The 10 most-cited articles in American Mathematical Monthly, 2019

The centers of circumscribed and inscribed circles to the triangles that are the 3-periodic orbits of an elliptic billiard are ellipses. In this article, we obtain the canonical equations of these ellipses. Moreover, we complement the previous results about incenters obtained by Romaskevich. Also the geometric locus defined by barycenters of the triangles of an elliptic billiard are ellipses as established by Schwartz and Tabachnikov and explicit equations of these ellipses are also given. In addition, we obtain that the geometric locus defined by the barycenters of the edges of billiard triangles is an ellipse and the canonical equation is obtained.

Every element u of [0,1] can be written in the form u=x2y, where x, y are elements of the Cantor set C. In particular, every real number between zero and one is the product of three elements of the Cantor set. On the other hand, the set of real numbers v that can be written in the form v = xy with x and y in C is a closed subset of [0,1] with Lebesgue measure strictly between 1721 and 89. We also describe the structure of the quotient of C by itself, that is, the image of C×(C∖{0}) under the function f(x,y)=x/y.

In 2018, a Pennsylvania court struck down the state’s congressional map due to unconstitutional partisan gerrymandering, and a new map was established in time for the midterm elections. Recent lawsuits alleging gerrymandering in Wisconsin, Maryland, and North Carolina have also caught the nation’s attention. The efficiency gap formula made headlines for the key role it played, particularly in the ongoing Wisconsin case. Since the introduction of this formula in 2014, the mathematics has moved forward more quickly than the courts, with alternative versions proposed, analyzed, and compared. Much of the relevant literature appears (or will appear) in political science and law journals, so we believe that the general mathematics audience might find benefit in a concise, self-contained overview of this application of mathematics that could have profound consequences for our democracy.

A classical result of Erdős and Turán states that if a monic polynomial has small size on the unit circle and its constant coefficient is not too small, then its zeros cluster near the unit circle and become equidistributed in angle. Using Fourier analysis we give a short and self-contained proof of this result.

The Markov, Chebyshev, and Chernoff inequalities are some of the most widely used methods for bounding the tail probabilities of random variables. In all three cases, the bounds are tight in the sense that there exists easy examples where the inequalities become equalities. Here we will show, through a simple smoothing using auxiliary randomness, that each of the three bounds can be cut in half. In many common cases, the halving can be achieved without the need for the auxiliary randomness.

The theorems of Hadamard and of Poincaré–Miranda give sufficient conditions for the existence of at least one zero for some continuous mappings from a Euclidean space into itself. The note gives elementary and brief derivations of these theorems from the Brouwer fixed point theorem.

During the last several years opinion dynamics, in particular the question of a consensus, have been studied by many researchers across various disciplines, ranging from physics, computer science, and electrical engineering to biology, economics, and sociology. Considering the mathematical analysis, the framework of multiagent systems has proven to be very useful. This article explores convergence behavior of d-dimensional multiagent systems x(t+1)=A(t)x(t) with particular emphasis on opinion dynamics under bounded confidence. Conditions are given that guarantee geometrically fast convergence to consensus, sharpening in particular a well-known theorem on Markov chains. A crucial condition to get consensus is what we call the principle of the third agent. In case overall consensus is not reachable, the process of fragmentation is analyzed and partial consensus among agents is proven given certain assumptions. Using these assumptions, the principle of the third agent holds for subgroups of agents.

We analyze the set of increasingly enumerable additive submonoids of R, for instance, the set of logarithms of the positive integers with respect to a given base. We call them ω-monoids. The ω-monoids for which consecutive elements become arbitrarily close are called tempered monoids. This is, in particular, the case for the set of logarithms. We show that any ω-monoid is either a scalar multiple of a numerical semigroup or a tempered monoid. We will also show how we can differentiate ω-monoids that are multiples of numerical semigroups from those that are tempered monoids by the size and commensurability of their minimal generating sets. All the definitions and results are illustrated with examples from music theory.

In 1997, Jean-Claude Hausmann and Allen Knutson introduced a natural and beautiful correspondence between planar n-gons and the Grassmann manifold of 2-planes in real n-space. This construction leads to a natural probability distribution and a natural metric on polygons which has been used in shape classification and computer vision. In this paper, we provide an accessible introduction to this circle of ideas by explaining the Grassmannian geometry of triangles. We use this to find the probability that a random triangle is obtuse, which was a question raised by Lewis Carroll. We then explore the Grassmannian geometry of planar quadrilaterals, providing an answer to Sylvester’s four-point problem, and describing explicitly the moduli space of unordered quadrilaterals.

Given the subjective preferences of n roommates in an n-bedroom apartment, one can use Sperner’s lemma to find a division of the rent such that each roommate prefers a distinct room. At the resulting rent division, no roommate has a strictly stronger preference for a different room. We give a new elementary proof that the subjective preferences of only n – 1 of the roommates actually suffice to achieve this envy-free rent division. Our proof, in particular, yields an algorithm to find such a division of rent. The techniques also give generalizations of Sperner’s lemma including a new proof of a conjecture of Meunier.