The American Mathematical Monthly
November 2011 Contents
In this month's issue, you'll learn about Ramanujan's friend and colleague S. Narayana Aiyar, a new kind of dissection problem, some enlightening examples of discontinuous functions, how graph theory can help your dean form compatible committees, and much more.
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ARTICLES
The Chief Accountant and Mathematical of Friend of Ramanujan—S. Narayana Aiyar
Bruce C. BerndtS. Narayana Aiyar was chief accountant at the Madras Port Trust office, where Ramanujan worked as a clerk in 1912–1914. In this article, a short biography of S. Narayana Aiyar is given, his mathematical contributions are discussed, and his personal and mathematical relationships with Ramanujan are examined and emphasized.
On Forming Committees
P. HaxellThis paper surveys a collection of results on finding special sets of vertices in graphs with vertex partitions, all of which can viewed as models for “committee-choosing” problems.
Complete Dissections: Converting Regions and Their Boundaries
Tom M. Apostol and Mamikon A. MnatsakanianClassical dissections convert any planar polygonal region onto any other polygonal region having the same area. If two convex polygonal regions are isoparametric, that is, have equal areas and equal perimeters, our main result states that there is always a dissection, called a complete dissection, that converts not only the regions but also their boundaries onto one another. The proof is constructive and provides a general method for complete dissection using frames of constant width. This leads to a new object of study: isoparametric polygonal frames, for which we show that a complete dissection of one convex polygonal frame onto any other always exists. We also show that every complete dissection can be done without flipping any of the pieces.
A “Bouquet” of Discontinuous Functions for Beginners in Mathematical Analysis
Giacomo Drago, Pier Domenico Lamberti, and Paolo ToniWe present a selection of a few discontinuous functions and we discuss some pedagogical advantages of using such functions in order to illustrate some basic concepts of mathematical analysis to beginners.
An Introduction to Gauss Factorials
John B. Cosgrave and Karl DilcherStarting with Wilson’s theorem and its generalization by Gauss, we define a Gauss factorial Nn! to be the product of all positive integers up to N that are relatively prime to n. We present results on the Gauss factorials ((n - 1)/M)n!, and more generally on partial products obtained when the product (n - 1)n! is divided into M equal parts, for integers M ≥ 2. Finally, extensions of the Gauss binomial coefficient theorem are presented in terms of Gauss factorials.
NOTES
Enumerating the Rationals from Left to Right
S. P. GlasbyThe rationals can be enumerated using Stern-Brocot sequences SBn, Calkin-Wilf sequences CWn, or Farey sequences Fn .We show that the rationals in SBn , CWn, and Fn can be computed using very similar second-order linear recurrence relations. (This, incidentally, obviates the need to compute previous sequences SBi , CWi , Fi with i < n.)
On the Banach-Steinhaus Theorem for Topological Groups
Semyon N. LitvinovWe prove the Banach-Steinhaus theorem and some of its consequences under the assumption that the underlying space is a topological group of second Baire category.
Ramanujan’s “Most Beautiful Identity”
Michael D. HirschhornWe give a simple proof of the identity which for Hardy represented the best of Ramanujan. On the way, we give a new proof of an important identity that Ramanujan stated but did not prove.
PROBLEMS AND SOLUTIONS
REVIEWS
Elementary Probability for ApplicationsRick Durrett
Reviewed by: Matthew Richey