Multiplications on spheres are studied in [11], [12] from the standpoint of homotopy theory. These multiplications are products with a unit element. The present paper deals with products in general. The investigation involves proving some results on the toric construction and the Whitehead product. These results also lead to theorems about the Stiefel manifold of unit tangent vectors to a sphere, originally proved by M. G. Barratt, which clear up some points in the homotopy theory of sphere bundles over spheres (see [15], [16]). They also enable us to prove that certain of the classical Lie groups are not homotopy-commutative.

In the present paper we consider a one-dimensional local ring Q with maximal ideal tn and residue field K = Q/m. It will be assumed that not every element of mis a zero-divisor but no other restricting hypothesis will be made. In particular Q and K may have unequal characteristics and K may be finite.

In a recent paper Northcott [3] introduced the notion of the reduction number of a one-dimensional local ring, and demonstrated its importance in the theory of abstract dilatations. In the present paper we define the reduction number of an ideal which is primary for the maximal ideal of a one-dimensional local ring, and show that under certain necessary and sufficient conditions the reduction numbers can take only a finite number of values.

Let R be a simple ring. If R contains at least one minimal nonzero one-sided ideal, then R has zero-divisors, unless R is a division ring. However, simple rings exist which are not division rings and have no zero-divisors. Our present object is to prove the following embedding theorem:

Theorem 1. Every ring R without zero-divisors may be embedded in a simple ring R* without zero-divisors. If there is a non-zero element ƒ in R satisfying ƒ2 = nƒ, where n is an integer, then R* necessarily has a unit-element; otherwise R* may be chosen to have no unit-element.

A polygon, with all sides and all diagonals of rational length, will be called a rational polygon.

Prof. I. Schoenberg has set a problem whether rational polygons are everywhere dense in the class of all polygons, that is, given an arbitrary polygon whether there exists a rational polygon whose sides and diagonals are of length arbitrarily near to those of the given one. Once set, the problem becomes very interesting both for its simplicity and for its fundamental nature. For obvious reasons the likely answer to the problem is in the negative. In this note I consider two problems simply related to the above problem.

In its simplest form, the theorem of Ascoli with which we are concerned is an extension of the Bolzano-Weierstrass theorem: it states that, if X and Y are bounded closed sets, of real or complex numbers, andis a sequence of equicontinuous functions mapping X into Y, thenhas a uniformly convergent subsequence. As is well known, this result has played a fundamental part in the development of several theories; it has also been widely generalized, by processes of abstraction and localization, and a very useful version of the theorem runs as follows (cf. [4], 233–234):

(A) Suppose that X is a locally compact regular space, and that Y is a Hausdorff space whose topology is determined by a uniform structure. Let YX be the space of all functions that map X into Y, with the topology of locally uniform convergence with respect to. Then a closed setin YX is compact if, at each point x of X, (i) the set(x) is relatively compact, in Y, and, (ii) is equicontinuous with respect to.

The Čech compactification of the set of integers is known [6] to have the remarkable property that it has no closed subsets of cardinal ℵ0 or c, every infinite closed subset of it having 2c points. The main object of the present paper is to investigate whether similar gaps in the cardinals of closed subsets can occur in metric spaces. We shall see that the situation there is rather different; if the generalized continuum hypothesis is assumed, there are no gaps, and in any case the missing cardinals, if any, must be big rather than small. The main results are obtained in §3; in particular, we completely determine the cardinals of the closed subsets of complete metric spaces, and also how many closed subsets of each cardinal there are. The methods depend on a study of the discrete subsets of metric spaces, which is carried out in §2, and which may be of independent interest. In conclusion, we briefly consider some fragmentary results for non-metric spaces, in §4. Throughout, we assume the axiom of choice but not the continuum hypothesis.

The main result concerning the stability of an inviscid liquid column, which is at rest, is due to Rayleigh [1], who showed that, taking account of capillarity at the surface, the column will be unstable to small axisymmetric disturbances whose wavelengths in the axial direction are greater than the circumference of the cross-section. The reason for such instabilities is simply that disturbances of these wavelengths decrease the surface area of the column and hence make available excess surface energy which goes into building up the disturbance. The effects of rotation on the stability of a column having a free surface do not appear to have been studied and this note establishes some simple results concerning the effect of plane two-dimensional disturbances on a rotating column. With regard to the stability of rotating fluids in general there is another result due to Rayleigh [2] stating that the fluid will be unstable if the numerical value of the circulation decreases with the radius at any point. But this result is again for axisymmetric disturbances, and hence does not necessarily bear any relationship to the results for plane disturbances which, in the case of an incompressible liquid as is assumed here, cannot be axisymmetric. A result more closely related to this work is that of Kelvin [3] who considered the stability of a column of uniform vorticity in a fluid otherwise free of vorticity. Such a column is stable to two-dimensional and to three-dimensional disturbances.

Let K be a n-dimensional convex body. A lattice Λ will be called a covering lattice for K (or simply a covering lattice), if each point of space has at least one representation in the form k + g where k ε K and g ε Λ. The density ϑ(K, Λ) of the resultant covering of the whole of space by the bodies of the form K + g with g ε Λ is, quite naturally, defined to be the ratio V(K)/d(Λ), where V(K) is the volume of K and d(Λ) is the determinant of Λ (i.e. the volume of its fundamental parallepipeds). Clearly this density is at least 1. A number of authors, in particular H. Davenport [3], R. P. Bambah and K. F. Roth [2], G. L. Watson [8], C. A. Rogers [4], [5] and W. Schmidt [7], have either constructed or proved the existence of lattices for which the density ϑ(K, Λ) is reasonably small. But, when n is large, all the densities obtained are of the form cn where c is a constant greater than 1. The object of this note is to obtain some rather stronger results. In the general case we prove the existence of covering lattices Λ with

Where

as n→∞.

The two centred expansion of the Coulomb Green's function arises naturally in discussing the static interaction energy of two charge distributions ρ1, and ρ2. This is given by the well-known expression

In this note those quotient groups of the absolute class group of number fields are to be studied which can be described in terms of absolutely Abelian fields. This investigation will be based on a suitable generalization of the classical concepts of the principal genus, the genus group and the genus field. One possible description of the genus group in a cyclic field is that as the maximal quotient group of the absolute class group which is characterized by rational congruence conditions, i.e. in terms of rational residue characters. From this point of view, however, the restriction to cyclic—or Abelian—fields is quite artificial; the given description can thus be taken as the definition of the genus group in any finite number field. In general the genus field will then no longer be absolutely Abelian; it can now be described as the maximal non-ramified extension obtained by composing the given field with absolutely Abelian fields.

In a recent paper Segedin [1] has derived a solution of the problem in which a perfectly rigid punch in the form of a solid of revolution of prescribed shape with axis along the z-axis bears normally on the boundary z = 0 of the semi-infinite elastic body z ≥ 0, so that the area of contact is a circle whose radius is a. Segedin solves the problem by building up the solution in a direct way which avoids both the use of dual integral equations and the introduction of an awkward system of curvilinear coordinates. By introducing a kernel function K(ξ), Segedin derives new potentials of the form

where U(r, z, a) is the solution of the simplest punch problem (namely that of a flat-ended punch) satisfying the mixed boundary conditions

on the boundary z = 0. It can then be easily shown that, under wide conditions on K, the function Φ (r, z, a) satisfies the boundary conditions

provided that K(ξ) is chosen so that

In the present paper a general solution of the equations of elasticity in complete aeolotropy is found under the assumption that the stresses and therefore the strains are linear in the third cartesian coordinate. This solution is applied to the elastic equilibrium of a completely aeolotropic cylinder, under a distribution of tractions on the lateral surface and resultant forces and couples on the end sections of the cylinder. The problem of extension of a completely aeolotropic cylinder by longitudinal lateral loading and end forces is solved with an application to the elliptic cylinder. The writer hopes to present in later communications applications of this general solution to the following problems: (i) Bending of a completely aeolotropic cylinder by longitudinal lateral loading and end bending couples, (ii) Torsion of a completely aeolotropic cylinder by transverse lateral loading and end twisting couples, (iii) Flexure with shear of a completely aeolotropic cylinder with free lateral surface. Particular cases of (ii) and (iii) were considered by Luxenberg [1] and Lechnitzky [2, 3].

Green and Zerna [1] have given a method of determining the electrostatic potential due to a circular disc maintained at a given axisymmetric potential, their method depending on the solution of a Volterra integral equation of the first kind and being a generalization of a method given by Love [2] for the determination of the electrostatic potential due to two equal co-axial circular discs maintained at constant potentials. In a recent paper [3], henceforth referred to as Part I, the author applied this method to the corresponding problem for a hollow spherical cap and also to the determination of the Stokes' stream-functions for perfect fluid flows past a cap and a disc. The method consists of expressing the potential as the real part of a complex integral of a real variable t, the integrand involving an unknown function g(t). The boundary condition on the disc or cap gives a Volterra integral equation of the first kind for g(t), it being possible to solve this equation and hence determine the potential by integration.

Extreme value problems and particularly those arising in the combinatorial field have a peculiar interest and challenge in that an exact solution is rarely possible. We discuss here four combinatorial extreme value problems each concerned with the distribution of the largest (or smallest) of a set of mutually dependent variables. These problems, widely different in character possess three points in common. First the probability distribution functions of the variables considered cannot be obtained explicitly, nor secondly can the moments of the variable, and thirdly the probability distribution functions are difficult to evaluate even for moderate sized samples. We shall treat the situation where the variables, or functions of them, have probability distribution functions which tend to an exponential limit, analogous to the well-known limit for extreme values in the case of independent events. The p.d.f.'s for the upper tails of the distributions which we consider are found to be very closely contained within a pair of inequalities (Bonferroni Inequalities), especially in the regions of statistical significance.

The receptance function is defined and constructions for it are given both for the general case when ω, the frequency of excitation, is not a natural frequency and for the special case when it is.

In the preceding paper of the same title (cf. [1]) I defined the notion of the principal genus GK of a finite number field K as the least ideal group, which contains the group IK of totally positive principal ideals and is characterized rationally. The quotient group of the group AK of ideals in K modulo GK is the genus group, its order (Ak: GK) = gK is the genus number, which is thus a factor of the class number hK (in the narrow sense). Associated with the genus group is the genus-field, of K, which is defined as the maximal non-ramified extension of K composed of K and of some absolutely Abelian field.

In the stream-line motion of fluid in a curved pipe the primary motion along the line of the pipe is accompanied by a secondary motion in the plane of the cross-section. The secondary motion decreases the rate of flow produced by a given pressure gradient and causes an outward movement of the region where the primary motion is greatest. It is difficult to deduce these consequences from the exact equations of motion, but it is easy to do so if it is assumed that the actual secondary motion is replaced by a uniform stream; conditions in the central part of the section mainly determines the motion and here the secondary motion is approximately a uniform stream. The appropriate velocity of the stream can be determined from the relation that has been found experimentally between the rate of flow in a curved pipe and the pressure gradient.

In a recent paper Rogers [13] has discussed packings of equal spheres in n-dimensional space and has shown that the density of such a packing cannot exceed a certain ratio σn. In this paper, we discuss coverings of space with equal spheres and, by using a method which is in some respects dual to that used by Rogers, we show that the density of such a covering must always be at least