The kth Fitting ideal of the Alexander invariant B of an arrangement [Ascr ] of n complex hyperplanes defines a characteristic subvariety, Vk([Ascr ]), of the algebraic torus ([Copf ]*)n. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of Vk([Ascr ]). For any arrangement [Ascr ], we show that the tangent cone at the identity of this variety coincides with [Rscr ]1k(A), one of the cohomology support loci of the Orlik–Solomon algebra. Using work of Arapura [1], we conclude that all irreducible components of Vk([Ascr ]) which pass through the identity element of ([Copf ]*)n are combinatorially determined, and that [Rscr ]1k(A) is the union of a subspace arrangement in [Copf ]n, thereby resolving a conjecture of Falk [11]. We use these results to study the reflection arrangements associated to monomial groups.

Calculations have been made of the momentum distribution and the shape of the Compton profile for gaseous methane, using the Dirac transformation theory and three distinct approximations (molecular orbital, electron-pair and self-consistent-field) to the true molecular wave functions. The bearing of the results upon the validity of the original wave functions is discussed; it is concluded that the exponents in molecular wave functions should be approximately 1·1 times greater than in corresponding atomic wave functions. Comparison is made with experimental electron scattering results.

Our thanks are due to Prof. Massey for providing us with the numerical tables of the self-consistent-field for methane in advance of publication, and to Mr E. G: Phillips of the Mathematics Department of Univeristy College of North Wales, for the loan of a calculating machine.

The motion of a fluid due to the rotation of an immersed plane disc about an axis perpendicular to itself is turbulent if ωa2/ν is greater than about 105, where ω is the angular velocity of the disc, a its radius, and ν the kinematic viscosity of the fluid. The torque opposing the rotation, due to the surface friction of the fluid, was calculated originally by Kármán, both when the motion of the fluid is steady and when it is turbulent.

Both the Ising theory of ferromagnetism and the theory of regular solutions are concerned with systems arranged on a lattice and make the assumption that each system interacts only with its nearest neighbours. Mathematically, there is a close parallel between the two problems (see, for instance, Rushbrooke (1)). In the first half of this present paper the partition functions for these two problems are examined in some detail. Power series expansions of the partition function of the Ising model, valid for low and high temperatures, are obtained. The terms obtained in the power series have been analysed and approximate numerical results obtained. It is hoped to publish these in a second paper.

This paper is concerned with the linear theory of the transmission of sound through a vortex sheet separating two fluids in relative motion, but with the same density and sound speed. For harmonic excitation asolution is determined, with particular attention to transition regions where large effects might be expected, and it is found that Helmholtz instability plays no role in this solution. However, this harmonic field does not lead to a solution of the initial value problem which satisfies causality. When causality is complied with an additional field must be superimposed which gives waves growing exponentially in space in the harmonic case and a singularity, which is more severe than has been previously encountered, in the time-dependent problem. The consequent effects of this instability are discussed.

A fundamental and interesting question in knot theory is:

Question 1. How many prime knots of n crossings are there ?

Over time, knot theorists have answered this question for n ≤ 13 by the method of exhaustion: one writes down a list of all possible knots of n crossings, and then works hard to eliminate duplications from the list [12]. A perhaps easier question is the following:

Infinite particle systems on lattices have been extensively studied in recent years. The main questions of interest concern the ergodic and limiting behaviour of these processes, and their relationship with the dimension of the underlying lattice. A comprehensive review is given by Durrett(6).

One of the more tractable of these processes is the voter model introduced by Clifford and Sudbury(3) and much studied since, see for example the monograph by Griffeath(8), or the papers by Harris(11), Holley and Liggett(13), Bramson and Griffeath(1) and (2) or, for a more general approach, Kelly(16).

In this paper we consider the case where the underlying spatial structure is finite and examine the transient behaviour of the voter process and also the infection process introduced by Williams and Bjerknes(21).

An approximate solution of the two-dimensional problem of reflexion of plane harmonic P and S V waves by an irregular boundary is obtained using a modification of Rice's perturbation method of approximation on the assumption that the curvature of the surface is everywhere small. The case of a periodic surface is treated in more detail. It is found that the reflected waves are composed of specularly reflected waves and various diffracted waves, propagating in both horizontal directions if the wavelength of the incident waves is long compared with that of the surface. If the wavelength of incident distortional waves is long compared with that of the surface, the amplitudes of some of the scattered waves decrease exponentially with depth. In general the phases of the waves change on reflexion and the phase angles of the reflected waves are functions of the wavelength of the corrugation and the angle of incidence. It is verified, in the case of zero angle of incidence, that the energy going into the scattered radiation is obtained at the expense of the energy of the specularly reflected waves.

Almost every species is, to a first approximation, in genetic equilibrium; that is to say no very drastic changes are occurring rapidly in its composition. It is a necessary condition for equilibrium that all new genes which arise at all frequently by mutation should be disadvantageous, otherwise they will spread through the population. Now each of two or more genes may be disadvantageous, but all together may be advantageous. An example of such balance has been given by Gonsalez(1). He found that, in purple-eyed Drosophila melanogaster, arc wing or axillary speck (each due to a recessive gene) shortened life, but the two together lengthened it.

This note contains a partial generalization of the results of (3). We extend to arbitrary n and k the complete determination of the cases in which cup length is strictly less than the maximum possible value dim Gn, k(R) = nk. We also show that the Lusternik–Schnirelmann category cat Gn, k(R) < nk in the same cases.

In this paper, a study is made of the properties of the motion due to a general localized disturbance in an infinite fluid initially at rest. Two invariants of the motion are found, which are identified with the net linear momentum, or effective dipole strength, and the net angular momentum imparted to the fluid, and an equation is derived expressing the rate of change of the effective quadrupole strength of the motion. Solutions are obtained for the final period of decay of such a motion, and it is shown that, when the net linear momentum of the fluid is non-zero, the motion corresponds to a type of viscous vortex ring. The length scale increases with time t as [ν(t – t0)]½, and the total energy decays as [v(t–t0)]-1, where ν is the kinematic viscosity and t0 is a virtual time origin. If the net momentum is zero, the total energy decays as , and expressions are given for the velocity field in this case also.

It is shown that, if it is attained, the final period of any turbulent motion can be found from the vortex ring type of solution described above. The final period of homogeneous turbulence is clearly one particular example of such a motion. The final period of an axially symmetrical turbulent wake is treated in this way and is shown to consist generally of the superposition of two types of motion: (i) a mean streaming motion whose energy per unit length of the wake decreases as [ν(t – t0)]-1, and (ii) a superimposed fluctuating motion whose energy per unit length decays as [v(t–t0)]-1. Expressions are given for the distribution of mean-square vorticity and the energy spectrum tensor in the wake.

We consider an ideal problem of adsorption of single and double particles upon a solid surface which has its sites of accommodation regularly arranged, and by comparing the equilibrium properties obtained by Bethe's method with the ordinary statistical formulae, we obtain approximate expressions for:

(1) g(N, n, X), the number of ways of arranging n particles upon N sites of a lattice so that the number of neighbouring sites occupied by the particles is X.

(2) g2(N, n, X), the number of ways of arranging n double particles upon N sites so that each of the double particles takes up two adjacent sites and the number of neighbouring sites occupied by two different particles is X.

Both these expressions are found to agree with the exact values when the N sites lie on a straight line. When we use the first expression to construct the configurational partition functions of certain physical assemblies and expand them in powers of 1/kT, they are found to agree with the corresponding rigorous expressions as far as (1/kT)3, which is the highest power which we can find rigorously at present. With the help of the first expression, formal equations for superlattice formation in an alloy with the composition 1: 1 and equations for the separation into phases of regular liquids are given. Lastly we show that atoms and molecules in a regular liquid may dissociate or recombine suddenly accompanied by a latent heat. This is a new cooperative phenomenon, which may bear some resemblance to the melting process between the solid and liquid states.

1. The wave function of an atom containing many electrons has not yet been solved completely, even that of helium being as yet unknown. In the absence of a direct solution of the Schrödinger equation for the electrons in an atom, various attempts have been made to devise approximate methods of solution in particular cases. The particular case of helium, being the easiest, has received considerable attention and a number of approximate wave functions appropriate to the normal state have been constructed. These functions usually contain empirical constants which are adjusted to make the energy of the system a minimum. Zener has attempted the more ambitious programme of finding the wave functions of all the atoms in the first period of the Periodic Table (Lithium to Neon), and has made interesting discoveries as to the way in which the wave functions differ from atom to atom. This work also is based on the variation of parameters.

The validity of Huygens' principle in the sense of Hadamard's ‘minor premise’ is investigated for scalar wave equations on curved space-time. A new necessary condition for its validity in empty space-time is derived from Hadamard's necessary and sufficient condition using a covariant Taylor expansion in normal coordinates. A two component spinor calculus is then employed to show that this necessary condition implies that the plane wave space-times and Minkowski space are the only empty space-times on which the scalar wave equation satisfies Huygens' principle.

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 state of a real fluid is completely specified by its velocity, density, pressure and temperature fields. When the fluid is in turbulent flow, all these quantities fluctuate in a disordered manner. The method of space Fourier spectra is used to show that these field variables separate into two physically distinct groups, one corresponding to fluctuating acoustical waves, or random noise, and the other to fluctuating vorticity, or eddy turbulence. The corresponding decomposition of the spectral and correlation tensors in a homogeneous field of turbulence is given. The noise Fourier components are shown to be coupled to the eddy Fourier components only through the non-linear inertia terms in the dynamical equations of the fluid; whereas the former propagate as acoustical waves, the wave character of the latter is due entirely to the mean motion of the fluid. The measurement of the noise component, its attenuation through absorption by walls and its effects on the eddy component are discussed. Finally, the dynamical equations for the eddy component of the velocity spectral tensor in a homogeneous field of turbulence are compared with the corresponding equations for an incompressible fluid.

Let x be a normally distributed variable, which may without loss of generality be measured from the mean of the distribution, so that E(x) = 0, E denoting mathematical expectation. Then x satisfies the differential relation

where k2 (otherwise σ2 or (2h2)−1) is the semi-invariant of order 2. Also k1 = 0, and we know that kr = 0 for r > 2.

For a long time there has been some interest in obtaining spherical harmonics with the symmetry of the regular polyhedrons, particularly for electrostatic problema involving polyhedral conductors. Work on the icosahedral group has been done, among others, by Meyer (7), Laporte (6), Hodgkinson (3) and Poole (8), apart from the classic work of Klein (4). In the several approaches of these authors, only spherical harmonics for the totally symmetric representation were obtained, the most complete table being that of Laporte who obtained the spherical harmonics up to l = 21. New interest in the icosahedral group has arisen in connexion with the structureof some proteins (5), and we obtain here, by the recently developed methodof Altmann(1), expansions in spherical harmonics for ali the representations of this group. This has been done up to and including l = 14. (For the totally symmetric representation we have also included l = 15.) In what follows we shall use Altmann's formulae and notation.

An orbit of the shift σ: t ↦ 2t on the circle = ℝ/ℤ is ordered if and only if it is contained in a semi-circle Cμ = [μ, μ+½]. We investigate the ‘devil's staircase’ associating to each μ ε the rotation number ν of the unique minimal closed σ-invariant set contained in Cμ; we present algorithms for μ in terms of ν, and we prove (after Douady) that if ν is irrational then μ is transcendental. We apply some of this analysis to questions concerning the square root map, and mode-locking for families of circle maps, we generalize our algorithms to orbits of the shift having ‘sequences of rotation numbers’, and we conclude with a characterization of all orders of points around realizable by orbits of σ.

1. The general theory underlying sampling inspection methods introduced during the war under the name of sequential analysis, applicable in cases where the sampling units can be taken serially, has mainly been developed by Wald and is now published (5). The original purpose of the present investigation was to exhibit the main structure of the distributional theory relating to the size of sample required by noting its relationship with the classical ‘random-walk’ problem. Thus in Part I of this paper are derived the distribution and characteristic functions of the absorption time for one-dimensional random-walk theory with constant drift and absorbing barriers. Attention is confined to the asymptotic case of numerous independent displacements, for which it is well known that the unrestricted total displacement becomes Gaussian; the motion may also be treated as continuous. Since these results were obtained Wald has independently given an alternative and more general discussion of this problem((4); see also Tweedie (3)), but the extension here of the method of images used by Chandrasekhar (1) still appears of interest, and has advantages over alternative direct methods of solution for the distribution function involving Fourier expansion (cf. 2).