In many applied probability models, one is concerned with a sequence {Xn: n > 1} of independent random variables (r.v.'s) with a common distribution function (d.f.), F say. When making statistical inferences within such a model, one frequently must do so on the basis of observations X1, X2,…, XN where the sample size N is a r.v. For example, N might be the number of observations that it was possible to take within a given period of time or within a fixed cost of experimentation. In cases such as these it is not uncommon for statisticians to use fixed-sample-size techniques, even though the random sample size, N, is not independent of the sample. It is therefore important to investigate the operating characteristics of these techniques under random sample sizes. Much work has been done since 1952 on this problem for techniques based on the sum, X1 + … + XN (see, for example, the references in (3)). Also, for techniques based on max(X1, X2, …, XN), results have been obtained independently by Barndorff-Nielsen(2) and Lamperti(9).

Some time ago, S. Bochner gave an interesting analysis of certain positive operators which are associated with the ultraspherical polynomials (1,2). Let {Pn(x)} denote these polynomials, which are orthogonal on [ − 1, 1 ] with respect to the measure

and which are normalized by settigng Pn(1) = 1. (The fixed parameter γ will not be explicitly shown.) A sequence t = {tn} of real numbers is said to be ‘positive definite’, which we will indicate by writing , provided that

Here the coefficients an are real, and the prime on the summation sign means that only a finite number of terms are different from 0. This condition can be rephrased by considering the set of linear operators on the space of real polynomials which have diagonal matrices with respect to the basis {Pn(x)}, and noting that

A new class of axi-symmetric stationary solutions of Einstein's empty space field equations is obtained. Non-existence of solutions of certain other classes is proved.

In the two-dimensional theory of diffraction by smooth curves there are certain canonical problems that can be solved explicitly in series form. The series converge slowly at short wavelengths but they can be transformed by Watson's transformation into another form (the residue series or creeping-mode expansion) which has been much used in shadow regions. It is found that throughout the shadow region the first few terms of the residue series are exponentially small and decrease rapidly, and these have often been used as an estimate of the wave potential without further justification. Leppington's recent work on the shadow of an ellipse has shown, however, that in part of the shadow some of the later terms of the residue series are exponentially large. In other words, the complete residue series in part of the shadow is even more slowly convergent than the original series.

In the present paper the Watson transformation is re-examined in the light of this result. The original series is expressed as the sum of a finite number of terms of the residue series and of a remainder. It is shown that throughout the shadow the remainder is at short wavelengths asymptotically of smaller order than the last term retained in the finite residue series. It follows that the residue series (which is a convergent infinite series) is also an asymptotic series, and this fact is sufficient to justify most of the usual applications. The proof is given in detail for the circle and in outline for the ellipse; it makes use of the theory of conformal mapping.

Recently, through a repeated application of one-dimensional finite integral transforms, Cinelli(1) gave a solution for the temperature distribution in a hollow circular cylinder of finite length. Since no new ideas or techniques are introduced, the extension claimed in (1) with regard to the finite Hankel transform technique employed in the transformation of the radial space variable in the hollow cylinder problem is trivial, in view of well-known works by Sneddon(2) and Tranter (3), to mention a few. The list of the finite Hankel transforms given in (1) for a variety of boundary conditions at r = a and r = b is the result of routine, algebraic manipulations well known from the general theory of eigenvalue problems specialized for the hollow cylinder. In this list a set of seemingly different series expansions is given for the inverse Hankel transform for each combination of boundary conditions at the two radial surfaces. In each case, the two expressions for inversion can readily be shown to be identical to each other when use is made of the frequency equation. One of the inversion forms is therefore unnecessary once the other is given. Furthermore, the general solution as given by equation (54) of Cinelli(1)does not satisfy his boundary conditions (27), (28), (29) and (30), unless these latter are homogeneous.

The electron avalanche that takes place within an internally amplifying semiconductor radiation detector has been analysed statistically, treating the process as an example of multi-dimensional branching processes. By extending the result obtained from Good's theorem to the case of continuous variable, an integral equation is derived for the generating function of the probability that a single primary electron-hole pair, formed at a depth x, induces an avalanche consisting of n pairs. On the basis of this equation, expressions are deduced for the mean value and the standard deviation σ of the total number n of pairs in an avalanche induced by n0 primary pairs produced at x, as well as the probability w(x, n0, n) of getting an avalanche of ‘size’ n. In particular cases these expressions lead to various simple relations of practical importance.

Eringen's theory(1) of homogeneous, isotropic, hyperelastic dielectric is applied to consider the effects of polarization on the surface tractions when a hollow cylindrical reinforced dielectric carrying a uniform surface charge at its inner surface is subjected to simultaneous large extension, inflation and torsion. New effects due to reinforcement and polarization have been observed separately. For vanishing charge, the results reduce to those of classical finite strain theory (2).

Following Heading (4) a systematic search is made for useful refractive index profiles in spherically stratified isotropic media which are capable of yielding wave functions expressed in terms of the standard transcendental functions. Previously known profiles are shown to emerge as special cases of the analysis. Subject to the criteria adopted all possible profiles are derived.

The problem of the diffraction of axisymmetric harmonic elastic waves by a rigid circular disc is reduced to finding the solution of a pair of integral equations of the second kind suitable for iteration at low frequencies. Using the principle of contraction mapping, the rate of convergence of the iteration procedure is discussed and the error caused by stopping at any particular stage estimated. Detailed calculations are given for plane compressional waves at normal incidence.