American Mathematical Monthly Contents—January 2013

ARTICLES

Factorization Theory and Decompositions of Modules

Nicholas R. Baeth and Roger Wiegand

Let $$R$$ be a commutative ring with identity. It often happens that $$M{1}\bigoplus\cdots\bigoplus M_{s}\cong N_{1}\bigoplus\cdots\bigoplus N_{t}$$ for indecomposable $$R$$-modules $$M_{1},\dots,M_{s}$$ and $$N_{1},\dots,N_{t}$$ with $$s\neq t$$. This behavior can be captured by studying the commutative monoid {[$$M$$]|$$M$$ is an $$R$$-module} of isomorphism classes of $$R$$-modules with operation given by $$[M]+[N]=[M\bigoplus N]$$. In this mostly self-contained exposition, we introduce the reader to the interplay between the study of direct-sum decompositions of modules and the study of factorizations in integral domains.

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N-Person Cake-Cutting: There May Be No Perfect Division

Steven J. Brams, Michael A. Jones, and Christian Klamler

A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake that is impossible to divide among three players, so that these three properties are satisfied, however many (finite) cuts are made. It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability, but not both). We prove that no perfect division exists for more than 4 cuts and for an extension of this example to more than three players.

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Transcendence of Generalized Euler Constants

M. Ram Murty and Anastasia Zaytseva

We consider a class of analogues of Euler’s constant $$\gamma$$ and use Baker’s theory of linear forms in logarithms to study its arithmetic properties. In particular, we show that with at most one exception, all of these analogues are transcendental.

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When Is $$L^{r}(\mathbb{R})$$ Contained in $$L^{p}(\mathbb{R})+L^{q}(\mathbb{R})$$?

Jean-Baptiste Hiriart-Urruty and Patrice Lassère

We prove a necessary and sufficient condition on the exponents $$p$$, $$q$$, $$r\geq1$$ such that $$L^{r}(\mathbb{R})\subset L^{p}(\mathbb{R})+L^{q}(\mathbb{R})$$. In doing so, we explore the structure of $$L^{p}(\mathbb{R})+L^{q}(\mathbb{R})$$ as a normed vector space.

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NOTES

A Use of Symmetry: Generalization of an Integral Identity Found by M. L. Glasser

Vinicius Nicolae Petre Anghel

The integral identity found by M. L. Glasser [3] is generalized using the permutation symmetry of coordinates of an $$n$$-spherical surface simplex. The first calculation technique is simple to apply, but the second technique allows further generalization of M. L. Glasser’s identity. Analogous results are discussed for the $$n$$-hemispherical surface of the unit $$n$$-sphere and for the entire surface of the $$n$$-sphere. The $$n$$-sphere surface result is used to generalize M. L. Glasser’s solution to a problem proposed by J. R. Bottiger [2].

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A Proof that Zeilberger Missed: A New Proof of an Identity by Chaundy and Bullard Based on the Wilf-Zeilberger Method

Yi Jun Chen

In this paper, a succinct new proof of an identity by Chaundy and Bullard is given, based on the Wilf-Zeilberger theory.

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Visual Angular Momentum: Mamikon Meets Kepler

L. P. Withers, Jr.

A new areal proof of Kepler’s second law of planetary motion is presented, based on Mamikon’s sweeping-tangent theorem.

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A Simple Characterization of Differentiation

Wlodzimierz Bak

In this short note we will give a result that can be treated as a characterization of differentiation.

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PROBLEMS AND SOLUTIONS

Problems 11684-11690
Solutions 11540, 11558, 11559, 11560, 11561, 11564, 11565, 11571

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REVIEWS

Elliptic Curves, Modular Forms, and Their L-functions (Student Mathematical Library IAS/Park City Subseries, vol. 58), by Alvaro Lozano-Robledo. American Mathematical Society, Providence, RI, 2011, 195 pp., ISBN 978-0-8218-5242-2, $29.60. Reviewed by Fernando Q. Gouvêa

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