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American Mathematical Monthly Contents—April 2012

American Mathematical Monthly

In April ’s American Mathematical Monthly, we will see how the geometry of submajorization has applications to diverse areas such as music and economics, why the "no cloning theorem" of quantum physics must be true, how a drill bit can drill a practically square hole, how to sum reciprocals of polynomials over finite fields, conditions under which the card game of "war" must be finite, and some limit representations of Riemann's zeta function. Our notes consider prime divisors of thin sequences, a familiar recurrence relation viewed from a different angle, how to toss coins to guess a secret number, a generalization of the Cayley-Hamilton Theorem, and a noncommutative Schur theorem. Bonnie Shulman reviews Population Games and Evolutionary Dynamics by William H. Sandholm, and, as always, our Problem Section will keep you thinking.

Vol. 119, No. 4, pp.263-348.

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Submajorization and the Geometry of Unordered Collections

Rachel Wells Hall and Dmitri Tymoczko

In this paper, we use submajorization to compare distances between either multisets of real numbers or multisets modulo translation on the real line. We provide a geometrical interpretation in which multisets are represented by points in an orbifold and bijections between multisets are represented by paths between these points. This interpretation shows that submajorization is closely related to the geometrical principle that the shortest path between two points is a straight line. Our results have applications to diverse problems from economics to music theory; moreover, they suggest generalizations of statistical measures of the center and spread of a distribution.

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Unknowable Matters: How Nature’s Speed Limit on Communication Relates to Quantum Physics

Stephen McAdam

It is shown that there is no way of identifying an unknown polarization state of a photon by proving that otherwise faster than light communication would be possible. It is similarly argued that the ‘no cloning theorem’ of quantum physics must be true.

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Drilling for Polygons

Barry Cox and Stan Wagon

We solve the problem of designing a simple device that uses rotary motion to drill a hole with a cross-section that is a regular polygon with an odd number of sides: the main idea is to use a polygonal trammel and a family of rotors. By using different rotors, one can produce a hole with a cross-section that is in any proportion of the trammel size from zero to exactly one. The key geometric idea is a result about the envelope of an edge of a triangle that rotates so that the other two edges maintain tangential contact with two fixed circles.

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Sums of Reciprocals of Polynomials over Finite Fields

Kenneth Hicks, Xiang-dong Hou, and Gary L. Mullen

We consider the sum of the reciprocals of all monic polynomials of a given degree over a finite field $$\mathbb{F}_{q}$$ each raised to the power of $$k$$. When $$k=q$$, the sum has a surprisingly simple result due to mysterious cancellations that occur in the sum. We discuss this interesting phenomenon and provide a new inductive proof.

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On Finiteness in the Card Game of War

Evgeny Lakshtanov and Vera Roshchina

The game of war is a popular international children’s card game. In the beginning of the game, the deck is split into two parts, then each player reveals their top card. The player having the highest card collects both and returns them to the bottom of their hand. The player left with no cards loses. It is often wrongly assumed that this game is deterministic and the result is set once the cards have been dealt. However, this is not so; the rules of the game do not prescribe the order in which the winning player will place their cards on the bottom of the hand. First, we provide an example of a cycling game with fixed rules and then assume that each player can seldom but regularly change the returning order. We have proved that in this case the mathematical expectation of the length of the game is finite. In principle it is equivalent to the graph of the game, which has edges corresponding to all acceptable transitions, having the following property: from each initial configuration there is at least one path to the end of the game.

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Limit Representations of Riemann’s Zeta Function

Djurdje Cvijovic and Hari M. Srivastava

In this article, it is shown that Riemann's zeta function $$\zeta(s)$$ admits two limit representations when $$\mathfrak{R}(s)>1$$. Each of these limit representations is deduced by using simple arguments based upon the classical Tannery's (limiting) theorem for series.

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Prime Divisors of Thin Sequences

Christian Elsholtz

For a large family of sequences, including quite thin sequences, we show that the set of primes dividing some member of the sequence is infinite.

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A Familiar Recurrence Occurs Again

I. E. Leonard and A. C. F. Liu

An explicit solution is given to a familiar third order recurrence relation

$$a_{n+3}=a_{n+1}+a_{n}$$, $$n\geq0$$, $$a_{0}=3$$, $$a_{1}=0$$, $$a_{3}=2$$

A proof using elementary number theory is given to show that $$a_{n}$$ is prime-divisible. That is, if $$n$$ is prime, then $$n|a_{n}$$.

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Tossing Coins to Guess a Secret Number

Robbert Fokkink

About fifty years ago Ed Gilbert proposed a guessing game that remains unsolved and has largely been forgotten. The aim of this note is to revive interest in the game.

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A Generalization of the Cayley-Hamilton Theorem

Lizhou Chen

Let $$A=[a_{ij}]_{n\times n}$$ and $$B=[b_{ij}]_{n\times n}$$ be two commuting square matrices of order $$n$$ over an arbitrary commutative ring. We prove that the determinant of the matrix $$[b_{ij}A-_{ij}B]_{n\times n}$$ which is regarded as an $$n\times n$$ block matrix with pairwise commuting entries, is exactly equal to the $$n\times n$$ zero matrix. If $$B$$ is the identity matrix, then the result is equivalent to the Cayley-Hamilton theorem.

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Monochromatic Permutation Quadruples—A Schur Thing in $$S_{n}$$

R. McCutcheon

Schur proved that for any finite partition of the naturals, some cell contains two numbers and their sum. We use Ramsey's theorem to prove a noncommutative Schur theorem for permutation quadruples $$\{x,y,xy,yx\}$$.

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Problems 11635-11641
Solutions 11487, 11488, 11490, 11496, 11498, 11500, 11503, 11505

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Population Games and Evolutionary Dynamics. William H. Sandholm. MIT Press, Cambridge, MA, 2011, xxv + 589 pp., ISBN 978-0-262-19587-4, $65.
Reviewed by Bonnie Shulman

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