In [4] Halmos considers the following situation. Let be a class of distribution functions over a given (Borel) subset E of the real line, and F a function over . He investigates which functions F admit estimates that are unbiased over and what are all possible such estimates for any given F. In particular he shows that on the basis of a sample (of size n) one can always obtain an estimate of the first moment which is unbiased in and that the central moments Fm of order m ≧ 2 have estimates which are unbiased in if and only if n ≧ m, provided satisfies the following properties: Fm exists and is finite for all distributions in and includes all distributions which assign probability one to a finite number of points of E. Halmos also finds that symmetric estimates which are unbiased on are unique1 and have smaller variances on than unsymmetric unbiased estimates.

Let (G) denote the lattice of all subgroups of a group G. By an -isomorphism (lattice isomorphism) of G onto a group H, we mean an isomorphism of (G) onto (H). By an -isomorphism (normaliser preserving -isomorphism) of G onto H, we mean an -isomorphism ø such that (Aø) = (A)ø for all A ∈ (G). In this paper, we study certain properties of groups which remain invariant under -isomorphisms.

Let {xt} (t = 0, ±1, ±2 …) be a stationary non-deterministic time series with E(x2t) < ∞, E(xt) = 0, and let its spectrum be continuous (strictly absolutely continuous) so that the spectral distribution function is the spectral density function. It is well known that {xt} then has a unique one-sided movingaverage representation where .

J. Gani and G. F. Yeo ([2] and [3]) have recently investigated certain age-distributions associated with a replacement model which serves, in particular, as a model for phage reproduction. In this paper, the characteristic functional for this model will be obtained explicitly.

Let L be a finite dimensional Lie algebra over the Field F. We denote by (L) the lattice of all subalgebras of L. By a lattice isomorphiusm (whicn we abbrevite to -isomorphism) of L onto a Lie algebra M over the same field F, we mean an isomorphism of (L) onto (M). It is possible for non-isomorphic Lie algebras to -isomorphic, for example, the algebra of real vectors with product the vector product is -isomorphic to any 2-dimensional Lie algebra over the field of real numbers.

In this paper results from Fluctuation Theory are used to analyse the imbedded Markov chains of two single server bulk-queueing systems, (i)with Poisson arrivals and arbitrary service time distribution and (ii) with arbitrary inter-arrival time distribution and negative exponential service time. The discrete time transition probailities and the equilibrium behaviour of the queue lengths of the systems have been obtained along with distributions concerning the busy periods. From the general results several special cases have been derived.

The functions where ξ > 0, are essentially Fourier convolutions of Gaussian and Cauchy type distributions, and have thus found application in many fields. In reactor theory, and no doubt elsewhere, the need has arisen to analyse complicated integrals and functional equations involving ψ and φ, the so-called Voigt profiles.

Felix Behrend was born at Berlin-Charlottenburg, Germany, on 23 April, 1911, the eldest of four children of Dr. Felix W. Behrend and his wife Maria, nee Zöllner. Felix Behrend senior was a mathematics and physics master at the Herderschule, a noted “Reform-Realgymnasium” in one of the western suburbs of Berlin; he was a widely known educationalist, and later headmaster of an important school elsewhere in Berlin, until demoted and finally dismissed by the Nazis, partly because of some Jewish ancestry, partly because of his liberal political views.

The paper considers the queueing system GI/G/1 with a type of customer impatience, namely, that the total queueing-time is uniformly limited. Using Lindiley's approach [10], an integral equation for the limiting waiting- time distribution is derived, and this is solved explicitly for M/G/1 using an expansion of the Pollaczek-Khintchine formula. It is also solved, in principle for Ej/G/l, and explicitly for Ej/Ek/l. A duality noted between GIA(x)/GB(x)/l and GIB(x)/GA(x)/l relates solutions for GI/Ek/l to Ek/G/l. Finally the equation for the busy period in GI/G/l is derived and related to the no-customer-loss distribution and dual distributions.

Kendall [4] has given for the distribution of the time to first emptiness in a store with an input process which is homogeneous and has non-negative independent increments and an output of one unit per unit time the formula . In this formula, z is the initial content of the store, g(t, z) is the density function of the time to first emptiness τ(z), defined by and k(t, x) is the density function of the input process ξ(t), defined by .

A great deal of attention has been given in the literature to the various properties of the simple binomial random walk. Explicit expressions are available for first passage times, absorption probabilities, average duration of the walk up to absorption and other quantities of interest. One aspect of the behaviour of this work which has, however, attracted little attention is the form of the distribution of occupation totals. This paper is devoted to the derivation of an explict expression for the joint probility generating function of the occupation totals up to absorption, for the binomial random walk in the presence of two absorbing points. The appropriate marginal form of this p.g.f. yields the distribution of the occupation total, and expected occupation total, at any particular lattice point. The limiting forms of these results provide explicit expressions for the corresponding quatities in the case of a binomial random walk having a single absorbing point and, where relevent, in the case of the unrestricted binomial random walk.