We study the question: Which ordinary irreducible representations of the symmetric group remain irreducible modulo a prime p?

Let Sλ be the Specht module corresponding to the partition λ of n. The definition of Sλ is ‘independent of the field we are working over’. When the field has characteristic zero, Sλ is irreducible, and gives the ordinary irreducible representation of corresponding to the partition λ. Thus we are interested in the problem of whether or not Sλ is irreducible over a field of characteristic p.

R denotes the set of real numbers. For any finite n, Rn denotes n-dimensional space with the usual (Hilbert space) metric, d.

This paper deals with the relationships of the maxima, minima and zeros of two random functions of known autocorrelations and cross-correlation, based on the work of Rice(4). Ratcliffe(3) and Briggs and Spencer(2) discuss a similar problem in connexion with a ‘Phillips Record’ of an experiment in ionospheric reflexion. In this experiment there are two highly correlated reflected signals, their maxima coming close together, and the record shows the time lags between a maximum on one signal and a maximum on the other. Briggs and Page (1) have made an experimental study of the distribution of the differences between the positions of the maxima of two highly correlated random functions, using EDSAC to construct the functions. In §§ 3–7 the frequency distributions of intervals between successive zeros and maxima, and of the lengths of intercepts by a horizontal line, are considered. This has applications to the study of the fading of long-wave radio signals, where the tune differences between successive maxima of the amplitude have been investigated.

We study the deformation of a long slender drop of viscosity ζμ suspended in another liquid of viscosity μ. Interfacial tension causes the drop to become spherical when there is no fluid motion. When the flow is weak the drop is slightly perturbed, and this case was studied by Taylor (7). Computing the flow around an exact sphere, he used the resulting imbalance in the normal stresses to predict the perturbed drop shape. When the drop is in viscid or slightly viscous (ζ ≪ 1), and when the flow is stronger, the drop becomes long and slender. Previous slender-body analyses (Taylor (8) Buckmaster (2, 3), Acrivos & Lo(1) and Hinch & Acrivos(5)) predict pointed ends, but break down in the neighbourhood of these ends. Here we adopt an approach similar to that of Taylor (7). The zero Reynolds number flow around a spindle-shaped drop with pointed ends is computed exactly. Interfacial tension does not quite balance the hydrodynamic stress, and the resulting imbalance in the normal stresses is used to predict a more accurate representation of the drop shape.

We establish growth tightness for a class of groups acting geometrically on a geodesic metric space and containing a contracting element. As a consequence, any group with non-trivial Floyd boundary are proven to be growth tight with respect to word metrics. In particular, all non-elementary relatively hyperbolic group are growth tight. This generalizes previous works of Arzhantseva-Lysenok and Sambusetti. Another interesting consequence is that CAT(0) groups with rank-1 elements are growth tight with respect to CAT(0)-metric.

A pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

A Noetherian integral domain R is said to be a splinter if it is a direct summand, as an R-module, of every module-finite extension ring (see [Ma]). In the case that R contains the field of rational numbers, it is easily seen that R is splinter if and only if it is a normal ring, but the notion is more subtle for rings of characteristic p>0. It is known that F-regular rings of characteristic p are splinters and Hochster and Huneke showed that the converse is true for locally excellent Gorenstein rings [HH4]. In this paper we extend their result by showing that ℚ-Gorenstein splinters are F-regular. Our main theorem is:

THEOREM 1.1. Let R be a locally excellent ℚ-Gorenstein integral domain of characteristic p>0. Then R is F-regular if and only if it is a splinter.

This paper is a continuation of four others under the same title†. The paragraphs are numbered following on to those of the fourth paper of the series. In § XXIV we show that, if λ(A) is a linear functional, then there exists a resolution Eμ such that λ(A) = ∫μdr(AEμ), and if B = ∫μdEμ is bounded, then λ(A) = τ(AB)‡ for all A, where τ is the trace. This implies that τ(A) is a linear functional, and that the conjugate space ℒ, i.e. the space of the linear functionals, has a subset ℒ′ which is in (1, 1) correspondence with the original set of operators, and that in this correspondence the linear functional τ(A) is associated with the unit operator.

Let A be a commutative ring. We denote by a standard A-algebra a commutative graded A-algebra U=[oplus ]n[ges ]0Un with U0=A and such that U is generated as an A-algebra by the elements of U1. Take x a (possibly infinite) set of generators of the A-module U1. Let V=A[t] be the polynomial ring with as many variables t (of degree one) as x has elements and let f[ratio ]V→U be the graded free presentation of U induced by the x. For n[ges ]2, we will call the module of effective n-relations the A-module E(U)n= ker fn/V1· ker fn. The minimum positive integer r[ges ]1 such that the effective n-relations are zero for all n[ges ]r+1 is known to be an invariant of U. It is called the relation type of U and is denoted by rt(U). For an ideal I of A, we define E(I)n= E([Rscr ](I))n and rt(I)=rt([Rscr ](I)), where [Rscr ](I)= [oplus ]n[ges ]0Intn⊂A [t] is the Rees algebra of I.

In this paper we give two descriptions of the A-module of effective n-relations. In terms of André–Quillen homology we have that E(U)n= H1(A, U, A)n (see 2·3). It turns out that this module does not depend on the chosen [x]. In terms of Koszul homology we prove that E(U)n= H1([x], U)n (see 2·4). Using these characterizations, we show later some properties on the module of effective n-relations and the relation type of a graded algebra. Our approach has connections with several earlier works on the subject (see [2, 5–7, 9, 10, 13, 14]).

When certain elements are bombarded by swift α particles, protons or hydrogen nuclei are liberated which are attributed to the disintegration of the nuclei of these elements. It is now generally assumed that in such an artificial disintegration of an atomic nucleus an α particle is captured by the nucleus. Experimental evidence for this capture of an α particle has been obtained by Blackett in the case of the disintegration of the nitrogen nucleus. Blackett photographed eight disintegration collisions of an α particle with a nitrogen nucleus; in each of these the track of the α particle divided into two branches, one of which was due to the residual nucleus set in motion in the collision, and the other was due to the ejected proton. No trace of a third track to correspond to the track of the α particle after the collision was observed. Blackett therefore concluded that the α particle was captured by the nitrogen nucleus.

In this paper we are concerned with the problem of counting rational points of bounded height on rational cubic surfaces. For most such surfaces this question appears much too hard for current methods. We shall therefore examine a particular example, namely the surface

formula here

for which the problem is tractable, though by no means trivial. This surface contains the lines X0=Xi=0, for i=1, 2, 3, which we shall exclude from consideration. We therefore define

formula here

where an integer vector is said to be primitive if its components have no common factor. The function [Nscr ](H) counts points with the most naive height available. This suffices, however, to bring out the key features of the problem. We may observe at once that the condition x0[les ]H is redundant, and that it suffices to assume that (x1, x2, x3) is primitive.

Our principal result is the following theorem.

Introduction. In this paper we consider the two cross-product combinations of Bessel functions

where δ = (k − 1) z and (') denotes differentiation with respect to the argument. Here Jν and Yν designate respectively the Bessel functions of the first and second kind of order ν.

Many years ago Rutherford remarked that the high energy lines of the radium C β-ray spectrum showed the existence of γ-rays of very high frequency, and later he pointed out that their frequencies must be closely connected with the β-ray energies through the quantum relation. Although the truth of Rutherford's point of view has become more and more obvious, no advance in the detailed knowledge of these lines has been made in the last seven years. This spectrum is very complicated and the lines faint.

1. A sketch of the first part of this paper, together with some of the second part, was first written out in October last, shortly after the publication of Mr Babbage's paper, A series of rational loci with one apparent double point. It is well known that the Del Pezzo quintic surface the only non-ruled quintic surface in [5], has one apparent double point; Babbage shows that this surface is a member of a series of loci in [2n + 1], each of which has one apparent double point; he establishes the existence of these loci from their representations on flat spaces ∑n, these representations being analogous to the plane representation of by means of cubic curves through four fixed points.

The S-matrix formulation of quantum electrodynamics, as developed by Feynman (3) and Dyson (1), expresses the matrix element for any process as a power series in the coupling constant, the coefficients of the series being, in general, rather complicated multiple integrals. These integrals contain singularities in their integrands, and in certain circumstances the coincidence of such singularities gives the S-matrix a non-analytic behaviour as a function of, for example, the total energy. The threshold behaviour of the S-matrix in the neighbourhood of energy values at which this phenomenon occurs has been investigated by Eden (2), who shows that the non-analytic behaviour is connected with the possible commencement of new physical processes, such as the creation of a particle.

1. This paper is supplementary to an article by M. H. A. Newman on the superposition of manifolds. Guided by his main idea, I extend Newman's theorem that two equivalent manifolds have a common subdivision from equivalent manifolds to equivalent complexes of an arbitrary nature. The theorem is also sharpened by the possibility of leaving a certain subcomplex unaltered, and by substituting a partition for a general subdivision.

We prove that every Lie derivation on a symmetrically amenable semisimple Banach algebra can be uniquely decomposed into the sum of a derivation and a centre-valued trace.

The elastic fields in an elastic circular inclusion and its surrounding infinite dissimilar elastic matrix, are determined when either the matrix or inclusion is subject to a concentrated force or edge dislocation.

1. The problem of the partition of numbers, first investigated in detail by Hardy and Ramanujan (1), has in recent years assumed importance on account of its application by Bohr and Kalckar (2) in evaluating the density of energy levels in heavy nuclei. A ‘physical approach’ to the partition theory has been made by Auluck and Kothari (3), who have studied the properties of quantal statistical assemblies corresponding to the partition functions familiar in the theory of numbers. The thermodynamical approach to the partition theory, apart from its intrinsic interest, draws attention to aspects and generalizations of the partition problem that would, otherwise, perhaps go unnoticed. Thus we are led to consider restricted partitions such as: partitions where the summands are repeated not more than a specified number of times; partitions where the summands are all different; partitions into summands which must not be less than a specified value; partitions into a prescribed number of summands, and so on. The generalization that seemed to us to be the most interesting is the extension of the partition concept to include partitions into non-integral powers of integers.