Volume 26 - Issue 1 - June 2001

Mathematics and the human genome project

W. J. Ewens

Abstract

The human genome project, and the parallel genome projects for other species, will soon produce data that will require entirely novel mathematical and statistical analyses, as well as new computer algorithms. This review discusses two aspects of the human genome project of interest to mathematicians. The first centers around the question of how the genome is assembled, while the second is concerned with methods that might be used to analyse human genome data once assembled. In particular the popular BLAST (Basic Local Alignment Search Tool) statistical analysis is discussed.

Tiling with Penrose rhomb clusters

Ellen Perstein

Abstract

This paper presents a formulation for the recursive use of four Penrose rhomb clusters, pentagon, quark, rhomb and star, to tile a plane aperiodically. All except quark take their name from the shapes they exhibit at higher levels of recursion. Rhombs and quarks have the property of reflection symmetry about one axis. Pentagons and stars have five-fold rotational and reflection symmetry. Repeated application of the appropriate recursive rule to any one of these clusters will tile a plane. Starting with a pentagon or star gives a tiling with fivefold symmetry.

Simple solutions for a special Riccati equation

Behzad Salimi

Abstract

The exact analytic solution of the special Riccati equation y' = ay2 + bx-1y + cx-2 in terms of elementary functions is derived using simple procedures. The solutions presented here are alternative forms of those of Polyanin and Zaitsev.

A logistic birth–death–immigration–emigration process

Randall J. Swift

Abstract

In this paper, the steady-state probabilities and moments of a logistic birth–death–immigration–emigration process are obtained in terms of a generalized hypergeometric function. These steady-state probabilities generalize those obtained by J. N. Kapur and S. Kapur (1978) for the birth–death–immigration–emigration process. The process is compared to the deterministic population model and some numerical results are presented.

Monotone decreasing distance between distributions of sums of unfair coins and a fair coin

Prem K. Goel and Chandra M. Gulati

Abstract

Distribution of sums of independent, discrete random variables on a finite group are shown to approach the uniform distribution monotonically, in terms of a large class of divergence/distance measures, as the number of terms in the sum increases. It is also shown that the Shannon entropy of the distribution of the sum increases monotonically with the number of terms summed.

Chaotic behaviour in a tatonnement process for a simple general equilibrium model of a pure exchange economy

David Chappell, Thomas W. Warke and Angela Anagnostou

Abstract

We examine a two-person, two-good model of a pure exchange economy. We propose a linear tatonnement process in a discrete time setting, and show that this may exhibit chaotic behaviour if there are relatively strong preferences by the two agents for a relatively scarce numeraire good.

On the asymptotic normality of the posterior distribution

W. F. Scott

Abstract

Let a random sample of size n be drawn from a variable with probability density depending on a vector of parameters, θ = (θ1, θ2, . . ., θr)T, and let us suppose that the prior distribution of θ has continuous, positive density π(θ) on an open set H. We give a proof of a theorem on the asymptotic normality of the posterior distribution of θ under certain conditions, which are similar to those often used to prove the asymptotic normality of the maximum likelihood estimator, \hat{θ}(n). The theorem implies that, in large samples, the posterior distribution of $\btheta$ is approximately normally distributed with mean \hat{θ}(n) and variance matrix equal to the inverse of the negative of the matrix of second partial derivatives of the log-likelihood function, evaluated at \hat{θ}(n). These results are frequently useful in practice, and are referred to by Lindley (1965), DeGroot (1975) and Cox and Hinkley (1974), among other writers. One of the most important practical applications is in the construction of approximate Bayesian confidence intervals (or, if this terminology is preferred, credibility intervals). Our work follows along the lines of Walker (1969), who mainly covered the one-dimensional case (r = 1). In the final section, we consider the important practical example of sampling from a normal distribution with unknown mean and standard deviation, and mention the James–Stein theorem.