In this paper we shall consider orthodox bands of commutative groups, together with a ring of endomorphisms. We shall generalize the concept of a left module by introducing orthodox bands of left modules; we shall also deal with linear mappings, the transpose of a linear mapping and with the dual of an orthodox band of left modules.

Necessary and sufficient conditions for the weak solvability of the Dirichlet problem for nonlinear differential equations of the second order are proved. The differential operators considered are in the form of a sum of a linear noninvertible operator, with the null-space generated by a positive function, and a monotone nonlinear perturbation, the growth of which is more than linear.

For a Hausdorff convergence space, necessary and sufficient conditions for its Richardson compactification to be the largest Hausdorff compactification are found and by modifying the convergence structure of the Richardson compactification, it is shown that the largest Hausdorff compactification, whenever it exists, is given by that modified Richardson compactification.

In this note a sufficient optimality condition is established for nonlinear programming problems over arbitrary cone domains. A Kuhn-Tucker type sufficient condition is established if the programming problem has a pseudoconvex objective function and a convex feasible region.

Two theorems are presented which characterize the existence of multiplicative left invariant means on a given algebra of unbounded continuous functions on a topological semigroup S in terms of certain common fixed point properties of actions of S on completely regular spaces. Also a lattice formulation of a related result of Theodore Mitchell for the case of bounded functions is shown to be equivalent to a certain common fixed point property on Bauer simplexes.

Dr M.F. Newman has asked whether in the absence of the Axiom of Choice it is possible to have two non-isomorphic elementary abelian groups of the same (finite) exponent and of the same (infinite) cardinality. By means of an example, I show that this is in fact possible, if the exponent is at least five; I do not know the answer in the remaining two cases. The example given requires the construction of a Fraenkel-Mostowski model of set theory, and for this purpose I draw upon the terminology, constructions, and results contained in the first two sections of a previous paper, “The construction of groups in models of set theory that fail the Axiom of Choice” (Bull. Austral. Math. Soc. 14 (1976), 199–232), with which I assume familiarity.

One form of Dirichlet's theorem on simultaneous diophantine approximation asserts that if α1, α2, …, αn are any real numbers and m ≥ 2 is an integer, then there exist integers q, p1, p2, …, pn such that 1 ≤ q < m and |qαi.-pi| ≤ m–1/n holds for 1 < i < n. The paper considers the problem of the extent to which this theorem can be improved by replacing m–1/n by a smaller number. A general solution to this problem is given. It is also shown that a recent result of Kurt Mahler [Bull. Austral. Math. Soc. 14 (1976), 463–465] amounts to a solution of the case n = 1 of the above problem. A related conjecture of Mahler is proved.

A group A is said to be SQ-universal if every countable group is embeddable in some quotient of A. It is well-known that the number of non-isomorphic factor groups of a countable sp-universal group B is the power of the continuum. Since B has such an abundance of non-isomorphic quotients it is natural to ask what types of quotients B must have and what types B need not have. This note is concerned with these questions.

This paper presents a necessary condition for a ribbon link to be an homology boundary link and gives a consequent simple counterexample to the conjecture of Smythe that the vanishing of the first Alexander polynomial characterizes homology boundary links among all 2-component links.

Let G be a finite group, p a prime divisor of |G|, and T a p–subgroup of G. Define σ(T) to be the number of Sylow p–subgroups of G containing T. Call T a central p–Sylow intersection if for some Σ ⊆ Sylp (G), T = ∩(S | S є Σ), and if, in addition, T contains the center of a Sylow p–subgroup of G. This work is inspired and motivated by work of G. Stroth [J. Algebra 37 (1975), 111–120]. Generalizing an argument of his we describe finite groups in which every central p–Sylow intersection T with p–rank(T) > 2 satisfies σ(T) ≤ p.

Related methods yield the description of finite groups in which every central p–Sylow intersection T with p–rank(T) ≥ 2 satisfies σ(T) ≤ 2p.

The characterizations of the closed ranges of the multiplication operators and the composition operators on L2 (λ) are reported in this paper.

In this note, Conjecture Cl of Toby Lewis (1976), concerning a reciprocal pair of characteristic variables, is shown to be false.

Converses are proved for the Osgood (the Principle of Uniform Boundedness), Dini, and other well known. theorems. The notion of a continuous step function on a topological space is defined and a class of spaces identified for which each lower semicontinuous function is the pointwise limit of a monotonically increasing sequence of step functions.

It is proved that every metrizable topological space without isolated points is the union of a continuum or fewer nowhere dense subsets.

A generating function is derived for the number of transitions to the first passage through a particular vertex of a random walk on a doubly regular tournament. As an application of this result, we obtain a generating function for the number of solutions (q1, …, qk) of qi ≡ r (mod p), where p is a prime of the form 4l + 3 and the qi are quadratic residues modulo p.

Let G be a compact Lie group and V and W be linear G spaces. A study is made of the canonical stratification of some algebraic varieties that arise naturally in the theory of C∞ equivariant maps from V to W. The main corollary of our results is the equivalence of Bierstone's concept of “equivariant general position” with our own of “G transversal”. The paper concludes with a description of Bierstone's higher order conditions for equivariant maps in the framework of equisingularity sequences.

This paper produces new types of designs, called product designs, which prove extremely useful for constructing orthogonal designs. An orthogonal design of order 2t and type

is constructed. This design often meets the Radon bound for the number of variables.

We also show that all orthogonal designs of order 2t and type (a, b, c, d, 2t-a-b-c-d), with 0 < a + b + c + d < 2t, exist for t = 5, 6, and 7.

Perhaps the simplest example is the lexicographic sum of copies Zn of the integers indexed by the integers. If one performs the shift in the 0-indexed summand m0 → (m+1)0 while leaving all the other summands fixed, and follows this with the shift Zn → Zn+1 in the index set, then the element 00. will be sent on 11; whereas under these automorphisms performed in the other order, it will be sent on 01. This non-commutativity contradicts Theorem 9 in [1].