A convolution formula is established for Bell polynomials. This is expressed in seven equivalent ways and used to derive further properties of these polynomials. The application of these results to some twenty-seven special polynomial sets is shown and illustrated in the case of binomial, Hermite, Gegenbauer and generalized Bernoulli sets.

The main purpose of the note is to compare necessary and sufficient conditions for weak ergodicity of finite inhomogeneous Markov chains given by Doeblin (3) and Hajnal (4), the former paper being little known; and more generally to expand on the nature and consequences of Doeblin's approach as compared to Hajnal's in some detail. A consequence is some insight into the relation between various ‘coefficients of ergodicity’.

Let S be n dimensional Euclidean space and let T be a division of S into cells. Assume that each cell must be either white or black at any time t. At time 0 the cell at the origin, α0, is black and all other cells are white. Let G be some stochastic growth process which tends to change white cells with black neighbours into black cells. Let C(t) be the black shape at time t. For a family, F, of such growth processes we prove the following theorem.

This paper deals with the influence of a vortex sheet separating two fluids in relative motion on the radiation from a point source of sound. Both the harmonic and impulsive sources are considered and it is found that waves due to Helmholtz instability must be included in order to ensure that there is no field before the source is excited. The instability wave is confined to a finite region and dominates other disturbances in that region. It is suggested that the instabifity wave is initiated by the unrestricted growth of the specularly reflected wave.

Let K be a closed, bounded, symmetric convex domain with centre at the origin O and gauge function F(x). By a homothetic translate of K with centre a and radius r we mean the set {x: F(x−a) ≤ r}. A family ℳ of homothetic translates of K is called a saturated family or a saturated system if (i) the infimum r of the radii of sets in ℳ is positive and (ii) every homothetic translate of K of radius r intersects some member of ℳ. For a saturated family ℳ of homothetic translates of K, let S denote the point-set union of the interiors of members of ℳ and S(l), the set S ∪ {x: F(x) ≤ l}. The lower density ρℳ(K) of the saturated system ℳ is defined by

where V(S(l)) denotes the Lebesgue measure of the set S(l). The problem is to find the greatest lower bound ρK of ρℳ(K) over all saturated systems ℳ of homothetic translates of K. In case K is a circle, Fejes Tóth(9) conjectured that

where ϑ(K) denotes the density of the thinnest coverings of the plane by translates of K. In part I, we state results already known in this direction. In part II, we prove that ρK = (¼) ϑ(K) when K is strictly convex and in part III, we prove that ρK = (¼) ϑ(K) for all symmetric convex domains.

The method of integral equations is the most familiar method of proving existence theorems for the Helmholtz equation of acoustics. The wave potentials are expressed as surface distributions of wave sources (for the Neumann problem) or wave dipoles (for the Dirichlet problem). By a wave source is meant the free-space wave source. The source and dipole strengths for the exterior potentials are found to be solutions of Fredholm integral equations of the second kind which are, however, singular at a certain discrete set of frequencies corresponding to eigensolutions of the interior problems. The existence of exterior solutions at the expected frequencies can still be shown, but the proof involves a detailed and complicated study of the interior solutions. It is physically evident that this difficulty arises from the method of solution and not from the nature of the problem.

In a Newtonian fluid, in which the body forces are conservative, in which the pressure is a function only of density and in which the kinematic viscosity v is uniform, the vorticity ω satisfies the equation

where u is the velocity field, so that ω = ▿ × u. This equation is normally derived by taking the curl of the Navier–Stokes equations. However, the vorticity has many interpretations in terms of the angular velocity of elements of fluid and it is natural to expect that (1) can be derived by equating the rate of change of the angular momentum of a small material volume element about its centre of mass with the moment of the forces acting on the element. Such a derivation is presented here in the hope that it may be of pedagogic interest.

Lagrangian coordinates are used in conjunction with the self-similarity hypothesis in order to examine the problem of the vertical entry at constant speed of a wedge and also of a cone into an incompressible fluid initially at rest. Certain known properties of the free surface are recovered in a very direct and simple manner and new exact results concerning the inclination of the free surface and the angle of contact with the rigid surface are obtained.

It is shown how the solution for the velocity potential may be determined when the effect of surface tension is included in the linearized theory of surface waves over a sloping beach. In particular, two independent standing wave solutions are found, both of which have finite amplitude at the shoreline. The results agree with those of previous writers when the surface tension force tends to zero.

A radiating point source is abruptly switched on and, thereupon, constantly maintained while immersed within a multi-dimensional, non-dispersive, originally uniform, unbounded, propagative medium with transverse isotropy, equivalent to an intrinsic axisymmetry. The number of spatial dimensions concerned is odd. The pertinent radiation problem is posed as a partial differential equation coupled with a preactivation zero condition. This problem is, first, resolved by a Fourier integral, which is then ‘frozen’ onto (i.e. evaluated at points on) the axis of symmetry. Postulating appropriate hyperbolic-type as well as elliptic-type conditions, a systematic reduction of the ‘frozen’ integral is carried out, enabling an analytic exact solution to be constructed with reference to certain phase curves in a phase plane. Away from the source, this solution is valid throughout the symmetry axis except at certain discrete, moving singularities. These fall into three different categories. The behaviour near each typical singularity is estimated. Three significant physical features are demonstrated (as corollaries); namely, the implicit compliance with a radiation principle, the linear interception of the symmetry axis by a Petrow sky's lacuna (associated with the instantaneous point impulse), and, the eventual attainment of either a steady or quiet state after an infinite period. Interpretations are provided in terms of the group velocity. To assist in probing into these various aspects, two appendices are separately included. An application is made to consider magnetically aligned, radiation reception within a field permeated, electrically conducting gas.

A unified variational approach to a class of second-order integral in-equalities is presented. A special case recently considered in a different manner by Anderson, Arthurs and Hall (1) is recovered.

Let p be a prime, and suppose that x1,…,xN are independent random variables which take the values 0, 1,…,p − 1 with probabilities s0, sl…,sp−1 where s0+…+sp−1 = 1 and 0 < sk < 1 for each k. PN(n) denotes the probability that the elementary symmetric function σr(x1,…,xN) = ∑x1…,xr of the rth degree in the variables x1,…,xN is congruent, modulo p, to a prescribed integer n.

Consider a queueing system M/G/s with the arrival intensity λ, the service time distribution function B(t) (B(0) < 1) having a finite mean and the waiting room size N ≤ ∞. If s < ∞ and N = ∞, we shall also assume that its relative traffic intensity is less than 1. Since the arrival process of this system is Poisson, it is immediate that in this case the distribution of the number of arrivals during an interval is infinitely divisible.

It is shown that the solutions of the field equations generated by the most general effective quadratic Lagrangian aR2 + bRklRkl which are static, singularity-free, of class C4 and asymptotically flat (of rank A3) are (i) conformally flat when 3a + b = 0, (ii) flat when 3a + b ≠ 0 (b ≠ 0), and (iii) have zero scalar curvature when b = 0.

In two previous papers (1, 2) representations of the unitary groups U4, U2 were found which described some of the properties of nucleons and electrons. In particular, the many electron wave functions were constructed from the irreducible representations of U2 restricted to the proper orthochronous Lorentz group Lp. In this paper the irreducible representations of U4 found in (1) will be shown to be also irreducible representations of the complete homogeneous Lorentz group L0 and the techniques of matrix contraction employed in (2) will be used to find the precise form of the matrices of the infinitesimal ring.