Vol. 121, No. 4, pp.283-376.

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## ARTICLES

### Periodicity Domains and the Transit of Venus

Andrew J. Simoson

A transit of Venus occurs when it passes directly between the Earth and the Sun. A straightforward linear algebraic model for the orbits of Earth and Venus—essentially using one parameter, namely, the relative angular velocity $$\sigma$$ of Venus—is powerful enough to generate respectable transit year predictions. We generalize, allowing $$\sigma$$ to vary; uncover an algebraic analog for predicting transits; and show that time cycles for transits are what they are because each $$\sigma$$ is sufficiently close to a suitably simple rational number, which for Venus is 13/8 , and which in turn induces a modulo 8 shuffling of successive transit years by a factor of 3.

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### A Drug-Induced Random Walk

Daniel J. Velleman

The label on a bottle of pills says “Take one half pill daily.” A natural way to proceed is as follows: Every day, remove a pill from the bottle at random. If it is a whole pill, break it in half, take one half, and return the other half to the bottle; if it is a half pill, take it. We analyze the history of such a pill bottle.

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### Analytical Solution for the Generalized Fermat–Torricelli Problem

Alexei Yu. Uteshev

We present an explicit analytical solution for the problem of minimization of the function $$F(x,y)=\sum_{j=1}^{3}m_{j}\sqrt{(x-x_{j})^{2}+(y-y_{j})^{2}}$$, i.e., we find the coordinates of the stationary point and the corresponding critical value as functions of $$\{m_{j},x_{j},y_{j}\}^{3}_{j=1}$$. In addition, we also discuss the inverse problem of finding such values for $$m_{1}$$, $$m_{2}$$, and $$m_{3}$$ for which the corresponding function $$F$$ possesses a prescribed position of stationary point.

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### Mathbit: A One-Sentence Line-of-Sight Proof of the Extreme Value Theorem

Samuel J. Ferguson

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### On the Proof of the Existence of Undominated Strategies in Normal Form Games

Martin Kovár and Alena Chernikava

In the game theory literature, there are two versions of the proof of the well-known fact that in a normal form game of $$n$$ persons with compact spaces of strategies and continuous utility functions, the sets of undominated strategies are nonempty. The older one, stated in the first edition of the well-known book by Herve Moulin, depends on certain, relatively nontrivial results from measure theory, metric topology, and mathematical analysis. The proof is valid only for metrizable topological spaces. The second, revised edition of the same book contains a simplified proof, which is, however, incorrect. The author implicitly assumes that any linearly ordered set contains a cofinal subsequence, which is certainly not true. In this paper we correct, simplify, and generalize the second proof of Moulin by its reformulation in terms of topological convergence of nets. This modified technique also yields a slightly better result than is stated in the original. The assertion now holds for almost compact spaces. The argument used is elementary and easily understandable to non-experts.

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### An Asymptotic Formula for $$(1 + 1/x)^{x}$$ Based on the Partition Function

Chao-Ping Chen and Junesang Choi

We present a method to produce estimations of the natural logarithmic constant $$e$$, accurate to as many decimal places as we desire. The method is based on an asymptotic formula for $$(1 + 1/x)^{x}$$, which uses the partition function.

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## NOTES

### Stirling’s Approximation for Central Extended Binomial Coefficients

Steffen Eger

We derive asymptotic formulas for central extended binomial coefficients, which are generalizations of binomial coefficients, using the distribution of the sum of independent discrete uniform random variables with the Central Limit Theorem and a local limit variant.

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### A New Proof of Stirling’s Formula

Thorsten Neuschel

A new, simple proof of Stirling’s formula via the partial fraction expansion for the tangent function is presented.

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### Zeta(2) Once Again

Ralph M. Krause

This note provides a strikingly efficient evaluation of zeta(2).

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### Polynomials $$(x^{3} – n)(x^{2} + 3)$$ Solvable Modulo Any Integer

Andrea M. Hyde, Paul D. Lee, and Blair K. Spearman

We give an infinite family of polynomials that are solvable modulo $$m$$ for every integer $$m > 1$$, yet have no roots in the rational numbers. Such polynomials are called intersective. Our classification uses only techniques available in an undergraduate course in number theory.

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### Macaulay Expansion

B. Sury

Given natural numbers $$n$$ and $$r$$ , the “greedy” algorithm enables us to obtain an expansion of the integer $$n$$ as a sum of binomial coefficients in the form $$\left(\begin{array}{c}a_{r}\\r\end{array}\right)+\left(\begin{array}{c}a_{r-1}\\r-1\end{array}\right)+\cdots+\left(\begin{array}{c}a_{1}\\1\end{array}\right)$$. We give an alternate interpretation of this expansion, which also proves its uniqueness in an interesting manner.

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### Evaluating Lebesgue Integrals Efficiently with the FTC

J. J. Koliha

This note addresses evaluation of Lebesgue integrals on the real line using the Fundamental Theorem of Calculus, without having to verify that the primitive is absolutely continuous.

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## PROBLEMS AND SOLUTIONS

Problems 11768-11774

Solutions 11628, 11631, 11633, 11637, 11643, 11645, 11646, 11648

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## REVIEWS

*Encounters with Chaos and Fractals*, 2nd edition. By Denny Gulick.

Reviewed by Jeffrey Nunemacher

To purchase from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.04.373