Volume 33 - Issue 2 - December 2008

An improved double-elimination tournament with application to the FIFA World Cup

Samuel S. Wu and Mark C. K. Yang

Abstract

A double-elimination tournament (DET) eliminates a team only when the team has lost twice. It is a viable alternative to a single-elimination tournament when we have time for more matches, or we wish every team to have more than one chance to demonstrate its ability. Despite its popularity, DETs have not been studied thoroughly. In this paper we propose a new DET as an alternative to the commonly used DET. This new format makes the tournament shorter, allows the better teams to play more matches, and makes most matches more competitive. Furthermore, the best team has a higher probability to win the championship in most cases. The new DET has many advantages when compared with the current FIFA World CupTM tournament which requires approximately the same number of matches.

Modeling the design of an airport runway

Paul J. Laumakis

Abstract

This paper describes a mathematical application project for a course in engineering mathematics. The particular course covers mathematical topics in multivariable and vector calculus, linear algebra, and ordinary differential equations. The project detailed here is an application associated with ordinary differential equations and can therefore also be used as an application project for a typical undergraduate mathematics course in differential equations. The overall intent of the project is to expose students to the mathematical modeling process while providing them with an interesting application of the mathematical techniques that they are learning in the classroom.

Dice sums

Amber Rosin, Mary Sharobiem and Randall Swift

Abstract

This paper is a celebration of the surprisingly rich structure associated with dice sums and dice labelings. Dice sums, though very elementary at first glance, have a deep and intriguing structure. Elements of abstract algebra and number theory arise in the analysis of dice sums and labels. The paper considers the classic question of equally likely sums for m n-sided dice, the Sicherman labelings of dice, and introduces the notion of Pythagorean dice. The paper surveys known results, presents new ideas, and poses several open problems for further study.

A class of Markov chains with beta ergodic distributions

Carlos G. Pacheco-Gonzalez and Jordan Stoyanov

Abstract

We study a class of discrete-time Markov chains describing random motions in the interval (0,1). They can be viewed as modelling the motion of particles, stock price ratios, or interest rate changes. At any step, both the direction in which the particle moves, up or down, and the new position are random. The Markov chains arising in such models are ergodic and one of the problems is to find their ergodic distributions explicitly. We show, for some models, that the ergodic distributions are beta distributions for which the parameters can be specified explicitly.

A Torricellian model and implicit functions

C. W. Groetsch

Abstract

A simple physical model, based on Torricelli's law, may be used as a motivator for topics in intermediate-level mathematical analysis courses. The particular mathematical topics that arise from the model involve the implicit function theorem and related issues. The model provides a concrete setting for highlighting the necessity of investigating the existence and character of implicitly defined functions, and serves as a platform for the use of graphical and numerical means to guide rigorous analysis. The analysis of the mathematical model relies on implicit differentiation and a simple fixed point theorem.

Euler, Lambert, and the Lambert W-function today

P. B. Brito, F. Fabião and A. Staubyn

Abstract

The Lambert W-function has found applications in an extraordinary number of scientific fields. In this paper we present a short historical review, a brief description of the function, and a survey of some of its applications.

Optimal constrained confidence estimation via tail functions

Borek Puza and Terence O'Neill

Abstract

This paper focuses on confidence estimation for constrained parameters. It is shown how the method of tail functions can be used to engineer a constrained confidence interval which is optimal in terms of prior expected length. The strategy is compared with the unified approach of Feldman and Cousins (1998) and illustrated by application to inference on the normal mean.

Filtering for some time series models by using transformation

H. Gong, A. Thavaneswaran and J. Singh

Abstract

This paper is concerned with filtering for various types of random coefficient times series models including the class of ARCH models and stochastic volatility models. We extend the results of Peiris and Thavaneswaran (2004) on filtering for some time series models using a transformation proposed by Shiryayev (1995). Approximate recursive filtering for asymmetric nonlinear volatility models are also discussed in some detail.