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.

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.

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 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→∞.

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 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].

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.

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.