The 10 most-cited articles in American Mathematical Monthly, 2016

It is not hard to prove that a uniformly continuous real function, whose integral up to infinity exists, vanishes at infinity, and it is probably little known that this statement runs under the name "Barbalat's Lemma." In fact, the latter name is frequently used in control theory, where the lemma is used to obtain Lyapunov-like stability theorems for non-linear and non-autonomous systems. Barbalat's Lemma is qualitative in the sense that it asserts that a function has certain properties, here convergence to zero. Such qualitative statements can typically be proved by "soft analysis", such as indirect proofs. Indeed, in the original 1959 paper by Barbalat, the lemma was proved by contradiction and this proof prevails in the control theory textbooks. In this short note we first give a direct, "hard analyis" proof of the lemma, yielding quantitative results, i.e. rates of convergence to zero. This proof allows also for immediate generalizations. Finally, we unify three different versions which recently appeared and discuss their relation to the original lemma.

Oftentimes the elements of a ring or semigroup can be written as finite products of irreducible elements. An element a can be a product of k irreducibles and a product of l irreducibles. The set L(a) of all possible factorization lengths of a is called the set of lengths of a, and the system consisting of all these sets L(a) is a well-studied means of describing the nonuniqueness of factorizations of a ring or semigroup. We provide a friendly introduction, which is largely self-contained, to what is known about systems of sets of lengths for rings of integers of algebraic number fields and for transfer Krull monoids of finite type as their generalization.

The authors wish to celebrate the centenary of Pólya's paper Ueber ganzwertige ganze funktionen where first explicitly appeared the term “integer-valued polynomials.” This survey is focused on the emblematic example of the ring Int(ℤ) formed by the polynomials with rational coefficients taking integer values on the integers. This ring has surprising algebraic properties, often obtained by means of analytical properties. Yet, the article mentions also several extensions, either by considering integer-valued polynomials on a subset of ℤ, or by replacing ℤ by the ring of integers of a number field.

We present a simple proof of Cayley's formula.

There have been many articles in the MONTHLY on quotient sets over the years. We take a first step here into the p-adic setting, which we hope will spur further research. We show that the set of quotients of nonzero Fibonacci numbers is dense in the p-adic numbers for every prime p.

The friendship paradox, proved by Feld in 1991, states that “on average, your friends have more friends than you do.” In fact, Feld proved two versions of the paradox. We discuss generalizations of each of them that talk about the average number of friends that, for instance, a friend of a friend of a friend of a friend has.

The Laplace operator is pervasive in many important mathematical models, and fundamental results such as the mean value theorem for harmonic functions, and the maximum principle for superharmonic functions are well known. Less well known is how the Laplacian and its powers appear naturally in a series expansion of the mean value of a function on a ball or sphere. This result is proven here using Taylor's theorem and explicit values for integrals of monomials on balls and spheres. This result allows for nonstandard proofs of the mean value theorem and the maximum principle. Connections are also made with the discrete Laplacian arising from finite difference discretization.

In this note we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic probability concepts.

We give a simple formula for the looping rate of a loop-erased random walk on a finite planar graph. The looping rate is closely related to the expected amount of sand in a recurrent sandpile on the graph. The looping rate formula is well suited to taking limits where the graph tends to an infinite lattice, and we use it to give an elementary derivation of the (previously computed) looping rate and sandpile densities of the square, triangular, and honeycomb lattices, and compute (for the first time) the looping rate and sandpile densities of many other lattices, such as the kagomé lattice, the dice lattice, the truncated hexagonal lattice (for which the values are all rational), and the square-octagon lattice (for which it is transcendental).

We extend the notion of parking functions to parking sequences, which include cars of different sizes, and prove a product formula for the number of such sequences.