The main results of the paper give necessary and sufficient conditions as well as sufficient conditions that continuous bijections of a manifold onto itself be homeomorphisms. Such conditions include the embedding of manifolds, preserving ends, preserving closed half-rays and restrictions on boundary components. A number of counterexamples are given to likely conjectures.

A class (called c-radicals) is defined such that, given a c-radical P, there is in any class M′ a certain internal criterion that its upper radical UM′ = P, For P a non-c-radical (called a q-radical) there exists no smallest class M such that UM = P, and P is a q-radical if and only if for some M with P = UM there exists 0 ≠ R ∈ M such that when an image ¯ of R has a non-zero image in M there exists an infinite chain of epimorphisms ¯ → R1 → R2 → … with all R1 ∈ M and no Ri, the image of any Rj, with j > i. Several examples of such rings are constructed including a ring all of whose images are primitive. Thus all radicals contained in the Jacobson radical are q-radicals.

The theory of groups acting on trees due to Bass and Serre (1969) is applied to simplify some results of Burns (1972, 1973) giving conditions under which an amalgamated free product or HNN extension has the properties that any finitely generated subgroup containing an infinite subnormal subgroup must have finite index and that the intersection of two finitely generated subgroups is finitely generated.

In this paper we consider the (non)oscillation properties of two general nonhomogeneous nonlinear delay differential equations of order 2n using as background and motivation the techniques previously applied to the associated homogeneous delay differential equations H+ and H−. The equations N+ and N− are each reduced to homogeneous form by the introduction of transformations u(t) = y(t) – R(t) and v(t) = R(t) — y(t), where R(t) is a solution of the associated nonhomogeneous differential equation (N). We first extend certain results for the equation H+ and then develop a classification of the positive solutions of equation H−. Using this classification and the one developed by Terry (1974) for H+ we develop a natural classification of the positive solutions of N+ and N− according to the sign properties of the derivatives of u(t) and v(t). For each choice of R(t), it is seen that there are 2n + 1 types of positive solutions of N+or N–. An intermediate Riccati transformation is employed to obtain integral criteriafor the nonexistence of some of these solutions. Analysis of the Taylor remainder results in sufficient conditions for the nonexistence of other such solutions.

We prove here that the multiplicator of certain simply described metabelian groups is not finitely generated

If (X, τ) is a topological space, is a family of subsets of X, then the expansion of τ by is the topology τ() on X having τ ∪ as a subbase. If (X, τ) has property , when does (X, τ()) have property ? In this paper we answer this question for the property of dimension. We apply our techniques to first countability and related properties.

We study the problem of representing a permutation C as a product of a minimum number, fk(C), of cycles of length k. Upper and lower bounds on fk(C) are obtained and exact results are derived for k = 2, 3, 4.

It is shown, by use of a Bochner-type condition for infinite divisibility, that the von Mises distribution is infinitely divisible for some values of the concentration parameter k, certainly for k < 0.16.

Some non-trivial real, symmetric square roots of the infinite identity matrix are exhibited. These may be found either from the use of involutory integral transforms and a set of real orthonormal functions or by an algebraic factorisation procedure. The two approaches are shown to be equivalent.

It is shown that a unital lattice-ordered ring in which the square of every element is positive is embeddable in a product of totally ordered rings provided it is archimedean, semiperfect, or π-regular. Also, some canonical examples of unital l-domains with squares positive that are not totally ordered are discussed.

The pointwise order makes the group C(X) of continuous real-valued functions on a topological space X a lattice-ordered group. We give a characterization of the compatible tight Riesz orders on C(X), and also of their maximal tangents, in terms of the zero-sets of X. The space of maximal tangents of a given compatible tight Riesz order T is studied, and consequently the concept of the T-radical of C(X) is introduced, the T-radical being the intersection of all the maximal tangents of T.

This paper gives theorems which encompass known characterizations of many of the generalized compactness properties.

My attention has been called to the fact that Theorem E1, quoted on p. 445 of my paper “On the embedding of a group in a join of given groups’, J. Austral. Math. Soc. 17 (1974), 434–495, is practically identical with Théoréme I (p. 47) of the second part of the three-part paper “Produit complet des groupes de permutations et probléme d'extension des groupes” by M. Krasner and L. Kaloujnine, Acta Sci. Math. Szeged 14 (1951), 39–66. My oversight in referring only to the third part of this important work, which contains the special case often quoted as the Kaloujnine-Krasner Theorem, is regrettable: it is the much more general and definitive Théoréme I which should be equally or better known by that name.