The groups considered are Λ = GL(2, Z), π = SL(2, Z) and Θ = PSL(2, Z). A presentation of Λ is obtained for which the word problem can be solved by a simple intrinsic algorithm. The presentation is modified to display other features of Λ, and to obtain related presentations of Λ and Θ. There is an algorithm which solves the conjugacy problem of Λ.

Consider the n-th order delay differential equation

In the last few years, the oscillatory behavior of delay differential equations has been the subject of intensive investigations. But much less is known about the equation (A) with small forcing term q(t). The only papers devoted to this problem are by Kartsatos, Kusano, and the present author. The purpose of this paper is to prove some new oscillation theorems which contain the previous results.

The regularity series, or briefly R-series, of a convergence space is an ordinal sequence of spaces leading to the regular modification of the space. The behavior of this series is studied relative to such basic constructs as products, subspaces, and various quotient maps. Upper bounds on the length of the R-series are obtained for several classes of spaces. This series can be employed to construct regular completions and compactifications.

Central limit theorems are obtained for martingale arrays without the requirement of uniform asymptotic negligibility. The results obtained generalise the sufficiency part of Zolotarev's extension of the classical Lindeberg-Feller central limit theorem [V.M. Zolotarev, Theor. Probability Appl. 12 (1967), 608–618] and also the main martingale central limit theorem (not functional central limit theorem however) of D.L. McLeish [Ann. Probability 2 (1974), 620–628.

The author obtains characterizations of the quotients, epimorphisms and extreme monomorphisms in the category of separated zero-set spaces and zero-set maps (defined by Hugh Gordon [Pacific J. Math. 36 (1971), 133–157]). The method employed, that of initiality constructions, is also used to elucidate the relationship between zero-set spaces and certain other topological structures by means of forgetful functors and their right inverses. Characterizations of pseudocompactness for zero-set spaces then follow.

For a Banach space X, smoothness at a point of the natural embedding ◯ in X**, is characterised by a continuity property of the support mapping from X into X*. It then becomes clear that there are many non-reflexive Banach spaces with smooth embedding, a matter of interest raised by Ivan Singer [Bull. Austral. Math. Soc. 12 (1975), 407–416].

In previous work, Page and Butson [Algebra Universalis 3 (1973), 112–126] characterized all equationally complete classes (atoms) of m–semigroups (universal algebras with one m–ary associative operation), and hence m–groups, in the commutative case. The further characterization of the non-commutative m-group atoms was thought to hinge upon a conjecture by Page [PhD thesis, University of Miami, 1973] that a weaker form of commutativity held true. In this paper that conjecture is proved and then used to study the special case m = 4. Two additional infinite sets of atoms are thereby determined, although it is not proved that these examples exhaust the remaining atoms for m = 4.

Any subset π* of the set of all planes through a line in a finite projective space PG(m, q) determines a subgeometry G(π*) of the combinatorial geometry associated with PG(m, q). In this paper the geometries G(π*) of rank greater than three in which every line contains at least four points, are characterized in terms of the existence of a certain set of automorphism groups Γ(C, X); where X is a copoint and C a point not in X, and each non-trivial element of Γ(C, X) fixes X and every copoint through C and fixes C and every point in X, but no other point; and where Γ(C, X) acts transitively on the points distinct from C and not in X of some line through C. As a corollary of the main theorem we obtain a characterization of the finite projective spaces PG(m, q) with m ≥ 3 and q ≥ 3.

This paper is concerned with projective metabelian -groups. where is the variety of all abelian groups, - of all abelian groups of exponent n. Let P be a projective -group. Since the group P/P′ is a free abelian group. Define rank P = rank P/P′. It is shown that for all numbers r, n > 1, except r = n = 2, there exists a protective nonfree -group of rank r with r + 1 generators.

It is well known that in the category of Hausdorff groups continuous homomorphisms having dense image are epic. That these homomorphisms are precisely the epics is a conjecture. If one could prove that proper closed subgroups (endowed with the subspace topology) cannot have epic inclusions, then the conjecture would be shown to be true. Closed subgroups fall into two classifications:

(a) open (and hence closed) subgroups and

(b) nowhere dense closed subgroups.

In this note it is shown that proper open subgroups cannot have epic inclusion. To do so an algebraic construction of Peter J. Hilton is topologized. The nowhere dense case remains open.

M.I.Graev has shown that subgroups of free topological groups need not be free. Brown and Hardy, however, have proved that any open subgroup of the free topological group on a kw-space is again a free topological group: indeed, this is true for any closed subgroup for which a Schreier transversal can be chosen continuously. This note provides a proof of this result more direct than that of Brown and Hardy. An example is also given to show that the condition stated in the theorem is not a necessary condition for freeness of a subgroup. Finally, a sharpened version of a particular case of the theorem is obtained, and is applied to the preceding example.

The velocity profiles in volumes of either passive resistance or active transport are calculated for both linear and non-linear resistive models. These include fluid flows containing particles in a channel or pipe subject to either a steady or oscillatory pressure gradient, cilia induced transport, and laminar flow in a non-linear resistive shear layer. Resistive elements tend to substantially reduce the inertial phase lag component in oscillatory flows. Only small concentrations of particles are needed to reduce the flow field to the Darcy approximation. An ‘active porous media’ model for a cilia sublayer predicts accurate voleoity profiles for dense concentrations of cilia (for example, on Opaline, and in the lung).

In [1], Theorem 6, a sufficient condition is given for a multifunction to be “of Souslin type”. However, the proof contains an error; we are required to prove that the multifunction Ω: S→ P×NN defined by

has values which are closed subsets of P×NN (the notation is explained in [1]). The “proof” of this fact given in [1] is manifestly incorrect as it appears to assume that NN has the discrete topology.