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American Mathematical Monthly - May 2017

As the academic year winds down, the May issue of the Monthly should provide you with plenty of mathematics to contemplate.

Pick a partner and hop like frogs from lily pad to lily pad so that the distance between you and your partner always strictly decreases; a player who cannot move is the loser of this game. What can be said about optimal strategies for this and similar games? If you aren’t feeling particularly jumpy, then consider exploring Holditch’s theorem about closed convex curves and the “mysterious” Holditch ellipse, or use the inclusion–exclusion principle to generalize the Steiner–Routh theorem about triangles to general simplicies, or understand the structure of the set of “openness points” of a continuous map between metric spaces.

In the Notes section, determine the greatest common divisor of the values of two monic integer polynomials, see how generalized Chebyshev polynomials can generate Stern’s diatomic sequence, construct rings lying between the polynomial rings R[x] and R[x,y] that are not finitely generated over R[x], and enjoy a new geometric proof of the Siebeck–Marden theorem relating the roots of a complex cubic polynomial to those of its derivative. There are, as ever, problems to solve, even a crossword puzzle. And Reuben Hersh reviews John Stilwell’s Elements of Mathematics: From Euclid to Gödel.

  — Susan Jane Colley, Editor

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Table of Contents

Friendly Frogs, Stable Marriage, and the Magic of Invariance

p. 387.

Maria Deijfen, Alexander E. Holroyd, and James B. Martin

We introduce a two-player game involving two tokens located at points of a fixed set. The players take turns moving a token to an unoccupied point in such a way that the distance between the two tokens is decreased. Optimal strategies for this game and its variants are intimately tied to Gale–Shapley stable marriage. We focus particularly on the case of random infinite sets, where we use invariance, ergodicity, mass transport, and deletion-tolerance to determine game outcomes.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.5.387

Holditch's Ellipse Unveiled

p. 403.

Juan Monterde and David Rochera

In plane geometry, Holditch’s theorem states that if a chord of fixed length is allowed to rotate inside a convex closed curve, then the locus of a point on the chord a distance p from one end and a distance q from the other is a closed curve whose area is less than that of the original curve by πpq. In this article we obtain, first, sufficient conditions to ensure the existence of the Holditch curve and, second, a version of Holditch’s theorem for convex polygons where the ellipse involved is explicitly shown.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.5.403

On the Steiner-Routh Theorem for Simplices

p. 422.

František Marko and Semyon Litvinov

It is shown in [28] that, using only tools of elementary geometry, the classical Steiner–Routh theorem for triangles can be fully extended to tetrahedra. In this article, we first give another proof of the Steiner–Routh theorem for tetrahedra, where methods of elementary geometry are combined with the inclusion–exclusion principle. Then we generalize this approach to (n − 1)-dimensional simplices. A comparison with the formula obtained using vector analysis yields an interesting algebraic identity.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.5.422

Local Extrema and Nonopeness Points of Continuous Functions

p. 436.

Marek Balcerzak, Michał Popławski, and Julia Wódka

We give a short argument showing that the set of openness points of a continuous function from a metric space X into a metric space Y is of type Gδ . If X is locally connected and Y := ℝ, the set of nonopenness points (of type Fσ ) coincides with the set of points of extrema of f . We discuss which Fσ sets can be equal to the set of points of extrema for a continuous function f from ℝ into ℝ, and we present a short survey of the known results on this topic.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.5.436

Notes

On the Greatest Common Divisor of the Value of Two Polynomials

p. 446.

Péter E. Frenkel and József Pelikán

We show that if two monic polynomials with integer coefficients have a square-free resultant, then all positive divisors of the resultant arise as the greatest common divisor of the values of the two polynomials at a suitable integer.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.5.446

The Stern Diatomic Sequence via Generalized Chebyshev Polynomials

p. 451.

Valerio De Angelis

Let a(n) be the Stern diatomic sequence, and let x1,..., xr be the distances between successive 1’s in the binary expansion of the (odd) positive integer n. We show that a(n) is obtained by evaluating generalized Chebyshev polynomials when the variables are given the values x1 + 1,..., xr + 1. We also derive a formula expressing the same polynomials in terms of sets of increasing integers of alternating parity and derive a determinant representation for a(n).

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.5.451

Subalgebras of a Polynomial Ring That Are Not Finitely Generated

p. 456.

Melvyn B. Nathanson

Let R1 be a commutative ring, let R2 be a finitely generated extension ring of R1, and let S be a ring that is intermediate between R1 and R2. For R1 = R[x] and R2 = R[x, y], there are simple combinatorial constructions of intermediate rings S that are not finitely generated over R[x].

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.5.456

A Geometric Proof of the Siebeck-Marden Theorem

p. 459.

Benjamin Bogosel

The Siebeck–Marden theorem relates the roots of a third degree polynomial and the roots of its derivative in a geometrical way. A few geometric arguments imply that every inellipse for a triangle is uniquely related to a certain logarithmic potential via its focal points. This fact provides a new direct proof of a general form of the result of Siebeck and Marden.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.5.459

Problems and Solutions

p. 465.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.5.465

Book Review

p. 475.

Elements of Mathematics: From Euclid to Gödel by John Stillwell

Reviewed by Reuben Hersh

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.5.475

End Notes

p. 480.

MathBits

100 Years Ago This Month in The American Mathematical Monthly

p. 421.

Mathematical Evolution

p. 444, 479.

A More Direct Proofs of the Mean Value Theorem

p. 464.

Dummy View - NOT TO BE DELETED