The 10 most-cited articles in American Mathematical Monthly, 2020

In this note, we present a new short proof of the classical Hermite–Hadamard inequalities.

In recent years, various notions of algebraic independence have emerged as a central and unifying theme in a number of areas of applied mathematics, including algebraic statistics and the rigidity theory of bar-and-joint frameworks. In each of these settings, the fundamental problem is to determine the extent to which certain unknowns depend algebraically on given data. This has, in turn, led to a resurgence of interest in algebraic matroids, which are the combinatorial formalism for algebraic (in)dependence. We give a self-contained introduction to algebraic matroids together with examples highlighting their potential application.

After reviewing the history of the crank of an integer partition, involving work of Freeman Dyson from 1944 and Frank Garvan and George Andrews in 1988, we connect this concept with more recent work of Andrews based on the smallest missing part of a partition. This alternative view of partitions having certain crank values allows us to strengthen a recent result of Yuefei Shen with a greatly simplified and more insightful proof.

Kirchhoff’s matrix-tree theorem asserts that the number of spanning trees in a finite graph can be computed from the determinant of any of its reduced Laplacian matrices. In many cases, even for well-studied families of graphs, this can be computationally or algebraically taxing. We show how two well-known results from linear algebra, the matrix determinant lemma and the Schur complement, can be used to count the spanning trees in several significant families of graphs in an elegant manner.

We provide a self-contained topological proof of the continuous dependence of the roots of a polynomial on its coefficients. We present it as a homeomorphism between the space of monic complex polynomials of degree n and the space of unordered n-tuples of complex numbers. Our proof uses bounds on the roots of polynomials to show that the map that associates to each monic polynomial its roots has an inverse that is a proper map.

We provide a simple proof for the extended Bertrand–De Morgan test that was earlier studied in [Ďuriš, F., (2009). Infinite series: Convergence tests. Bachelor thesis, Univerzita Komenského, Bratislava, Slovakia] and [Tabatabai Adnani, A. A., Reza, A., Morovati, M. (2013). J. Lin. Topol. Algebra. 2(3): 141–147] and demonstrate an application of that test to the theory of birth-and-death processes.

We study variants of the Frobenius coin-exchange problem: Given n positive relatively prime parameters, what is the largest integer that cannot be represented as a nonnegative integral linear combination of the given integers? This problem and its siblings can be understood through generating functions with 0/1 coefficients according to whether or not an integer is representable. In the 2-parameter case, this generating function has an elegant closed form, from which many corollaries follow, including a formula for the Frobenius problem. We establish a similar closed form for the generating function indicating all integers with exactly k representations, with similar wide-ranging corollaries.

According to Mantel’s theorem, a triangle-free graph on n points has at most n2/4 edges. A linear k-graph is a set of points together with some k-element subsets, called edges, such that any two edges intersect in at most one point. The k-graph Fk, called a fan, consists of k edges that pairwise intersect in exactly one point v, plus one more edge intersecting each of these edges in a point different from v. We extend Mantel’s theorem as follows: fan-free linear k-graphs on n points have at most n2/k2 edges.This extension nicely illustrates the difficulties of hypergraph Turán problems. The determination of the case of equality leads to transversal designs on n points with k groups—for k = 3 these are equivalent to Latin squares. However, in contrast to the graph case, new structures and open problems emerge when n is not divisible by k.

Zero divided by zero is arguably the single most important concept underlying calculus. For functions of more than one variable, methods of proof for indeterminate limits are not as familiar as for functions of a single variable. We present a l’Hôpital’s rule that provides a way to simplify and resolve a wide variety of zero-over-zero limits in terms of quotients of their derivatives.

We define a triangular array closely related to Stern’s diatomic array and show that for a fixed integer r≥1, the sum ur(n) of the rth powers of the entries in row n satisfy a linear recurrence with constant coefficients. The proof technique yields a vast generalization. In certain cases, we can be more explicit about the resulting linear recurrence.