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American Mathematical Monthly - April 2018

The April issue of the Monthly blooms with all sorts of interesting mathematics.

Inspired by Chicago’s Cloud Gate, David Broaddus, Stephen Lovett, Dawson Miller, and Caitlin Smith investigate the degree to which one can reconstruct the principal curvatures of a surface from its reflection. Card puzzles and strategic prisoners meet in Larry Carter and Stan Wagon’s Mensa Correctional Institute. Maria Bras-Aromós discusses Reed–Solomon codes and presents a new decoding method for them. Chungwu Ho and Seth Zimmerman partition the real numbers into finitely many uncountable, dense subsets such that every real numbers is a condensation point for each of the subsets.

In the Notes, Matthias Keller, Yehuda Pinchover, and Felix Pogorzelski improve a Hardy inequality; Brian Lubeck and Vadim Ponomarenko determine convergence conditions for harmonic subsums; Vaibhav Pandey, Sagar Shrivastava, and Balasubramanian Sury show that real-analytic functions defined on the unit circle form a Dedekind domain whose class group (i.e., fractional ideals modulo principal fractional ideals) is of order 2; Mark Villarino gives a new proof of a refined estimate of the error (due to R. Johnsonbaugh) in an alternating series; and José Adell and Alberto Lekuona use an identity involving moments of random variables to provide proofs of three binomial identities.

Prove that you’re no April fool by solving some Monthly problems. And get started making mathematical physical and tactile for yourself (and your students) with Laura Taalman’s reviews of Henry Segerman’s Visualizing Mathematics With 3D Printing and George Legendre’s Pasta by Design.

— Susan Jane Colley, Editor


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Table of Contents

Seeing Curvature on Specular Surfaces

p. 291.

David Broaddus, Stephen Lovett, Dawson Miller & Caitlin Smith

Reflections on a specular surface distort their surrounding environments in ways that give visual clues about the surface's principal and Gaussian curvatures. This article mathematically examines this phenomenon by introducing a calibrated environment and using standard photography to record the distortions of that environment on the specular surface. We provide a novel approach to this examination which ultimately succeeds in demonstrating the availability of this curvature information in the reflections.

DOI: 10.1080/00029890.2018.1418124

The Mensa Correctional Institute

p. 306.

Larry Carter & Stan Wagon

We investigate the following puzzle and several variations, some of which have quite surprising answers. Alice and Bob are in prison under the care of warden Charlie. Alice will be brought into Charlie’s office and shown 52 cards, face-up in a row in an arbitrary order. Alice can interchange two cards. Charlie then turns all cards face-down in their places and Alice leaves the room. Then Bob is brought in and Charlie calls out a random target card. Bob can turn over cards, one after another, at most 26 times as he searches for the target. Both prisoners are freed if Bob finds the target. Find a strategy that never fails.

DOI: 10.1080/00029890.2018.1418100

A Decoding Approach to Reed–Solomon Codes from Their Definition

p. 320.

Maria Bras-Amorós

Because of their importance in applications and their quite simple definition, Reed–Solomon codes can be explained in any introductory course on coding theory. However, decoding algorithms for Reed–Solomon codes are far from being simple and it is difficult to fit them in introductory courses for undergraduates. We introduce a new decoding approach, in a self-contained presentation, which we think may be appropriate for introducing error correction of Reed–Solomon codes to nonexperts. In particular, we interpret Reed–Solomon codes by means of the degree of the interpolation polynomial of the code words and from this derive a decoding algorithm. Compared to the classical algorithms, our algorithm appears to arise more naturally from definitions and to be easier to understand. It is related to the Peterson–Gorenstein–Zierler algorithm.

DOI: 10.1080/00029890.2018.1420333

On Certain Dense, Uncountable Subsets of the Real Line

p. 339.

Chungwu Ho & Seth Zimmerman

It is a challenge to define an uncountable set of real numbers that is dense in the real line and whose complement is also uncountable and dense. In this article we specify, for any positive integer k, a partition of the real line into k + 1 sets that are uncountable and dense; moreover, every real number is a condensation point for each of these sets. We show that one of these sets has measure one when restricted to the unit interval, while each of the others has measure zero. We then define a function that is continuous on the set of measure 1 but discontinuous elsewhere.

DOI: 10.1080/00029890.2018.1420334


An Improved Discrete Hardy Inequality

p. 347.

Matthias Keller, Yehuda Pinchover & Felix Pogorzelski

In this note, we prove an improvement of the classical discrete Hardy inequality. Our improved Hardy-type inequality holds with a weight w which is strictly greater than the classical Hardy weight wH(n) ≔ 1/(2n)2, where ∈ ℕ.

DOI: 10.1080/00029890.2018.1420995

Subsums of the Harmonic Series

p. 351.

Brian Lubeck & Vadim Ponomarenko

We consider subsums of the harmonic series, and determine conditions for their convergence. We apply these conditions to determine convergence for a family of series that generalizes Kempner’s series.

DOI: 10.1080/00029890.2018.1420996

A Dedekind Domain with Nontrivial Class Group

p. 356.

Vaibhav Pandey, Sagar Shrivastava & Balasubramanian Sury

We show that the ring of real-analytic functions on the unit circle is a Dedekind domain with class number two.

DOI: 10.1080/00029890.2018.1424429

The Error in an Alternating Series

p. 360.

Mark B. Villarino

We present a new proof of Johnsonbaugh’s estimates for the error in an alternating series based on an idea of R. M. Young. We also use this same idea to prove the convergence of the Euler transform and a corresponding error estimate.

DOI: 10.1080/00029890.2017.1416875

Binomial Identities and Moments of Random Variables

p. 365.

José A. Adell & Alberto Lekuona

We give unified simple proofs of some binomial identities, by using an elementary identity on moments of random variables.

DOI: 10.1080/00029890.2018.1424428

Problems and Solutions

p. 370.

DOI: 10.1080/00029890.2018.1438001

Book Review

p. 379.

Visualizing Mathematics With 3D Printing by Henry Segerman and Pasta by Design by George Legendre

Reviewed by Laura Taalman

DOI: 10.1080/00029890.2018.1424427


√2 is Not 1.41421356237 or Anything of the Sort

p. 346.

100 Years Ago This Month in The American Mathematical Monthly

p. 369.