Professor Sir Thomas Cherry, F.A.A., F.R.S., died at his home in Melbourne on 21st November 1966 at the age of 68. He was widely known and highly respected as Australia's most distinguished mathematician and a leader in university affairs. He was associated with the University of Melbourne for most of his life, and latterly with La Trobe University as first chairman of its Academic Planning Board. He was a foundation member and a president of both the Australian Academy of Science and the Australian Mathematical Society. His greatest contributions to knowledge were probably made in the mathematics of air flow in trans-sonic flight, simultaneously with Lighthill in Britain; but he also made major contributions to global differential equation theory and general dynamics, and solved some difficult special problems in various branches of applied mathematics. He was a most distinguished teacher, amongst whose students are numbered two Fellows of the Royal Society and several professors in Australian and overseas universities. He was a man of wide interests and great ability, of keen insight and broad vision. He knew much more than he ever wrote, and his influence will live on in the minds of innumerable people with whom he worked.

This note shows that the set of bare points of a compact convex subset of a normed linear space is, in general, a proper subset of its set of exposed points.

In his paper [1], W. D. Munn determines the irreducible matrix representations of an arbitrary inverse semigroup. Munn also gives a necessary and sufficient condition upon a 0-simple inverse semigroup for it to have a non-trivial matirx representation and for such semigroups gives a complete account of their representations. Munn's results rest upon the earlier work of Clifford [2] in which the representations of Brandt semigroups were determined. An alternative account of such representations was given by Munn in [3]. This earlier work is presented in Sections 5.2 and 5.4 of [4].

In this paper the Toeplitz determinant of order s ≧ 1 generated by the rational function , with , and , is evaluated exactly for all values of s ≧ m, as , where in with and , thus proving Szegö's formula for the function fm, n(z).

By forming the rational approximation of the generating function the formula is then extended to enabling the evaluation of the limit of Toeplitz determinants generated by certain classes of complex valued functions.

By the geometric moving average of the independent, identically distributed random variables {Xn}, we mean the stochastic process , where a is a real number such that 0≦a≦1.

Throughout this paper F is an algebraically closed field of characteristic p (≠ 0) and g is a finite group whose order is divisible by p. We define in the usual way an F-representation of g (or F G-representation) and its corresponding module. The isomorphism class of the, F G-representation module M is written {M} or, where no confusion arises, M. A (G) denotes the F-representation algebra of G over the complex field G (as defined on pages 73 and 82 of [6]).

As the applications of category theory increase, we find ourselves wanting to imitate in general categories much that was at first done only in abelian categories. In particular it becomes necessary to deal with epimorphisms and monomorphisms, with various canonical factorizations of arbitrary morphisms, and with the relations of these things to such limit operations as equalizers and pull-backs.

It is a fascinating problem in the axiomatics of any mathematical system to reduce the number of axioms, the number of variables used in each axiom, the length of the various identities, the number of concepts involved in the system etc. to a minimum. In other words, one is interested finding systems which are apparently ‘of different structures’ but which represent the same reality. Sheffer's stroke operation and. Byrne's brief formulations of Boolean algebras [1], Sholander's characterization of distributive lattices [7] and Sorkin's famous problem of characterizing lattices by means of two identities are all in the same spirit. In groups, when defined as usual, we demand a binary, unary and a nullary operation respectively, say, a, b →a·b; a→a−1; the existence of a unit element). However, as G. Rabinow first proved in [6], groups can be made as a subvariety of groupoids (mathematical systems with just one binary operation) with the operation * where a * b is the right division, ab−1. [8], M. Sholander proved the striking result that a mathematical system closed under a binary operation * and satisfying the identity S: x * ((x *z) * (y *z)) = y is an abelian group. Yet another identity, already known in the literature, characterizing abelian groups is HN: x * ((z * y) * (z * a;)) = y which is due to G. Higman and B. H. Neumann ([3], [4])*. As can be seen both the identities are of length six and both of them belong to the same ‘bracketting scheme’ or ‘bracket type’.

Milne-Thomson's well-known circle theorem [1] gives the stream function for steady two-dimensional irrotational flow of a perfect fluid past a circular cylinder when the flow in the absense of the cylinder is known. Butler's sphere theorem [2] gives the corresponding result for axially symmetric irrotational flow of a perfect fluid past a sphere. Collins [3] has obtained a sphere theorem for axially symmetric Stokes flow of a viscous liquid which gives a stream function satisfying the appropriate viscous boundary conditions on the surface of a sphere when the stream function for irrotational flow in the absence of the sphere is known.

Let be a given series with its partial sums {Sn} and {Pn} a sequence of real or complex parameters. Write. The transformation given by defines the Nörlund means of {Sn} generated by {Pn}. The series Σann is said to be absolutely summable (N, pn) or summable ∣N, pn∣, if {tn} is of bounded variation, i.e., Σ|tn—tn−1| converges.

The problem of determining the class of homomorphic images of a given class of topological semigroups seems to have received little attention in the literature. In [4] Cohen and Krule determined the homomorphic images of a semigroup with zero on an interval. Anderson and Hunter in [1] proved several theorems in this direction. In general, the problem seems to be rather difficult. However, the difficulty is lessened somewhat if all of the homomorphisms of the semigroups in question must be monotone. Phillips, [7], showed that every homomorphism of a standard thread is monotone and hence every homomorphic image of a standard thread is either a standard thread or a point. In this paper a larger class of topological semigroups which admit only monotone homomorphisms is given. These results are used to determine the topological nature of the homomorphic images of certain classes of topological semigroups. These include products of standard threads with min threads, certain semilattices on a two-cell, and compact connected lattices in the plane.

The generalized Pexider equation where f and g are unknown and x, y, are real, has been discussed by J. Aczél [1] and J. Aczé and M. Hosszú [2]. In [2] it is shown that if F is continuous and F and H are strictly increasing in their first variables and strictly decreasing in their second variables, then two initial conditions suffice to determine at most one continuous solution f of (1). We extend these results to strictly increasing and strictly decreasing functions F and derive results for strictly monotonic F and H.

In [1], the following theorem is proved:THEOREM. If f ∈C2π, α is a positive integer, and then Where , .

A lattice ordered group(‘l-group’) is called complete if each set of elements that is bounded above has a least upper bound (and dually). A complete l-group is archimedean and hence abelian, and each archimedean l-group has a completion in the sense of the following theorem.

The double-six configuration in classical 3-dimensional projective geometry has been discussed by a number of authors. It consists of two sets a1,…, a6 and b1, …, b6 of six lines such that no two lines of the same set intersect, and ai meets bi if and only if i †j. The existence of a doublesix in the 3-dimensional projective geometry over a field F has been proved by Hirschfeld in [2] for all fields F except those of 2, 3 and 5 elements. For an arbitrary 3-dimensional projective geometry in which the number of points on a line is at least 5 but is not 6, the existence of a double-six follows from the fact that the geometry is a geometry over a division ring D with a subfield F satisfying the conditions of Hirschfeld's theorem.

Extensions of semigroups have been studied from two points of view; ideal extensions and Schreier extension. In this paper another type of extension is considered for the class of inverse semigroups. The main result (Theorem 2) is stated in the form of the classical treatment of Schreier extensions (see e.g.[7]). The motivation for the definition of idempotentseparating extension comes primarily from G. B. Preston's concept of a normal set of subsets of a semigroup [6]. The characterization of such extensions is applied to give another description of bisimple inverse ω-semigroups, which were first described by N. R. Reilly [8]. The main tool used in the proof of Theorem 2 is Preston's characterization of congruences on an inverse semigroup [5]. For the standard terminology used, the reader is referred to [1].

The object of this note is to study two properties of groups, which we will denote by (*) and (**). The property (*) is possessed by solvable groups (and in fact, by groups which have a solvable invariant system) and the property (**) is possessed by nilpotent groups (and in fact, by groups which have a central system).

Ricci identities in a Finsler space have been given by C. I. Ispas [1], H. Rund [2], R. S. Mishra and H. D. Pande [3] and others. Here we shall prove some identities using the principle of mathematical induction. Considering a second order contravariant tensor depending on the element of support , we have the following theorems.

In [1], Aull and Thron introduce several separation axioms between T0 and T1. In particular, they define a space X to be TD if for each x ∈ X, {x}' is a closed set.

It is intuitively obvious that if z is a complex number such that ∣1–zv∣ ≤ b ≺ 1 for all positive integers p and some real number b, then z = 1. The purpose of this note is to exhibit a proof of the following generalisation of this observation: THEOREM. Let A be continuous linear operator on a reflexive Banachspace B. If there exists a continuous linear operator T on B, a real number b, and a positive integer p' such that, p an integer and , then A = I. Moreover, in this case ∥I—T∥.