We show that if the normal, closure of an element a, of an orderable group, G, is abelian, then G can be embedded in an orderable group, G#, which contains an n-th root of a for every positive integer, n. Furthermore, every order of G extends to an order of G#.

This paper is concerned with the stability, boundedness, and asymptotic behavior of solutions of integrodifferential systems of the form

We shall also investigate the behavioral relationships between the solutions of two integrodifferential systems related to this system.

We display the relevance of Martin's Axiom to suitable forms of some long-known results in classical measure and category theory (for example, outer-measure-preserving partitions of sets of positive outer measure). While our theorems are at best fragmentarily new, the proofs are very simple, and, we think, instructive as regards the force of the axiom.

A question of L.G. Kovács, Joachim Neubüser, B.H. Neumann (J. Austral. Math. Soc. 12 (1971), 287–300) on the existence of ‘secretive’ prime-power groups of large rank is settled affirmatively by proving the following result: given a prime p and integer d ≥ 2, there exists a finite p–group P with cyclic centre and minimal number of generators d and having the property that every element not in its Frattini subgroup has a non-trivial power in its centre.

Analogues of Artin-Wedderburn and Goldie structure theorems are obtained for a class of halfrings which includes the semisubtractive ones. In the semisubtractive case, precise results are obtained which show that non-ring examples of these structures are relatively limited.

An elementary proof is given of a result obtained by concerning the sum of an infinite series involving a 4F3.

A construction due to Reilly is extended to show that there is a correspondence between compatible tight Riesz orders on ZX and non-principal filters on X. The maximal compatible tight Riesz orders are in one-to-one correspondence with non-principal ultra-filters, and are dual prime subsets of the positive set of ZX. Conversely every dual prime algebraic Riesz order is maximal.

If an optimal solution exists for a nonlinear fractional programming problem, then this solution is shown to be obtainable by solving two associated programming problems whose objective functions are no longer fractional. A certain restriction is assumed on the constraint sets of the latter problems. This result includes various known theorems as special cases.

We announce a number of results on the existence of solutions of nonlinear elliptic boundary value problems in the case where the dominating linear part is not invertible. Our theorems improve recent results of Landesman and Lazer, and Williams.

We prove that if E is a Banach space which has a subspace G such that the conjugate space G* contains a proper norm closed linear subspace V of characteristic 1, then E** is not smooth and there exist in πE(E) points of non-smoothness for E**, where πE: E → E** is the canonical embedding. We show that the spaces E having such a subspace G constitute a large proper subfamily of the family of all non-reflexive Banach spaces.

This paper is a continuation of the study of the boundedness operator δ. By determination of the congruences (that is, collapsings) of the smallest lattice containing δ and closed under application of δ, a nev classification of all topological spaces is obtained according to boundedness criteria.

The oscillatory behaviour of the solutions of a functional differential equation, of more general form than an equation arising from in industrial problem, is examined. The question of whether one can maintain the oscillatory properties of this equation by adding a forcing term is also answered. The results obtained extend already known results on the subject and complete the relevant literature.

It is shown that if q is a prime power then there exists a circulant weighing matrix of order q2 + q + 1 with q2 nonzero elements per row and column.

This result allows the bound N to be lowered in the theorem of Geramita and Wallis that “given a square integer k there exists an integer N dependent on k such that weighing matrices of weight k and order n and orthogonal designs (1, k) of order 2n exist for every n > N”.

In the study of the theory of rings, matrix rings, group rings, algebras, and so on, play a very important role. However, the analogous systems may not exist in the theory of near-rings. Recently Ligh obtained a necessary and sufficient condition for the set of n × n matrices with entries from a near-ring to be a near-ring. This opens the door for the study of other structures such as group near-rings, algebras, and so on. In this paper we initiate a study of the basic properties of pseudo-distributive near-rings, which is exactly the class of near-rings needed to carry out the construction of matrix near-rings, group near-rings, polynomials with near-ring coefficients, and so on.

In 1965, D.L. Berman established an interesting divergence theorem concerning Hermite-Fejér interpolation on the extended Chebyshev nodes. In this paper it is shown that this phenomenon is not an isolated incident. A similar divergence theorem is proved for a higher order interpolation process. The paper closes with a list of several related open problems.

The object of this paper is to prove two new results on the nature of the spectrum of a class of nonlinear elliptic eigenvalue problems. In the first case, sufficient conditions are given for which the spectrum is bounded and, in the second case, conditions are given for which the spectrum is open.