The 10 most-cited articles in American Mathematical Monthly, 2024

The Euler characteristic transform (ECT) is a simple to define yet powerful representation of shape. The idea is to encode an embedded shape by tracking how the Euler characteristic, a simple integer-valued topological invariant, changes as the shape is built up in a particular direction. Because the ECT has been shown to be injective on the space of embedded simplicial complexes, it has been used for applications spanning a range of disciplines, including plant morphology and protein structural analysis. In this survey article, we present a comprehensive overview of the Euler characteristic transform, highlighting the main idea on a simple leaf example, and surveying its key concepts, theoretical foundations, and available applications.

Quicksort is a classical divide-and-conquer sorting algorithm. It is a comparison sort that makes an average of 2(n+1)Hn−4n comparisons on an array of size n ordered uniformly at random, where Hn:=∑i=1n1i is the nth harmonic number. Therefore it makes n![2(n+1)Hn−4n] comparisons to sort all possible orderings of the array. In this article, we prove that this count also enumerates the parking preference lists of n cars parking on a one-way street with n parking spots resulting in exactly n−1 lucky cars (i.e., cars that park in their preferred spot). For n≥2, both counts satisfy the second order recurrence relation fn=2nfn−1−n(n−1)fn−2+2(n−1)! with f0=f1=0.

Let k be a field of any characteristic, V a finite-dimensional vector space over k, and Sd(V*) be the d-th symmetric power of the dual space V*. Given a linear map φ on V and an eigenvector w of φ, we prove that the pair (φ,w) can be used to construct a new Lie algebra structure on Sd(V*). We prove that this Lie algebra structure is solvable, and in particular, it is nilpotent if φ is a nilpotent map. We also classify the Lie algebras for all possible pairs (φ,w), when k=C and V is two-dimensional.

The basic problem of the calculus of variations consists of finding a function that minimizes an energy, like finding the fastest trajectory between two points for a point mass in a gravity field or finding the best shape of a wing. The existence of a solution may be established in quite abstract spaces, much larger than the space of smooth functions. An important practical problem is that of being able to approach the value of the infimum of the energy. However, numerical methods work with very “concrete” functions and sometimes they are unable to approximate the infimum: this is the surprising Lavrentiev phenomenon. The papers that ensure the nonoccurrence of the phenomenon form a recent saga, and the most general result formulated in the early ’90s was actually fully proved just recently, more than 30 years later. Our aim here is to introduce the reader to the calculus of variations, to illustrate the Lavrentiev phenomenon with the simplest known example, and to give an elementary proof of the nonoccurrence of the phenomenon.

The intricacies of our world often lead to nuanced data, which has significant implications for data analysis. Data analysts must be wary of moving too quickly to final conclusions. Behind an insight might lurk initially misleading results. In the context of sports analytics, specifically basketball, this article demonstrates the intrinsic need for the sports analyst to pose a question and then prod and investigate the insights offered by the data. The article analyzes public datasets with less than a dozen data points and more than 400,000 rows of data. Further, the article offers an entryway to the field of data analytics via sports.

We present five trigonometric proofs of the Pythagorean theorem, and our method for finding proofs (Section 5) yields at least five more.

An algorithm is given to express a proper rational function (i.e., the degree of the denominator exceeds that of the numerator) over any field as an alternating sum of reciprocals of polynomials of increasing degree. This is analogous to existing algorithms for representing proper rational numbers.

A strange title, might you say: Answer is in the question! However, contrary to popular belief and numerous citations in the literature, the image of the snowflake curve is not present or even mentioned in Helge von Koch’s original articles. So, where and when did the first snowflake fall? Join us on a journey back in time to discover the snowflake curve.

This paper draws attention to an underappreciated method for evaluating certain types of definite integrals. The method relies on a substitution in the Eulerian integral for the Legendre gamma function, and has become known in some quarters as a Schwinger parametrization. We present some examples to illustrate the utility of this technique in the hope that by doing so we may convince the reader that it makes a valuable addition to one’s integration toolkit.

We give a characterization of rational functions among meromorphic functions in the complex plane, which extends a characteristic of polynomials and also the Fundamental Theorem of Algebra to rational functions.