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American Mathematical Monthly Contents—June/July 2013

Need some reading material for the beach? Look no further than the friendly green cover appearing soon in your mailbox (or on your computer screen)! In the June/July issue of the Monthly, check out Andrew Hwang's paper on paper geometry construction models, Papakonstantinou and Tapia's discussion of the evolution of the secant method, and Tadashi Tokieda's perpetual motion machine. Don't forget that the Monthly is not only about articles and problems: David Bressoud reviews not one, but FIVE calculus texts.

Stay tuned for the August/September issue: David Aldous explains how to use market data to illustrate undergraduate probability. Enjoy the beach! —Scott Chapman

Vol. 120, No. 6, pp.487-580.

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Paper Surface Geometry: Surveying a Locally Euclidean Universe

Andrew D. Hwang

The concepts of parallel transport and intrinsic (Gaussian) curvature arising in the differential geometry of surfaces may be pleasantly and concretely investigated using paper models and familiar notions of length and angle. This paper introduces parallel transport and curvature in the context of "locally Euclidean" surfaces: polyhedra, for which curvature is concentrated at isolated points, and "polycones," for which curvature is concentrated along circular arcs. The geometry of polycones is used to recover a strikingly simple intrinsic formula for the Gaussian curvature of a surface of rotation. We give instructions for building paper models of the catenoid and surfaces of constant Gaussian curvature.

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Origin and Evolution of the Secant Method in One Dimension

Joanna M. Papakonstantinou and Richard A. Tapia

Many in the mathematical community believe that the secant method arose from Newton's method using a finite difference approximation to the derivative, most likely because that is the way that it is taught in contemporary texts. However, we were able to trace the origin of the secant method all the way back to the Rule of Double False Position described in the 18th-century B.C. Egyptian Rhind Papyrus, by showing that the Rule of Double False Position coincides with the secant method applied to a linear equation. As such, it predates Newton's method by more than 3,000 years. In this paper, we recount the evolution of the Rule of Double False Position as it spanned many civilizations over the centuries leading to what we view today as the contemporary secant method. Unfortunately, throughout history naming confusion has surrounded the Rule of Double False Position. This naming confusion was primarily a product of the last 500 years or so and became particularly troublesome in the past 50 years, creating confusion in the use of the terms Double False Position method, Regula Falsi method, and secant method. We elaborate on this confusion and clarify the names used.

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Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials

Nathan Bliss, Ben Fulan, Stephen Lovett, and Jeff Sommars

Almost all algebra texts define cyclotomic polynomials using primitive $$n$$th roots of unity. However, the elementary formula $$\gcd(x^{m}-1,x^{n}-1)=x^{\gcd(m,n)}-1$$ in $$\mathbb{Z}[x]$$ can be used to define the cyclotomic polynomials without reference to roots of unity. In this article, partly motivated by cyclotomic polynomials, we prove a factorization property about strong divisibility sequences in a unique factorization domain. After illustrating this property with cyclotomic polynomials and sequences of the form $$A^{n}-B^{n}$$, we use the main theorem to prove that the dynamical analogues of cyclotomic polynomials over any unique factorization domain $$R$$ are indeed polynomials in $$R[x]$$. Furthermore, a converse to this article's main theorem provides a simple necessary and sufficient condition for a divisibility sequence to be a rigid divisibility sequence.

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Several Proofs of the Irreducibility of the Cyclotomic Polynomials

Steven H. Weintraub

We present a number of classical proofs of the irreducibility of the $$n$$th cyclotomic polynomial $$\Phi_{n}(x)$$. For n prime we present proofs due to Gauss (1801), in both the original and a simplified form, Kronecker (1845), and Schönemann/Eisenstein (1846/1850). For general $$n$$, we present proofs due to Dedekind (1857), Landau (1929), and Schur (1929).

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The Equation $$x^{2}+\overline{m}y^{2}=\overline{k}$$ in $$\mathbb{Z}/p\mathbb{Z}$$

Kurt Girstmair

We present a simple and character-free approach to the determination of the number of nontrivial solutions of the equation $$x^{2}+\overline{m}y^{2}=\overline{k}$$ in $$\mathbb{Z}/p\mathbb{Z}$$, where $$m$$ and $$k$$ are integers not divisible by $$p$$.

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Coxeter Friezes and Triangulations of Polygons

Claire-Soizic Henry

We study relations between Coxeter's frieze patterns and discrete Sturm-Liouville equations. We present a short proof of the Coxeter-Conway classification theorem.

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A Probabilistic Proof of a Binomial Identity

Jonathon Peterson

We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity.

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The Euclidean Algorithm and the Linear Diophantine Equation $$ax+by=\gcd(a,b)$$

S. A. Rankin

In this note, we prove that for any positive integers $$a$$ and $$b$$, with $$d=\gcd(a,b)$$, among all integral solutions to the equation $$ax+by=d$$, the solution $$(x_{0},y_{0})$$ that is provided by the Euclidean algorithm lies nearest to the origin. In fact, we prove that $$(x_{0},y_{0})$$ lies in the interior of the circle centered at the origin with radius $$\frac{1}{2d}\sqrt{a^{2}+b^{2}}$$.

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A Buoyancy-Driven Perpetual Motion Machine

Tadashi Tokieda

A perpetual motion machine driven by buoyancy is presented. We see that it does not quite work, and we also discuss how it would move in reality. Debunking the trick behind most such machines is a nice exercise in exact and non-exact differentials.

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Example of a Monotonic, Everywhere Differentiable Function on $$\mathbb{R}$$ Whose Derivative Is Not Continuous

Manas R. Sahoo

We construct an example of a monotonic function which is differentiable everywhere, but the derivative is not continuous. This is done using a nonnegative discontinuous integrable function whose every point is a Lebesgue point.

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Problems 11712-11718
Solutions 11595, 11605, 11607, 11608, 11612, 11614

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Calculus by Michael Spivak; Calculus Deconstructed: A Second Course in First-Year Calculus by Zbigniew Nitecki; Approximately Calculus by Shahriar Sharhriari; A Guide to Cauchy's Calculus: A Translation and Analysis of Calcul Infinitesimal by Dennis M. Cates; and The Calculus Integral by Brian S. Thomson

Reviewed by David Bressoud

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