Volume 39 - Issue 1 - June 2014

Polish mathematics in the first half of the 20th century

Robert A. Beeler and Rick Norwood


In this article we present a short survey of the contributions of Polish mathematicians in the first half of the 20th century, with particular emphasis on the period including the two world wars. Major Polish mathematicians of this period include Kuratowski, Steinhaus, Sierpinski, Rejewelski, and Banach. We also discuss the University of Lwów and Jagiellonian University.

A cluster of anniversaries

Oscar Sheynin


John Graunt originated population and medical statistics and discovered the life table. The beginning of the 18th century had seen an upsurge of probability. Its principles and main notions were formulated anew (after Pascal, Fermat, and Huygens) and the first limit theorems proved. A century later Laplace summarized the work of his predecessors and his own previous achievements, and supplied probability with new mathematical armoury. However, his theory of probability belonged to applied mathematics, he did not raise its level of abstraction and it barely yielded to development.

A multiple scales approach to a wrist oscillation model

Joe Latulippe and Randy Sierra


Wrist dislocations can occur from the tearing of ligaments in the wrist. In order for the wrist to heal, surgery is often performed. During the post-operative healing process the repaired ligament will stretch causing the range of motion of the wrist to vary with time. To better understand this healing process, a mathematical model that treats the wrist as an oscillator is introduced. A corresponding weakly nonlinear differential equation is investigated using perturbation methods and an asymptotic approximation of the solution is found. A numerical solution to the model is then compared to the asymptotic approximation under various parameter regimes.

Looking for the integrating factor: the Darboux approach

Y. T. Christodoulides and P. A. Damianou


We present a method for the solution of first-order ordinary differential equations based on a class of polynomials known as Darboux polynomials. Integrating factors, used to convert the equations in exact form, and constants of motion, which give the general solution, are constructed in this way. Several examples illustrating the method are provided.

Probabilities involving directional similarity

John F. Newell


In this paper, we explore the mathematics behind a method for discovering, comparing, or computing the degrees of similarity or dissimilarity between the orientations of two Euclidean vectors. The method has been in use since the 1800s, but only as a formula whose mathematical origins have been unexplained. The absence of an explanation has led to confusion and speculation regarding causal links. The explanation offered here involves projecting the components of a vector back onto it, thus forming constituent parts which reflect the influence of the components. This allows for the formation of probabilities which reflect degrees of similarity between the vector and its components. In the related experiments, the outcomes stochastically reflect changes in the orientations of the detection equipment, and so the search for causal links is shown to be unnecessary.

Pell and Pell–Lucas trees

Thomas Koshy


We present two sets of graph-theoretic models for Pell and Pell–Lucas numbers by constructing rooted binary trees.

Neoconjugate norms and a generalized ladder problem

Dan Kalman, Alan Krinik and Chaitanya K. Rao


In the first quadrant of the plane, we consider the L-shaped region R = {(x, y) | x <= a or y <= b}. For p not equal to 0 and for a vector (x, y) in the plane, define |(x, y)|p = (xp + yp)1/p, called the p-length of the vector. We find the maximum L such that a segment of fixed p-length L can go around the corner within R. This is |(a, b)|q, where 1/q - 1/p = 1. This is equal to the q-length of the segment across the corner in R from (0, 0) to (a, b). With p = 2 and q = 2/3, this solves a classical optimization problem of calculus. Our analysis uses envelopes of families of curves, as well as elementary inequality methods.

On the need for a new playing die

Andrew D. Irving and Ebrahim L. Patel


We model the rolling of a standard die using a Markov matrix. Though a die may be called fair, its initial position influences a roll's outcome. This being undesirable, a simple solution is proposed.

Estimating deaths in an animal herd

J. Gani and R. J. Swift


In a recent paper, Huso (2011) estimated the number of deaths in a wildlife population by the count of animal carcasses left on the ground.  We present some simple models, both deterministic and stochastic, for such a population. The results obtained will allow a ranger to estimate the total number of deaths suffered by the herd.