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American Mathematical Monthly Contents—October 2013

Frost on the pumpkin means more time reading in front of the fire. October’s Monthly is exactly what you need to get your fall off to a fast start. Not only does this issue feature our annual review of the Putnam Examination, but Alan Tucker offers us an in depth view of the history of the undergraduate mathematics major in American universities. In our Notes Section, Jean Van Schaftingen gives a direct proof of the existence of eigenvalues and eigenvectors using Weierstrass’s Theorem. 

Stay tuned for November when Jacques Lévy Véhel and Franklin Mendivil analyze “Christiane’s Hair.” Scott Chapman

Vol. 120, No. 8, pp.679-768.


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The Seventy-Third William Lowell Putnam Mathematical Competition

Leonard F. Klosinski, Gerald L. Alexanderson, and Mark Krusemeyer

The results of the Seventy-Third William Lowell Putnam Mathematical Competition.

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The History of the Undergraduate Program in Mathematics in the United States

Alan Tucker

This article describes the history of the mathematics major and, more generally, collegiate mathematics, in the United States. Interestingly, the Mathematical Association of America was organized about 100 years ago, around the same time that academic majors came into existence.

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Solitaire Mancala Games and the Chinese Remainder Theorem

Brant Jones, Laura Taalman, and Anthony Tongen

Mancala is a generic name for a family of sowing games that are popular all over the world. There are many two-player mancala games in which a player may move again if their move ends in their own store. In this work, we study a simple solitaire mancala game called Tchoukaillon that facilitates the analysis of “sweep” moves, in which all of the stones on a portion of the board can be collected into the store. We include a self-contained account of prior research on Tchoukaillon, as well as a new description of all winning Tchoukaillon boards with a given length. We also prove an analogue of the Chinese Remainder Theorem for Tchoukaillon boards, and give an algorithm to reconstruct a complete winning Tchoukaillon board from partial information. Finally, we propose a graph-theoretic generalization of Tchoukaillon for further study.

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Frobenius’ Result on Simple Groups of Order $$\frac{p^{3}-p}{2}$$

Paul Monsky

The complete list of pairs of non-isomorphic finite simple groups having the same order is well known. In particular, for $$p>3$$, $$PSL_{2}(\mathbb{Z}/p)$$ is the “only” simple group of order $$\frac{p^{3}-p}{2}$$. It’s less well known that Frobenius proved this uniqueness result in 1902. This note presents a version of Frobenius’ argument that might be used in an undergraduate honors algebra course. It also includes a short modern proof, aimed at the same audience, of the much earlier result that$$PSL_{2}(\mathbb{Z}/p)$$ is simple for $$p>3$$, a result stated by Galois in 1832.

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Higher-Dimensional Analogues of the Map Coloring Problem

Bhaskar Bagchi and Basudeb Datta

After a brief discussion of the history of the problem, we propose a generalization of the map coloring problem to higher dimensions.

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Stirling’s Formula and Its Extension for the Gamma Function

Dorin Ervin Dutkay, Constantin P. Niculescu, and Florin Popovici

We present new short proofs for both Stirling’s formula and Stirling’s formula for the Gamma function. Our approach in the first case relies upon analysis of Wallis’ formula, while the second result follows from the log-convexity property of the Gamma function.

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A Direct Proof of the Existence of Eigenvalues and Eigenvectors by Weierstrass’s Theorem

Jean Van Schaftingen

The existence of an eigenvector and an eigenvalue of a linear operator on a complex vector space is proved in the spirit of Argand’s proof of the fundamental theorem of algebra. The proof relies only on Weierstrass’s theorem, the definition of the inverse of a linear operator, and algebraic identities.

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A Variational Principle and Its Application to Estimating the Electrical Capacitance of a Perfect Conducto

A. G. Ramm

Assume that $$A$$ is a bounded self-adjoint operator in a Hilbert space $$H$$. Then, the variational principle,

$$\text{max}_{v}\frac{|(Au,v)|^{2}}{(Av,v)}=(Au,u), (*)$$          

holds if and only if $$A\geq0$$; that is, if $$(Av,v)\geq0$$ for all $$v\in H$$. We define the left-hand side in $$(*)$$ to be zero if $$(Av,v)=0$$. As an application of this principle, a variational principle for the electrical capacitance of a conductor of an arbitrary shape is derived.

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Proving the Pythagorean Theorem via Infinite Dissections

Zsolt Lengvárszky

Novel proofs of the Pythagorean Theorem are obtained by dissecting the squares on the sides of the abc triangle into a series of infinitely many similar triangles.

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Problems 11726-11732
Solutions 11557, 11582, 11583, 11585, 11587, 11590, 11591

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An Introduction to Measure Theory, by Terence Tao

Reviewed by Takis Konstantopoulos

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