Let {sn} be the sequence of partial sums of the series and let

If , we say that the series is summable (l) to s, and if , the series is said to be summable (l2) to s. It is clear that .

The following generalization of a theorem of Weyl appeared in part I of this series of papers. Let G be a locally compact Abelian group with dual group ĝ. Let be a sequence in ĝ, not too slowly growing in a certain precise sense. Then, provided ĝ has ‘not too many’ elements of finite order, the sequences

are uniformly distributed on the circle, for almost all x in G.

A class of objects, sometimes known as weighted tensors, are investigated. Their relation to a Lie group acting as a transform group on a manifold is analysed. An equivalence relation between weighted tensors is established along with a method for their generation. Several examples are given.

1. We suppose that f(t) is integrable L and periodic with period 2π and we denote the Fourier series of f(t) at the point t = x by

We denote the nth partial sum of the Fourier series by sn. We will prove here a general result (i.e Theorem 1) and some interesting results in connexion with associated series of a Fourier series of the type

We write

The object of the present paper is to prove the following theorems. (In all the following cases k is any constant > π if nothing else is mentioned.)

If F(f) denotes the Fourier transform of a generalized function f and f * g denotes the convolution product of two generalized functions f and g then it is known that under certain conditions

Jones (2) states that this is not true in general and gives as a counter-example the case when f = g = H, H denoting Heaviside's function. In this case

and the product (x−1 – iπδ)2 is not defined in his development of the product of generalized functions.

Wald(4) proved that under rather weak conditions, maximum-likelihood estimators are consistent. In an earlier paper (3) he had promised to publish conditions under which they are uniformly consistent but he did not do so. As the uniformity of consistence of maximum-likelihood estimators is important in studying the asymptotic power of certain tests, particularly when the true value of the parameter lies on the boundary of the parameter space, the purpose of the present note is to fill this gap. The method of proof follows that of Wald(4) very closely.

Motivated by the Iwasawa decomposition and its geometrical interpretation, two new decompositions of the de Sitter group are obtained. The first is applied to construct the representations of the de Sitter group in a form immediately comparable with those of the Poincaré group. In particular they act on functions over an hyperboloid like the momentum hyperboloid of the Poincaré group, although they require both positive and negative mass shells of that hyperboloid. Using the second decomposition it is shown that the representations of the de Sitter group are localizable in the sense of Mackey and Wightman. Position operators are exhibited.

The origin of the present paper is the desire to study the asymptotic behaviour of certain tests of significance which can be based on maximum-likelihood estimators. The standard theory of such problems (e.g. Wald(4)) assumes, sometimes tacitly, that the parameter point corresponding to the null hypothesis lies inside an open set in the parameter space. Here we wish to study what happens when the true parameter point, in estimation problems, lies on the boundary of the parameter space.

In a recent paper, R. F. Churchhouse has studied the function b(n) giving the number of partitions of n into powers of 2. The generating function of b(n) is

so that

Using this relation, Churchhouse has shown that

and

where

with

Moreover

and b(n) is even for n ≥ 2, while b(2n) ≡ (mod 4), for n = 22m−l(2k + 1); and, it is not ≡ (mod 8) for any n. He has conjectured that for k ≥ 1 and n odd,

and

The object of this note is to prove these conjectures.

The velocity and pressure fields for Stokes's flow due to a point force (‘stokeslet’) in the vicinity of a stationary plane boundary are analysed, using Fourier transforms, to obtain the image system required to satisfy the no-slip condition on the boundary. The image system, which is illustrated by diagrams, is found to consist of a stokeslet equal in magnitude but opposite in sign to the initial stokeslet, a stokes-doublet and a source-doublet, the displacement axes for the doublets being in the original direction of the force. The influence of the wall on the near and far fields is discussed. In the far field it is found that a stokeslet aligned parallel to the wall produces a stokes-doublet far-field, whereas a stokeslet normal to the wall produces a combination of a source-doublet and a stokes-quadrupole far-field. Although results can be alternatively derived by the method of Lorentz (7) using a reciprocal theorem, the present method yields much more clearly the form of the image system.

In this note a brief description is given of a method for reducing a type of singular integro-differential equation to both an integral equation with a quasi-regular kernel and an integral equation with a regular kernel. A particular integro-differential arising in aerofoil theory is used as illustration.

In this paper we obtain the short-wave asymptotic value of the force coefficient for a heaving body partially immersed in the free surface of water which has finite constant depth. The body is required to intersect the free surface normally and have a vertical axis of symmetry. The motion of the water is thus axisymmetric with short outgoing cylindrical waves at a distance, and these must be calculated. For this we use a non-rigorous argument. The potential problem discussed is analogous to the two-dimensional problem afready investigated involving a heaving surface cylinder. Numerical values are obtained in the case of a half-immersed sphere in conclusion.

In this paper we first demonstrate how a certain formula, which expresses (n − 1 )th divided difference in the form of a multiple integral, may be used to obtain the density function of a suitable random variable and then apply this to obtain the density of a useful variate.

A generalization is given of the notion of stochastic monotonicity. This is used in setting up, and showing how to obtain approximate solutions for, an integral equation formulation of GI/G/l queues, when access to the waiting-line is subject to periodic interruptions.

The classical wave-maker problem to determine the forced two-dimensional wave motion with outgoing surface waves at infinity generated by a harmonically oscillating vertical plane wave-maker immersed in water was solved long ago by Sir Thomas Havelock. In this paper we reinvestigate the problem, making allowance for the presence of surface tension which was excluded before, and obtain a solution of the boundary-value problem for the velocity potential which is made unique by prescribing the free surface slope at the wave-maker. The cases of both infinite and finite constant depth are treated, and it is essential to employ a method which is new to this problem since the theory of Havelock cannot be extended in the latter case of finite depth. The solution of the corresponding problem concerning the axisymmetric wave motion due to a vertical cylindrical wave-maker is deduced in conclusion.

The solution of problems involving the propagation of discontinuities and other singularities for hyperbolic partial differential equations by means of progressing wave expansions is discussed in the book by Courant(l). He also refers to the work of Hadamard, Friedlander, Ludwig and others on this subject. More recently, Ludwig (2), Lewis(3) and others have considered 'uniform' progressing wave expansions for various problems. These expansions are valid in regions where the standard expansions are not suitable and they can be re-expanded in the standard form outside these regions. Examples of such regions are given by envelopes of bicharacteristic curves or, equivalently, caustics and by shadow boundaries such as occur in diffraction problems. In each of these regions, which we term 'transition regions' different types of uniform expansions are required.

The problem of the steady motion of a heavy cylinder, of infinite length, over the surface of an elastic half-space has been examined by Craggs and Roberts(1), Clements(2) and Roberts (3). In papers (1) and (2) the surfaces of both the cylinder and the half-space are assumed to be smooth so that the shear stress over the region of contact is zero. With this assumption Craggs and Roberts analysed the motion for an isotropic half-space while Clements considered the case of an anisotropic half-space. In the paper by Roberts (3) various problems concerning the rolling of rigid and non-rigid cylinders over an isotropic elastic half-space are discussed. In this paper we consider a similar class of rolling cylinder problems for an anisotropic half-space. We begin, in section 2, by deriving the relevant basic equations for the stress and displacement while in section 3 some properties of certain constants occurring in the equations of section 2 are derived for use in later sections. The problem of the steady motion of an inflated tyre is discussed in section 4 under the assumptions that the tyre exerts a constant pressure over the contact region which is instantaneously at rest. The solution shows that, with these assumptions, the shear stress is infinite near the ends of the region of contact. This indicates that there will be slipping of the tyre over the half-space at both ends of the contact region. If these zones of slip are large compared with the total contact area then the assumption of a contact region which is instantaneously at rest is invalidated so that the solution may only be regarded as being applicable when the zones of slip are small. An expression to determine the width of the zones of slip is derived and this expression is used, together with the numerical results of section 5, to obtain information about the influence of anisotropy on the width of the zones of slip. In section 6 the problem of the inflated tyre is discussed with allowance made for a zone of slip at both ends of the region of contact. The results of this section could be expected, particularly when the zones of slipping are wide, to more accurately predict the response of the anisotropic half-space to the rolling tyre than the results of section 4. Unfortunately, in order to satisfactorily complete the analysis, it is necessary to restrict the class of anisotropic materials for which the results are applicable. As a consequence, the work of section 4 is in many ways preferable to that of section 6 in indicating the influence which elastic anisotropy has upon the results.

We consider the equation:

where a: [0, ∞) → R1, a(t) > 0, a'(t) is continuous,

f:( −∞, +∞) → R1, f is continuous, and xf(x) > 0 for x ≠ 0. The problem is to give conditions on a(t) and f(x) to ensure that all solutions of (1) tend to zero as t → ∞. First, however, we give some sufficient conditions and some necessary and sufficient conditions to ensure that all solutions of (1) are oscillatory or bounded.

In a recent paper (3) I discussed the slow steady rotation of an insulated axisymmetric solid in a bounded viscous fluid of finite conductivity, there being a uniform magnetic field applied parallel to the axis of rotation. By reducing the small Hartmann number problem to a set of potential type Fredholm integral equations of the first kind, expressions were derived for the frictional couple G on the solid in a number of different geometrical situations. The formulae obtained were in the form of power series in the Hartmann number M together with the lowest order wall correction, the zero-order term in G being the frictional couple G0 for the non-magnetic infinite-fluid problem.

Representations of the four-dimensional unitary group U4 were considered long ago by Wigner(1)as a model for nuclear isobaric multiplets. However, the Hamiltonian did not include the components of isospin which together with the spin coordinates are known to label the nuclear states. In this paper we shall find representations characterized by the eigenvalues of angular momentum J, isospin T and parity π, and will find mass relations which give good agreement with the experimental energy levels of Li6 and Be8 labelled by the same quantum numbers. The representations found by Wigner give good results for the ground state energies, or masses, of all the nuclei up to a mass number of A = 110(2), and we shall derive Wigner's representations as a special case. In fact, unless these are satisfied it is impossible for particle-like representations of U4 to exist!