An n × n–matrix on n signed variables is called Hadamard of Williamson type if each variable occurs exactly once in each row, and the inner product of any pair of distinct rows is zero. We show here that these matrices correspond in a natural way to rational formulas for products of sums of n squares, shown by Hurwitz to exist only for n = 1, 2, 4, and 8. Hurwitz' arguments contain an implicit proof that this correspondence is one-to-one (we show this directly) and hence that Hadamard matrices of Williamson type exist for orders 1, 2, 4 and 8 only.

Given qo ∈ [1, 2), functions and are constructed such that fgqAq for every q ∈ [1, qo]. In particular, if p ∈ (1, 2), AP is not an algebra.

Let p be a prime and let Q be a centre-by-finite p′-group. It is shown that the ZQ-modules which satisfy the minimal condition on submodules and have p–groups as their underlying additive groups can be classified in terms of the irreducible ZpQ-modules. If such a ZQ-module V is indecomposable it is either the ZpQ-injective hull W of an irreducible ZpQ-module (viewed as a ZQ-module) or is the submodule W[pn] of such a W consisting of the elements ω ∈ W which satisfy pnw = 0.

This classification is used to classify certain abelian-by-nilpotent groups which satisfy Min-n, the minimal condition on normal subgroups. Among the groups to which our classification applies are all quasi-radicable metabelian groups with Min-n, and all metabelian groups which satisfy Min-n and have abelian Sylow p-subgroups for all p.

It is also shown that if Q is any countable locally finite p'-group and V is a ZQ-module whose additive group is a p-group, then V can be embedded in a ZQ-module whose additive group is a minimal divisible group containing that of V. Some applications of this result are given.

The paper extends a theorem given by Entringer on the asymptotic expansion of a composite function.

Let S be a compact connected semigroup. If the decomposition space of S, under the action of a compact connected subgroup at the identity, is a plane continuum then this decomposition is a congruence.

The following convergence criterion of Fourier series is due to M. Izumi, S. Izumi and the author:

THEOREM. Let Δ ≥ 1. If

(i) , and

(ii) as t → 0

for an a, 0 < a < 1 and for a δ, 0 < δ < π, then the Fourier series of φ(t) is convergent at the origin.

The object of this paper is to generalize the above theorem in the Hardy-Littlewood direction.

In this paper we discuss the interrelations among various spaces of distributions and show that none of them can be linearly and differentiably homeomorphic to the space of Mikusiński operators. It is also shown that the distributions of Mikusiński-Sikorski can also be defined by the method described by Temple as the completion of the space of continuous functions after introducing a weaker notion of convergence in this space.

One condition on D was omitted from the statement of Theorem B in the paper [1], namely that (and the corresponding condition in Theorem A). The proof of the theorem clearly requires that this be so and it is not difficult to see that such a D can always be chosen. However, Dr R.M. Bryant has pointed out that not only is the theorem true as it stands, but indeed the conditions can be relaxed slightly to allow D to be any group such that

.

Through an editorial error, the second sentence in the penultimate paragraph of [1] does not make sense. The first line in that paragraph should read:

“It now follows that K/G separates the plane, Where K is, as usual,”.

This paper contains some results on order paracompact spaces. Some of the results offer improvements of some of the results of McCandless in Canad. J. Math. 21 (1969). Some other results of McCandless are deduced as corollaries from our results. The concepts of order closure preserving and order cushioned collections are introduced and using these, characterizations of paracompactness in regular spaces are obtained.