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American Mathematical Monthly Contents—January 2014

Ring in 2014 the right way with the January issue of the American Mathematical Monthly. Why the purple cover?  

Find out why in editor Scott Chapman's issue opening letter. Learn more about British number systems in Mel Nathanson's article titled "Additive Systems and a Theorem of de Bruijn." While no polynomial produces only prime values over the integers, Stephen Weintraub shows us in "Values of Polynomials over Integral Domains" that such polynomials can be constructed over particular localizations of the integers.  

Ken Ross reviews In the Dark on the Sunny Side: a Memoir of an Out-of-Sight Mathematician by Larry Baggett, and our Problem Section is as enticing as ever. Stay tuned for the February issue where Chi and Lange construct a very fast algorithm for solving the Euclidean version of the generalized Heron problem. Scott Chapman

Vol. 121, No. 1, pp.3-92.


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Additive Systems and a Theorem of de Bruijn

Melvyn B. Nathanson

This paper proves a theorem of de Bruijn that classifies additive systems for the nonnegative integers, that is, families $$\mathcal{A}=(A_{i})_{i\in I}$$ of sets of nonnegative integers, each set containing $$0$$, such that every nonnegative integer can be written uniquely in the form $$\sum_{i\in I}a_{i}$$, with $$a_{i}\in A_{i}$$ for all $$i$$, and $$a_{i}\neq0$$ for only finitely many $$i$$.

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On Equidistant Sets and Generalized Conics: The Old and the New

Mario Ponce and Patricio Santibáñez

This article is devoted to the study of classical and new results concerning equidistant sets, both from the topological and metric point of view. We include a review of the most interesting known facts about these sets in Euclidean space and we prove two new results. First, we show that equidistant sets vary continuously with their focal sets. We also prove an error estimate result about approximative versions of equidistant sets that should be of interest for computer simulations. Moreover, we offer a viewpoint in which equidistant sets can be thought of as a natural generalization for conics. Along these lines, we show that the main geometric features of classical conics can be retrieved from more general equidistant sets.

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Guessing Games

Anthony Mendes and Kent E. Morrison

In a guessing game, players guess the value of a random real number selected using some probability density function. The winner may be determined in various ways; for example, a winner can be a player whose guess is closest in magnitude to the target, or a winner can be a player coming closest without guessing higher than the target. We study optimal strategies for players in these games and determine some of them for two, three, and four players.

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Using Difference Equations to Generalize Results for Periodic Nested Radicals

Chris D. Lynd

We investigate sequences of nested radicals where the coefficients and radicands are periodic sequences of real numbers, and the indices are periodic sequences of integers greater than one. We show that we can determine the end behavior of a periodic nested radical by analyzing the basin of attraction of each equilibrium point, and each period-2 point, of the corresponding difference equation. Using this method of analysis, we prove a few theorems about the end behavior of nested radicals of this form. These theorems extend previous results on this topic because they apply to large classes of nested radicals that contain arbitrary indices, negative radicands, and periodic parameters with arbitrary periods. In addition, we demonstrate how to construct a periodic nested radical, of a general form, that converges to a predetermined limit; and we demonstrate how to construct a nested radical that converges asymptotically to a periodic sequence.

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When the Identity Theorem “Seems” to Fail

J. A. Conejero, P. Jiménez-Rodríguez, G. A. Muñoz-Fernández, and J. B. Seoane-Sepúlveda

The Identity Theorem states that an analytic function (real or complex) on a connected domain is uniquely determined by its values on a sequence of distinct points that converge to a point of its domain. This result is not true in general in the real setting, if we relax the analytic hypothesis on the function to infinitely many times differentiable. In fact, we construct an algebra of functions $$\mathcal{A}$$ enjoying the following properties: (i) $$\mathcal{A}$$ is uncountably infinitely generated (that is, the cardinality of a minimal system of generators of $$\mathcal{A}$$ is uncountable); (ii) every nonzero element of $$\mathcal{A}$$ is nowhere analytic; (iii)$$\mathcal{A}\subset\mathcal{C}^{\infty}(\mathbb{R})$$; (iv) every element of $$\mathcal{A}$$ has infinitely many zeros in $$\mathbb{R}$$; and (v) for every $$f\in A\setminus\{0\}$$ and $$n\in\mathbb{N}$$, $$f^{(n)}$$ (the $$n$$th derivative of $$f$$) enjoys the same properties as the elements in $$\mathcal{A}\setminus\{0\}$$.

This construction complements those made by Cater and by Kim and Kwon, and published in the American Mathematical Monthly in 1984 and 2000, respectively.

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Peano’s Unnoticed Proof of Borel’s Theorem

Ádám Besenyei

In 1876, P. du Bois-Reymond gave an example of an infinitely differentiable function whose Taylor series diverges everywhere except one point. By generalizing this example, G. Peano proved a theorem in 1884, often credited to É. Borel, which states that every power series is a Taylor series of some infinitely differentiable function. The aim of the paper is to recall Peano’s unnoticed contributions to this result.

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Values of Polynomials over Integral Domains

Steven H.Weintraub

It is well known that no nonconstant polynomial with integer coefficients can take on only prime values. We isolate the property of the integers that accounts for this, and give several examples of integral domains for which there are polynomials that only take on unit or prime values.

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A Direct Proof and a Transcendental Version of the Fundamental Theorem of Algebra via Cauchy’s Theorem

Bao Qin Li

We will show that Cauchy’s theorem implies the Fundamental Theorem of Algebra in a direct and elementary manner. Furthermore, the argument can be used to give a transcendental version of the theorem for holomorphic functions or transcendental entire functions.

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An Inductive Proof of Hex Uniqueness

Samuel Clowes Huneke

A short, inductive proof is presented of the fact that a Hex board cannot be colored such that winning conditions are satisfied for both players.

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Real-Valued Ring Homomorphisms on $$C(\Omega)$$

Karim Boulabiar

Let $$\Omega$$ be a nonempty open set in $$\mathbb{R}^{n}$$. This note provides an elementary and accessible proof of the result that every real-valued ring homomorphism on $$C(\Omega)$$ is an evaluation at some point of $$\Omega$$.

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Problems 11747-11753
Solutions 11626, 11630, 11632, 11634, 11635, 11636, 11638

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In the Dark on the Sunny Side: A Memoir of an Out-of-Sight Mathematician, by Larry Baggett

Reviewed by Kenneth Ross