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.
Vol. 121, No. 1, pp.3-92.
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ARTICLES
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 $$i$$.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.01.005
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.01.018
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.01.033
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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NOTES
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.01.069
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.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.01.075
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 $$\Omega$$.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.01.081
PROBLEMS AND SOLUTIONS
Problems 11747-11753
Solutions 11626, 11630, 11632, 11634, 11635, 11636, 11638
To purchase from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.01.083
REVIEWS
In the Dark on the Sunny Side: A Memoir of an Out-of-Sight Mathematician, by Larry Baggett
Reviewed by Kenneth Ross
JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.01.091