We shall be concerned with the behaviour of the fractional iterates of analytic functions which have a fixpoint ζ with multiplier 11. The general form of such a function is If ζ is finite, if ζ = ∞.

An analytic function f(z) is said to have a fixpoint ξ ≠ ∞ of multiplier 1 if f(ξ) = ξ, f'(ξ) = 1. The function then has an expansion .

In the present paper we are concerned with Schröder's functional equation , where ϕ(x) is the unknown function and is a number between 0 and 1: . We shall prove a theorem which generalizes some earlier results on convex solutions of the Schröder equation [4], [5].

Let be a finite group, a field. A twisted group algebra A() on over is an associative algebra whose elements are the formal linear combinations and in which the product (A)(B) is a non-zero multiple of (AB), where AB is the group product of A, B ∈: . One gets the ordinary group algebra () by taking each fA, B ≠ 1.

A quasi-permutation group of degree n was defined in [3] to be a finite group with a faithful representation of degree n whose character has only non-negative rational integral values. If G is such a group, then the following simple properties of permutation groups of degree n were proved to hold also for G:

(i) the order of G is a divisor of the order of the symmetric group Sn of degree n; and (ii) if G is a p-group and n < p2, then G has exponent at most p and derived length at most 1 (i.e. G is elementary Abelian).

An (explicit) Runge-Kutta process is a means of numerically solving the differential equation , at the point x = x0+h, where y, f may be vectors.

By an ∮-related family ∮ we mean a non-empty family ∮ of elements such that to each element F ∈ ∮ is associated a set R(F) of elements of ∮, called the R-class of F, which contains F. An element G ∈ R(F) is said to be R-related to F. By an R-section S of ∮ we mean a set of elements of ∮ such that for any elements F1, F2 of S either F1 ∈ R(F2) or F2 ∈R(F1). If R(F) = {F} for each F ∈ ∮ then the only R-Sections are the sets {F}. The interesting applications of the lemma proved below are to those cases when there exist R-sections which do not contain a finite number of elements.

Let an abstract space of points x and let ℳ be a σ-field of subsets of , that is a class of subsets of such that (i) ∈ ℳ, (ii) if M ∈ ℳ then — M ∈ ℳ and (iii) if {Mj} is any sequence of elements of ℳ then UjMj ∈ ℳ.

Let Xi, i = 1, 2, 3,··· be a sequence of independent and identically distributed random variables and write Sn = X1+X2+…+Xn. If the mean of Xi is finite and positive, we have Pr(Sn ≦ x) → 0 as n → ∞ for all x1 – ∞ < x < ∞ using the weak law of large numbers. It is our purpose in this paper to study the rate of convergence of Pr(Sn ≦ x) to zero. Necessary and sufficient conditions are established for the convergence of the two series where k is a non-negative integer, and where r > 0. These conditions are applied to some first passage problems for sums of random variables. The former is also used in correcting a queueing Theorem of Finch [4].

Let X1, X2,…Xn, … be independent and identically distributed random variables, and write . In [2] Chung and Fuchs have established necessary and sufficient conditions for the random walk {Zn} to be recurrent, i.e. for Zn to return infinitely often to every neighbourhood of the origin. The object of this paper is to obtain similar results for the corresponding process in continuous time.

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.

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.

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.

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.