The August-September* Monthly* takes a new look at two classic results. Picard's Theorem for holomorphic functions on the complex plane is considered in detail in Lawrence Zalcman's "A Tale of Three Theorems," and a fundamental result of geometry is celebrated in Dennis Ivanov and Serge Tabachnikov's "The Six Circle Theorem Revisited." Matthew Richey reviews "An Introduction to Statistical Learning with Applications in R" by Gareth James, Daniela Witten, Trevor Hastie, and Robert Tibshirani, and our Problem Section is as challenging as ever. Stay tuned for the October* Monthly*, where Jeffrey Lagarias will offer us an up to date view of the "3x+1 Problem."

- *Scott T. Chapman, Editor*

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## Table of Contents

### The Range of a Rotor Walk

Laura Florescu, Lionel Levine and Yuval Peres

In *rotor walk *on a graph, the exits from each vertex follow a prescribed periodic sequence. We show that any rotor walk on the *d*-dimensional lattice ℤ^{d} visits at least on the order of *t*^{d/}^{(d+1)} distinct sites in *t* steps. This result extends to Eulerian graphs with a volume growth condition. In a *uniform* rotor walk, the first exit from each vertex is to a neighbor chosen uniformly at random. We prove a shape theorem for the uniform rotor walk on the comb graph, showing that the size of the range is of order *t*^{2/3} and the asymptotic shape of the range is a diamond. Using a connection to the mirror model, we show that the uniform rotor walk is recurrent on two different directed graphs obtained by orienting the edges of the square grid: the Manhattan lattice and the *F*-lattice.We end with a short discussion of the time it takes for rotor walk to cover a finite Eulerian graph.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.7.627

### A Tale of Three Theorems

Lawrence Zalcman

This article presents a bird′s eye view of the celebrated theorems of Picard and their close relatives, with particular emphasis on some surprising developments of the past half century.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.7.643

### An Electric Network for Nonreversible Markov Chains

Márton Balázs and Áron Folly

We give an analogy between nonreversible Markov chains and electric networks much in the flavor of the classical reversible results originating from Kakutani and later Kem´eny-Snell-Knapp and Kelly. Nonreversibility is made possible by a voltage multiplier—a new electronic component. We prove that absorption probabilities, escape probabilities, expected number of jumps over edges, and commute times can be computed from electrical properties of the network as in the classical case. The central quantity is still the effective resistance, which we do have in our networks despite the fact that individual parts cannot be replaced by a simple resistor. We rewrite a recent nonreversible result of Gaudilli`ere-Landim about the Dirichlet and Thomson principles into the electrical language. We also give a few tools that can help in reducing and solving the network. The subtlety of our network is, however, that the classical Rayleigh monotonicity is lost.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.7.657

### A Characteristic Averaging Property of the Catenary

Vincent E. Coll and Jeff Dodd

It is well-known that the catenary is characterized by an extremal centroidal condition: It is the shape of the curve whose centroid is the lowest among all curves having a prescribed length and specified endpoints. Here, we establish a broad characteristic averaging property of the centenary that yields two new centroidal characterizations.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.7.683

### The Six Circles Theorem Revisited

Dennis Ivanov and Serge Tabachnikov

The six circles theorem of C. Evelyn, G. Money-Coutts, and J. Tyrrell concerns chains of circles inscribed into a triangle: the first circle is inscribed in the first angle, the second circle is inscribed in the second angle and tangent to the first circle, the third circle is inscribed in the third angle and tangent to the second circle, and so on, cyclically. The theorem asserts that if all the circles touch the sides of the triangle, and not their extensions, then the chain is 6-periodic. We show that, in general, the chain is eventually 6-periodic but may have an arbitrarily long pre-period.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.7.689

## Notes

### Cayley′s Formula: A Page From The Book

Arnon Avron and Nachum Dershowitz

We present a simple proof of Cayley′s formula.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.7.699

### On Tangents and Secants of Infinite Sums

Michael Hardy

We prove some identities involving tangents, secants, and cosecants of infinite sums.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.7.701

### Factorization of a Matrix Differential Operator Using Functions in Its Kernel

Alex Kasman

Just as knowing some roots of a polynomial allows one to factor it, a well-known result provides a factorization of any scalar differential operator given a set of linearly independent functions in its kernel. This note provides a straightforward generalization to the case of matrix coefficient differential operators.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.7.704

### Computing ζ (2m) by using Telescoping Sums

Brian D. Sittinger

In this article, we give another proof for the closed form of ζ (2m) inspired by the elementary telescoping sum proof for ζ (2) given by Daners [**3**]. This proof, which begins with recurrence relations derived from certain integrals by using integration by parts, yields a identity giving the value of ζ (2m) in terms of ζ (2), ζ (4), …, ζ (2*m* − 2). A quick proof by induction yields the closed form of ζ (2*m*).

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.7.710

### Generating Iterated Function Systems for the Vicsek Snowflake and the Koch Curve

Yuanyuan Yao and Wenxia Li

We determine all generating iterated function systems for certain self-similar sets such as the Vicsek snowflake and the Koch curve.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.7.716

## Problems and Solutions

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.7.722

## Book Review

*An Introduction to Statistical Learning with Applications in R* by Gareth James, Daniela Witten, Trevor Hastie, and Robert Tibshirani

Reviewed by Matthew Richey

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.7.731

## MathBits

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

Edited by Vadim Ponomarenko