2 wr … wr Bn, where the constituent groups Ai., Bi are each either cyclic of prime power order or quasicyclic. Here n may be chosen arbitrarily, and the choice of constituent groups is subject only to a condition on the sequences of prime numbers that may occur as orders of elements in the groups

respectively. The construction is applied to give certain examples which illustrate the limitations of results on particular classes of soluble groups satisfying the minimal condition for normal subgroups obtained in recent papers by Hartley, McDougall, and the present author.

We consider a method by which players in a continuous N-person game can arrive at a Pareto optimal solution by a trial process. The process has a number of novel features. Firstly, it is assumed that the players do not know the payoff functions. Secondly, the players are assumed to act quite independently. In spite of this lack of information and lack of cooperation, the players eventually arrive at what is usually regarded as a cooperative solution. The process is a model of the accounting procedures used by firms, and the results predict approach to an equilibrium state of a market model. Proofs are given only in outline here.

D.L. Berman has proved several divergence theorems about “extended” Hermite-Fejér interpolation on the Chebyshev nodes of the first kind. These are surprising in light of the classical convergence theorem of L. Fejér concerning ordinary Hermite-Fejér interpolation on these nodes. However there is one case which has been neglected so far: the case of quasi-Hermite-Fejér interpolation on these nodes. In this paper it is proved that quasi-Hermite-Fejér interpolation polynomials on the Chebyshev nodes converge uniformly to the continuous function being interpolated. In addition, an estimate for the rate of convergence is established.

Let the group T be the direct product of groups Si (i = 1, … r) where for a given group Ai, Si is the direct product of ni factors Ai × Ai × … × Ai. Let B be a group that has a faithful permutation representation Γi of degree ni. (i = 1, …, r). Consider G, the split extension of T by B defined by letting B act on T as follows.

Each Si is normal in G. If and b ∈ B then where . It is proved that if T is an M-group and all subgroups of B are M-groups, then G is an M-group. This is a generalisation of a result of Gary M. Seitz, Math. Z. 110 (1969), 101–122, who proved the particular case where r = 1 and Γ1 is the regular representation of B.

The classical formulae for Fredholm integral equations, including expansions in terms of eigenfunctions such as Mercer's Theorem are extended to square-integrable kernels on an arbitrary measure space.

An explicit construction is given for a primitive group ring, together with an explicit construction of a faithful irreducible module for it.

Let G be a Hausdorff locally compact abelian group. The author has shown (Bull. Austral. Math. Soc. 10 (1974), 59–66) that, given ε > 0 and a certain base {Vi}i∈I of symmetric open neighbourhoods of zero, the algebra L1(G) admits a bounded positive approximate unit {ki}i∈I such that for every p–th integrable function f on G,

where ω(p; f; Vi) denotes the mean modulus of continuity with exponent p of f. The purpose of this paper is to obtain {ki}i∈I (as above) with a simple dependence of supp on {ki}i∈I on Vi; this is achieved for finite products and homomorphic images of groups for which such a simple dependence is known. The results obtained are used to give a simplified proof of the classical Jackson's Theorem for the circle group, and an analogue of this theorem for the a-adic solenoid.

Each semiprime ring admits a unique projectable, strongly projectable, laterally complete and orthocomplete hull. Almost all of the theory for X–hulls of lattice-ordered groups in Paul Conrad, “The hulls of representable l-groups and f-rings”, J. Austral. Math. Soc. 16 (1973), 385–415, has a counterpart for semiprime rings. The proofs of these results will appear elsewhere. They come in a large part directly from the corresponding theory for lattice-ordered groups. There is also a feedback from the rings to the groups.

There is a small error in the proof of Lemma 1 [1], namely, that there may not be a subsequence of the sequence {Φn} converging to Φ in the weak*-topology. Thus equation (5) should read: In view of continuity of Pk we have

The same error occurs in the proof of Theorem 1 and Lemma 2 also. Therefore, equation (7) should read

and this leads to lim . Similarly, equation (8) should read

which implies that Ψn (x) → ‖x‖. Then there is no need of taking a subsequence.

All the same the results are not affected by the above error.