The present note is concerned with the proof and applications of the following theorem: Let A be a commutative ring, N be an A-module, and g1, …, gk be elements of A such that (g1,…, gi−1) A:gi = (g1,…, gi−1) N:gi = (g1,…, gi−1) N (i = 1,…, k), so that 0:g1 = 0 in N. Let g denote the ideal (g1 …, gk), let B = A/g and let M be any A-module such that gM = 0. then

where, on the right-hand side of (ii), M, N/gN are considered as B-modules.

1. Introduction. 1·1. Notation. Let

and

be, respectively, the upper and lower central series of a group G. By definition, Zi+1/Zi is the centre of G/Zi and Γj+1 = [Γj, G] is the commutator subgroup of Γj with G. When necessary for clearness, we write ZiG) for Zi and Γj(G) for Γj.

The present paper is a sequel to a previous study (7) of the completely regular threefolds which possess anticanonical systems, i.e. for which the virtual canonical system, reversed in sign, is effective of positive order. On any such threefold the process of successive adjunction, applied to any linear system of surfaces, must terminate; we have thus to deal with a special case of the adjunction problem for the regular threefolds. By making certain simplifying hypotheses (such as irreducibility and absence of base elements) concerning the anticanonical systems, one can classify the threefolds in broad outline and show that, provided the anticanonical systems are sufficiently ample, the corresponding threefolds are either unirational or birational.

The recent attempt at a physical interpretation of non-Riemannian spaces by Einstein (1, 2) has stimulated a study of these spaces (3–8). The usual definition of a non-Riemannian space is one of n dimensions with which is associated an asymmetric fundamental tensor, an asymmetric linear affine connexion and a generalized curvature tensor. We can also consider an n-dimensional space with which is associated a complex symmetric fundamental tensor, a complex symmetric affine connexion and a generalized curvature tensor based on these. Some aspects of this space can be compared with those of a Riemann space endowed with two metrics (9). In the following the fundamental properties of this non-Riemannian manifold will be developed, so that the relation between the geometry and physical theory may be studied.

This paper contains a discussion of various points which arise in the numerical evaluation of the small real zeros of the confluent hypergeometric function

where

There are two distinct problems, first the determination of those values of x for which M(a, b; x) = 0, given a and b, and secondly the study of the curves represented by M (a, b; x) = 0, for fixed values of x. These curves all lie on the surface M(a, b; x) = 0, of course.

In numerical solution of a Sturm-Liouville system, it is necessary to determine an eigenvalue by a method of successive approximation. A relation is derived between the estimated accuracy of an approximate eigenvalue and the accuracy at every point of its corresponding eigenfunction. A method is also described whereby the correction to a trial eigenvalue, required for convergence to its true value, can be automatically determined. This method has been successfully used in solving radial wave-function equations, both with and without ‘exchange’, arising from the Hartree-Slater-Fock analysis of Schrödinger's equation.

Ward(1) has discussed a method, introduced by Tricomi (6), of calculating transients by means of series involving Laguerre functions which in some cases makes it unnecessary to determine poles of the relevant characteristic function. This method is here investigated with special reference to conditions for convergence and adjustments for improving convergence; some of the examples discussed by Ward are reconsidered.

Both in Ward's paper and here the location of poles of the characteristic function is assumed to be approximately known. In some cases the determination of poles of outstandingly small or large modulus and their separation from the remainder may be the most satisfactory procedure. Lin's method (2) for determining a quadratic factor of a polynomial is more widely applicable than has previously been supposed, and this is discussed in bare outline, without proof, here, but in detail, with adequate numerical examples, elsewhere (3).

1. degenerate iterative processes. A well-known class of iterative processes is specified by the equation

1.The study of the numerical transformation of slowly convergent series and sequences permits of a certain unity of approach, since the partial sums of the former may be regarded as members of a slowly convergent sequence, while the differences of the latter may be treated as terms in a slowly convergent series. In the ensuing discussion, no essential distinction between the two problems is made, and methods devised for one relate with equal facility to the other.

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.

A method is described for improving the approximate calculation of energy levels for atomic configurations consisting of a set of closed shells and a single ‘series’ electron. This involves the inclusion of a ‘polarization’ potential in the Hartree-Slater-Fock equation, caused by the series electron. The form of potential found in two cases investigated, namely, Si+3 and Ca+, is described and discussed.

The interaction of conduction electrons with a vibrating metallic lattice has been considered by Ziman in terms of non-adiabatic perturbations to the adiabatic approximation of the Born-Oppenheimer theory. This programme is continued to calculate the self-energy of the electrons. When account is taken of the dependence of the energy levels of the adiabatic Hamiltonian itself upon the electron configuration, the final result is the same as the standard one.

Preliminary self-consistent field calculations without exchange have been performed for the ground states of the structures In3+ and Sb3+. The results are presented.

An attempt is made to calculate the first few angular terms in an expansion of the electron density for the phosphine molecule in Legendre polynomials. Such an expansion is appropriate for a model in which the three hydrogen nuclei are smeared to form a circular line charge. The Thomas–Fermi approximation has been used in conjunction with the variational method. The variational density employed includes p and f angular terms. An approximate charge density map is constructed for a plane containing the molecular axis in order to demonstrate the effect of the angular terms.

A very simple model, consisting of N particles moving in a one-dimensional assembly divided by potential ‘humps’ into M cells, is studied. The process of passing from a quantum-mechanical description of such an assembly to the equation of diffusion type that governs it in practice is shown to consist of at least three separate steps: ‘averaging over phases’, and letting N and M become large. The effects of these steps are considered separately. Strict irreversibility in time appears after the first step, but the assembly remains ergodic until after the second step and fluctuations persist until after the third step.

This paper describes an attempt to formulate quantum field theory, in particular quantum electrodynamics, in terms of Hilbert space theory. The work of Cook (1) is extended to give a precise description of non-interacting electrons and positrons. The hole interpretation is not required in this extension, and no subtraction formalism is required. It is shown that the formalism can never reduce to that of intuitive quantum field theory except by an abuse of language associated with the δ-function. Interaction cannot be introduced in a simple manner into the rigorous formalism, so it seems extremely difficult to develop the Hilbert space formalism for quantum field theory in any useful manner.

These difficulties indicate that an investigation of the Hilbert space basis of simple quantum theory is necessary before a rigorous mathematical formalism for intuitive quantum field theory can be developed.

The complex potentials and deflexion at any point of a thin circular plate with a normal linearly varying load over an eccentric circle are determined under a general boundary condition including the usual clamped and hinged boundaries.

Complex variable methods have been applied to isotropic and aelotropic plate problems by several authors. The notation used here is that of Stevenson(14). Dawoud(5) has expressed the continuity conditions between two differently loaded regions in terms of the complex potentials and the particular integrals for the two regions.

The problem of a transverse load at any point of a clamped circular plate was solved by Clebsch(4), Michell(11), Melan(10) and Flügge(6). A series solution for the simply supported circular plate under the same load was given by Foeppel(8). Using Stevenson's tentative method Dawoud(5) applied complex potentials to solve the problem of an eccentric isolated load under certain boundary conditions. Applying Muskhelishvili's method, Washizu(15) obtained the same results for clamped and simply supported boundaries.

It is easy to get solutions for a circular plate concentrically and uniformly loaded. For non-uniform loadings there are the solutions found by Sen (13) for certain distributions of normal thrust over the complete plate or over a concentric circle and the solution of Flügge (7) for a linearly varying load over the simply supported circular plate. The present author and Dawoud(3) obtained the solutions for a circular plate with the load over the complete plate or over a concentric circle, under a general boundary constraint including as special cases the usual clamped and hinged boundaries. Ghose (9) worked out the problem of a clamped circular plate when the load is uniformly distributed between two concentric circles and two radii. Schmidt (12) found the solution for a clamped circular plate uniformly loaded over an eccentric circle. The complex variable method was applied by the author and Dawoud(2) to obtain the solutions for a circular plate having an eccentric circular patch symmetrically loaded with respect to its centre under the general boundary condition mentioned before. The author (1) also found the solution for a linearly varying load over an eccentric circle under the same boundary condition. In this paper the power of the complex variable method is exhibited by rinding the appropriate complex potentials corresponding to the load

over an eccentric circular patch, where R, θ are measured from the centre of the patch and the common diameter of the plate and the patch. Since the two cases n = 0, 1 require special consideration and were dealt with separately (in (2) and (1) respectively), we see that this paper completes the solution of the problem of a circular plate with an eccentric circular patch symmetrically loaded with respect to the common diameter of the plate and patch, the load being in this case expressible in the form .

For a clamped boundary the solution is obtained in finite terms.