- *Scott T. Chapman, Editor*

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Volume 122, Issue 10, pp. 916 - 1028

## Table of Contents

## Articles

### The Exponential Map Is Chaotic: An Invitation to Transcendental Dynamics

Zhaiming Shen and Lasse Rempe-Gillen

We present an elementary and conceptual proof that the complex exponential map is chaotic when considered as a dynamical system on the complex plane. (This was conjectured by Fatou in 1926 and first proved by Misiurewicz 55 years later.) The only background required is a first undergraduate course in complex analysis.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.919

### Hanging Around in Non-Uniform Fields

Fred Kuczmarski and James Kuczmarski

We define a family of curves, the n-catenaries, parameterized by the nonzero reals. They include the classical catenaries (n = 1), parabolas (n = 1/2), cycloids (n = −1/2), and semicircles (n = −1). A chain of uniform density in the shape of an n-catenary hangs in equilibrium in the upper half-plane of the nonuniform gravitational field Fn, where the force is parallel to the y-axis and has magnitude proportional to yn−1. An n-catenary is a brachistochrone in F−2n and a trajectory in F2n. For n > 0 the —n-catenary solves a modified isoperimetric problem; it is the shortest of all curves enclosing with the x-axis a region of fixed mass in the upper half-plane of the density field ∑n, where the density is proportional to yn−1. The surface of revolution generated by rotating the —n-catenary in ∑n about the x-axis has a property analogous to the equal area zones property of the sphere; equally spaced planes perpendicular to the x-axis cut out zones of equal mass.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.941

### Strong Divisibility and LCM-Sequences

Andrzej Nowicki

We give a complete characterization of strong divisibility sequences, and we present some consequences of this characterization.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.958

### On Convex Curves Which Have Many Inscribed Triangles of Maximum Area

Jesús Jerónimo Castro

Let K be a convex figure in the plane such that every point x ∈ ∂ K serves as a vertex of an inscribed triangle with maximum area. In this note, we prove a conjecture due to Genin and Tabachnikov that says where T is a triangle with maximum area inscribed in K. Moreover, we prove that the bounds in the left side and the right side of the inequality are obtained only for ellipses and parallelograms, respectively.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.967

### A Graph Partition Problem

Sebastian M. Cioabă and Peter J. Cameron

Given a graph G on n vertices, for which m is it possible to partition the edge set of the m-fold complete graph mKn into copies of G? We show that there is an integer m0, which we call the partition modulus of G, such that the set M(G) of values of m for which such a partition exists consists of all but finitely many multiples of m0. Trivial divisibility conditions derived from G give an integer m1 that divides m0; we call the quotient m0/m1 the partition index of G. It seems that most graphs G have partition index equal to 1, but we give two infinite families of graphs for which this is not true. We also compute M(G) for various graphs and outline some connections between our problem and the existence of designs of various types.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.972

### What Does “Less Than or Equal“ Really Mean?

Guram Bezhanishvili and David Pengelley

The Cantor–Bernstein theorem is often stated as ‘a ≤ b and b ≤ a imply a = b’ for cardinalities. This suggestive form of the theorem may lead to a trap, into which many early 20th century mathematicians fell, unless we are very careful in interpreting ≤. The key is the subtle interplay between < and ≤. Originally, following Cantor, < was considered the primary relation, and ≤ was defined as the disjunction of < and =. However, the above suggestive form of the Cantor–Bernstein theorem requires the modern definition of ≤. The uncertainty, sometimes confusion, and evolution due to these subtleties can fascinate and motivate both us and our students today.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.983

## Notes

### On Products of Uniquely Geodesic Spaces

Mehmet Kılıç and Şahin Koçak

After introducing the geodesic spaces, we prove that the product of two complete, uniquely geodesic spaces with respect to the product metric dp is again a uniquely geodesic space for 1 < p < ∞.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.990

### Equicevian Points of a Triangle

Sadi Abu-Saymeh, Mowaffaq Hajja and Hellmuth Stachel

A point P in the plane of a given triangle ABC is called equicevian if the cevians AAP, BBP, and CCP through P are of equal length. In this note we prove that apart from points on the side lines of ABC, the real and the imaginary focal points of the Steiner circumellipse are the only equicevian points.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.995

### A Generalized Liouville Theorem for Entire Functions

Weimin Peng

Let be a holomorphic function such that for any . We show that if is a complete Riemannian metric, then f must be a constant. As a corollary we give a new proof of the classical Liouville theorem.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.1001

### Yet Another Proof of Poincaré′s Theorem

Atsushi Yamamori

This note gives a concise proof of a classical Poincaré′s theorem which asserts that the unit ball and the polydisk are not holomorphically equivalent.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.1003

### Polya′s Random Walk Theorem Revisited

Kenneth Lange

This note uses a Poisson process embedding to give a simple intuitive proof of Polya′s random walk theorem.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.1005

### A Semi-Finite Proof of Jacobi′s Triple Product Identity

Jun-Ming Zhu

Jacobi′s triple product identity is proved from one of Euler′s q-exponential functions in an elementary way.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.1008

## Problems and Solutions

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.1010

## Book Review

**Birth of a Theorem.** By **Cédric Villani**

Reviewed by Michael Harris

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.10.1018

**Manifold Mirrors: The Crossing Paths of the Arts and Mathematics** By ** Felipe Cucker**

Reviewed by Michael Henle

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.912

## MathBits

### All Triangles at Once

Jaime Gaspar

### A Simple Proof of the Uniform Continuity of Real-Valued Continuous Functions on Compact Intervals

Haryono Tandra

### Newton′s Interpolation Polynomial for the Sums of Powers of Integers

José Luis Cereceda

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

Edited by Vadim Ponomarenko