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.
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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.
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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 $$\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6}$$.
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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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PROBLEMS AND SOLUTIONS
Problems 11719-11725
Solutions 11584, 11588, 11592, 11611, 11615, 11616, 11621, 11603 (correction)
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REVIEWS
The Mathematical Coloring Book by Alexander Soifer
Reviewed by Joseph Malkevitch
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