David J. Gates

Clustering estimates for spatial point distributions with unstable potentials

SummarySeveral authors have tried to model highly clustered point patterns by using Gibbs distributions with attractive potentials. Some of these potentials violate a stability condition well known in statistical mechanics. We show that such potentials produce patterns which are much more tightly clustered than those considered by the authors. More generally, our estimates provide a useful test for rejecting unsuitable potentials in models for given patterns. We also use instability arguments to reject related approximations and simulations.

On the Distributions of Scan Statistics

Abstract This article considers the distribution of the scan statistic over some interval of the real line, both for a fixed number of independent uniform points and for a Poisson process. It provides a new series method of calculating this distribution in the Poisson case and a simple self-contained recursive approximation in the fixed case. These results work best at low densities, which is precisely the case when the closed formulas are computationally impractical. The results are compared with values tabulated by Neff and Naus (1980) and with a recent approximation due to Naus (1982). There is also an explanation of why Nauss approximation is good in the Poisson case and a derivation of recursive bounds in the fixed case.

Negative skewness and negative correlations in spatial competition models

We consider a population of identical biological entities that compete with each other for space and derive, under rather weak assumptions, some statistical properties of the sizes (e.g. biomasses) of the individuals. It is proved, for example, that, during the early stages of competition, their size distribution becomes negatively skewed and the sizes of neighbouring individuals become negatively correlated. The results are proved for a regular periodic array of centres of the individuals where they can be compared with data on plantations.

On the fluctuations in levels of closed lakes

Abstract A stochastic differential equation model of closed lakes is introduced and analysed with particular reference to Lake George in New South Wales, Australia. Our results indicate that the standard deviation in water level at the prehistoric 30-m level is approximately equal to its present value at the 3-m level. At low water levels, the theory predicts that the expected level is slightly higher than that predicted by a purely deterministic theory. The model provides a more fundamental and complementary approach to the fluctuation calculations of Langbein. It also provides some support for the work of Galloway, predicting a dry late-Quaternary climate, although the question of beach formation remains unresolved.

A threshold theorem for the general epidemic in discrete time.

SummaryA theorem, analogous to the continuous time Threshold Theorem of Kermack and McKendrick, is proved for a certain discrete time epidemic model. This model, in contrast to its continuous time analogue, leads to some solutions in which the total population of susceptibles may become infected in a finite time.

SYMMETRIC SOLUTIONS FOR TWO-BODY DYNAMICS IN A COLLISION PREVENTION MODEL

This paper presents the first analytical solutions for the three-dimensional motion of two idealized mobiles controlled by a particular guidance law designed to avoid a collision with minimal path deviation. The mobiles can be regarded as particles, and guidance can be interpreted as complex forces of interaction between the particles. The motion is then a generalized form of two-body Newtonian dynamics. If the mobiles have equal speeds, the relative motion is determined through various transformations of the differential equations. Solvability relies on congruence and symmetries of the paths, which is exploited to reduce the original twelve first-order differential equations to three first- order equations for the relative motion. The resulting state space is partitioned into five invariant subsets, with various symmetries and stabilities. One of these sets describes planar motion, where simple explicit solutions are given. In nonplanar motion, the solution is formally reduced to quadrature. A numerical calculation gives the separation at the closest point of approach, which provides control over minimum separation. The results should be of interest because of their application, which includes, most importantly, the prevention of midair collisions between aircraft, but also potential application to land, water and space vehicles. The solutions should be of interest to mathematical specialists in dynamical systems, because of some novel constants of the motion, novel symmetries, and the associated reducibility of the equations. doi:10.1017/S1446181111000691

Replacement of train wheels: an application of dynamic reversal of a Markov process

A problem of regrinding and recycling worn train wheels leads to a Markov population process with distinctive properties, including a product-form equilibrium distribution. A convenient framework for analyzing this process is via the notion of dynamic reversal, a natural extension of ordinary (time) reversal. The dynamically reversed process is of the same type as the original process, which allows a simple derivation of some important properties. The process seems not to belong to any class of Markov processes for which stationary distributions are known. INVENTORY PROCESS; MIGRATION PROCESS AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60J20 SECONDARY 60K30; 82C20