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American Mathematical Monthly Contents—May 2013

In the May Monthly, we honor Nobel Prize winning author Lloyd Shapley by reprinting in its entirety his classic 1962 Monthly paper "College Admissions and the Stability of Marriage," which was co-authored with David Gale and cited by the Nobel Committee in its award announcement. The paper is preceded by a foreword from Ehud Kalai of Northwestern University.

The remainder of the issue is highlighted by Mel Nathanson's study of fractional parts of roots of positive real numbers and a tribute by Stephen Buckley and Desmond MacHale to a famous problem from Herstein's quintessential text Topics in Algebra.

Stay tuned for the June/July issue, in which David Bressoud reviews five different calculus texts. —Scott Chapman

Vol. 120, No. 5, pp.383-483.

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A Letter from the Editor

Scott Chapman

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Foreword: The High Priest of Game Theory

Ehud Kalai

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College Admissions and the Stability of Marriage

D. Gale and L. S. Shapley

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Real Analysis in Reverse

James Propp

Many of the theorems of real analysis, against the background of the ordered field axioms, are equivalent to Dedekind completeness, and hence can serve as completeness axioms for the reals. In the course of demonstrating this, the article offers a tour of some less-familiar ordered fields, provides some of the relevant history, and considers pedagogical implications.

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On the Fractional Parts of Roots of Positive Real Numbers

Melvyn B. Nathanson

Let $$[\theta]$$ denote the integer part and $$\{\theta\}$$ the fractional part of the real number $$\theta$$. For $$\theta>1$$ and $$\{\theta^{1/n}\}\neq0$$, define $$M_{\theta}(n)=[1/\{\theta^{1/n}\}]$$. The arithmetic function $$M_{\theta}(n)$$ is eventually increasing, and the limit as $$n$$ goes to infinity of $$M_{\theta}(n)/n=1/\log\theta$$. Moreover, $$M_{\theta}(n)$$ is "linearly periodic" if and only if $$\log\theta$$ is rational. Other results and problems concerning the function $$M_{\theta}(n)$$ are discussed.

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Variations on a Theme: Rings Satisfying $$x^{3}=x$$ Are Commutative

Stephen M. Buckley and Desmond MacHale

A ring satisfying $$x^{3}=x$$ is necessarily commutative. We consider a variety of weaker forms of this condition and show that many, but not all of them, imply commutativity. We also present a variety of elementary proofs of the fact that $$x^{3}=x$$ implies commutativity.

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Greedy Galois Games

Joshua Cooper and Aaron Dutle

We show that two duelers with similar, lousy shooting skills (a.k.a. Galois duelers) will choose to take turns firing in accordance with the famous Thue-Morse sequence if they greedily demand their chances to fire as soon as the other's a priori probability of winning exceeds their own. This contrasts with a result from the approximation theory of complex functions, which says what more patient duelers would do, if they really cared about being as fair as possible. We note a consequent interpretation of the Thue-Morse sequence in terms of certain expansions in fractional bases close to, but greater than, 1.

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Geometric Multiplicities and Geršgorin Discs

Rachid Marsli and Frank J. Hall

If $$A$$ is an $$n\times n$$ complex matrix and $$\lambda$$ is an eigenvalue of $$A$$ with geometric multiplicity $$k$$, then $$\lambda$$ is in at least $$k$$ of the $$n$$ Geršgorin discs of $$A$$.

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A Note on the Cauchy-Schwarz Inequality

Jim X. Xiang

The Cauchy-Schwarz inequality is one of most widely used and most important inequalities in mathematics. The aim of this note is to show a new inequality that improves the Cauchy-Schwarz inequality.

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A Class of Continued Radicals

Costas J. Efthimiou

We compute the limits of a class of continued radicals, extending the results of a previous note in which only periodic radicals of the class were considered.

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A Binomial Identity via Differential Equations

D. Aharonov and U. Elias

In the following we discuss a well-known binomial identity. Many proofs by different methods are known for this identity. Here we present another proof, which uses linear ordinary differential equations of the first order.

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Another Proof for Non-Supercyclicity in Finite Dimensional Complex Banach Spaces

F. Galaz-Fontes

We give an elementary proof, based on linear algebra and on a simple and well-known technique from the theory of dynamical systems, for the non-existence of supercyclic linear operators defined on a finite dimensional complex Banach space with dimension greater than or equal to two.

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Problems 11705-11711
Solutions 11574, 11576, 11577, 11596, 11597, 11600, 11603, 11604

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Symmetry: A Mathematical Exploration. By Kristopher Tapp. Reviewed by Kevin Woods.

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