If R is any ring then the set of isomorphism types of finitely generated projective right R-modules may be made into a semigroup by defining

.

If one of the locally compact groups H and K is of type I, the reduced dual of their product is shown to be homeomorphic to the product of the reduced duals.

The abstract group

is finite for n = 4,6,7,8, and the relations are incompatible for n = 1,2,3,5. A criterion of Coxeter ((1)) suggests that (2,3,7; n) should be infinite for all n ≥ 9, but its applicability to these groups is unproved, and it is not known whether there are any further examples of finite groups (2,3,7; n). However, (2,3,7; 9) has been proved infinite by Sims ((3)), and it follows at once that (2,3,7; n) is infinite whenever n is a multiple of 9 as it then has an infinite factor group.

1. Let E be a finite non-null set and write (E) for the family of all permutations of E. Let be a non-null subset of (E) and write () for the subgroup of (E) generated by the members of . For any α ∈ we put

so that () is a subgroup of () and is independent of the choice of α in . We suppose that E splits into k disjoint transitivity sets (orbits) Ei(1 ≤ i ≤ k) with respect to (); thus σEi = Ei for all σ ∈ ().

This note gives homotopy-theoretic criteria in the metastable range for an (n + l)-manifold with boundary to be diffeomorphic to the total space of an l-disc bundle over a closed n-manifold and for two such structures to be equivalent. The results are similar to some theorems of (8) and the main technique used in the proofs is surgery on a map ((l), pp. 42–46, (7), (10)). All manifolds will be smooth.

Theorem 1. Let M3be a compact 3-manifold and let M3 = E3 ∪ R be a standard decomposition where R is a subcomplex in a triangulation of M3((2)). If M3is a homology 3-sphere while R contains no arc that is wild in M3then M3 = S3

If X is a uniformizable space any uniform structure compatible with its topology will give a uniform structure on the collection H(X) of all non-empty closed sets of X ((1), Section 2, Exercises 7, 8, definition 1·6): for each entourage U ∈ we define an entourage U on H(X) by putting U = {(A, B); A ⊂ U(B) and B ⊂ U(A)}. It has been conjectured ((6), p. 35, Exercise 17) that if 1 and 2 are distinct uniform structures on X, 1 and 2 induce indifferent topologies on H(X). Unfortunately the method proposed in (6) fails, even for metrizable structures, as we shall show by an example. We shall, however, establish a number of partial results tending to support the conjecture; in particular we shall establish the result when both the uniform structures are metrizable, or when one is precompact.

An error in Dieudonné's paper, Sur les espaces de Köthe, published in Journal d' Analyse Mathématique, 1 (1951), 81–115, is pointed out and a correction put forward.

If α is a positive real number then, for each set E in R3, we define

where U(ρ, E) is any countable collection of convex sets, each of diameter at most ρ, whose union covers E, and Aα is a positive real number. Then the convex α-dimensional measure Λα(E), of E, is . In this paper we shall only be considering the cases α = 1,2 where, as is usual, we take A1 = 1 and A2 = ¼π. The symbols ,Λs(E) will denote the spherical (circular) measure of E, i.e. when the coverings are restricted to being collections of spheres (circles), and E(z) is the intersection of a set E (in R3) with the plane (x, y) at z.

Small letters denote non-negative integers and capital letters denote sets. The set theoretic difference of A and B is A − B. The empty set is ø and |A| denotes the cardinal of A. We say that A is an immediate extension of B if B ⊂ A and |A − B| = 1. If C is a set of sets we put C(n) = {X: X ∈ C, |X| = n}. A set of sets C, has the property P(k, n) if (i) ø ∈ C and (ii) if X ∈ C(i) and i < n, then C(i + 1) contains at least k immediate extensions of X. We assume throughout that k ≥ 1.

1. We work in the category of compact polyhedral spaces and polyhedral maps ((5)). Given spaces M and Q and a map f: M → Q, we define Ur(f) to be the set of points x ∈ M such that f−1f(x) contains at least r points. Sr(f) is the closure of Ur(f) in M. f is an embedding if it is a homeomorphism into, and an immersion if it is locally an embedding. We shall call f a simple immersion if S3(f) = ø and the connected components of S2(f) are individually embedded by f. Obviously a simple immersion is an immersion. If M and Q are manifolds (as they will be for the rest of the paper)f: M → Q is proper if it takes ∂M, the boundary of M, into ∂Q. In (2) the following result was proved:

Theorem 1. Let Mmbe(2m−q)-connected and Qq be(2m−q + 1)-connected, m ≤ q−3. Then any proper map f: M → Q which embeds ∂M is homotopic rel ∂M to an embedding.

The purpose of this paper is to prove theorems for subharmonic functions analogous to those stated below for regular functions. We make the following definition: D(∈, R) is the domain in the Euclidean plane defined by

1. The determination of potentials and wave functions defined in regions bounded by the natural coordinate surfaces of a cylindrical polar coordinate system leads to eigenvalue problems connected with Bessel's differential equation

We shall consider the annulus 0 < a ≤ r ≤ b and take first the boundary condition to be that of vanishing on both surfaces. It is then convenient to introduce the function

This function will automatically satisfy the condition of vanishing for r = a. It will also vanish on r = b if u and k are related by the condition that ψ(u, k, b) = 0. If we regard u as being fixed and real then there will be an infinite number of zeros kn(u)(n = 1,2,…).

This is one of a number of papers in which the author studies the properties of the transform Hα(f) and allied operators. In this case, identities involving these operators and the Fourier transform are obtained, and these are applied in giving alternative proofs to certain inversion processes derived by the author elsewhere.

The evaluation of the eigenvalues and corresponding solutions of a Schrödinger equation is an ever-present problem in atomic and nuclear physics. For the numerical evaluation of the Schrödinger equation for a particle in a central potential, a useful corrector formula was derived by Douglas and Ridley ((5)) some years ago. Given a first estimate of the eigenvalue, this formula enables one to obtain a better estimate using any standard integration procedure for the differential equation. In a particular example, it was found ((5)) that it was necessary to start with an initial value which was accurate to about 10% in order that the procedure should converge to the desired eigenvalue, and that few iterations were required to obtain the eigenvalue with good precision.

1. Let (S, S,μ) be a measure-space, and let f(x) and ø(x) be two μ-integrable functions defined in S. We shall assume that f(x) is a ‘difficult’ function, of which all that is known is how to calculate f(x), generally by a rather complicated process, for any given x∈S; and that ø(x) is an ‘easy’ function, whose value for any x, and whose integral over any subset T of S belonging to the class S of μ-measurable sets, are relatively simple to obtain. We shall define the sets

and assume that

It will be of interest to consider those functions ø(x) which may be regarded as approximations to f(x) in S.

A solution of Einstein's vacuum field equations, apparently new, is exhibited. The metric, which is homogeneous (that is, admits a three-parameter group of motions transitive on space-like hypersurfaces), belongs to Taub Type V. The canonical form of the Riemann tensor, which is of Petrov Type I, is determined.

The propagation of magneto-elastic plane waves is a topic which has recently aroused much interest. Thus Paria ((l)), for example, and the present author ((2)), have studied in some detail the interaction between the magneto-elastic field and the thermal field. Analyses published so far, however, have neglected the effects due to the passage of the wave upon the magnetic permeability of the medium. It is the object of the present paper to consider these effects and it will be shown that although the elastic strains induced cause but a small change in the numerical values of the permeability, nevertheless the effects upon wave propagation may be very considerable. The principal result is the demonstration that the effects considered are equivalent so far as the dispersion equation is concerned, to an anisotropic rescaling of the primary magnetic field, the direction of propagation of the wave being a preferred direction. For simplicity, thermal effects are not considered here.

This note supplements our paper ((4)) on the hydromagnetics of an ellipsoid moving in a cross-field. Familiarity with this paper will be assumed. Together they account for creeping motion in an arbitrary direction when the applied magnetic field is parallel to a principal axis.

Within the limitations of the classical small deflexion theory of thin plates and using complex variable methods, exact expressions are obtained in series form for the deflexion at any point of a thin isotropic circular plate simply supported along a concentric circle and subject to loading symmetrically distributed over an eccentric circular patch which lies inside the circle of support. In special and limiting cases the solutions reduce to those obtained before.