In the classical theory of representations of a finite group by matrices over a field , the concept of the group algebra (group ring) over is of fundamental importance. The chief property of such an algebra is that it is semi-simple, provided that the characteristic of is zero or a prime not dividing the order of the group. As a consequence of this, the representations of the algebra, and hence of the group, are completely reducible.

In most theories for the construction of finite groups with given properties a major difficulty is the ‘isomorphism problem’, which consists of specifying how one representative of each class of isomorphic groups may be selected from the totality of groups constructed by the process laid down. To do this we need a practical criterion for the isomorphism of two constructed groups. The main object of the present paper is to establish such a criterion in a particular case, which in spite of its simplicity is important because it gives a method for the construction of A-groups (i.e. soluble groups whose Sylow subgroups are all Abelian). All soluble groups of cube-free order are included in this class of groups, and to exemplify the application of the criterion we summarize that part of our unpublished dissertation (7) which deals in detail with the groups of order 22.32.52.

It is well known that a characteristically-simple finite group, that is, a group having no characteristic subgroup other than itself and the identity subgroup, must be either simple or the direct product of a number of isomorphic simple groups. It was suggested to the author by Prof. Hall that finite groups possessing exactly one proper characteristic subgroup would repay attention. We shall call a finite group having a unique proper characteristic subgroup a ‘UCS group’. In the present paper we first give some results on direct products of isomorphic UCS groups, and then we consider in more detail one of the types of UCS groups which can exist, that consisting of groups whose orders are divisible by exactly two distinct primes.

In general, the class of a nilpotent linear algebra of dimension n is at most n + 1, and the index, or derived length, of a solvable linear algebra of dimension n is at most n. In this note it is shown that, for a nilpotent linear algebra of dimension n satisfying x2 = 0 for all x, the class is at most n; and bounds are obtained for the indices of solvable Lie algebras.

The method of solving equations by iteration is very old and is discussed in many well-known books. But conditions for its validity have never been properly formulated. In the first place, it is necessary to know that the method will not carry us outside the domain of definition of our functions, and that the ‘approximations’ will not converge to something which is not a solution of the equations. These difficulties are easily forestalled; it is more difficult to ensure that the process really will converge. Our object will be to find the most general conditions under which we can set off with the certainty that we will ultimately arrive at a root, in the case of one equation with one unknown.

1·1. Throughout this paper, all functions considered are assumed to be measurable, and an equation between two functions indicates equality almost everywhere. When the space Lp(a, b) is considered, the interval (a, b) may be finite, semi-infinite, or infinite.

1. In the theory of conformal mapping, there are many problems associated with domains of infinite connectivity which have not yet been fully explored. We consider, in this paper, some of these problems in relation to domains in the closed complex plane. In the following all domains are assumed to be plane domains and all mappings to be (1, 1) and conformal in the domains in which they are defined.

In his paper ‘Some Tauberian theorems connected with the prime number theorem’, Ingham(10) discusses the method of summation of the series defined by

1. Kepler's equation

has, when θ, x are real with 0 < x < 1, just one real root ξ = ξ*(θ, x). For this root there are the well-known formulae, dating from Lagrange and Bessel,

The objects of this paper are (i) to obtain analogous formulae for the unreal roots ξ of the equation, and (ii) to sum the conjugates (as Fourier series in θ) of the series on the right of (2) and (3).

In a previous paper (4) central limit theorems were obtained for sequences of m-dependent random variables (r.v.'s) asymptotically stationary to second order, the sufficient conditions being akin to the Lindeberg condition (3). In this paper similar theorems are obtained for sequences of m-dependent r.v.'s with bounded variances and with the property that for large n, where s′n is the standard deviation of the nth partial sum of the sequence. The same basic ideas as in (4) are used, but the proofs have been simplified. The results of this paper are examined in relation to earlier ones of Hoeffding and Robbins(5) and of the author (4). The cases of identically distributed r.v.'s and of vector r.v.'s are mentioned.

Consider a Markoff chain (by which is meant a stochastic process defined for discrete values of both variable and parameter, and whose probability dependence does not extend to more than a unit interval) with completely specified transition probability matrix P = (pij), where pij is the conditional probability that the (r + 1)th observation belongs to the state Ei, given that the rth observation belongs to the state Ei for r≥ I. We assume that there are a states E1, E2, …, Ea, where a is finite. We also assume an ergodic property for the stochastic process, that is, we consider non-periodic chains for which all the possible initial states remain permanently available. This defines what has been called the positively regular case. Also under the above assumption the moduli of all the latent roots λr of the transition probability matrix P = (pij) other than the first simple latent root λ1 = 1 are less than one (see Bartlett(1)).

Eigenvalues and eigenvectors of the total Hamiltonian in field theory are determined by the method of direct diagonalization, using canonical transformations. The difficulties of this method are discussed in detail.

The S matrix is expressed in terms of momentum-space operators and its Cayley transform exhibited. A new graphical formalism called O-graphs is developed in which the vertices are temporally ordered and the lines represent real particles. This is used to prove a conjecture of Gupta concerning the relation of real and virtual processes to the Cayley transform. The transformation operator for a semi-infinite time interval is also considered and some of its properties demonstrated. In conclusion, another graphical formalism with temporally ordered vertices is developed and renormalized.

The usual formula for the rate of loss of energy by a fast particle contains a factor z2, the square of the charge, which might suggest that He+ should lose energy at a quarter of the rate that He++ does. It is shown that when Born's approximation is applicable the energy-loss is reduced instead by a factor of about ⅝.

There has recently been some interest in atomic wave functions for the Mn+2 ion in two different physical contexts, namely neutron diffraction (Shull, Straussen and Wollan (12), and Halpern (3)) and luminescence in solids (Williams (14)). In the neutron diffraction context the particular substance concerned is MnF2, for which the nuclear scattering is small so that the main contribution to the diffraction of slow neutrons is through the magnetic scattering by the incomplete (3d)5 group of the Mn+2 ion.

The Feynman probability amplitudes for interactions between spin ½ fields and fields of integral spin quantized according to the generalized method of Green are investigated. It is shown that the scattering amplitudes are independent of the method of quantization though the amplitudes of the vacuum fluctuations are not. The question of distinguishing particles satisfying the generalized statistics, if they exist in nature, from Fermi or Bose particles is considered. It is shown that the statistical thermodynamics of the particles depends on the statistics, but as this requires a large number of particles in thermodynamical equilibrium, it offers no means of distinction for particles with short lifetimes. In practice, an examination of particles in bound states would identify particles obeying generalized statistics.

The consideration of the statistical aspect of the second law of thermodynamics of a previous paper is extended. The influence of fluctuations in the wind-wood model of the Ehrenfests is investigated, and the relative occurrence, mean life and time of recurrence of such fluctuations are calculated. These are found to agree with the results obtained for the simpler models considered previously.

In 1897 Rayleigh(14) pointed out that it is possible to obtain a family of solutions to certain problems in diffraction theory, notably those involving plane obstacles, by the simple operation of differentiation of some one solution. This, of course, alters the nature of the solution, especially so in the neighbourhood of a sharp edge. For, at an edge the parent solution may be finite and yet its derivative can become infinite. Further, such a differentiated solution may produce an infinite total energy in the neighbourhood of the edge. It is, therefore, natural to ask what conditions are required to make the solutions of these problems physically and mathematically acceptable and thus define the boundary-value problem uniquely.

This paper is devoted to a consideration of the following problem: Given a static state in which an incompressible viscous fluid is arranged in horizontal strata and the density is a function of the vertical coordinate only, to determine the initial manner of development of an infinitesimal disturbance. The mathematical problem is reduced to one in characteristic values in a fourth-order differential equation and a variational principle characterizing the solution is enunciated. The particular case of two uniform fluids of different densities (but the same kinematic viscosity) separated by a horizontal boundary is considered in some detail. The mode of maximum instability in case the upper fluid is more dense and the manner of decay in case the lower fluid is more dense are determined; and the results of the calculations are illustrated graphically. Gravity waves (obtained in the limit when the density of the upper fluid is zero) are also treated.

The equilibrium of a layer of a viscous incompressible fluid of density ρ variable in the vertical (z) direction, the fluid extending indefinitely in the horizontal directions and being confined between rigid horizontal boundaries, is examined by the usual method of studying the initial behaviour of a small disturbance. Diffusion effects are ignored. Chandra-sekhar has shown quite generally that a variational procedure for solving the resulting characteristic value problem is admissible, suggesting a method whereby an approximate solution can be obtained analytically. In this paper, by making use of appropriate approximate methods of solution, three types of density configuration, ρ(z), are studied. The first corresponds to two superposed fluids of great depth, the upper and lower fluids haying densities and coefficients of viscosity ρ2 and μ2, and ρ1 and μ1, respectively. Chandrasekhar has given an exact numerical solution to this problem for the case where μ2/ρ2 = μ1/ρ1 The results of the approximate theory, by which not all the boundary conditions can be satisfied, and the exact theory are in good agreement. Expressions for the wave-length and growth rate of the mode of maximum instability in the unstable case (ρ2 > ρ1) are found. The stable case (ρ2 > ρ1) is also discussed in detail; the formulae obtained for the phase and group velocities of gravity waves in deep viscous fluids are especially interesting. There is an upper limit to the natural frequency of oscillation of the system which would be well within the range of experimental investigation.

The second problem considered is that of two superposed fluids of small depth. Formulae for the phase and group velocities of ‘long gravity waves’ (which are not equal when viscous effects are included) are obtained.

Finally, a continuously stratified fluid of finite depth d in which ρ(z) = ρ0eβz is investigated. The character of the equilibrium depends on a Grashof number G = (gβd4)/(π4ν2) where g is the acceleration of gravity and ν is the coefficient of kinematical viscosity, assumed for simplicity to be constant. Expressions for the wave-length and growth rate of the mode of maximum instability in the unstable case (G > 0) are obtained. In the stable case (G < 0) all modes are aperiodically damped if – G < . Otherwise, the possibility of a disturbance being propagated as a horizontal wave does arise, but only within a limited wave-length range.