American Mathematical Monthly - January 2018

Happy New Year to you from the Monthly! We start off the first issue of 2018 with Adrian Rice’s celebration of the centennial of Hardy and Ramanujan’s asymptotic formula for the partition function, followed by Harold Boas’s modern treatment of Cauchy’s original vision of the residue calculus. Kristen Mazur, Mutiara Songjaja, Matthew Wright, and Carolyn Yarnall consider approval voting in situations where voters make multiple decisions simultaneously, and David Treeby revisits the classic problem of “harmonic blocks” and their overhang, as well as generalizations thereof.

In the Notes, ranks of sums of matrices are considered, a brief proof of Cayley’s formula for the number of labeled trees on n vertices is provided, subsets of the unit sphere in Cⁿ are determined that are mapped bijectively onto the numerical range of a complex n × n matrix, Giuseppe De Marco provides an example of a function that that is almost everywhere constant yet surjective on every nonempty open set, and the Lebesgue decomposition theorem for nonnegative finite measures is reconsidered.

Begin the new year by resolving to solve some problems, and enjoy Geoffrey Dietz’s review of Edward Scheinerman’s The Mathematics Lover’s Companion: Masterpieces for Everyone.

Finally, with the January 2018 issue, the Monthly (and the Monthly’s sister journals) commences publication in partnership with Taylor & Francis. While the character of the Monthly will remain unchanged, we look forward to various improvements and new capabilities, including a better experience for online reading and additional features for authors and referees.

— Susan Jane Colley, Editor

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Table of Contents

Partnership, Partition, and Proof: The Path to the Hardy–Ramanujan Partition Formula

p. 3.

Adrian Rice

The year 2018 marks 100 years since the publication of one of the most startling results in the history of mathematics: Hardy and Ramanujan’s asymptotic formula for the partition function. To celebrate the centenary, this paper looks at the creation of their remarkable theorem: where it came from, how it was proved, and how the assistance of a third contributor helped to influence its ultimate form.

DOI: 10.1080/00029890.2017.1389178

Cauchy’s Residue Sore Thumb

p. 16.

Harold P. Boas

Cauchy's method from two centuries ago for computing integrals along the real axis by passing into the complex plane is not rigorous by present-day standards. Yet when properly formulated, his original approach is simpler than modern presentations of the residue calculus.

DOI: 10.1080/00029890.2017.1389200

Approval Voting in Product Societies

p. 29.

Kristen Mazur, Mutiara Sondjaja, Matthew Wright & Carolyn Yarnall

In approval voting, individuals vote for all platforms that they find acceptable. In this situation it is natural to ask: When is agreement possible? What conditions guarantee that some fraction of the voters agree on even a single platform? Berg et al. found such conditions when voters are asked to make a decision on a single issue that can be represented on a linear spectrum. In particular, they showed that if two out of every three voters agree on a platform, there is a platform that is acceptable to a majority of the voters. Hardin developed an analogous result when the issue can be represented on a circular spectrum. We examine scenarios in which voters must make two decisions simultaneously. For example, if voters must decide on the day of the week to hold a meeting and the length of the meeting, then the space of possible options forms a cylindrical spectrum. Previous results do not apply to these multi-dimensional voting societies because a voter's preference on one issue often impacts their preference on another. We present a general lower bound on agreement in a two-dimensional voting society, and then examine specific results for societies whose spectra are cylinders and tori.

DOI: 10.1080/00029890.2018.1390370

Further Thoughts on a Paradoxical Tower

p. 44.

David Treeby

What is the maximum overhang that can be obtained when a set of blocks of variable width are stacked so that there is one block at each level? First, we stack these blocks in a prescribed order, and describe infinite sets of blocks that yield towers with arbitrarily large overhang. We then stack the blocks in any order, and describe how they should be stacked to maximize their overhang.

DOI: 10.1080/00029890.2018.1390375

NOTES

Rank My Update, Please

p. 61.

Carl D. Meyer

Updating a given a matrix A_{m × n} by a rank-one matrix B = cd^T, where c and d are appropriately sized column vectors, is a common practice throughout all applied areas of mathematics, science, and engineering. Because rank is often tied to the number of degrees of freedom or the level of independence in underlying models or data, it can be imperative to know exactly how the update term affects rank. While it is well known that a rank-one update can only increase or decrease rank by at most one, there is not a widely known formula for exactly how this occurs. This note presents an expression in simply stated terms for the exact rank of a rank-one updated matrix.

DOI: 10.1080/00029890.2017.1389199

A Short Proof of Cayley's Tree Formula

p. 65.

Alok Shukla

Cayley's tree formula is one of the most beautiful results in enumerative combinatorics with a number of well-known proofs. A fascinating and recurring theme in mathematics is the existence of rich and surprising interconnections between apparently disjoint domains. We will, hopefully, see one such instance in this note, in which we give a short proof of Cayley's tree formula for counting the number of distinct labeled trees on n vertices by employing a combinatorial argument. In fact, we deduce a nonlinear recursive relation for the number of labeled trees on n vertices and then we show that this recursion is same as the celebrated Cayley tree formula. One interesting feature of this proof is that a discrete counting problem is solved by going from a recursion, to a differential equation, to complex analysis, and then back to a discrete formula.

DOI: 10.1080/00029890.2018.1392750

Preimages of the Numerical Range

p. 69.

John Clifford, Kelly Jabbusch & Michael Lachance

For a nonnormal two-by-two matrix B, there exist two open disjoint subsets of the complex unit sphere from which the mapping z*Bz is a bijection onto the interior of the numerical range of B. Moreover, the union of the closures of these two subsets, up to rotation, yields the entire unit sphere.

DOI: 10.1080/00029890.2018.1392737

An Almost Everywhere Constant Function Surjective on Every Interval

p. 75.

Giuseppe De Marco

We describe an almost everywhere constant function surjective on every nonempty open set.

DOI: 10.1080/00029890.2017.1389204

The Singular Part as Fixed Point

p. 77.

Tamás Titkos

The aim of this note is to investigate the Lebesgue decomposition theorem of nonnegative finite measures from a new point of view. Using a standard iteration scheme, we identify the singular part as a fixed point of a nonnegative finite measure-valued map.

DOI: 10.1080/00029890.2018.1393219

Problems and Solutions

p. 81.

DOI: 10.1080/00029890.2018.1397465

Book Review

p. 90.

The Mathematics Lover’s Companion: Masterpieces for Everyone by Edward Scheinerman

Reviewed by Geoffrey Dietz

DOI: 10.1080/00029890.2018.1392785

MathBits

100 Years Ago This Month in The American Mathematical Monthly

p. 28.

A Noninductive Proof of de Moivre's Formula

p. 80.