You are here

American Mathematical Monthly - January 2015

The January Monthly begins our year-long celebration of the MAA’s Centennial. We kick it off fittingly with David Zitarelli’s “The First 100 years of the MAA.” What’s a Napoleon Polygon? Find out in the paper “Napoleon Polygons,” by Titu Andreescu, Vladimir Georgiev, and Oleg Mushkarov. Edward F. Schaefer reviews “The Mathematics of Encryption: An Elementary Introduction,” by Margaret Cozzens and Steven J. Miller. Visit our first 2015 Problem Section so that you can keep that New Year’s resolution to work more problems. Stay tuned for our February issue when Chris O’Neill and Roberto Pelayo will show us “How Do You Measure Primality?” —Scott T. Chapman, Editor

JOURNAL SUBSCRIBERS AND MAA MEMBERS:

To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.

Table of Contents

A Letter from the Editor: 2015 is the Centennial Year of the MAA

Scott T. Chapman

DOI: http://dx.doi.org/10.4169/amer.math.monthly.122.01.3

The First 100 Years of the MAA

David E. Zitarelli

Why was the MAA founded? What role has the Association played in American mathematics? What were its primary activities? We answer these questions in this overview of the MAA over its 100-year history from its founding in 1915. Along the way, we describe MAA sections, governance, meetings, prizes/awards, and headquarters. The account of MAA activities is divided into two periods, 1916–1955 and 1955–2014 and contains a discussion for the critical role played by the Committee on the Undergraduate Program in Mathematics in this division.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.122.01.4

Napoleon Polygons

Titu Andreescu, Vladimir Georgiev, and Oleg Mushkarov

An n-gon is called Napoleon if the centers of the regular n-gons erected outwardly on its sides are vertices of a regular n-gon. In this paper we obtain a new geometric characterization of Napoleon n-gons and give a new proof of the well-known theorem of Barlotti–Greber ([1], [4]) that an n-gon is Napoleon if and only if it is affine-regular. Moreover, we generalize this theorem by obtaining an analytic characterization of the n-gons leading to a regular n-gon after iterating the above construction k times.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.122.01.24

Historical Remark on Ramanujan’s Tau Function

Kenneth S. Williams

It is shown that Ramanujan could have proved a special case of his conjecture that his tau function is multiplicative without any recourse to modularity results.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.122.01.30

Somewhat Stochastic Matrices

Branko Ćurgus and Robert I. Jewett

The standard theorem for stochastic matrices with positive entries is generalized to matrices with no sign restriction on the entries. The condition that column sums be equal to 1 is kept, but the positivity condition is replaced by a condition on the distances between columns.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.122.01.36

Notes

Stereographic Trigonometric Identities

Michael Hardy

We show that trigonometric identities arising from the most well-known alternative to the arc-length parametrization of the circle share some of the same elaborate nature as the more familiar identities involving sines, tangents, etc.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.122.01.43

Infinitely Many Primes in the Arithmetic Progression kn-1

Xianzu Lin

In this paper we give a simple and elementary proof of the infinitude of primes in the arithmetic progression kn − 1, n > 0.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.122.01.48

A Point of Tangency Between Combinatorics and Differential Geometry

Francis C. Motta, Patrick D. Shipman, and Bethany Springer

Edges of de Bruijn graphs, whose labeled vertices are arranged in sequential order on a circle, envelop epicycloids.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.122.01.52

The Axiom of Choice, Well Ordering, and Well-Classification

Hossein Hosseini Giv

Let X be a nonempty set and P∗ (X) denote the set of all nonempty subsets of X. In this note, we give a simple characterization of those choice functions on P∗ (X) that are induced by a well-ordering on X. The arguments involved in this result lead us to an interesting theorem, which we refer to as the well-classification theorem. This is finally proved to be an equivalent form of the axiom of choice.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.122.01.56

Kronecker Square Roots and the Block Vec Matrix

Ignacio Ojeda

Using the block vec matrix, I give a necessary and sufficient condition for factorization of a matrix into the Kronecker product of two other matrices. As a consequence, I obtain an elementary algorithmic procedure to decide whether a matrix has a square root for the Kronecker product.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.122.01.60

On a Formula of S. Ramanujan

Pablo A. Panzone

Certain finite families of rational functions have the property that the coefficients of the expansions in power series of their elements are related by simple algebraic expressions. We prove an identity in the spirit of S. Ramanujan using a result of T. N. Sinha.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.122.01.65

Continuity is an Adjoint Functor

Edward S. Letzter

We explain how the definition of continuity for functions between topological spaces can be rephrased as an adjointness condition between naturally arising functors.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.122.01.70

Problems and Solutions

Book Review

The Mathematics of Encryption: An Elementary Introduction by Margaret Cozzens and Steven J. Miller

Reviewed by Edward F. Schaefer

Dummy View - NOT TO BE DELETED