Van der Corput has shown (2), using a general criterion of Weyl (1), that a necessary and sufficient condition, that a sequence of points Pn = (αn, βn) (n = 1, 2,…) in two-dimensional space be uniformly distributed modulo 1, is that for all pairs of integers (u, v) other than u = v = 0 the one-dimensional sequence (uαn + vβn) (n = 1, 2,…) is uniformly distributed modulo 1. The object of this paper is to give a quantitative form to the sufficiency part of this qualitative criterion.

Throughout this paper, the concept of a ‘freely generated algebra’ will be meant in the sense of (3), Definition 2·1. It is understood that all conventions concerning terminology and notation introduced in (3) hold in this paper. In particular, it is understood that the sign denotes a freely generated algebra, and that the definition-set and the free basis of are denoted by C and D respectively. The letter q will denote a cardinal number.

It is well known that if K is a commutative field then the polynomial ring K[x1, x2, …, xn] has global dimension (in the sense of homology theory) equal to the number of variables. This, of course, is not an isolated result but, in view of the special interest which attaches to such a familiar object as a polynomial ring, a short and comparatively elementary proof of this particular result may be of interest. The present paper is devoted to such a proof and therefore it is proper that this first section should indicate, in some detail, what knowledge of homological algebra is presupposed.

The statistical operation of multiple linear regression by least squares is equivalent to the orthogonal projection of vectors of observations on a space spanned by vectors of observations; and a partial regression can be similarly represented as an oblique projection. This connexion between a statistical and a formal algebraical operation gives the main source of interest for this investigation. Its object is to develop algebraical theory which supplies terms necessary for a unified algebraical and geometrical formulation of concepts in multivariate analysis. An exposition of the statistical theory is to be made in a separate paper.

In the theory of vector-spaces an ordered, ortho-normal set of r vectors is called an r-frame. Let Sn denote the unit sphere in euclidean (n+ 1)-space, where n ≥ 1. By an r-field on Sn is meant a continuous function which assigns to each point of Sn an r-frame in the tangent space at that point. If q < r we obtain a q-field from an r-field by suppressing the first r – q vectors of each r-frame. Certainly Sn admits a 0-field, and does not admit an (n+ 1)-field, since the tangent space is n-dimensional. An n-field on Sn is called a parallelism. Notice that an (n − 1)-field on Sn can always be extended to an n-field, since spheres are orientable. The problem is to determine the greatest value of r such that Sn admits an r-field.

The theory of distributions, systematized by L. Schwartz (3), has many applications in pure mathematics and physics. A number of different approaches to the theory are possible. Schwartz's own account requires for its comprehension a familiarity with the topology of general spaces. I give in this note another approach to the theory based on a process of integration of the Stieltjes type.

Consider a Markov chain with an enumerable infinity of states, labelled 0, 1, 2, …, whose one-step transition probabilities pij are independent of time. Then

I write

and, departing slightly from the usual convention,

Then it is known ((1), pp. 324–34, or (6)) that the limits πij always exist, and that

The canonical form of the Riemann tensor is found for certain non-flat empty space-times. Some invariant physical properties of these space-times are investigated. The status of Mach's principle in general relativity theory is briefly discussed in the light of these examples.

The Yang-Feldman formalism vising the Feynman-like Green's functions is set up. The corresponding free fields have non-trivial commutation relations and contain information about the scattering. S-matrix elements are simply the matrix elements of anti-normal products of the field φF′(x). These are evaluated, and they give directly expressions used in the theory of causality and dispersion relations. It is possible to formulate field theory in a form in which the fields obey free field equations and the effects of interaction are contained in their commutation relations.

A variational principle is obtained for the Boltzmann transport equation in the presence of a magnetic field and modified by boundaries, when the standard formulation of Kohler breaks down because of the lack of microscopic reversibility. The connexion with entropy production is discussed.

In (3), some of us proved that Brownian paths in n-space have double points with probability 1 if n = 2 or 3; but, for n ≥ 4, there are no double points with probability 1. The question naturally arises as to whether or not Brownian paths in n-space (n = 2 or 3) have triple points. The case of paths in the plane is settled by (4), where it is shown that, with probability 1, Brownian paths in the plane have points of multiplicity k (k = 2, 3,4, …). The purpose of the present paper is to settle the remaining case, n = 3. We prove that, with probability 1, Brownian paths in 3-dimensional space have no triple points. The general idea behind our proof is to show that there are not too many double points. That is, we show that the set of double points has sigma-fmite linear measure, and therefore zero capacity, with probability 1.

The Ising model of a two-dimensional ferromagnetic crystal containing defects is considered. It is shown that the combinatorial method is particularly appropriate to the investigation of the two-dimensional defect crystal and how it may be readily modified to calculate the partition function for such a crystal. It is found that the cooperative behaviour of the infinite perfect crystal, as indicated by the occurrence of a logarithmic singularity in the specific heat as a function of temperature, is not destroyed in the defect crystal; the singularity persists, though at a progressively lower temperature as the density of defects is increased, until a high density of defects is reached. The value of this limiting density of defects, above which the cooperative behaviour is destroyed, is calculated.

A brief discussion is given of attempts that have been made to justify, in terms of electrostatic principles, the conjecture that surface charge on a conductor tends to infinity towards a convex sharp point and to zero towards a concave point, and it is concluded that the problem has not hitherto been solved. A new attempt is then made using potential-theoretic methods, and the problem is solved with a fair degree of generality. The limitations are of a kind familiar in this type of analysis and, roughly speaking, concern the degree of convexity of the conducting surface.

A new set of stability criteria for linear systems is derived. This shows that about half of the Hurwitz criteria are redundant when certain of the coefficients of the characteristic equation are known to be positive. The theory is applied to obtain a very short derivation of the known aperiodicity criteria. The conditions for realizability of RC networks are shown to be closely related to the stability and aperiodicity criteria, and are stated as sets of criteria in terms of the polynomial coefficients. Two basic theorems are involved which give the necessary and sufficient conditions for the roots of two polynomial equations to be real and separated.

Consideration of the direction of the normal at any point on a continuous sheet of a surface yields a sufficient condition for the existence of parabolic points on that sheet. This condition has been used to derive some simple inequalities between elastic constants, whose fulfilment determines the existence of parabolic points upon the inverse surfaces of media of orthorhombic, tetragonal, cubic or hexagonal symmetry; in virtue of the polar reciprocal relation between inverse and wave surfaces, the existence of cusp points on the latter is thereby simultaneously established. It is also pointed out that conditions for external conical refraction may prevail in hexagonal media.

General methods of successive approximations for problems in the theory of large elastic deformations have been proposed by Signorini (9), Misicu (5), Rivlin (6), Green and Spratt (4) and Rivlin and Topakoglu (7). Rivlin has applied his method to the problem of torsion of cylinders as far as terms of the second order. A solution of the problem of second order effects in the torsion of incompressible cylinders of arbitrary cross-section has been given by Green and Shield (3) in terms of complex potential functions, and the complete details of the stress distribution can be obtained once certain integral equations are solved. Blackburn and Green (l) have considered the second order effects in the deformation of compressible cylinders by forces on the ends. For the problems of torsion and bending by couples they obtained details of the displacements and stress distributions in terms of two complex potential functions which satisfy a certain boundary condition. The second order torsion problem has also been discussed by Sheng(8).