"Can you hear the November Monthly?" This question is motivated by our lead article "The Sound of Symmetry" by Zhiqin Lu and Julie Rowlett. This paper explores many open problems inspired by the classic 1966 Monthly paper by M. Kac “Can one hear the shape of a drum?” Do you think zero isn't interesting? Zbigniew Nitecki may change your mind in his paper "Cantorvals and Subsum Sets of Null Sequences." In our book reviews, Robin Hartshorne takes a look at Leila Schneps' *Alexandre Grothendieck: A Mathematical Portrait*. Our Problem Section will keep you busy until final exams next month. We will end 2015 with a "bang" as Zhaiming Shen and Lasse Rempe-Gillen give us a new elementary view of chaos in "The Exponential Map Is Chaotic: An Invitation to Transcendental Dynamics." - *Scott T. Chapman, Editor*

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Volume 122, Issue 09, pp. 813 - 915

## Table of Contents

## Articles

### The Sound of Symmetry

Zhiqin Lu and Julie Rowlett

The inverse spectral problem was popularized by M. Kac's 1966 article in THIS MONTHLY “Can one hear the shape of a drum?” Although the answer has been known for over twenty years, many open problems remain. Intended for general audiences, readers are challenged to complete exercises throughout this interactive introduction to inverse spectral theory. The main techniques used in inverse spectral problems are collected and discussed, then used to prove that one *can*hear the shape of: parallelograms, acute trapezoids, and the regular *n*-gon. Finally, we show that one can *realistically* hear the shape of the regular *n*-gon amongst all convex *n*-gons because it is uniquely determined by a finite number of eigenvalues; the sound of symmetry can really be heard!

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.815

### Convex Polygons and Common Transversals

Péter Hajnal, László I. Szabó and Vilmos Totik

It is shown that if two planar convex n-gons are oppositely oriented, then the segments joining the corresponding vertices have a common transversal. A different formulation is also given in terms of two cars moving along two convex curves in opposite directions. Some possible analogs in 3-space are also considered, and an example is shown that the full analog is not true in this setting.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.836

### Cayley–Bacharach Formulas

Qingchun Ren, Jürgen Richter-Gebert and Bernd Sturmfels

The Cayley–Bacharach theorem states that all cubic curves through eight given points in the plane also pass through a unique ninth point. We write that point as an explicit rational function in the other eight.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.845

### Kronecker's Diophantine Approximation and the Asymptotics of Solutions of Difference Equations

J. Roberto Hasfura-Buenaga

The asymptotic properties of the solutions of a class of linear, difference equations have been described—in increasing level of refinement—by many authors including H. Poincaré, O. Perron, and, finally, M. Pituk. In this paper we provide a proof of the constantcoefficient case of Pituk's theorem based on the recurrence properties of rotations of the torus.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.855

### Cantorvals and Subsum Sets of Null Sequences

Zbigniew Nitecki

A sequence of real numbers converging to zero need not be summable, but it has many summable subsequences. The set of sums of all summable (infinite, finite, or empty) subsequences is a closed set of real numbers which we call the *subsum* set of the sequence. When the sequence is not absolutely summable, its subsum set is an unbounded closed interval which includes zero. The subsum set of an absolutely summable sequence is one of the following: a finite union of (nontrivial) compact intervals, a Cantor set, or a “symmetric Cantorval,” a hybrid Cantor-like set with both trivial and nontrivial components.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.862

### Sets Closed Under the Division Algorithm

Robert O. Stanton

This paper examines subsets of the nonnegative integers that are closed under the division algorithm.We determine all infinite sets of this form. Although classification of finite sets remains open, we make progress in this direction.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.871

### Characterizing the Number of *m*-ary Partitions Modulo *m*

George E. Andrews

Motivated by a recent conjecture of the second author related to the ternary partition function, we provide an elegant characterization of the values *b*_{m}(*mn*) modulo *m* where *b**m*(*n*) is the number of*m*-ary partitions of the integer *n* and *m* ≥ 2 is a fixed integer.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.880

## Notes

### A Product of Nested Radicals for the AGM

Thomas J. Osler

The arithmetic-geometric mean of two positive numbers *a* and *b* (*AGM* (*a*, *b*)) is the common limit of two sequences generated by an iterative process. This has proven to be an important device for calculating numbers and function in recent years. In this paper, we derive an infinite product representation for the *AGM*. The factors of this product are nested radicals resembling Vieta's famous product for pi.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.886

### Computing Some Integrals by Computing Derivatives

Rafael de la Llave

We show how to compute the standard integrals ∫ *e*^{at}cos(*bt*) *dt*, ∫ *e*^{at}sin(*bt*) *dt*, and others just by computing derivatives..

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.888

### A Lemma of Nakajima and Süss on Convex Bodies

Dmitry Ryabogin

Let *K* and *L* be two convex bodies in ℝ^{n} such that their projections onto every (*n* − 1)-dimensional subspace are translates of each other. Then *K* is a translate of *L*. We give a very simple analytic proof of this fact.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.890

### A Limit Comparison Test for General Series

Nguyen S. Hoang

The well-known limit comparison test is only applicable for series with nonnegative terms. Thus, it can be used only for proving or disproving the absolute convergence of a series. In this note we formulate and justify a modified version of the limit comparison test for general series. The test can be used to prove the convergence of conditionally convergent series.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.893

### A New Proof of ℵ_{0}-Resolvability of a Metric Space Without an Isolated Point

Hamid Reza Daneshpajouh

A topological space is *k*-resolvable if *X* has *k* disjoint dense subsets. In this paper we shall give a new proof for ℵ_{0}-resolvability of each metric space without an isolated point.

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.897

## Problems and Solutions

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.899

## Book Review

**Alexandre Grothendieck: A Mathematical Portrait** Ed. By ** Leila Schneps**

Reviewed by Robin Hartshorne

To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.9.907

**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

### Correction to “Napoleon Polygons

Titu Andreescu, Vladimir Georgiev and Oleg Mushkarov

### A New Proof of Euler's Inequality

Elias Lampakis

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

Edited by Vadim Ponomarenko