Volume 33 - Issue 1 - June 2008

In vino veritas, in dolio calamitas

François Jongmans

Abstract

We review five classical geometrical models for the volume of a barrel, four of which go back at least to Johannes Kepler in the 17th century. The fifth model, proposed by Charles Camus in 1741, indicated a fruitful new direction, but was superseded by a number of `empirical' formulae for volume, some still used by wine gaugers today. These are generally inaccurate and/or do not permit a logical extension to partly filled barrels. We propose three new geometrical models motivated by Camus' ideas. All eight volume expressions are shown, using integral calculus, to have a common structure. The physical measurement is most easily made by a rod inserted vertically through the bunghole, followed by a simple calculation.

Approximating the path of a celestial body with a circular orbit from two close observations

Thomas J. Osler and Joseph Palma

Abstract

Data from at least three observations of an asteroid or planet are required to calculate the true elliptical orbit. Using ecliptic longitudes from only two close observations, we try to compute a circle that approximates the elliptical orbit. Our calculations result in not one, but two circular orbits. In some cases, we can use physical arguments to decide which of these two orbits is correct; otherwise, more observations may be necessary. The argument is elementary as only trigonometry and Kepler's laws of planetary motion are used. An example is given using data from a real asteroid. From such meager data, we can only expect a rough approximation.

Reflections on The Cauchy–Schwarz Master Class

Kenneth Lange

Abstract

This brief commentary on the recent book The Cauchy–Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by Michael Steele is meant to engage the interest of students of elementary probability. Probability theory provides a setting for proving and extending several of the concrete inequalities discussed in Steele's book. Our treatment of six problems illustrates the probabilistic perspective. One of these problems forges a connection between the arithmetic–geometric mean inequality and the MM algorithm, a general-purpose method of optimization.

Skew Laplace distributions. I. Their origins and inter-relations

Tomasz J. Kozubowski and Krzysztof Podgórski

Abstract

There are numerous asymmetric extensions of the classical Laplace distribution scattered in the literature. In this survey we discuss their origins and inter-relations. In particular, we point out which types of skew Laplace distributions are essentially the same, described in different, albeit equivalent, parametrizations. In a companion paper, Kozubowski and Podgórski (2008), the authors reviewed the properties of classical and geometric infinite divisibility as well as self-decomposability, which are crucial in extending univariate Laplace models to stochastic processes.

Skew Laplace distributions II. Divisibility properties and extensions to stochastic processes

Tomasz J. Kozubowski and Krzysztof Podgórski

Abstract

This paper is a continuation of Kozubowski and Podgórski (2008), where the origins and inter-relations of major types of skew Laplace distributions were discussed. Here, we review the properties of classical and geometric infinite divisibility as well as self-decomposability, which are crucial in extending univariate Laplace models to stochastic processes. General schemes based on these properties lead to several new non-Gaussian stationary autoregressive processes and continuous-time Lévy processes having potential use in stochastic modeling.

A proof of the shape of the Birnbaum–Saunders hazard rate function

Mark Bebbington, Chin-Diew Lai and Ričardas Zitikis

Abstract

We prove that the Glaser's η-function of the Birnbaum–Saunders distribution is upside-down bathtub-shaped. This implies – as a consequence of Glaser's general result – that the Birnbaum–Saunders hazard rate function is upside-down bathtub-shaped. To the best of our knowledge, this is the first proof confirming what has long been believed about the Birnbaum–Saunders hazard rate function, on the basis of computer visualization and/or its similarity to other hazard rate functions whose shapes are upside-down bathtub-shaped.

On some explicit results for the balanced generalized Pólya urn

Stefano Giordano and Fausto Camboni

Abstract

This article describes an explicit approach to urn models of the balanced generalized Pólya type (with two types of balls). The treatment starts by obtaining the difference equations, describing the discrete time behavior of the expected value and of the variance of the selection probability for a given type of ball. The explicit solutions of such difference equations have been found in terms of gamma and psi (digamma) functions. This unified approach is useful in didactics in order to present a general method that leads to the final results without using complicated analytical tools. The more advanced mathematical procedure utilized is the solution of a first-order difference equation. All the theoretical results have been confirmed by a series of Monte Carlo simulations in order to clarify and better explain the behavior of the system.