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
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