If a class Z of morphisms in a monoidal category A is closed under tensoring with the objects of A then the category obtained by inverting the morphisms in Z is monoidal. We note the immediate properties of this induced structure. The main application describes monoidal completions in terms of the ordinary category completions introduced by Applegate and Tierney. This application in turn suggests a “change-of-universe” procedure for category theory based on a given monoidal closed category. Several features of this procedure are discussed.

It is shown that a homomorphism from a commutative, idempotent and medial groupoid with a reducible set of generators into another medial groupoid may be characterized by certain simultaneous equations. This result is used to characterize the arithmetic mean without introducing either continuity or order.

We give some sufficient conditions to guarantee the uniqueness of certain mean boundary value problems for a circle. Also we show that, in general, we cannot expect uniqueness of the problem for an arc unless the function is analytic in a neighborhood of the unit circle or some shifted means of the function are also known.

Finitely presented groups having word, problem solvable by functions in the relativized Grzegorczyk hierarchy, {En(A)| n ε N, A ⊂ N (N the natural numbers)} are studied. Basically the class E3 consists of the elementary functions of Kalmar and En+1 is obtained from En by unbounded recursion. The relativization En(A) is obtained by adjoining the characteristic function of A to the class En.

It is shown that the Higman construction embedding, a finitely generated group with a recursively enumerable set of relations into a finitely presented group, preserves the computational level of the word problem with respect to the relativized Grzegorczyk hierarchy. As a corollary it is shown that for every n ≥ 4 and A ⊂ N recursively enumerable there exists a finitely presented group with word problem solvable at level En(A) but not En-1(A). In particular, there exist finitely presented groups with word problem solvable at level En but not En-1 for n ≥ 4, answering a question of Cannonito.

The work continues some earlier investigations into dynamic properties of prestressed incompressible elastic materials. Whereas the material was previously assumed to be a Mooney material, it is here allowed to have any strain energy function. Plane wave propagation and the motions caused by an impulsive line of traction are examined. The results obtained are compared with the earlier work.

A study is made of the secular equation governing the propagation of plane harmonic waves of small amplitude in a continuum which is able to conduct heat. This equation defines an algebraic function, called the modal function, whose regular branches specify the slownesses of the possible modes of harmonic wave propagation as functions of the frequency. At each extreme of the frequency range one mode is diffusive in type and the others wave-like, and we suppose here that there is a single wave-like mode which produces changes of temperature. In this case the mode which is diffusive in type at low frequencies is wave-like or diffusive in the high-frequency limit according as the thermoelastic coupling constant (a dimensionless measure of the strength of thermo-mechanical interaction in the continuum) does or does not exceed unity. This property is shown to have a simple interpretation in terms of a Riemann surface of the modal function. The results obtained are quite general, referring to principal longitudinal waves in a homogeneously deformed isotropic heat-conducting elastic material, to dilatational disturbances of a stress-free configuration of such a material, and to acoustic waves in a heat-conducting inviscid fluid.

Let φ: M → N be a submersion from a metrizable manifold to any (topological) manifold, let B ⊂ M be compact, y є N and C ⊂ φ−1(y) be a compact neighbourhood (in φ−1(y)) of B ∩ φ−1(y). It is proven that there is a neighbourhood U of y in N and an embedding ε: U × C → M such that φε is projection on the first factor, ε(y, x) = x for each x ε C, and B ∩ φ−1 (U) ⊂ ε(U×C). The main application given is to topological foliations, it being shown that if C is a compact regular leaf of a foliation F on M then every neighbourhood of C contains a saturated neighbourhood which is the union of compact regular leaves of F.

Each metacyclic p–group has a natural canonical presentation which is easily derived from the usual presentation. Parameters in the canonical presentation measure how far the group is from splitting, and from being either commutative or dihedral. The structure of the groups is discussed.

A condition which was previously found to be sufficient for global existence and uniqueness of solutions of an ordinary differential equation is shown herein to be necessary, if it is also required that solutions are exponentially bounded.

Let p be a prime and G a group with a p–reduced nilpotent normal subgroup N such that G/N is a nilpotent p–group. It is shown that if G has the subnormal intersection property and if G/N is finite or N is p–torsion-free, then G is nilpotent. This result is used to prove that an abelian-by-finite group has the subnormal intersection property if and only if it has a bound for the subnormal indices of its subnormal subgroups.

We prove that if π is a generalized Hall plane of odd order with associated Baer subplane π0 then π is a Hall plane if and only if there is a collineation σ of π such that π0σ ∩ π0 is an affine point.