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.

It is shown that the Wilks large sample likelihood ratio statistic λn, for testing between composite hypotheses Θ0 ⊂ Θ1 on the basis of a sample of size n, behaves as n varies like a diffusion process related to an equilibrium Ornstein-Uhlenbeck process, whenever the null hypothesis is true. This fact is used to construct large sample sequential tests based on λn, which are the same whatever the underlying distributions. In particular, the underlying distributions need not belong to an exponential family.

We prove that for a 7-dimensional manifold M with cylindrical ends the moduli space of exponentially asymptotically cylindrical torsion-free G2-structures is a smooth manifold (if non-empty), and study some of its local properties. We also show that the holonomy of the induced metric of an exponentially asymptotically cylindrical G2-manifold is exactly G2 if and only if the fundamental group π1(M) is finite and neither M nor any double cover of M is homeomorphic to a cylinder.

The following is a preliminary account of experiments made in an attempt to obtain information about the relation between the energy of a positive ion and the effect produced on a surface which it bombards. The energy relations which hold in collisions between rapidly moving positive ions and the molecules of a gas or of a solid are of extreme importance in the theory of gaseous discharges at low pressures, in particular of the initiation of the self-sustained discharge.

Photographs have been taken under controlled illumination of the tracks of α particles in a cloud expansion chamber and a calibration of the photographic plates employed has been carried out. Systematic photometry of the track images has made possible the calculation of the variation of the light scattering power of an α particle track over the last two centimetres of its length in standard air, and the variation of this quantity has been identified with the variation of ionisation along the track.

Photographs of tracks in air, helium and hydrogen have been examined. In these three gases the maximum ionising efficiency of the α particle occurs when it possesses the velocity respectively appropriate to the distances 3·0, 2·55 and 2·25 mms. from the end of its path in standard air. This common velocity parameter has been employed throughout the discussions which are appended to the experimental results.

We classify finite Moufang loops which are centrally nilpotent of class 2 in terms of certain cubic forms, concentrating on small Frattini Moufang loops, or SFMLs, which are Moufang loops L with a central subgroup Z of order p such that L/Z is an elementary abelian p-group. (For example, SFM 2-loops are precisely the class of code loops, in the sense of Griess.)

More specifically, we first show that the nuclearly-derived subloop (normal associator subloop) of a Moufang loop of class 2 has exponent dividing 6. It follows that the subloop of elements of p-power order is associative for p > 3. Next, we show that if L is an SFML, then L/Z has the structure of a vector space with a symplectic cubic form. We then show that every symplectic cubic form is realized by some SFML and that two SFMLs are isomorphic in a manner preserving the central subgroup Z if and only if their symplectic cubic spaces are isomorphic up to scalar multiple. Consequently, we also obtain an explicit characterization of isotopy in SFM 3-loops. Finally, we extend many of our results to all finite Moufang loops of class 2.

Recently Kohler, on the basis of some experiments by Verleger, has questioned the validity of the usual assumption that the linear Hall effect is entirely perpendicular to the current. The available experimental data are however contradictory, so a simple magneto-resistance experiment with a bismuth crystal was made, which suggested that the usual assumption was, after all, valid. The disymmetry of the Hall effect is discussed, and some of Kohler's results are generalized.

Let ξ, η, ζ be linear forms in u, v, w with real coefficients and determinant Δ ≠ 0. A conjecture of Minkowski, which was subsequently proved by Remak, tells us that for any real numbers a, b, c there exist integral values of u, v, w for which

and the constant ⅛ on the right is best possible.

In the course of researches on the formation of galaxies one meets the following idealized problem. What is the form of the collapse under gravitational forces of a uniformly rotating spheroidal gas cloud? In the special case where initially the gas is absolutely cold and of uniform density within the spheroid, we show that the collapse proceeds through a series of uniform, uniformly rotating spheroids until a disk is formed. This demonstration is followed by some remarks and conjectures as to what will happen in the physical case when the initial temperature is not necessarily zero.

The existence of non-homogeneous modes of deformation in an equibiaxially stretched elastic half-space is investigated. The surface of the half-space is assumed to be traction-free.

It is shown that three different eigenmodal deformations are possible, according as the cube of the critical stretch is greater than, less than, or equal to ¾.

For the class of strain-energy functions due to Ogden (9), necessary conditions for bifurcation are deduced-for equibiaxial tension. Results are given and discussed for single-term, as well as two- and three-term Ogden strain-energy functions, for equibiaxial tension and compression.

We prove that if q ≥ 5 is an integer, then every qth power of an integer contains at least 5 nonzero digits in its binary expansion. This is a particular instance of one of a collection of rather more general results, whose proofs follow from a combination of refined lower bounds for linear forms in Archimedean and non-Archimedean logarithms with various local arguments.

Experiments carried out with the aim of elucidating the action of the α-particle counter are described. A normal counting circuit including an ionization chamber with point and plane electrodes was used. The “wave-form” and intensity of the transient impulses resulting from discharges stimulated by single electrified particles have been determined using a cathode-ray oscillograph. It is found that the nature of the impulse is determined largely by the capacity and resistance of the system, and not by the nature of the electrical stimulus. The potential impulse normally detected by string electrometer methods is shown to be proportional to the current responsible for the recharging of the electrostatic capacity of the counter after the rapid discharge of the capacity initiated by the electrified particle.

The Fibonacci and Lucas polynomials Fn(z) and Ln(z) are denned. These reduce to the familiar Fibonacci and Lucas numbers when z = 1. The polynomials are shown to satisfy a second order linear difference equation. Generating functions are derived, and also various simple identities, and relations with hypergeometric functions, Gegenbauer and Chebyshev polynomials.

Given Banach spaces X and Y let (X, Y) be the space of germs at 0 of C∞ Y-valued mappings defined on neighbourhoods of 0 in X. Let j(X, Y) be those f ∈ (X, Y) such that the ith derivative f(i)(0) = 0 for i = 0,1,…, j – 1. Abbreviations: (X) for (X, ℝ) and j(X) for j(X, ℝ). For ease of exposition germs will often be replaced by mappings which ‘represent’ them.

By a finite lattice† is meant any finite class having the following properties:

For a long time, the main activity in knot theory, we would even say the only one for the problems related to knot cobordism, has been focused onto the development and the analysis of various algebraic invariants. The present paper intends to illustrate some geometric techniques, and to advertise a recent theorem of Casson and Gordon (8) which provides a necessary condition for a fibred classical knot to be ribbon (see definition below) in terms of a cobordism property of its monodromy. We want to show how this last result, combined with some previous work of ours on the cobordism of surface diffeomorphisms (4) (see also (10)) and Thurston's theory of pseudo-Anosov diffeomorphisms (28), (12), can effectively be used to show that a knot is not ribbon.

A positive integer is square-full if each prime factor occurs to the second power or higher. Each square-full number can be written uniquely as a square times the cube of a square-free number. The perfect squares make up more than three-quarters of the sequence {si} of square-full numbers, so that a pair of consecutive square-full numbers is a pair of consecutive squares at least half the time, with

Nunn May (1) has developed the theory of the mode of action of the Geiger-Müller counter in terms of the photoemission produced in the electron avalanches constituting the discharge of the tube. An avalanche of electrons in the counter produces excitation and ionization of atoms and molecules in the neighbourhood of the positive central wire: the photons liberated by these in their turn release from the walls of the counter the photoelectrons which initate the following avalanche. Direct experimental evidence that such a sequence of events may occur is to be found in the work of Greiner (2).

The calculation of the diffraction pattern for an out-of-focus line source viewed through an axially symmetrical lens involves the evaluation of the integral

for a constant value of β (related to the numerical aperture of the lens) and a range of values of x and y. For small values of x and y this can best be done by quadrature; for larger values of x or y or both it is more practicable to use numerical evaluation of an appropriate ordinary differential equation. It is shown that the variation of

with x for y = 0, and with y for x = const., can both be expressed by second-order ordinary differential equations of a convenient form for numerical integration; the evaluation of I1(x, y) from solutions of these equations is straightforward. For comparison with experiment, results are required in the form of I1(x, y) and a derived quantity as functions of y, at relatively widely spaced values of x; numerical work for β = 0.72 has been carried out on the EDSAC and has been organized to produce the results in the required form directly, without interpolation.

Comparison of results obtained by using different Gauss quadrature formulae and different interval lengths show some interesting features of the errors of high-order integration formulae, and show that care is necessary in estimating these errors from the differences between quadratures carried out with intervals of different lengths.

Suppose that A is a Banach algebra and let A be the second dual algebra of A equipped with the first Arens product 3. In this paper we characterize compact and weakly compact multipliers of A, when A possesses a bounded approximate identity and is a two sided ideal in A. We use this to study the isomorphisms between second duals of various classes of Banach algebras satisfying the above properties.