An elementary proof is given for the Lp conjecture, p > 2, which states that for a locally compact group G, Lp (G) (p > 2) is closed under convolution if and only if G is compact.

Apart from some (insoluble) subvarieties of , the jnC (just-non-Cross) varieties known so far comprise the following list: , , , , where p, q and r are any three distinct primes. In a recent paper I gave a partial confirmation of the conjecture that the soluble jnC varieties all appear in this list. Here I show that a jnC variety is reducible if and only if it is soluble of finite exponent; this reduces the problem of classifying jnC varieties to finding the irreducibles of finite exponent. I observe that these fall into three distinct classes, and show that the questions of whether or not two of these classes are empty have some bearing on some apparently difficult problems of group theory.

We prove a partial confirmation of Kovács and Newman's conjecture that a just-non-Cross variety is soluble if and only if it is in the following list: , , , , where p, q and r are any three distinct primes.

This paper deals with the motion of incoherent matter, and hence of test particles, in the presence of fields with an arbitrary energy-momentum tensor. The equations of motion are obtained from Einstein's field equations and are written in the form of geodesic equations of an affine connection. The special cases of the electromagnetic field, the Proca field and a scalar field are discussed.

The method of interlacing of modules, like amalgamation of groups, is a way of getting new objects from old. Briefly, the interlacing module we consider is a certain factor module of a direct sum of copies (finite or infinite) of an original module M. The conditions given in a previous paper by the first author in order that the interlacing module (using finitely many copies) be indecomposable are here greatly weakened, and we further allow the number of copies of the original to be infinite. R. Colby has shown that if R is a left artinian ring, the existence of a bound on the number of generators required for any indecomposable finitely-generated left R-module implies that R has a distributive lattice of two-sided ideals. This result is extended to rings whose identity is a sum of orthogonal local idempotents.

For these rings the same distributivity is proved in case every indecomposable interlacing module of the above type which begins with an indecomposable projective M is finitely-generated. A consequence is that any finite-dimensional algebra over a field having infinitely many two-sided ideals has infinite-dimensional indecomposables.

In 1942 F.W. Levi described those groups in which any two inner automorphic images of an arbitrary element commute. Until recently it was not known whether there existed non-abelian groups with the property that any two endomorphic images of an arbitrary element commute. Now, R. Faudree has given examples of finite p–groups having this property. In this paper we give a necessary and sufficient condition that a torsion group which contains no elements of order 2 has this property. The technique of proof involves looking at certain near rings of functions on the group. The inspiration for the theorem comes from the “halving” technique used by B.H. Neumann in 1940.

The method of Lagrange multipliers for solving a constrained stationary-value problem is generalized to allow the functions to take values in arbitrary Banach spaces (over the real field). The set of Lagrange multipliers in a finite-dimensional problem is shown to be replaced by a continuous linear mapping between the relevant Banach spaces. This theorem is applied to a calculus of variations problem, where the functional whose stationary value is sought and the constraint functional each take values in Banach spaces. Several generalizations of the Euler-Lagrange equation are obtained.

Two theorems for Lebesgue integrals, namely the Gauss-Green Theorem relating surface and volume integrals, and the integration-by-parts formula, are shown to possess generalizations where the integrands take values in a Banach space, the integrals are Bochner integrals, and derivatives are Fréchet derivatives. For integration-by-parts, the integrand consists of a continuous linear map applied to a vector-valued function. These results were required for a generalization of the calculus of variations, given in another paper.

A refinement of the Sobolev imbedding theorem, due to Trudinger, is shown to be optimal in a natural sense.

Let v be a symmetric monoidal closed category with equalizers. The v–triples T, T′, … in the enriched category A, together with suitably defined morphisms form a category v–Trip(A). The v–categories AT, AT′, … and the v–functors R: AT′ → AT which are compatible with the forgetful functors form a category V–Alg(A).

In the subsequent note it is shown that V–Trip(A) is isomorphic to the dual of V–Alg(A) and that the morphisms of V–Alg(A) are inverse limit preserving V–functors.

If E, F are regularly ordered vector spaces the tensor product E ⊗ F can be ordered by the conic hull Kπ of tensors, x ⊗ y with x ≥ 0 in E and y ≥ 0 in F, or by the cone K⊗ of tensors Φ ∈ E ⊗ F such that Φ(x′, y′) ≥ 0 for positive linear functionals x′, y′ on E, F.

If E, F are locally convex spaces the tensor product can te given the π-topology which is defined by seminorms pα ⊕ qβ where {pα}, {qβ} are classes of seminorms defining the topologies on E, F. The tensor product can also be given the ε-topology which is the topology of uniform convergence on equicontinuous subsets J x H of E′ x F′. The main result of this note is that if the regularly ordered vector spaces E, F carry their order topologies then the order topology on E ⊕ F is the π-topology when E ⊕ F is ordered by kπ, and the ε-topology when E ⊕ F is ordered by K⊕.

For three possible types of tide-well systems the non-dimensional head response, Y(τ), to a sinusoidal fluctuation of the sea-level is given by the differential equation Estimates of the non-dimensional well response Z(τ) = sinτ - Y(τ) are found by considering the steady state solutions of the above equation. With n = 2 the equation is linear and an exact solution can be found; for n ≠ 2 the equation is non-linear and several methods which give approximate solutions are described. The methods used can be extended to cover other values of n; for example, with n = 4 the equation corresponds to one governing oscillations near resonance in open pipes.

The fact that continued fractions can be described in terms of Farey sections is used to obtain a generalised continued fraction algorithm. Geometrically, the algorithm transfers the continued fraction process from the real line R to an arbitrary rational line l in Rn. Arithmetically, the algorithm provides a sequence of simultaneous rational approximations to a set of n real numbers θ1, …, θn in the extreme case where all of the numbers are rationally dependent on 1 and (say) θ1. All but a finite number of best approximations are given by the algorithm.

The Editor regrets that two errors have appeared in [1], through no fault of the author's. On page 325 the result (3.5) and the two lines following it should read:

where

On page 328 the result (4.3) and the two lines following it should read:

where