Volume 25 - Issue 2 - December 2000

Dirac operators — from differential calculus to the index theorem

David Applebaum

Abstract

This article is an extended version of the text of the author's inaugural lecture as Professor of Mathematics at the Nottingham Trent University, given on 16 February 2000. The aim is to illustrate the intriguing relationship between beautiful mathematics and physical applications using the Dirac equation for relativistic electrons as a case study. We trace the antecedents of this equation from the differential calculus of the 18th century and the Clifford algebras of the 19th century, examine its emergence in physics through the need to combine special relativity with quantum mechanics and briefly outline its recent incarnation in index theory.

On a certain kind of generalized number-theoretical Möbius function

Tom C. Brown, Leetsch C. Hsu, Jun Wang And Peter Jau-Shyong Shiue

Abstract

The classical Möbius function appears in many places in number theory and in combinatorial theory. Several different generalizations of this function have been studied. We wish to bring to the attention of a wider audience a particular generalization which has some attractive applications. We give some new examples and applications, and mention some known results.

Some gambling strategies: an evaluation

Surekha Mudivarthy and M. Bhaskara Rao

Abstract

The focal point of this paper is an evaluation of some gambling strategies designed to obtain approximately an eleven percent return on a given capital. A random walk with uneven steps arises in one of the strategies. The Dubins and Savage strategy is adapted for various odds. The strategies are discussed as they apply to the games of roulette and craps. The Dubins and Savage strategy gives the highest probability among the strategies evaluated to reach a target capital starting with a given initial capital.

Birth and death processes, orthogonal polynomials and limiting conditional distributions

Wim Schoutens

Abstract

We use the Karlin–McGregor spectral representation for birth and death processes based on orthogonal polynomials, for the analysis of the (doubly) limiting conditional distributions. These distributions can be used to gain insight into the asymptotic behaviour of absorbing and transient processes. The examples treat the linear birth and death process.

Some results for a chain binomial model

G. Jones, C. D. Lai and J. C. W. Rayner

Abstract

A simple chain binomial model is presented and possible applications discussed. We derive some results that provide interesting illustrations of some general probabilistic and statistical procedures.

Power surfaces

Mary C. Phipps

Abstract

The P-value analogue of the classical Neyman–Pearson power curve is defined in this paper as a power surface. The power surface allows us to view power from a new perspective, and provides a fresh interpretation of uniform q–q plots of bootstrap P-values. An illustration is provided.

Sample size and the accuracy of a consistent estimator

Paul van der Laan and Constance van Eeden

Abstract

Birnbaum (1948) introduced the notion of peakedness about θ of a random variable T, defined by P(|T-θ|< ε), ε>0. What seems not to be well known is that, for a consistent estimator Tn of θ, its peakedness does not necessarily converge to 1 monotonically in n. In this article some known results on how the peakedness of the sample mean behaves as a function of n are recalled. Also, new results concerning the peakedness of the median and the midrange are presented.

Reducing the pairing effect in 4-team double-elimination tournaments

Peter Thompson, Eric Jaryszak and John Wamil

Abstract

The initial pairings in standard 4-team elimination tournaments can play a large role in determining who wins the tournament. Two new tournaments are introduced which address this problem.