January 2013 Contents
We start the New Year out with a bang at the MONTHLY! In our lead articles for January, you will learn the connection between solutions of Diophantine equations and the decompositions of modules, how cake cutting may not be perfect among 3 or more persons, why Euler’s constant is most likely transcendental, and an interesting containment relation involving Lp spaces. Our notes feature a look at an integral identity found by M. L. Glasser, a proof that Zeilberger missed, what happens when Mamikon meets Kepler, and a simple characterization of differentiation. Fernando Gouvêa reviews Elliptic Curves, Modular Forms, and Their L-functions by Álvaro Lozano-Robledo. Finally, don’t forget our world-famous Problem Section.
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ARTICLES
Factorization Theory and Decompositions of Modules
Nicholas R. Baeth and Roger WiegandLet R be a commutative ring with identity. It often happens that
for indecomposable R-modules 
and 
with s != t. This behavior can be captured by studying the commutative monoid 
is an R-module} of isomorphism classes of R-modules with operation given by 
. 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.
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.
Transcendence of Generalized Euler Constants
M. Ram Murty and Anastasia Zaytseva
We consider a class of analogues of Euler’s constant 
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.
When Is 
Contained in 
?
Jean-Baptiste Hiriart-Urruty and Patrice Lassère We prove a necessary and sufficient condition on the exponents p, q, r >= 1 such that
. In doing so, we explore the structure of as a normed vector space.
NOTES
A Use of Symmetry: Generalization of an Integral Identity Found by M. L. Glasser
Vinicius Nicolae Petre AnghelThe 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].
A Proof that Zeilberger Missed: A New Proof of an Identity by Chaundy and Bullard Based on the Wilf-Zeilberger Method
Yi Jun ChenIn this paper, a succinct new proof of an identity by Chaundy and Bullard is given, based on the Wilf-Zeilberger theory.
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.
A Simple Characterization of Differentiation
Wlodzimierz BakIn this short note we will give a result that can be treated as a characterization of differentiation.