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.
Journal subscribers and MAA members: Please login into the member portal by clicking on 'Login' in the upper right corner.
ARTICLES
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.263
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.284
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.300
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.313
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.318
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.324
NOTES
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.331
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 $$n|a_{n}$$.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.333
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.337
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.340
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 $$\{x,y,xy,yx\}$$.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.342
PROBLEMS AND SOLUTIONS
Problems 11635-11641
Solutions 11487, 11488, 11490, 11496, 11498, 11500, 11503, 11505
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.344
REVIEWS
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
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.04.352