The August/September *Monthly* will not let you down as you prepare for the fall semester. All the terms you learned in your first real analysis course are explained in "A Pedagogical History of Compactness" by Manya Raman-Sundström. Also, don't miss "Divisors of the Middle Binomial Coefficient," by Carl Pomerance. Joseph Malkeitch reviews "The Mathematical Coloring Book" by Alexander Soifer, and our Problem Section will get you up to speed after your summer vacation. WOW, hang on to your hats as "California" Jack Cassidy shows us in October how to bluff in Texas hold-em. - *Scott T. Chapman, Editor*

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Volume 122, Issue 07, pp. 617 - 712

## Table of Contents

## Articles

### A Pedagogical History of Compactness

Manya Raman-Sundström

This paper traces the history of compactness from the original motivating questions through the development of the definition to a characterization of compactness in terms of nets and filters. The goal of the article is to clarify the central concepts of open-cover and sequential compactness, including details that a standard textbook treatment tends to leave out.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.7.619

### Divisors of the Middle Binomial Coefficient

Carl Pomerance

We study some old and new problems involving divisors of the middle binomial coefficient .

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.7.636

### Isosynchronous Paths on a Rotating Surface

Neil Ashby

In special relativity, clock networks may be self-consistently synchronized in an inertial frame by slowly transporting clocks, or by exchanging electromagnetic signals between network nodes. However, clocks at rest in a rotating coordinate system—such as on the surface of the rotating earth—cannot be self-consistently synchronized by such processes, due to the Sagnac effect. Discrepancies that arise are proportional to the area swept out by a vector from the rotation axis to the portable clock or electromagnetic pulse, projected onto a plane normal to the rotation axis. This raises the question whether paths of minimal or extremal length can be found, for which the Sagnac discrepancies are zero. This paper discusses the variational problem of finding such “isosynchronous” paths on rotating discs and rotating spheres. On a disc, the problem resembles the classical isoperimetric problem and the paths turn out to be circular arcs. On a rotating sphere, however, between any two endpoints there are an infinite number of extremal paths, described by elliptic functions.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.7.645

### Parking Functions, Shi Arrangements, and Mixed Graphs

Matthias Beck, Ana Berrizbeitia, Michael Dairyko, Claudia Rodriguez, Amanda Ruiz, and Schuyler Veeneman

The *Shi arrangement* is the set of all hyperplanes in ℝ^{n} of the form *x*_{j} − *x*_{k} = 0 or 1 for 1 ≤ *j* < *k* ≤ *n*. Shi observed in 1986 that the number of regions (i.e., connected components of the complement) of this arrangement is (*n* + 1)^{n−1}. An unrelated combinatorial concept is that of a *parking function*, i.e., a sequence (*x*_{1}, *x*_{2}, …, *x*_{n}) of positive integers that, when rearranged from smallest to largest, satisfies *x*_{k} ≤ *k*. (There is an illustrative reason for the term *parking function*.) It turns out that the number of parking functions of length *n* also equals (*n* + 1)^{n−1}, a result given by Konheim and Weiss in 1966. A natural problem consists of finding a bijection between the *n*-dimensional Shi arrangement and the parking functions of length *n*. Pak and Stanley (1996) and Athanasiadis and Linusson (1999) gave such (quite different) bijections. We will shed new light on the former bijection by taking a scenic route through certain mixed graphs.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.7.660

### A Family of Eventually Expanding Piecewise Linear Maps of the Interval

Pablo G. Barrientos

We show that for every 0 < *s* ≤ 1 < *p* the piecewise linear map *f*_{s,p} on the interval [0, 1] given by an expanding branch *px* for 0 ≤ *x* ≤ 1/*p*, and a contracting branch *sx* − *s*/*p* for 1/*p* < *x* ≤ 1, is eventually piecewise expanding and exact.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.7.674

## Notes

### Deleting Edges from Ramsey Minimal Examples

Robert Cowen

If the Ramsey number *r*(*s*, *t*) = *p*, we shall call the complete graph *K*_{p}, a *Ramsey minimal example*. We prove that deleting an edge from a Ramsey minimal example destroys the Ramsey property; that is, the edges of the reduced graph can be colored, red or blue, without having a red *K*_{s} or a blue *K*_{t}.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.7.681

### A Double Binomial Sum: Putnam 35-A4 Revisited

Hans J. H. Tuenter

We show that a double binomial sum identity that arises in the context of Hadamard matrices can be reduced to a convolution over a simpler binomial sum that was featured in the 1974 Putnam Mathematical Competition. The proof uses the fact that these binomial sums can be interpreted as moments of a symmetric Bernoulli random walk.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.7.684

### An Elementary Treatment of the Construction of the Free Product of Groups

James E. McClure and Alec McGail

We consider the elementary part of the theory of free products of groups.We review the history and then give an elementary proof that the multiplication in the free product is associative. This proof originated in a suggestion by the second author and has not been noticed, to our knowledge, by the experts.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.7.690

### Euler’s Product Expansion for the Sine: An Elementary Proof

Óscar Ciaurri

We provide, by using elementary tools, a new proof of Euler's product expansion for the sine.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.7.693

### On Xiang’s Observations Concerning the Cauchy-Schwarz Inequality

Edward Omey

We give a simple proof of an improved version of the Cauchy–Schwarz inequality for vectors and for functions, first observed by Xiang. Also we improve an inequality of Minkowski.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.7.696

## Problems and Solutions

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.7.700

## Book Review

*The Mathematical Coloring Book *By Alexander Soifer

Reviewed by Joseph Malkevitch

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.7.708

## MathBits

### A New Fibonacci–Lucas Relation

Diego Marques

### Wallis's Product and the Central Binomial Coefficient

Michael D. Hirschhorn

### The Dramatic and Ultimate Shortening of a Doctoral Dissertation in Mathematics

Samuel G. Moreno

### 100 Years Ago This Month in *the American Mathematical Monthly*

Edited by Vadim Ponomarenko