American Mathematical Monthly Contents—August/September 2013

While many of you enjoy some vacation time during the late summer, the Monthly cruises into August and September with an outstanding collection of five articles and five notes. Key amongst our articles is a piece by David Aldous on how to illustrate undergraduate probability issues using live market data. Also, Peter Kuchment and Sergey Lvin offer an interesting array of identities involving sin x which resulted from work in medical imaging. In our notes section, Ram Murthy and Jaban Meher rewrite Ramanujan's proof of Bertrand's postulate without the use of Stirling's formula. We close with our Problem Section and Joseph Malkevitch's review of The Mathematical Coloring Book, by Alexander Soifer.

Check back for the October issue when Allan Tucker will offer us a review of the history of undergraduate mathematics in the United States. —Scott Chapman

Vol. 120, No. 7, pp.583-674.

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ARTICLES

Using Prediction Market Data to Illustrate Undergraduate Probability

David J. Aldous

Prediction markets provide a rare setting, where results of mathematical probability theory can be related to events of real-world interest and where theory can be compared to data. The paper discusses two simple mathematical results—the halftime price principle and the serious candidates principle—and corresponding data from baseball and the 2012 Republican presidential nomination race.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.07.583

Pi, the Primes, Periodicities, and Probability

Stephen D. Casey and Brian M. Sadler

The theory of numbers has repeatedly shown itself to be both practical and beautiful. This paper gives an example of this duality. We present a very efficient (and practical) algorithm for extracting the fundamental period from a set of sparse and noisy observations of a periodic process. The procedure is computationally straightforward, stable with respect to noise, and converges quickly. Its use is justified by a theorem, which shows that for a set of randomly chosen positive integers, the probability that they do not all share a common prime factor approaches one quickly as the cardinality of the set increases. The proof of this theorem rests on a (beautiful) probabilistic interpretation of the Riemann zeta function.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.07.594

Identities for $$\sin x$$ that Came from Medical Imaging

Peter Kuchment and Sergey Lvin

The article describes interesting nonlinear differential identities satisfied by standard exponential and trigonometric functions. They appeared as byproducts of medical imaging research and look like some type of non-commutative binomial formulas. A brief description of the origin of these identities is provided, as well as their direct algebraic derivation. Relations with separate analyticity theorems in several complex variables and some open problems are also mentioned.

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Subdivision Using Angle Bisectors Is Dense in the Space of Triangles

Steve Butler and Ron Graham

Starting with any nondegenerate triangle, we can use an interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show that starting with any arbitrary triangle, the resulting set of triangles formed by this process contains triangles arbitrarily close (up to similarity) to any given triangle when the point that we use to subdivide is the incenter.

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Commuting and Noncommuting Infinitesimals

Mikhail G. Katz and Eric Leichtnam

Infinitesimals are natural products of the human imagination. Their history goes back to Greek antiquity. Their role in the calculus and analysis has seen dramatic ups and downs; they have stimulated strong opinions and even vitriol. Edwin Hewitt developed hyperreal fields in the 1940s. Abraham Robinson’s infinitesimals date from the 1960s. A noncommutative version of infinitesimals, due to Alain Connes, has been in use since the 1990s. We review some of the hyperreal concepts, and compare them with some of the concepts underlying noncommutative geometry.

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NOTES

Another Proof of $$\zeta(2)=\frac{\pi^{2}}{6}$$ Using Double Integrals

Daniele Ritelli

Starting from the double integral $$\int_{0}^{\infty}\int_{0}^{\infty}\frac{dxdy}{(1+y)(1+x^{2}y)}$$ we give another solution to the Basel Problem $$\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6}$$.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.07.642

Simple Approximation Formulas for the Birthday Problem

Matthias Arnold and Werner Glaß

We approximate the probability for at least $$k=2$$ or $$3$$ of $$n$$ persons having the same birthday if we ignore leap years and assume that each day is equally likely. Although there are exact formulas available in the literature, a possible drawback to using them in the classroom is that these formulas are comparatively cumbersome. As an alternative, we suggest an easily computable approximation that still provides a close fit. Furthermore, the approximation allows an easy answer to the question of how many people are needed, so that the probability of at least two or three of them having the same birthday exceeds a given value.

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Another Irreducibility Criterion

Anthony J. Bevelacqua

For any prime $$p$$ and integers $$a_{1},\dots,a_{n}$$ such that $$p\geq a_{1}\geq\cdots\geq a_{n}\geq1$$, the polynomial $$f=p+a_{1}x+\cdots+a_{n}x^{n}$$ is irreducible in $$\mathbb{Z}[x]$$ if and only if the list $$(p,a_{1},\dots,a_{n})$$ does not consist of $$(n+1)/d$$ consecutive constant lists of length $$d>1$$.

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Ramanujan’s Proof of Bertrand’s Postulate

Jaban Meher and M. Ram Murty

We present Ramanujan’s proof of Bertrand’s postulate and in the process, eliminate his use of Stirling’s formula. The revised proof is elegant and elementary so as to be accessible to a wider audience.

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A Simple Proof of Vitali’s Theorem for Signed Measures

T. Samuel

Several theorems are named after the Italian mathematician Vitali. In this note, we provide a simple proof of an extension of Vitali’s Theorem on the existence of nonmeasurable sets. Specifically we show, without using any decomposition theorems, that there does not exist a nontrivial, atom-less, $$\sigma$$-additive and translation invariant set function $$\mathcal{L}$$ from the power set of the real line to the extended real numbers with $$\mathcal{L}([0,1])$$. (Note that $$\mathcal{L}$$ is not assumed to be nonnegative.)

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