February 2013 Contents
What do twins do at twin conventions? Martin Griffiths doesn't know, but he has an idea for what a combinatorialist could do. Of course, it involves Stirling Numbers.
Also in this issue: flight plans, infinite products, and power series, and a proposal to save Pi Day.
—Walter Stromquist, Editor
Be Careful What You Assign: Constant Speed Parametrizations
Michelle L. Ghrist and Eric E. Lane
We explore one aspect of a multivariable calculus project: parametrizing an elliptical path for an airplane that travels at constant speed. We find that parametrizing a constant speed elliptical path is significantly more complicated than a circular path and involves taking the inverse of the elliptic integral of the second kind. We compare our constant-speed parametrization to the standard ellipse parametrization (x(t) = a cos wt, y(t) = b sin (wt) and generalize to parametrizing other constant-speed curves.
New Infinite Products of Cosines and Vičte-Like Formulae
Samuel G. Moreno and Esther M. García
In this article, the authors show that Vičte's formula is only the tip of the iceberg. Under the surface, they search for curious and interesting Vičte-like infinite products, rare species made of products of nested square roots of 2, but here with some minus signs occurring inside. To explore this fascinating world, they only use the simple trigonometric identity cos x = 2 cos((π+ 2x)/4) cos((π-2x)/4), combined with a recent formula by L. D. Servi.
Sitting Down for Dinner at a Twin Convention
In this article we consider a particular combinatorial scenario involving n sets of identical twins. We show how, subject to various assumptions and conditions on the possible groupings, formulas may be obtained in terms of n for the number of ways in which these 2n individuals can be seated at k tables for any fixed value of k. This is achieved by deriving recurrence relations and subsequently employing exponential generating functions.
The Lah Numbers and the nth Derivative of e1/x
Siad Daboul, Jan Mangaldan, Michael Z. Spivey, and Peter J. Taylor
We give five proofs that the coefficients in the nth derivative of e1/x are the Lah numbers, a triangle of integers whose best-known applications are in combinatorics and finite difference calculus. Our proofs use tools from several areas of mathematics, including binomial coefficients, Faŕ di Bruno's formula, set partitions, Maclaurin series, factorial powers, the Poisson probability distribution, and hypergeometric functions.
The Equation (2x) = 2(x)'(x)
A. F. Beardon
We discuss solutions of the equation (2x) = 2(x)'(x), which is essentially a delay differential equation, with the boundary condition '(0)=1, on the interval [0,∞). In particular, we note that the only known solutions of this type that are positive when x is positive are the functions c-1sinh(cx), where c>0, and the function x.
In 1932, Stanislaw Golab proved that, for a large class of metrics in the plane, the perimeter of the unit disk can vary from 6 to 8. Hence, the ratio corresponding to pi can vary from 3 to 4. We illustrate this result by defining a family of metrics that can be understood easily by any student of linear algebra.
Proof Without Words: Fibonacci Triangles and TrapezoidsAngel Plaza and Hans R. Walser
The Scope of the Power Series Method
This note examines the application of the power series method to the solution of second order homogeneous linear ODEs, and shows that it works in a straightforward way only under restrictive conditions—in cases that reduce to the hypergeometric equation or the confluent hypergeometric equation. On the other hand, it is noted that these equations account for most "special functions."
NEWS AND LETTERS
73rd Annual William Lowell Putnam Examination