1. Introduction: If G is a group, Z(G) its integral group-ring and AG the augmentation ideal, then we can form the Abelian groups

In (5) we have studied the structure of these Abelian groups which we called polynomial grouups. If C denotes the category of Abelian groups, then Pn and Qn are functors from C into C. We call these functors polynomial functors. The object of this work is to study the nature of these funtors. Except for n = 1, these functors are non-additive. In fact, in the sense of Eilenberg–Maclane (4) these are functors of degree exactly n (Theorem 2·3). Because of their non-additive nature, their derived functors cannot be calculated in the traditional Cartan–Eilenberg(1) method. We have to make use of the more recent theory of Dold–Puppe (3).

The growth of transversal theory has been greatly enriched by the study of abstract independence. In this development, Rado's theorem on independent transversals (4) has played a prominent role, and recent work (e.g. by Welsh(7)) suggests that its applications are by no means exhausted. Rado's theorem can be extended in many directions, and my object is to describe a further generalization (Theorem 2 below) which does not seem to have been noted in the literature.

The main result proved in this paper can be stated as follows:

Theorem. Let Mnbe a closed l-connected topological manifold of dimension n ≥ 5. Then M × R carries a differentiable (respectively aPL) structure if and only if M × S1carries a differentiable (respectively aPL) structure.

Some topological identifications are introduced into the DS-spaces of Robinson and Trautman, and the symmetry and asymptotic symmetry groups of the resulting spaces are calculated. The latter have structures similar to the GBM group, each being the semi-direct product of a group of supertranslations by a group of classical type.

In a recent paper (4) concerned with the study of ‘fundamental operators’, we have obtained results involving the resolvents of the generators of some semi-groups of operators in Lp (− ∞, ∞). In this paper we consider those results which, under suitable conditions, can be extended to cases where the resolvents do not necessarily form a class of fundamental operators. The main results of this type, given in Theorems 2·7 and 2·8, involve inequalities between members of the semi-group and the resolvent of the generators. The semi-groups to which the main results apply include those denned by the Poisson and the Gauss–Weierstrass operators.

For r > 0 a non-empty subset U of a linear space is said to be absolutely r-convex if x, y ∈ U and |λ|r + |μ|r ≤ 1 together imply λx + μy∈ U, or, equivalently, xl, …, xn∈ U and

It is clear that if U is absolutely r-convex, then it is absolutely s-convex whenever s < r. A topological linear space is said to be r-convex if every neighbourhood of the origin θ contains an absolutely r-convex neighbourhood of the origin. For the case 0 < r ≤ 1, these concepts were introduced and discussed by Landsberg(2).

Let G be a locally compact Abelian group; Γ the dual group of G; C0(Γ) the algebra of continuous functions on Γ which vanish at infinity; CB(Γ) the continuous, bounded functions on Γ; M (G) the algebra of bounded Borel measures on G; L1(G) the algebra of absolutely continuous measures; and M(G)∩ the algebra of Fourier–Stieltjes transforms.

In (2) I described a canonical isometric representation of an arbitrary real Banach space X by vector-valued functions (with the uniform norm) on a compact Hausdorif space ω with the following properties: (1) the representing function space is invariant under multiplications by continuous real functions on ω; (2) the norm of each representing function, as a real non-negative function on ω, is upper semicontinuous; and (3) this decomposition of X is maximally fine. I called attention to the class of spaces X for which at every point of ω the component space of this representation is one-dimensional or 0, so that the representing functions are in effect real valued. I propose to call such Banach spaces square, because of the shape of the unit ball in the two-dimensional case. In (2) I stated without proof, erroneously as it turns out, that the class of square spaces coincides with what Lindenstrauss in (4) called G-spaces. The primary purpose of this paper is to show that the class of square spaces is actually properly contained in that of G-spaces. It is known ((2), p. 620, Example 1) that it contains properly the class of continuous function spaces C(ω). Among G-spaces are the M-spaces treated by Kakutani as Banach lattices (3). I shall show further that neither class, square spaces or M-spaces (regarded now purely as Banach spaces), contains the other.

Let X be a real Banach space which is partially ordered by a closed cone C. Let X* denote the Banach dual space, and let

X is called a simplex space (2) if (X*, C*,‖·‖) is an AL-space. We proved in (3) that X is a simplex space if and only if X is an approximate order-unit-normed space and has the Riesz decomposition property. When the given norm on X is an order-unit norm, some representation theorems are obtained by Effros ((2), theorem 4·8) and Sterrmer ((6), theorem 4·3). In this note, we shall study the representation problem for more general simplex spaces. One of our aims is to prove that X is equivalent to C0(ω) if and only if X has the Urysohn property (for definitions, see below).

Segal(1) in his paper ‘Summability by Dirichlet convolutions’ makes direct use (in the proofs of Theorems 4 and 6) of a result on the Dirichlet product of two series which may be stated as follows:

is (C, k)-summable to the sum α (k≥0) and is (C, l)-summable to the sum β, (l ≥ 0), then the Dirichlet product

is (C, k + l + 1)-summable to the sum αβ.’

The hypergeometric transformation formulae (6) and the one that follows on p. 766 in my paper (1) should be read as

and

respectively, where 2F1[a,b; c; z] denotes Gauss's hypergeometric function, and ((2), p. 211)

with, as usual,

the conditions of convergence for the double series being

In this paper, it will be shown how it is possible, using Bernoulli polynomials of the first kind, to derive correction terms for any quadrature approximation in the form of a weighted sum over a finite interval. The terms derived will involve either the difference or the sum of derivatives taken at the ends of the range of integration, and they will be described as being of either Type A or Type B respectively. There will also be some discussion of the uses of such terms.

Introduction and summary: In the distribution of the sum of n random variables

for which we suppose , the tail probability will be called an extreme tail probability if the normal approximation

is poor, when is asymptotically normal and n is large. This paper concerns the specialization of (1.1) in which

with a random permutation of nλ ones and n − nλ zeros, that is, in which T is the total for a simple random sample from the population with sampling fraction λ. In this case, we will always write T as Tλ. For brevity, we may write {c1,…, cn} for .

An account is given of the seiche motion in a curved lake. It is shown that the periods of the first two modes of oscillation are effectively unchanged in comparison with those for a straight lake having a length equal to that of the median line along the curved lake. Near to the nodal positions, the surface elevation is uniform across the lake. In the neighbourhood of the anti-nodes, which includes the ends of the lake, the wave amplitude is shown to be a maximum at the outside of the bend.

The linear fluid dynamics is considered when infinite vertical boundaries are set in oscillatory vertical motion. The case of exponential stratification with constant kinematic viscosity is explicitly studied. When the forcing frequency equals the Brunt–Vaisälä frequency for the fluid, the customary boundary layers are absent in the steady-state oscillation, however small be the kinematic viscosity; for a semi-infinite fluid the corresponding horizontal extent of the region influenced by the boundary motion is then of the order of the stratification length. The sign of the phase angle is everywhere dependent on whether the magnitude of the forcing frequency is greater than or less than that of the Brunt–Vaisäla frequency.

The general solution of Chaplygin's equation for the motion of an inviscid compressible fluid is obtained for ‘simple wedge flows’. Some particular examples are given and a discrepancy in earlier Russian papers explained. After a discussion of the properties of the velocity potential and the expression for the physical coordinates in terms of the hodograph variables, a general theorem on sonic jets is proved, namely, that the physical changes due to the presence of solid boundaries in the flow are completed within a finite distance in those directions in which the flow becomes finally a sonic jet.

The general solution of the Chaplygin equation for ‘simple wedge flows’ is used for the particular case of Réthy flows. The drag coefficient is evaluated and comparisons made with the results obtained from the approximate hodograph equations of Tricomi and of Tomotika and Tamada. A distinction is made between subsonic and transonic régimes and the limitations on the lengths and pressure differences permissible in the various ranges are discussed.

This paper is concerned with the problem of the flow of an incompressible electrically conducting fluid along a rectangular duet under a transverse magnetic field. The case in which the walls perpendicular to the field are perfectly conducting and those parallel to the field are non-conducting has been considered by Hunt (1), who derives the solution in two ways; as the limiting cases of the flows with (a) non-conducting walls parallel and thin walls of arbitrary conductivity perpendicular to the field, and (b) thin walls of arbitrary conductivity parallel and perfectly conducting walls perpendicular to the field. We show that these two limiting solutions derived by Hunt are in fact equivalent. In addition, we extend the solution of case (b) above by removing the thin wall restriction.

Riley (1) has considered a magnetohydrodyriamic Stokes flow in which the modifications to the well-known Stokes flow past a fixed sphere were investigated for the case in which there is a magnetic pole situated at its centre. Again, Barthel and Lykoudis (2) and Ludford and Murray (3) have considered the low Reynolds number flow and inviscid flow respectively, past a sphere in the presence of a magnetic field due to a dipole situated at the centre of the sphere and directed parallel to the uniform stream. For the viscous problem, the velocity field and induced magnetic field were calculated for small values of the Hartmann number M and an expression found for the total drag on the sphere. Tamada and Sone (5) have also considered the same problem for a viscous conducting fluid at small Reynolds numbers and calculated the solution for small values of M while an asymptotic solution was found for large values of M. For small values of M they show that, since the Lorentz force retards the fluid motion, the viscous forces are reduced and hence the drag on the sphere is also reduced. However, the magnetic stresses contribute a force on the boundary which actually results in a net increase for the total drag on the sphere.

A systematic analysis is conducted for refractive index profiles capable of yielding exact solutions in terms of hypergeometric or confluent hypergeometric functions for the electromagnetic field of vertically polarized waves propagating in horizontally stratified isotropic media. Previously known profiles emerge as special cases of the analysis. Profiles suitable for use with isotropic lossless ionospheres and of prescribed form n2 = 1 − T(z)/k2, with T(z) independent of k, do not arise in this paper.