Necessary and sufficient conditions for an ordered semigroup to admit an isotone homomorphic Boolean image are given together with a complete description of how all such images are obtained. Also discussed are the situations arising from a strengthening of the notion of a homomorphism.

In an attempt to evaluate the integral (5) below, using a decomposition of an orthogonal matrix (Jack 1968), the author is led to define a set of polynomials, one for each partition of an integer k, which are invariant under the orthogonal group and which depend on a real parameter α. An explicit representation of these polynomials is given in an operational form. When α = − 1, these polynomials coincide with the augmented monomial symmetric functions. When α = 1, a systematic way of taking linear combinations of these polynomials is explained and it is shown that the resulting polynomials coincide with the Schur functions from the representation theory of the symmetric group. A similar procedure in the case α = 2 then appears to give the zonal polynomials as defined by James (1964, p. 478).

A new approach to the analysis of transport processes in a turbulent fluid is presented In this approach a model is used to represent the detailed fluid behaviour and it is shown that the model has similarities with a Fourier integral representation of the flow field. The model assumes that the turbulent motions can be represented by fluid entities of random size, shape and velocity, and that the large-scale transport processes are the consequences of the creation, decay and mutual interaction of the individual entities. The effects of this are analysed and it is shown how the diffusivities for vector and scalar quantities can be determined in terms of properties of the turbulence. The theory is applied in both bounded and free turbulent flows and it is shown that the predicted diffusivity ratios compare favourably with experimental data. Relaxation phenomena are also investigated and the memory function for stress and thermal relaxation is determined. It is concluded that the model provides a most useful framework for the analysis of turbulence phenomena and that its diverse and accurate predictions make it a powerful research tool.

A study is undertaken of the field excited in a semi-infinite cylindrical dielectric rod having small conductivity when plane harmonic electromagnetic waves are incident obliquely upon its end. All back-scattering and the complicated edge effects due to the end of the rod are neglected. Expressions for the normalized power absorbed in the rod are obtained by an approximation technique from the corresponding results for a similar rod whose conductivity is zero, assuming that the radiation through the walls of the rod may be ignored. Selected numerical results are presented and the relevance of these to models of a retinal rod of the human eye is discussed.

In a recent paper, Jyoti Chaudhuri and W. N. Everitt linked the spectral properties of certain second order ordinary differential operators with the analytic properties of the solutions of the corresponding differential equations. This paper considers similar properties of the spectrum of the corresponding partial differential operators.

The number of different connected graphs (with some property P) on n labelled nodes with q edges is fnq. Again Fnq is the number of graphs on n labelled nodes with q edges, each of whose connected components has property P. We consider 8 types of graph for which . We use a known relation between the generating functions of fnq and Fnq to find an asymptotic expansion of fnq in terms of binomial coefficients, valid if (q – ½n log n)/n→∞ as n→∞. This condition is also necessary for the existence of an asymptotic expansion of this kind.

The acyl groups in aromatic ketones with hydroxyl or methoxyl groups in the ortho or para positions are readily removed by hydrobromic acid in acetic acid. The results are correlated with the fine structure of the parent hydrocarbons. 5,8-Dimethoxy-I-tetralone and 1,3-dihydroxyxanthone fail to undergo ring-fission with the reagent. The mechanism is discussed and it seems probable that fission occurs by direct proton attack at nucleophilic carbon atoms bearing the acyl groups.

The solution of certain integral equations, which are related to the Hilbert transform, is discussed from the viewpoint of generalized functions. The behaviour of the corresponding transforms in the complex plane is examined for generalized functions which effectively vanish at infinity. A space of generalized functions which need not be small at infinity is introduced and the integral equations solved in this space. Various possible extensions of the theory are also mentioned.

It is shown that the essential condition, energy-wise, for the emission of a light particle from a binary fission fragment, in the immediate post-scission stage, is that the fragment deformation must be capable of sudden partial collapse so as to release energy equal to the binding energy of the particle in the undeformed (unexcited) fragment together with an amount cancelling the mutual (electrostatic) potential energy of the particle and the residual deformed (excited) fragment at the instant of separation. On this basis the mean energy required for alpha-particle release in the thermal-neutron-induced ternary fission of 235U is estimated as 16·5 MeV, and it is concluded that those binary fissions which develop into alpha-particle-accompanied ternary fissions in this case are those in which, on the average, the total deformation-excitation energy of the fragments is some 10 MeV greater than the most probable value for all binary fissions. A qualitative discussion of the relative yields of other light-particle-accompanied modes is also given.

An Appendix contains a summary of comparable calculations based on the one-stage view of the ternary fission process.

When the Helmholtz equation ∇2V + k2V = o is separated in the general paraboloidal coordinate system, the three ordinary differential equations obtained each take, after a suitable change of variable, the form of the Whittaker Hill equation. For the case k2 < o, a considerable amount is known about the periodic solutions of this equation. The theory for k2 < o does not carry over immediately to the case k2 < o, and so far only perturbation solutions have been obtained. This paper gives, in sections 1–5, explicit solutions for the case k2 < o, in the form of trigonometric series determined by three term recurrence relations. In sections 6, 7 some important relations and orthogonality properties are discussed, which are of particular significance in respect of application to boundary value problems. Section 8 discusses some degenerate cases, and in the Appendix an important property is established of continuity of the solutions with respect to the parameters of the equation.

In this paper the response of an Euler-Bernoulli viscoelastic beam to impulsive excitation is obtained using Volterra's model for the stress-strain relationship. In order to achieve a better approximation of actual materials a series of parallel connections of one or more basic models must be used. However, the analytic solutions to most problems then become very difficult. Therefore an alternative approach is to formulate such a problem in terms of the hereditary integral as proposed by Vito Volterra [1].

A hemisphere, resting on a horizontal plane, is initially at rest relative to an incompressible, inviscid, non-diffusive fluid whose density is vertically stratified. The hemisphere is then given, impulsively, a small constant horizontal velocity which is maintained thereafter. Assuming that the Froude number is small, and using the Boussinesq approximation, the equations of motion are linearised and solved using a Laplace transform. The disturbance in the fluid is analysed for large times and is found to contain a steady component of purely horizontal flow, an internal wave field and internal oscillations at the Brunt-Väisälä frequency, together with their various interactions. The effects of viscosity and diffusivity are discussed qualitatively by considering their effects on an internal wave.

In general the terminology and notation of [1] is used throughout. A correspondence for topological spaces is a triple f: P → Q where P and Q are topological spaces and f is a subset of P × Q, the graph of f: P → Q. A correspondence f: P → Q will be called graph-compact, or graph-closed, or graph-Souslin, or graph-analytic if f is, respectively, compact or closed or Souslin or analytic in P × Q.