Showing papers in "American Mathematical Monthly in 2019"
TL;DR: The canonical equations of the centers of circumscribed and inscribed circles to the triangles that are the 3-periodic orbits of an elliptic billiard are ellipses and the geometric locus defined by the barycenters of the edges of billiard triangles is an ellipse.
Abstract: 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 o...
37 citations
TL;DR: In particular, every real number between zero and one is the product of three elements of the Cantor set C as discussed by the authors, where x, y are elements of C and x is the smallest element in C.
Abstract: 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 th...
28 citations
TL;DR: Using Fourier analysis, a short and self-contained proof of a classical result of Erdős and Turán 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 unitcircle and become equidistributed in angle.
Abstract: A classical result of Erdős and Turan 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 ...
17 citations
TL;DR: 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.
Abstract: The theorems of Hadamard and of Poincare–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 el...
11 citations
TL;DR: 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.
Abstract: 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 laws...
11 citations
TL;DR: The Markov, Chebyshev, and Chernoff inequalities are some of the most widely used methods for bounding the tail probabilities of random variables as mentioned in this paper, and they are tight in all three cases.
Abstract: 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 se...
11 citations
TL;DR: 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 as discussed by the authors.
Abstract: 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 resulti...
11 citations
TL;DR: Convergence behavior of d-dimensional multiagent systems with particular emphasis on opinion dynamics under bounded confidence is explored, sharpening in particular a well-known theorem on Markov chains.
Abstract: 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, ...
10 citations
TL;DR: The question: which pairs of knots are related to each other by a knotoid is asked and the two most natural ways of defining such maps give an interesting relation between pairs of Knots.
Abstract: Turaev recently introduced the concept of a knotoid as a particular sort of knotted arc. There are several maps from knotoids to ordinary knots, or knotted circles. The two most natural way...
9 citations
TL;DR: This article investigates how to deterministically perturb a class of unstable Hamiltonian systems in such a way that the qualitative behavior is preserved, but that enables the systems to exhibit noise-induced stabilization.
Abstract: Noise-induced stabilization is the phenomenon in which the addition of randomness to an unstable system of ordinary differential equations results in a stable system of stochastic differential equa...
8 citations
TL;DR: In this paper, the Grassmann manifold of 2-planes in real n-space has been constructed for planar n-gons, and a natural and beautiful correspondence between planar gons and Grassmann manifolds has been established.
Abstract: 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 lea...
TL;DR: This work considers the convergence properties of series that are formed from the harmonic series by removing some of the terms, and focuses on those integers that contain a given digit in their decimal representation.
Abstract: We consider the convergence properties of series that are formed from the harmonic series by removing some of the terms. Our primary focus concerns the removal of those integers that contain a give...
TL;DR: Though Siebeck’s theorem is a geometric statement about complex functions, it is used linear algebra and the numerical range of a matrix to provide a proof of the theorem.
Abstract: We consider several seemingly unrelated theorems, each with the word ellipse in the statement: Siebeck’s theorem, the elliptical range theorem, Poncelet’s theorem, and the Blaschke ellipse theorem....
TL;DR: It is shown that a group with a finite number of elements of maximal order must be finite, in other words, there are no infinite groups with finitely many elements ofmaximal order.
Abstract: In this note we show that a group with a finite number of elements of maximal order must be finite. In other words, there are no infinite groups with finitely many elements of maximal order. In par...
TL;DR: This work gives a self-contained proof that selection along deterministic sequences preserves normality and provides several interesting examples of nontrivial deterministic sequence.
Abstract: If a sequence of numbers behaves like a random sequence, do we expect subsequences to also behave like a random sequence? Wall proved that normality of a base-b expansion is preserved along arithme...
TL;DR: The distribution of the number of incorrectly seated passengers is studied, and the case of several absent-minded passengers is discussed.
Abstract: Passengers board a fully booked airplane in order. The first passenger picks one of the seats at random. Each subsequent passenger takes his or her assigned seat if available, otherwise takes one o...
TL;DR: In this paper, it was shown that there exists a geodesic trajectory on the dodecahedron from a vertex to itself that does not pass through any other vertex, and that such a trajectory can be obtained from any vertex to any vertex.
Abstract: We prove that there exists a geodesic trajectory on the dodecahedron from a vertex to itself that does not pass through any other vertex.
TL;DR: A natural definition of the product of infinite matrices mimics the usual formulation of multiplication of finite matrices with the caveat (in the absence of any sense of convergence) that the inte...
Abstract: A natural definition of the product of infinite matrices mimics the usual formulation of multiplication of finite matrices with the caveat (in the absence of any sense of convergence) that the inte...
TL;DR: This article answers the question of whether 73 is the unique Sheldon prime in The Big Bang Theory and concludes that it is in the affirmative.
Abstract: In [3], the authors introduce the concept of a Sheldon prime, based on a conversation between several characters in the CBS television situation comedy The Big Bang Theory. The authors of [3] leave...
TL;DR: This work tells the story of Erdős' attempt to give as many points in as possible such that all angles formed by the points are acute, before a number of twists took place recently.
Abstract: Erdős asked to give as many points in Rd as possible such that all angles formed by the points are acute. For decades the progress had been slow before a number of twists took place recently. We te...
TL;DR: This work presents a new solution to the question of how many incongruent triangles with integer sides have perimeter n based on group actions and orbit enumeration.
Abstract: The following question has been considered by several mathematicians over the last half century:How many incongruent triangles with integer sides have perimeter n?We present a new solution based on...
TL;DR: The history of the twisted cubic curves in three-dimensional affine and projective spaces is traced to serve as primary objects of study, as motivating examples, and as hidden underlying structures for objects in mathematics and its applications.
Abstract: We trace some of the history of the twisted cubic curves in three-dimensional affine and projective spaces. These curves have reappeared many times in different guises. They have served as primary ...
TL;DR: In this article, it was shown that the primes do not contain arbitrarily long arithmetic progressions, and a straightforward argument was provided to show that the primitives do not have arbitrarily long progressions.
Abstract: A celebrated and deep result of Green and Tao states that the primes contain arbitrarily long arithmetic progressions. In this note, I provide a straightforward argument demonstrating that the prim...
TL;DR: A polyhedron has the Rupert property if a straight tunnel can be made in it, large enough so that a copy of can pass through this tunnel.
Abstract: A polyhedron P has the Rupert property if a straight tunnel can be made in it, large enough so that a copy of P can pass through this tunnel. Eight Archimedean polyhedra are known to have the Ruper...
TL;DR: In this article, a survey of all available information on a remarkable problem in number theory proposed by Moser in 1957 is presented. The general question is: Can a collection of n numbers be uniquely restored given a given set of numbers?
Abstract: This is a survey of all available information on a remarkable problem in number theory proposed by Leo Moser in 1957. The general question is: Can a collection of n numbers be uniquely restored giv...
TL;DR: In this article, it was shown that there is a constant θ∈(0, 1) such that, for each l>0, t t ≥ 1/2, t is divergent.
Abstract: Let (xn) be a positive real sequence decreasing to 0 such that the series ∑nxn is divergent and liminfnxn+1/xn>1/2 . We show that there exists a constant θ∈(0,1) such that, for each l>0 , t...
TL;DR: This class of game is analyzed using Arthur Engel’s stochastic abacus—a chip firing algorithm—revealing a natural relationship with the Fibonacci numbers.
Abstract: A “pass-the-buck” game begins with n players arranged as the vertices in a graph. One player has a dollar bill. The game is played in turns where, at random, the current bill-holder either wins out...
TL;DR: A direct algebraic method is used that yields all rational solutions of Newton’s equation, albeit parameterized in a way very different from that obtained by Barnett (and Hajja), and an explicit, nonobvious, birational correspondence between the two parameterizations is established.
Abstract: A Newton quadrilateral is a convex quadrilateral inscribed in a circle, with one side a diameter. For diameter d and the other side lengths a,b, and c , Newton arrived at the cubic equation d3−(a2+...
TL;DR: The question of when is a function from an abelian group to itself expressible as a difference of two bijections for finite cyclic groups is answered and the question is extended beyond the realm of group theory, where the expressibility of a function translates to questions about partial transversals in Latin squares.
Abstract: When is a function from an abelian group to itself expressible as a difference of two bijections? Answering this question for finite cyclic groups solves a problem about juggling. A theorem of Mars...
TL;DR: An elementary transformation formula is derived, allowing double integrals of the type introduced by F. Beukers to be reduced, and new ones to be constructed.
Abstract: An elementary transformation formula is derived, allowing double integrals of the type introduced by F. Beukers to be reduced, and new ones to be constructed.