Professor Herbert Melvin Lieberstein died at Royal Newcastle Hospital on 18th August,1973, at the age of 47. As an applied mathematician, Professor Lieberstein was devoted tohis research and teaching in the application of mathematics to other disciplines; his most recent book entitled “Mathematical Physiology”, was published just before his death. Because his interests coincided so closely with the aims and aspirations of theFaculty of Mathematics at Newcastle, his appointment to that Faculty in 1971 was particularly appropriate.He lived less than two years in Australia, but during that time he consolidated his reputation as a most versatile and successful applied mathematician. His work is well known throughout Australia and, indeed, throughout the world, especially to those who work in fields to which mathematics may be profitably applied. However, his versatility and ability were not confined to mathematics, for he also made substantial contributions to a number of community projects; these contributions will be long remembered by those interested in the welfare of the Australian Aborigines, problems of the environment and associated projects.

It is of course well known that within the framework of any reasonable set theory whose axioms include that of choice, we can characterize well-orderings in two different ways: (1) a total order for which every nonempty subset has a minimal element; (2) a total order in which there are no infinite descending chains.

Now the theory of well-ordered sets and their ordinals that is expounded in various texts takes as its definition characterization (1) above; in this paper we commence an investigation into the corresponding theory that takes characterization (2) as its starting point. Naturally if we are to obtain any differences at all, we must exclude the axiom of choice from our set theory. Thus we state right at the outset that we are working in Zermelo-Fraenkel set theory without choice.

In this paper the method of conformal transformation is used to consider a class of crack problem in linear isotropic elasticity. The conformal transformation used is the Joukowski transformation so that in general the crack profile is an aerofoil. This is a useful profile to consider since as well as providing information about the general aerofoil crack it includes as special cases the circular arc crack, the straight line crack and, as will be shown, can also be used to obtain similar results to those obtained by Bowie (1956) for the crack originating at the boundary of a circular hole. Thus by varying the parameters in the one elastic solution it is possible to obtain the elastic solution for a number of crack profiles. An examination of the stresses at the crack tip is made since this gives useful information about the direction of crack extension and also gives information necessary for the application of the fracture criterion which is derived in section 6. Some particular crack profiles are considered in detail in section 7.

A Dedekind-finite set is one not equinumerous with any of its proper subsets; it is well known that the axiom of choice implies that all such sets are finite. In this paper we show that in the absence of the axiom of choice it is possible to construct Dedekind-finite sets which are large, in the sense that they can be mapped onto large ordinals; we extend the result to proper classes. It is also shown that the axiom of choice for countable sets is not implied by the assumption that all Dedekind-finite sets are finite.

In a series of papers ((1967), (1967a) and (1967b)) Magill has considered the semigroups J(X, Y; θ) (definition below), a natural, but extensive, generalization of the usual transformation semigroup J(X). They have also been studiedin Sullivan (to appear). Under the assumption that θ be onto Magill described their automorphisms and determined when one J(X, Y; θ) is isomorphic to another.

This paper concerns the completions of partially ordered groups introduced by Fuchs (1965a) and the author (to appear); the p.o. groups under consideration are, generally, abelian tight Riesz groups, and so, throughout, the word “group” will refer to an abelian group.

In section 3 we meet the cornerstone of the work, the central product theorem, by means of which we can interpret the Cauchy completion of a tight Riesz group in terms of the completion of any of its o-ideals; one particularly important case arises when the group has a minimal o-ideal. Such a minimal o-ideal is o-simple, and in section 6 the completion of an isolated o-simple tight Riesz group is shown to be a tight Riesz real vector space.

A further development of the concept of contractors for non-linear functionals is presented by introducing two-point contractors. On this basis two-point secant methods are proposed for finding roots of non-linear functionals in Banach spaces.

Products of positive measures play a very important role in analysis. The purpose of this paper is to construct a theory of products of two measures taking values in two (possibly different) Banach spaces. A Fubini theorem is obtained which generalizes the Fubinitheorem for the Bochner integral (Dunford and Schwartz (1958), Theorem 9, page 190), and hence also the classical result.

We use the theory of vector integration presented in Dinculeanu (1967). Our arguments rely upon a standard sort of application of the dominated convergence theorem (cf. Dunford and Schwartz (1958), Theorem 9, page 190), and therefore do not appear to generalize to any theory of integration where this theorem is lacking (e.g. Bartle (1956)).

Let and .

In 1951 Mohanty proved the following theorem: .

In this paper a general theorem on summability |R,l (w), 1 | of Σ An(x) has been given which improves upon Mohanty's result in different ways (see Corollaries 1, 2 and 3) and it is also shown that some of the results of this note are the best possible.

Výborný and I (1972) proved maximum principles for a quasilinear elliptic operator where the boundary satisfied a smoothness condition weaker than the interior sphere property. In this paper I extend these to parabolic operators of a similar form and through a simple device to weakly coupled systems of such operators. Finally, I extend all of these results to an operator similar to the “parabolic” case of an operator introduced by Redheffer (1971). His conditions on the coefficients are replaced by conditions analogous to those Dow and Výborný (1972).

A formula is established for differentiating the composite F(ξ)○G(ξ) with respect to the parameter ξ where F and G are maps which assumetheir values in function spaces. This composite is denoted by . Use of this notation, together with the tangent functor T, enables the formula to be written in the variable-free form where π1 is a projection map and TF(ξ) = T(F(ξ)). Formal verificationof this formula is straightforward. It has application to many small divisor problems such as those which occur in celestial mechanics.