Abstract The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings

(2021). 100 Years Ago This Month in The American Mathematical Monthly. The American Mathematical Monthly: Vol. 128, No. 6, pp. 499-499.

Abstract Call a reduced fraction with denominator q a bronze, silver, or gold approximation with respect to a given irrational number ω if they differ by less than the reciprocal of the product of the square of q and 1, 2, or 5, respectively. Suppose A and B are neighboring Farey fractions sandwiching ω, where A’s denominator is at least as large as B’s denominator. Then B is bronze; either A or B

Abstract Given two d × d matrices, say A and B, when do p(A) and p(B) have the same “size” for every polynomial p? In this article, we provide definitive results in the cases d = 2 and d = 3 when the notion of size used is the spectral norm.

(2021). Grades Are Not Money. The American Mathematical Monthly: Vol. 128, No. 6, pp. 524-524.

Abstract A new root and ratio test for bicomplex power series produces the bicomplex extension theorem, where any complex power series with positive radius of convergence R extends, via a simple change of the complex domain variable z to the bicomplex variable ζ, to a bicomplex function analytic in a four real-dimensional ball with the same positive radius R. In this way, any complex-analytic function

Abstract It is a 300-year-old counterintuitive observation of Prince Rupert of the Rhine that in the cube a straight tunnel can be cut through which a second congruent cube can be passed. A hundred years later P. Nieuwland strengthened Rupert’s problem and asked for the largest aspect ratio so that a larger homothetic copy of the same body can be passed through it. We show that cubes and, in fact all

Abstract It is well known that a line can intersect at most 2n−1 unit squares of the n × n chessboard. Here we consider the three-dimensional version: how many unit cubes of the 3-dimensional cube [0,n]3 can a hyperplane intersect?

(2021). 169 and Sums of Positive Squares. The American Mathematical Monthly: Vol. 128, No. 6, pp. 553-553.

Abstract We give a short proof—not relying on ideal classes or the geometry of numbers—of a known criterion for quadratic orders to possess unique factorization.

Abstract We show that a real bounded sequence (xn) is Cesàro convergent to l if and only if the sequence of averages with indices in [αk,αk+1) converges to l for all α>1. If, in addition, the sequence (xn) is nonnegative, then it is Cesàro convergent to 0 if and only if the same condition holds for some α>1.

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by December 31, 2020, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions

(2021). x + y: A Mathematician’s Manifesto for Rethinking Gender. The American Mathematical Monthly: Vol. 128, No. 6, pp. 572-576.

Abstract The mathematics of crystalline structures connects analysis, geometry, algebra, and number theory. The planar crystallographic groups were classified in the late 19th century. One hundred years later, Bérard proved that the fundamental domains of all such groups satisfy a very special analytic property: the Dirichlet eigenfunctions for the Laplace eigenvalue equation are all trigonometric

Abstract Let f(x)=αx+β mod1 for fixed real parameters α and β. For any positive integer n, define the Sós permutation π to be the lexicographically first permutation such that 0≤f(π(0))≤f(π(1))≤⋯≤f(π(n))<1. In this article, we give a bijection between Sós permutations and regions in a partition of the parameter space (α,β)∈[0,1)2. This allows us to enumerate these permutations and to obtain the following

(2021). 100 Years Ago This Month in The American Mathematical Monthly. The American Mathematical Monthly: Vol. 128, No. 5, pp. 422-422.

Abstract We give an explicit geometric proof that for each n there exists a cycle of n independent random variables such that for each random variable from this cycle the probability that it takes a value smaller than the next random variable from the cycle is at least 1−1/(4 cos 2πn+2). We also show that this bound cannot be replaced by any larger number. This generalizes the famous paradox of nontransitive

Abstract Ceva’s theorem, which concerns triangles, is a central result of post-Euclidean plane geometry. The three-dimensional generalization of a triangle is a tetrahedron, and the n-dimensional generalization of these is an n-simplex. We extend Ceva’s theorem to n-simplices and in doing so illustrate the considerations and choices that can be made in generalizing from plane geometry to high-dimensional

Abstract This article proposes a method for computing the Moore–Penrose inverse of a complex matrix using polynomials in matrices. Such a method is valid for all matrices and does not involve spectral calculation, which could be infeasible when the size of the matrix is large. We first study under which conditions the Moore–Penrose inverse of a square matrix A is a polynomial in A. As an application

(2021). Box-Object Redistribution Problem. The American Mathematical Monthly: Vol. 128, No. 5, pp. 456-456.

Abstract S. Marcus used a fat Cantor set to show that there is a quasicontinuous function from the interval [0,1] to R that is not Lebesgue measurable. However, his article is not very well known. We give a very simple example of a quasicontinuous non-Lebesgue measurable function from [0,1] to R and we show that there are 2c quasicontinuous non-Lebesgue measurable functions and 2c quasicontinuous Lebesgue

Abstract For positive integers n and p, we give a short proof of a formula expressing np as a linear combination of figurate numbers with coefficients given by the numbers cp,l of (p−l)-dimensional facets of p-dimensional simplices obtained by cutting a p-dimensional cube, where l=0,1,…,p−1. This formula was formulated as Conjecture 16 in F. Marko and S. Litvinov (2020), Geometry of figurate numbers

Abstract We give a simple geometric interpretation of an algebraic construction of Wenger that gives n-vertex graphs with no cycle of length 4, 6, or 10 and close to the maximum number of edges.

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by September 30, 2021, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions

(2021). Reviews. The American Mathematical Monthly: Vol. 128, No. 5, pp. 476-478.

(2021). Editor’s Endnotes. The American Mathematical Monthly: Vol. 128, No. 5, pp. 479-480.

Abstract In 1734, Leonhard Euler summed the infinite series of reciprocals of the squares, thereby solving a challenge known as the “Basel problem.” He later extended his method to find closed-form sums for the reciprocals of 4th, 6th, and other even powers. But those techniques did not yield a value for the sum of the reciprocals of the cubes. Here, we show how Euler tried to evaluate this series

Abstract A Puiseux monoid is an additive submonoid of the nonnegative rational numbers. If M is a Puiseux monoid, then the question of whether each nonunit element of M can be written as a sum of irreducible elements (that is, M is atomic) is surprisingly difficult. For instance, although various techniques have been developed over the past few years to identify subclasses of Puiseux monoids that are

Abstract A polynomial with nonnegative coefficients does not have zeros in a sector around the positive real axis. It is shown that other polynomials with different sign restrictions on their coefficients satisfy similar properties, which is then applied to derive conditions that widen the angle of the zero exclusion sector for polynomials with positive coefficients. A geometric approach is used for

(2021). Complex Numbers as Real Continued Fractions. The American Mathematical Monthly: Vol. 128, No. 4, pp. 336-336.

Abstract We provide elementary proofs of several results concerning the possible outcomes arising from a fixed profile within the class of positional voting systems. Our arguments enable a simple and explicit construction of paradoxical profiles, and we also demonstrate how to choose weights that realize desirable results from a given profile. The analysis ultimately boils down to thinking about positional

(2021). 100 Years Ago This Month in The American Mathematical Monthly. The American Mathematical Monthly: Vol. 128, No. 4, pp. 351-351.

Abstract Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the hypercube’s 2-skeleton. For odd dimension n, the entire 2-skeleton decomposes into (n−1)/2 copies of the surface, and the intersection of any two copies is

Abstract The ring Int(Z) of integer-valued polynomials has the two-generator property, which means that every finitely generated ideal may be generated by two elements. As the known proofs of this fact are rather complicated using strong topological arguments, we propose here a constructive proof obtained by means of elementary tools. Along the way, we also obtain constructive proofs of two other well-known

Abstract We point out a curious coincidence: two famous series of Leibniz and Newton may be also computed, via a simple change of variable, by Cauchy’s residue theorem.

Abstract Hölder’s inequality receives a variety of proofs in the literature. This note gives a new derivation, interpreting the inequality as the tendency of still water to settle in the lowest potential energy.

Abstract Choose a random linear operator on a vector space of finite cardinality N; then the probability that it is nilpotent is 1/N. This is a linear analogue of the fact that for a random self-map of a set of cardinality N, the probability that some iterate is constant is 1/N. The first result is due to Fine, Herstein, and Hall, and the second is essentially Cayley’s tree formula. We give a new proof

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by August 31, 2021, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions

(2021). Yeuh-Gin Gung and Dr. Charles Y. Hu Award for 2021 to Deanna Haunsperger for Distinguished Service. The American Mathematical Monthly: Vol. 128, No. 3, pp. 195-197.

ABSTRACT Following on from our recent investigation of series and products using the Euler–Maclaurin formula, we show how the trapezoidal rule can be used to obtain the same asymptotic expansions and can also produce exact transformations into equivalent series with different convergence properties.

(2021). 100 Years Ago This Month in The American Mathematical Monthly. The American Mathematical Monthly: Vol. 128, No. 3, pp. 213-213.

ABSTRACT By combining theoretical and computational techniques from geometry, calculus, group theory, and Galois theory, we prove the nonexistence of a closed-form algebraic solution to a Japanese geometry problem first stated in the early nineteenth century. This resolves an outstanding problem from the sangaku tablets which were at one time displayed in temples and shrines throughout Japan.

(2021). Composites in the Ulam Spiral. The American Mathematical Monthly: Vol. 128, No. 3, pp. 238-238.

ABSTRACT We give a definition of a class of Dedekind domains which includes the rings of integers of global fields and give a proof that all rings in this class have finite ideal class group. We also prove that this class coincides with the class of rings of integers of global fields.

(2021). A Pi in the Face of Tau. The American Mathematical Monthly: Vol. 128, No. 3, pp. 249-249.

ABSTRACT We show that Fermat’s last theorem and a combinatorial theorem of Schur on monochromatic solutions of a + b = c implies that there exist infinitely many primes. In particular, for small exponents such as n = 3 or 4 this gives a new proof of Euclid’s theorem, as in this case Fermat’s last theorem has a proof that does not use the infinitude of primes. Similarly, we discuss implications of Roth’s

ABSTRACT What is the probability two distinct ranks (say, a king and a five) will appear within a card of one another in a shuffled deck? An old bar bet is predicated on it being nearly guaranteed. By modeling the outcome using a finite automaton, we construct the probability generating function for card rank separation. We then derive the corresponding probability mass function, mean, and variance

ABSTRACT This note provides a short, self-contained proof of the famous fact that any countable metric space without isolated points is homeomorphic to the space of rational numbers. The discussion is carried out entirely in the language of metric spaces.

ABSTRACT We determine necessary conditions for when powers corresponding to positive/negative coefficients of Φ n are in arithmetic progression. When n = pq for any primes q > p > 2 , our conditions are also sufficient. Finally, we generalize the result when n = pq to the so-called inclusion-exclusion polynomials first introduced by Bachman.

ABSTRACT We offer an alternative method for evaluating Dirichlet’s integral, i.e., the integral of the sinc function sin   ( t ) / t over the positive real line. Our method employs the fundamental theorem of calculus for complex path integrals, but avoids the customary invocation of Cauchy’s theorem.

(2021). Widder-style Real Analytic Functions. The American Mathematical Monthly: Vol. 128, No. 3, pp. 275-275.

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by July 31, 2021, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions.

(2021). Reviews. The American Mathematical Monthly: Vol. 128, No. 3, pp. 285-288.

Abstract The purpose of this article is to take a fresh look at the problem of finding an arithmetic representation of Hilbert’s curve, building on the same initial observations as many have done previously. The approach we take is an iterated function view and this view leads in somewhat distinct directions. This is the first of what will be several papers on the subject and focuses on a basic approach

Abstract–We explore the applications of the Euler–Maclaurin formula in analyzing functions expressed as infinite series and products. Three illustrative examples show the difficulties that may be encountered and the means by which these can be overcome.

Abstract We study directions along which the norms of vectors are preserved under a linear map. In particular, we find families of matrices for which these directions are determined by integer vectors. We consider the two-dimensional case in detail, and also discuss the extension to three-dimensional vector spaces.

(2021). 100 Years Ago This Month in The American Mathematical Monthly. The American Mathematical Monthly: Vol. 128, No. 2, pp. 139-139.

Abstract–“From a bag containing red and blue balls, two are removed at random. The chances are 50-50 that they will differ in color. What were the possible numbers of balls initially in the bag?” This problem appeared in the National Museum of Mathematics’ Varsity Math puzzle, week 117. It is quite easy to solve, but what if we generalize to arbitrary odds? In this article, we characterize the solutions

(2021). A Cauchy–Schwarz Type Inequality for Differences. The American Mathematical Monthly: Vol. 128, No. 2, pp. 150-150.

Abstract The Moufang loop named for Richard Parker is a central extension of the extended binary Golay code. It the prototypical example of a general class of nonassociative structures known today as code loops, which have been studied from a number of different algebraic and combinatorial perspectives. This expository article aims to highlight an experimental approach to computing in code loops, by