Christine W. Ayoub has defined diagrams for abelian groups and has shown that the structure of an abelian group A is determined by certain subgroups Ti of A which arise naturally from a given diagram for A. In this paper it is shown that a given diagram for A is formed from certain elementary diagrams for the Ti.

The asymptotic behavior of nonoscillatory solutions of the second order functional differential equation

is studied. First, in the case when a(t)is oscillatory, sufficient conditions are given in order that all bounded non-oscillatory solutions of (*) approach zero as t → ∞. Secondly, in the case when a(t) is nonnegative, conditions are provided under which all nonoscillatory solutions of (*) tend to zero as t → ∞.

If (A′, B′), (B′, C′) and (A, B), (B, C) are torsion-torsion free theories on RM and MR respectively which are generated by an idempotent ideal I of R, then M ∈ RM is said to be relatively flat if (·) ⊗ RM preserves short exact sequences 0 → L → X → N → 0 in MR with N ∈ B. Several characterizations of relatively flat modules are given and it is shown that any module M ∈ RM which is codivisible with respect to (A′, B′) is relatively flat. In addition, when (A′, B′) is hereditary, it is proven that M ∈ RM is relatively flat if and only if M/IM is a flat R/I-module. Finally, a relatively flat dimension for M ∈ RM and a left global relatively flat dimension for R are defined and it is shown, again when (A′, B′) is hereditary, that the left global relatively flat dimension of R coincides with the left global flat dimension of R/I.

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.

Let a(z) and b(w) be two rational functions in z or W with algebraic coefficients, where a(0) = 0 and let

Assume that 0 < |z| < 1, that z is not a pole of for n ≥ 0, that w is neither a pole nor a zero of b(w, n) for n ≥ 1, and that the series

for fixed w is a transcendental function of z. Then, if z and W are algebraic numbers, f(z, w) is a transcendental number.

The object of this paper is to study common fixed points of mappings of a complete metric space into itself. The results obtained are generalizations of theorems of Iséki and Ray. An example is given to show that our results are indeed extensions of the theorem of Iséki.

The sufficient optimality conditions, of Fritz John type, given by Gulati for finite-dimensional nonlinear programming problems involving polyhedral cones, are extended to problems with arbitrary cones and spaces of arbitrary dimension, whether real or complex. Convexity restrictions on the function giving the equality constraint can be avoided by applying a modified notion of convexity to the other functions in the problem. This approach regards the problem as optimizing on a differentiable manifold, and transforms the problem to a locally equivalent one where the optimization is on a linear subspace.

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.

In this note a converse duality theorem is proved for a class of nonlinear programming problems over polyhedral cones in finite dimensional complex space by a direct use of a Kuhn-Tucker type necessary and sufficient condition for constrained optimization in complex space.

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).

A system of set theory which appears as an extension of Ackermann set theory is introduced. In this sytem we construct a syntactic model for a theory proposed by Feferman for the development of category theory.

Let Γ be a family of isoclinic finite groups, and let acl(Γ) be the group of autoclinisms of Γ. In this paper we prove the following formula:

where the sum is taken over a complete system of stem groups S in Γ. This result is due to P. Hall, who outlined a proof in his paper, “On groups of automorphisms” (J. reine angew. Math). 182 (1940), 194–204, using presentations, whereas in this paper we consider stem groups in terms of central extensions.

Recently, Eklund (Proc. Amer. Math. Soc. 47 (1975), 137–142) has shown that to each continuous function F on ∂pQ −⊲ {∂Ω × [0, T]} ∪ {Ω × (0)} there is a unique solution to the boundary value problem

where L is a linear second order parabolic operator in divergence form, Ω ⊂ Rn is a bounded domain with compact closure and ∂Ω denotes its boundary. In this note, it is shown that the existence theorem of Eklund remains valid for the following boundary problem