In this paper models for diffusion in one dimension are obtained which are based on correlated random walks. The equations for diffusion with drift can be transformed into the equations for diffusion without drift (and conversely) by the transformations of Special Relativity Theory. The relationship of these equations to Maxwell's equations for electromagnetic phenomena is discussed.

The restoration of crystal lattices which have been jittered presents some problems that do not seem to have been previously studied. This is true especially in the multi-dimensional case. It is shown that a statistic suggested by Bartlett can be used for this purpose, and this is done both for Gaussian and general deformation mechanisms. The results are accompanied by some numerical experiments.

Representing a rotation in three dimensions by a unit tensor quaternion and supposing that the errors in the measurements of a number of vector directions follow Fisher's distribution, the maximum likelihood estimator of a rotation is obtained. It is shown that the natural Haar measure for three-dimensional rotations is mapped 1:2 onto the natural Haar measure for random directions in 4-space. The natural Haar measure for rotations in 4-space is also mappable onto the product measure of the Haar measures for two separate random directions in 4-space.

The trajectory of a car traveling at a constant speed on an idealized infinite highway can be viewed as a straight line in the time-space plane. Entry times are governed by a Poisson process with intensity parameter A leading to all trajectories as random lines in a plane. The Poisson distribution of number of encounters of cars on the highway is developed through random line models and non-homogeneous Poisson fields, and its parameter, which depends on the specific random measure employed, is obtained explicitly.

This paper studies a Markov chain model for type counts {Xn} in a literary text. First, a homogeneous Markov chain in discrete time is considered. This is then embedded in a continuous time Poisson process; the probability generating function for the resulting continuous time Markov chain is obtained. Expectations and variances of type counts are found for different values of the token count and various sizes M of an author's vocabulary; these results are finally tested against known data for three of Shakespeare's plays.

After a general review of symmetric reversibility for countable-state continuous-time Markov chains the author shows that the birth-death-and-immigration process is symmetrically reversible and further that it remains so even when the description of the present state is refined to include a list of the sizes of all ‘families’ alive at the epoch in question. This result can be useful in genealogy because the operational direction of time there is the negative one. In view of the symmetric reversibility, some of the questions which face the genealogist can be answered without further calculation by quoting known results for the process with the usual (‘forward’ instead of ‘backward’) direction of time.

Further topics discussed include social mobility matrices, surname statistics, and Colin Rogers' ‘problem of the Spruces’.

The use of the cross-ratio as a measure of association in a 2×2 table is closely related to Bartlett's (1935) definition of interaction in a higher-order table. Inference about aetiological associations from case-control studies is most naturally done in terms of the cross-ratio, as a measure of relative risk. Standard methods of statistical analysis, for the comparison and combination of relative risks and for matched pairs, are reviewed, and some new results noted.

Maurice Bartlett's work on the mathematical theory of epidemics, recurrent epidemics and endemicity over the years 1953–66 has helped to stimulate a wide range of applied studies of practical importance. This paper reviews contemporary trends in the control of infectious disease. Historically, the subject started in response to very practical problems, but subsequent developments showed an increasingly marked divergence between general theory and practical applications. In recent years, however, improvements in parameter estimation, asymptotic and stochastic approximation, the modelling of individual diseases, advances in computerized simulations, the construction of resource allocation models, the use of control theory, etc., have been gradually leading to a synthesis of the utmost importance to public health action. Models can now be fitted to specific field data; alternative intervention strategies involving immunization, prophylaxis or treatment can be evaluated; and the incorporation of realistic epidemiological models in a wider decision-oriented system dynamics setting may soon help to solve broader strategic problems on the policy level.

An approximation technique is developed for studying the deterministic spread of a simple epidemic which avoids a difficulty inherent in the diffusion approximation. Travelling wave solutions are found and calculated for various contact distributions. The solution developing from an isolated infective is also briefly discussed.

This paper illustrates a simple method of deriving distributions for samples from cell populations in exponential growth. The distribution of the elapsed proportion of the DNA-synthetic phase of the cell cycle is derived and used to find the distribution of DNA content in a random sample of cells. Allowing for a normally-distributed measurement error, the resulting distribution agrees well with DNA distributions observed empirically. The statistical analysis of DNA distributions, and the interpretation of the distributional parameters in terms of cell population kinetics, are discussed and illustrated with an example.

A stochastic tessellation automaton (STA) is introduced and analysed as an analogue to a stochastic lattice process, called the Markov configuration process. The STA is considered as an (infinite regular) array with interconnected Moore-type automata, each of these representing a B-object interacting with its neighbours. The objective of the paper is to examine some consequences of the analogy between an STA and the Markov configuration process. In addition, the possibility of finding a suitable stochastic grammar arising from the study of this configuration model is briefly considered.