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American Mathematical Monthly Contents—December 2012

This month’s highlights include the article “Killer Problems,” which examines a select group of problems used on entrance exams at Moscow State University to keep students of certain ethnic groups from gaining admission. Income inequality is addressed in another article that explains how to split the Gini index in two. Notes concern hyperplane sections of the $$n$$-dimensional cube, tilings of Hamiltonian cycles, and a toy weather forecast model. Gerald Alexanderson reviews Transcending Tradition: Jewish Mathematicians in German Speaking Academic Culture, by Birgit Bergmann. —Scott Chapman

Vol. 119, No. 10, pp.815-906.

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Killer Problems

Tanya Khovanova and Alexey Radul

This is a special collection of problems that were given to select applicants during oral entrance exams to the Department of Mechanics and Mathematics of Moscow State University. These problems were designed to prevent Jewish candidates and other “undesirables” from getting a passing grade, thus preventing them from studying at MSU. Among problems that were used by the department to blackball unwanted candidate students, these problems are distinguished by having a simple solution that is usually difficult to find. Using some problems with a simple solution protected the administration from extra complaints and appeals. This collection, therefore, has mathematical as well as historical value.

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On the Mathematics of Income Inequality: Splitting the Gini Index in Two

Robert T. Jantzen and Klaus Volpert

Income distribution is described by a two-parameter model for the Lorenz curve. This model interpolates between self-similar behavior at the low and high ends of the income spectrum, and naturally leads to two separate indices describing both ends individually. These new indices accurately capture realistic data on income distribution, and give a better picture of how income data is shifting over time.

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The Extraordinary SVD

Carla D. Martin and Mason A. Porter

The singular value decomposition (SVD) is a popular matrix factorization that has been used widely in applications ever since an efficient algorithm for its computation was developed in the 1970s. In recent years, the SVD has become even more prominent due to a surge in applications and increased computational memory and speed. To illustrate the vitality of the SVD in data analysis, we highlight three of its lesser-known yet fascinating applications. The SVD can be used to characterize political positions of congressional representatives, measure the growth rate of crystals in igneous rock, and examine entanglement in quantum computation. We also discuss higher-dimensional generalizations of the SVD, which have become increasingly crucial with the newfound wealth of multidimensional data, and have launched new research initiatives in both theoretical and applied mathematics. With its bountiful theory and applications, the SVD is truly extraordinary.

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An Elementary View of Weyl’s Theory of Equal Distribution

William F. Trench

Suppose that $$-\infty<a<b<\infty$$, $$a\leq u_{1n}\leq u_{2n}\leq \cdots \leq u_{nn}\leq b$$, and $$a\leq v_{1n}\leq v_{2n}\leq \cdots \leq v_{nn}\leq b$$ for $$n\geq1$$. We simplify and strengthen Weyl’s definition of equal distribution of $$\{\{u_{in}\}^{n}_{i=1}\}^{\infty}_{n=1}$$ and $$\{\{v_{in}\}^{n}_{i=1}\}^{\infty}_{n=1}$$by showing that the following statements are equivalent:

(i) $$\lim_{n\to\infty}\frac{1}{n}\displaystyle\sum_{i=1}^{n}(F(u_{in})-F(v_{in}))=0$$ for all $$F\in C[a,b]$$,
(ii) $$\lim_{n\to\infty}\frac{1}{n}\displaystyle\sum_{i=1}^{n}|u_{in}-v_{in}|=0$$,
(iii) $$\lim_{n\to\infty}\frac{1}{n}\displaystyle\sum_{i=1}^{n}|F(u_{in})-F(v_{in})|=0$$ for all $$F\in C[a,b]$$. 

We relate this to Weyl’s definition of uniform distribution and Szegö’s distribution formula for the eigenvalues of a family of Toeplitz matrices $$\{[t_{r-s}]^{n}_{r,s=1}\}^{\infty}_{n=1}$$ where $$t_{r}=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-irx}g(x)dx$$ and $$g$$ is real-valued and continuous on $$[-\pi,\pi]$$.

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On the Elementary Symmetric Functions of $$1,1/2,\dots,1/n$$

Yong-Gao Chen and Min Tang

In 1946, P. Erdös and I. Niven proved that there are only finitely many positive integers $$n$$ for which one or more of the elementary symmetric functions of $$1,1/2,\dots,1/n$$ are integers. In this paper we prove that if $$n\geq4$$, then none of the elementary symmetric functions of $$1,1/2,\dots,1/n$$ are integers.

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Hyperplane Sections of the $$n$$-Dimensional Cube

Rolfdieter Frank and Harald Riede

We deduce an elementary formula for the volume of arbitrary hyperplane sections of the $$n$$-dimensional cube and show its application in various dimensions.

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Tiling Hamiltonian Cycles on the 24-Cell

Jacob A. Siehler

We present a construction for tiling the 24-cell with congruent copies of a single Hamiltonian cycle, using the algebra of quaternions.

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A Toy Forecast Model

Witold Sadowski

We present a deterministic toy forecast model and give an elementary proof of why it is not reliable when used to make long-term predictions.

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Problems 11677-11683
Solutions 11530, 11533, 11537, 11538, 11539, 11545, 11546, 11553

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Transcending Tradition: Jewish Mathematicians in German-Speaking Academic Culture. By Birgit Bergmann, Moritz Epple, and Ruti Ungar, editors. Springer Verlag, Berlin and Heidelberg, 2012, xi + 289 pp., ISBN 978-3-642-22463-8, $49.95.

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