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Showing papers in "American Mathematical Monthly in 2017"


Book ChapterDOI
TL;DR: This paper aims to explore how the deeply human themes that drive us to do mathematics can be channeled to build a more beautiful and just world in which all can truly flourish.
Abstract: How can the deeply human themes that drive us to do mathematics be channeled to build a more beautiful and just world in which all can truly flourish?

51 citations


Journal ArticleDOI
TL;DR: A numerical semigroup is an additive submonoid of the natural numbers with finite complement as discussed by the authors, and the size of the complement is called the genus of the semigroup, which is defined as a function of the number of natural numbers in a semigroup.
Abstract: A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical semigroups have gen...

35 citations


Journal ArticleDOI
TL;DR: In this paper, a survey of extremal regular graphs with respect to the number of independent sets and graph homomorphisms is presented, in particular in the family of d-regular graphs.
Abstract: This survey concerns regular graphs that are extremal with respect to the number of independent sets and, more generally, graph homomorphisms. More precisely, in the family of of d-regular graphs, ...

30 citations


Journal ArticleDOI
TL;DR: A different approach using the space of two-dimensional Euclidean lattices shows that there are at most three distinct gap lengths in the fractional parts of the sequence α, 2α,.
Abstract: The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence α, 2α,.…., Nα, for any integer N and real number α. This statement was proved in the 1950s independently by various authors. Here we present a different approach using the space of two-dimensional Euclidean lattices.

25 citations


Journal ArticleDOI
TL;DR: Chen, Hou, and Zeilberger as discussed by the authors developed an algorithm for finding congruence identities (modulo primes) of indefinite sums of many combinatorial sequences, namely those (like the Catalan and Motzkin sequences) that are expressible in terms of constant terms of powers of Laurent polynomials.
Abstract: Recently, William Y.C. Chen, Qing-Hu Hou, and Doron Zeilberger developed an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences, namely those (like the Catalan and Motzkin sequences) that are expressible in terms of constant terms of powers of Laurent polynomials. We first give a leisurely exposition of their approach and then extend it in two directions. The Laurent polynomials may be of several variables, and instead of single sums we have multiple sums. In fact, we even combine these two generalizations. We conclude with some super-challenges.

24 citations


Journal ArticleDOI
TL;DR: In this article, the authors give the flavor of the subject of self-similar tilings in a relatively elementary setting and provide a novel method for the construction of such polygonal tilings.
Abstract: The purpose of this paper is to give the flavor of the subject of self-similar tilings in a relatively elementary setting and to provide a novel method for the construction of such polygonal tilings.

13 citations


Journal ArticleDOI
TL;DR: Asquarefree product formula for the denominators of the Bernoulli polynomials is derived and it is shown that such a denominator equals n + 1 times the squarefree product of certain primes p obeying the condition that the sum of the base-p digits of n +1 is at least p.
Abstract: The power sum 1n + 2n +… + xn has been of interest to mathematicians since classical times. Johann Faulhaber, Jacob Bernoulli, and others who followed expressed power sums as polynomials in x of de...

13 citations


Journal ArticleDOI
TL;DR: Geometric proofs of Menger's results on isometrically embedding metric spaces in Euclidean space are presented.
Abstract: We present geometric proofs of Menger's results on isometrically embedding metric spaces in Euclidean space.

12 citations


Journal ArticleDOI
TL;DR: In this paper, the authors recast Euclid's proof of the infinitude of prime numbers as a Euclidean criterion for a domain to have infinitely many atoms and showed that their criterion applies even in certain domains in which not all nonzero nonunits factor into products of irreducibles.
Abstract: We recast Euclid's proof of the infinitude of prime numbers as a Euclidean criterion for a domain to have infinitely many atoms. We make connections with Furstenberg's “topological“ proof of the infinitude of prime numbers and show that our criterion applies even in certain domains in which not all nonzero nonunits factor into products of irreducibles.

12 citations


Journal ArticleDOI
TL;DR: There is a mathematical explanation for the auditory illusion in my example, sometimes called the missing fundamental: the perceived pitch is the greatest common divisor of the frequencies of the sine waves present.
Abstract: (2017). From Music to Mathematics: Exploring the Connections. By Gareth E. Roberts. The American Mathematical Monthly: Vol. 124, No. 10, pp. 979-982.

12 citations


Journal ArticleDOI
TL;DR: It is shown that if two monic polynomials with integer coefficients have a square-free resultant, then all positive divisors of the resultant arise as the greatest common divisor of the values of the two polynomers at a suitable integer.
Abstract: We show that if two monic polynomials with integer coefficients have a square-free resultant, then all positive divisors of the resultant arise as the greatest common divisor of the values of the t...

Journal ArticleDOI
TL;DR: A few relations satisfied by Chebyshev polynomials of the first and second kind and the minimal polynomial of cos(2π/n) are presented and the proof of the main theorem shows how cyclotomic polynmials can be used to link these two kinds of polynomsials.
Abstract: The minimal polynomial of cos(2π/n) allows one to realize the value of cos(2π/n) as the root of a polynomial with rational coefficients. These polynomials prove to be instrumental in expressing som...

Journal ArticleDOI
TL;DR: An elementary treatment for the case that the circular arc is a semicircle is presented and a new proof of Lexell's theorem is presented, without using Girard's theorem.
Abstract: Lexell's theorem states that two spherical triangles ABC and ABD have the same area if C and D lie on the same circular arc with endpoints A* and B*, which are the antipodal points of A and B, respectively. We present an elementary treatment for the case that the circular arc is a semicircle. In addition, a new proof of Lexell's theorem is presented, without using Girard's theorem. Finally, we give an improved version of Lexell's theorem in terms of the chord-tangent angle.

Journal ArticleDOI
TL;DR: Simple and intuitive proofs deriving matrix versions of Fermat's little theorem and the Gauss congruence to integer square matrices from the corresponding number-theoretic results are given.
Abstract: In recent years, several papers appeared on generalizations of Fermat's little theorem and the Gauss congruence to integer square matrices. We will give simple and intuitive proofs deriving these matrix versions from the corresponding number-theoretic results.

Journal ArticleDOI
TL;DR: This paper gives an elementary way of filling the gap in Gauss's proof of the fundamental theorem of algebra and shows that Gauss' proof contained a significant gap.
Abstract: Carl Friedrich Gauss is often given credit for providing the first correct proof of the fundamental theorem of algebra in his 1799 doctoral dissertation. However, Gauss's proof contained a signific...

Journal ArticleDOI
TL;DR: The general context of the separation between roots of a polynomial is presented and tight bounds for the case of real roots are given.
Abstract: While the separation (the minimal nonzero distance) between roots of a polynomial is a classical topic, its absolute counterpart (the minimal nonzero distance between their absolute values) does no...

Journal ArticleDOI
TL;DR: This paper cast a glance back one hundred years to the early part of the twentieth century, focusing on the following question: What does it mean to assert a mathematical claim, for example that there is a prime between 5 and 10? And how do we come to know it in the first place?
Abstract: What does it mean to assert a mathematical claim, for example that there is a prime between 5 and 10? If the claim is true, then what makes it true? And how do we come to know it in the first place? It is apparently basic questions such as these that drive the field of philosophy of mathematics. That these questions arise for even the most elementary mathematical propositions makes the philosophical project to elucidate the nature of mathematics accessible to nonspecialists. It also makes it frustratingly inconclusive. Before delving into contemporary philosophy of mathematics, let us begin by casting a glance back one hundred years to the early part of the twentieth century. At this time, philosophers of mathematics were focused on the following question

Journal ArticleDOI
TL;DR: In this paper, it was shown that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrary high bridge distance, and every knot of arbitrary high number of crossings is at least one bridge crossing away from any given bridge.
Abstract: We show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance.

Journal ArticleDOI
TL;DR: As a consequence of this approach, this work is able to deal with other problems such as factorization properties of Chebyshev polynomials of the first and second kind and with the classical problems of computing minimal polynomsials of algebraic values of trigonometric functions.
Abstract: Fil: Cafure, Antonio Artemio. Universidad de Buenos Aires. Ciclo Basico Comun; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

Journal ArticleDOI
TL;DR: A short probabilistic proof of Hoffman's and Sidorenko's results is given, using the basic identity of importance sampling, of the bound on the number of walks in a graph on the basis of matrix inequality.
Abstract: Hoffman [7] proved a matrix inequality that yields a useful upper bound on the number of walks in a graph. Sidorenko [14] extended the bound on the number of walks to a bound on the number of homom...

Journal ArticleDOI
TL;DR: This work provides a new direct proof of a general form of the result of Siebeck and Marden that every inellipse for a triangle is uniquely related to a certain logarithmic potential via its focal points.
Abstract: The Siebeck—Marden theorem relates the roots of a third degree polynomial and the roots of its derivative in a geometrical way. A few geometric arguments imply that every inellipse for a tr...

Journal ArticleDOI
TL;DR: In this article, the players take turns moving a token to an unoccupied point in such a way that the distance between the two tokens to the same point is smaller than the distance from the two to a fixed set.
Abstract: We introduce a two-player game involving two tokens located at points of a fixed set. The players take turns moving a token to an unoccupied point in such a way that the distance between the two to ...

Journal ArticleDOI
TL;DR: A review of five recent papers in this area by undergraduates, ranging from generating functions and modular forms to more combinatorial tools such as abaci, posets, and lattice paths, give a flavor of the richness of the subject.
Abstract: The theory of s-core partitions, integer partitions whose hook sets avoid hooks of length s, lies at the intersection of a surprising number of fields, including number theory, combinatorics, and representation theory. A more recent trend has been to study partitions whose hook sets avoid multiple lengths, known as simultaneous core partitions. This paper, divided into five sections, is a review of five recent papers in this area by undergraduates ([3], [4], [5], [18], [67]). All of the authors surveyed conducted their research while participating in the University of Minnesota Duluth REU.In the first section, we introduce partitions, the abacus, s-core partitions, and their connections to several fields. In the second section, we turn to self-conjugate s-core partitions and discuss several theorems of L. Alpoge on their asymptotic behavior and their connection, for small s, with points on curves. In the third section, we discuss simultaneous (s, t)-core partitions and the work of A. Aggarwal and ...

Journal ArticleDOI
TL;DR: It turns out that the authors can derive an explicit formula for the solution to the differential equation, and from that solution, they see that the effect of gravity exactly counteracts the tension in the Slinky.
Abstract: It is an interesting and counterintuitive fact that a Slinky released from a hanging position does not begin to fall all at once but rather each part of the Slinky only starts to fall when the coll...

Journal ArticleDOI
TL;DR: The unbiased extension identifies mixtures of subsets from both sides such that their expansions imply the standard conditions of Hall's perfect matching theorem.
Abstract: The standard conditions in Hall's perfect matching theorem for a bipartite graph G require that all subsets from one side of G are expanding. The unbiased extension identifies mixtures of subsets f...

Journal ArticleDOI
TL;DR: In this paper, the authors studied the length of curves passing through a fixed number of points on the boundary of a convex shape in the plane and showed that any curve passing through these points is at least half of the perimeter of the shape.
Abstract: We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that, for any convex shape K, there exist four points on the boundary of K such that the length of any curve passing through these points is at least half of the perimeter of K. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of K. Moreover, the factor ½ cannot be achieved with any fixed number of extreme points. We conclude the paper with a few other inequalities related to the perimeter of a convex shape.

Journal ArticleDOI
TL;DR: In this paper, it was shown that among all equilateral polygons with a given number of sides and the same diameter, the regular polygon has the maximal area, and that the polygon with the largest diameter has the largest area.
Abstract: We show that among all equilateral polygons with a given number of sides and the same diameter, the regular polygon has the maximal area.

Journal ArticleDOI
TL;DR: Brunn-Minkowski theory and well-known integration tools are used to prove Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.
Abstract: We use Brunn-Minkowski theory and well-known integration tools to prove Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its su...

Journal ArticleDOI
TL;DR: A new proof that there are infinitely many primes is given, relying on van derWaerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an arithmetic progression.
Abstract: We give a new proof that there are infinitely many primes, relying on van derWaerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an arithmetic pro...

Journal ArticleDOI
TL;DR: In this paper, Fuchs posed the following problem: determine whether a given abelian group can occur as the group of units in a commutative ring, which remains open.
Abstract: Laszlo Fuchs posed the following problem in 1960, which remains open: determine whether a given abelian group can occur as the group of units in a commutative ring. In this note, we provide...