August/September 2012 Contents
In the August/September Monthly articles, we open with a piece by David Borwein, Jon Borwein, and Armin Straub on sinc integral evaluation that solves an open problem first posed in the Monthly in 1967. A paper by Lionel Levine and Katherine Stange follows which explains how to get the most out of a meal by analyzing a game known as Ethiopian Dinner. How rare is the occurrence of consecutive strings of letters in randomly generated words? Kai Kristensen answers this question in a paper which uses only Linear Algebra. We round the articles out with Franz Lemmermeyer’s analysis of algebraic curves using only Number Theory, and a paper by Peter Borwein and Joe Hobart which examines the power of division in straight line programs. Our notes examine a binomial like matrix equation, a generalization of Gauss’s Cyclotomic Formula, linear dependencies among p-norms of vectors, a generalization of a problem that appeared on the 70th Putnam Examination, and close by finding an upper bound for the measure of spherical caps. Marion Cohen reviews Change Is Possible: Stories of Women and Minorities in Mathematics, by Patricia Clark Kenschaft, and as always, our Problem Section marches onward.
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A Sinc that SankDavid Borwein, Jonathan M. Borwein, and Armin Straub
We resolve and further study a sinc integral evaluation, first posed in this MONTHLY in [1967, p. 1015], which was solved in [1968, p. 914] and withdrawn in [1970, p. 657]. After a short introduction to the problem and its history, we give a general evaluation which we make entirely explicit in the case of the product of three sinc functions. Finally, we exhibit some more general structure of the integrals in question.
How to Make the Most of a Shared Meal: Plan the Last Bite FirstLionel Levine and Katherine E. Stange
If you are sharing a meal with a companion, then how is it best to make sure you get your favourite mouthfuls? Ethiopian Dinner is a game in which two players take turns eating morsels from a common plate. Each morsel comes with a pair of utility values measuring its tastiness to the two players. Kohler and Chandrasekaran discovered a good strategy—a subgame perfect equilibrium, to be exact—for this game. We give a new visual proof of their result. The players arrive at the equilibrium by figuring out their last move first and working backward. We conclude that it’s never too early to start thinking about dessert.
A Nearest Integer Solution to the Longest Run of Randomly Generated WordsKai F. Kristensen
How rare is the event of observing more than a certain number of consecutive and identical letters of any kind somewhere in a randomly generated word? No one can deny that the use of generating functions is crucial for giving answers to questions like this. This paper, however, gives an answer, essentially based on elementary linear algebra. The derived formula is nevertheless simpler, has computational advantages and gives rise to a ‘nearest integer’ representation with an improved analytical range, as compared to earlier results.
Parametrization of Algebraic Curves from a Number Theorist’s Point of View
We present the technique of parametrization of plane algebraic curves from a number theorist’s point of view and present Kapferer’s simple and beautiful (but little known) proof that nonsingular curves of degree > 2 cannot be parametrized by rational functions.
The Extraordinary Power of Division in Straight Line Programs
Peter Borwein and Joe Hobart
A lovely circle of ideas due primarily to Shub, Smale and Shamir says that if it is possible to divide quickly, then it is possible to factor quickly. Here, dividing quickly means modular division over a straight line program. In this context, quickly means actual computations done quickly. The point of this note is to advertise this lovely circle of ideas. The language of complexity theory sometimes clouds the underlying simplicity of the ideas. It is our hope to provide a straight forward explanation of these intrinsically simple ideas.
A Binomial-like Matrix Equation
Alin Bostan and Thierry Combot
We show that a pair of matrices satisfying a certain algebraic identity, reminiscent of the binomial theorem, must have the same characteristic polynomial. This is a generalization of Problem 4 (11th grade) from the Romanian National Mathematical Olympiad 2011.
Generalizing Gauss’s Gem
Ezra Brown and Marc Chamberland
Gauss’s Cyclotomic Formula is extended to a formula with p variables, where p is an odd prime. This new formula involves the determinant of a circulant matrix. An application involving the Wendt determinant is given.
Norms as a Function of p Are Linearly Independent in Finite Dimensions
We show that there are no non-trivial linear dependencies among p-norms of vectors in finite dimensions that hold for all p. The proof is by complex analytic continuation.
Functions with Constant Mean on Similar Countable Subsets of
Cédric De Groote and Mitia Duerinckx
We prove the following generalization of a problem proposed at the 70th William Lowell Putnam Mathematical Competition. Given a nonempty finite set E of n points in and a function such that the arithmetic mean of the values of f at the n points of every image of E by a direct similarity is equal to a constant, then f is constant on . This result is extended to nonempty countable sets, and its validity is discussed in a more general context.
An Upper Bound for Spherical Caps
We prove a useful upper bound for the measure of spherical caps.
PROBLEMS AND SOLUTIONS
Change Is Possible: Stories of Women and Minorities in Mathematics.
By Patricia Clark Kenschaft, American Mathematical Society, Providence, RI, 2005, ix + 211 pp., ISBN 0-8218- 3748-6, $32.00
Reviewed by Marion Cohen