Scoins(1) has shown that if Π1 = {(1), …, (n)} and Π2 = {(n + 1), …, (n + m)} are two sets of points, there are exactly mn−1nm−1 trees of alternate parity connecting the points of Π1 ∪ Π2, where each tree consists of n + m − 1 segments and each segment joins a point of Π1 to a point of Π2. Another proof based on the three following results is given here.

The second minimum of the form x2 − 23y2 is evaluated and shown to be non-isolated, thus exhibiting a possibility conjectured by Barnes and Swinnerton-Dyer in 1952.

A general method is described in this paper for the minimization of a function subject to a set of non-linear constraints. This method has been used to solve some problems in the geometry of numbers. Critical lattices of several regions have been computed and some results are given.

Ascending series. Let G be a group and λ an ordinal number. An ascending series of G of type λ is a set of subgroups Gα of G, defined for all α ≤ λ, and such that (i) G0 = 1, Gλ = G; (ii)Gα is a proper normal subgroup of Gα+1 for all α < λ (iii) for all limit ordinals μ ≤ λ.

In this note a simple principle is explained for constructing a transformation group which is a free product of given transformation groups. The principle does not seem to have been formulated explicitly, though it has been used in a more or less vague form in the theory of discontinuous groups (see, for instance, L. R. Ford, Automorphic functions, vol. I, pp. 56–59). It is perhaps of interest that the formulation given here is purely set-theoretic, without any topology, and that it can apply to any free product, whatever the cardinal number of the set of factors. The principle is used to establish the closure under the formation of countable free product of the family of groups which can be represented as discontinuous subgroups of a certain group of rational projective transformations. (The word ‘discontinuous’ is used here in a weak sense, defined later.) Finally, these results are applied to give a new proof of the theorem of Gruenberg that a free product of residually finite groups is itself residually finite (K. W. Gruenberg: Residual properties of groups, Proc. London Math. Soc. (3), 7 (1957), 29–62. See Corollary (ii) of Theorem 4.1, p. 44). The present proof is completely different from Gruenberg's and seems to be of interest for its own sake, though it does not appear to lead to Gruenberg's other results in this connexion. I have to thank Mr W. J. Harvey and Mr C. Maclachlan for checking a first draft of this paper and pointing out a few errors.

Let α be a prime element of the ring of integers of an algebraic number field, R. Mr C. Sudler verbally raised the question as to how many prime ideal factors α can have. This is equivalent to a problem on the group of ideal classes of R, as we now show. If there is a prime α = p1p2 … Pk, where the prime ideal pi, is in the ideal class Pi, then P1P2…Pk equals the identity class, but no subproduct has this property. The converse holds since every ideal class contains prime ideals. So the problem is equivalent to one on finite Abelian groups, which we now write additively.

It was remarked in (1) that many of the results valid for compact (this term is taken to imply Hausdorff) semigroups with jointly continuous multiplication also hold if the multiplication is only separately continuous. It is the purpose of this note to show that this is true of the structure theorem for compact simple semigroups. Any unsubstantiated facts about semigroups which we use may be found in the second section of (1) (the results given there do not depend on the assumption that a semigroup should contain an identity).

Given an ‘extraordinary cohomology theory’, that is, a cohomology theory satisfying all the axioms of Eilenberg and Steenrod (7) except the dimension axiom, it is well known that there exists a spectral sequence relating the ordinary cohomology of a space with the extraordinary theory (see, for example, (3) in the case of K*(X)). Obviously, it would be useful to know the differentials in this spectral sequence, and it is the purpose of this paper to identify them in terms of cohomology operations defined by certain k-invariants. We shall make use of E. H. Brown's recent theorem (5) on the representability of extraordinary cohomology theories, to construct a second spectral sequence, in which the differentials are readily identifiable, which we shall prove is isomorphic to the usual one.

Let A be a right Λ-complex and C be a left Λ-complex where Λ is a ring which is both left and right hereditary; a principal ideal domain is a special case of such a ring. This paper is concerned with the Künneth theorem expressing the homology H(A⊗C) of the product complex A⊗C (= A⊗ΛC) in terms of H(A) and H(C). Any notations we do not explain are those of (1).

It was pointed out by Copson(1) in 1947 that the solution of the integral equation for the electrostatic problem for a circular disc could be reduced to the solution of Abel integral equations and hence a solution obtained in a fairly elementary manner. This result was obtained independently by Lebedev(2), who also obtained a similar result for the electrostatic problem for the spherical cap. The solution for the spherical cap was also obtained independently by Collins (3). In view of the relative simplicity of the approach it seems to be of interest to examine whether there exist other integral equations which can be treated in a similar fashion. In the present paper one such class of integral equations will be considered.

Let gn(z) be a polynomial of degree n in z. Then there exist constants Πk, nsuch that

.

The propagator for a particle in the presence of a one-dimensional square-well potential is calculated. The related t-function is written down in Fourier transform. The analytic properties of the propagator and t-function are discussed.

The title problem for a perfectly conducting sphere was considered by Stewartson(6). Here we use a different method and a more realistic assumption concerning the sphere. Although the flow is still ultimately cylindrical, there is no longer a column of fluid which moves with the sphere as if solid. There is no reason to expect that in practice the flow will break away into eddies. In fact the same is true in Stewartson's case, once his results are corrected.

Errors in a previous discussion (4) of the title problem for a perfectly conducting sphere are corrected. An alternative method of determining the ultimate motion is described for both a perfectly conducting and a non-conducting sphere moving along the magnetic field.

The wave equation with axial symmetry is reduced, by the assumption of dynamic similarity, to an equation in two variables, r/t and θ A basic solution, having some of the attributes of a source, is introduced, and it is shown that, by use of this solution and a suitable contour integral representation, the solution of any appropriate boundary-value problem may be reduced to the determination of an analytic function, with boundary conditions given by an integral equation.

The method of solution is illustrated by reference to some basic problems in the theory of linearized compressible flow.

The method of Part I(3) is applied to problems involving boundary conditions on an equatorial plane.

The planetary equations for natural elements, which do not depend upon the choice of a particular frame of reference, are derived by using the theory of Poisson brackets. These natural elements include vectorial elements which lead to the introduction of vector and dyadic Poisson brackets. The use of vectorial elements can also lead to redundancy in the set of elements; this idea is extended to show that as many redundant elements as desired can be used and that the evaluation of the derivatives of the disturbing function can be simplified thereby.

One of the classical results of ring theory is due to Levitzki ((3), p. 199) and asserts the equivalence of the notions of nil and nilpotent one-sided ideals in a ring satisfying the ascending chain condition on left ideals. Recent, distinctive proofs of this result have been given by Goldie (1) and Herstein (2). We present below yet a fourth way of proving Levitzki's theorem.

Let G be a bounded region in k-dimensional space, with boundary Γ, such that the Laplace equation,

is uniquely soluble (to within an added constant) under the Neumann boundary conditions

where ∂/∂n denotes outward normal differentiation on Γ, and it is assumed that h is a function in G ∪ ∂, and thus that g is a function on ∂. In what follows, we shall assume certain properties of the solution h: these are all well known (see, for example, Osgood(l) or Courant(2)).