Showing papers in "American Mathematical Monthly in 2020"
TL;DR: Mantel’s theorem is extended as follows: fan-free linear k-graphs on n points have at most edges, which nicely illustrates the difficulties of hypergraph Turán problems.
Abstract: According to Mantel’s theorem, a triangle-free graph on n points has at most n2/4 edges. A linear k-graph is a set of points together with some k-element subsets, called edges, such that any two ed...
17 citations
TL;DR: It is shown that any ω-monoid is either a scalar multiple of a numerical semigroup or a tempered monoid, and how it can be differentiated from those that are tempered monoids by the size and commensurability of their minimal generating sets.
Abstract: 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. T...
16 citations
TL;DR: In recent years, various notions of algebraic independence have emerged as a central and unifying theme in a number of areas of applied mathematics, including algebraic statistics and the r....
Abstract: In recent years, various notions of algebraic independence have emerged as a central and unifying theme in a number of areas of applied mathematics, including algebraic statistics and the r...
14 citations
TL;DR: A new short proof of the classical Hermite–Hadamard inequalities is presented, and it is shown that the inequality between Hermite and Hadamard is equivalent to 1-2 and 2-3.
Abstract: Abstract In this note, we present a new short proof of the classical Hermite–Hadamard inequalities.
13 citations
TL;DR: This alternative view of partitions having certain crank values allows us to strengthen a recent result of Yuefei Shen with a greatly simplified and more insightful proof.
Abstract: After reviewing the history of the crank of an integer partition, involving work of Freeman Dyson from 1944 and Frank Garvan and George Andrews in 1988, we connect this concept with more recent wor...
12 citations
TL;DR: It is shown how two well-known results from linear algebra, the matrix determinant lemma and the Schur complement, can be used to count the spanning trees in several significant families of graphs in an elegant manner.
Abstract: Kirchhoff’s matrix-tree theorem asserts that the number of spanning trees in a finite graph can be computed from the determinant of any of its reduced Laplacian matrices. In many cases, even for we...
12 citations
TL;DR: A variety of new proofs of Campbell’s hypergeometric formula for the parbelos constant S are offered, including a creative proof that makes use of a Fourier–Legendre expansion.
Abstract: In the December 2013 issue of this Monthly, Sondow introduced a parabolic analog of the classical arbelos figure called the parbelos. He found that the ratio of the perimeter of an arbitrar...
10 citations
TL;DR: In this article, the Frobenius coin exchange problem was studied in terms of the largest integer that cannot be represented as a nonnegative integral linear combinatorial combinatoria.
Abstract: We study variants of the Frobenius coin-exchange problem: Given n positive relatively prime parameters, what is the largest integer that cannot be represented as a nonnegative integral linear combi...
10 citations
TL;DR: A self-contained topological proof of the continuous dependence of the roots of a polynomial on its coefficients is provided as a homeomorphism between the space of monic complex polynomials of degree n and thespace of unordered n-tuples of complex numbers.
Abstract: We provide a self-contained topological proof of the continuous dependence of the roots of a polynomial on its coefficients. We present it as a homeomorphism between the space of monic complex poly...
10 citations
TL;DR: A geometric proof of Fermat’s fundamental formula for figurate numbers is given and a formula—motivated by the inclusion-exclusion principle—for as a linear combination of figurateNumbers is presented.
Abstract: First, we give a geometric proof of Fermat’s fundamental formula for figurate numbers. Then we use geometrical reasoning to derive weighted identities with figurate numbers and observe some of thei...
9 citations
TL;DR: In this paper, the authors provided a simple proof for the extended Bertrand-De Morgan test that was earlier studied in [Ďuris, F., (2009). Infinite series: Convergence tests.
Abstract: We provide a simple proof for the extended Bertrand–De Morgan test that was earlier studied in [Ďuris, F., (2009). Infinite series: Convergence tests. Bachelor thesis, Univerzita Komenskeho, Bratis...
TL;DR: If the group G acts on the set X, the orbit of the element x∈X is defined as orb(x)={gx:g∈G}.
Abstract: If the group G acts on the set X, we define the orbit of the element x∈X as orb(x)={gx:g∈G}. The orbits partition X, and the orbit-counting lemma (see Neumann [2] for the history of its name) shows...
TL;DR: A l’Hôpital’s rule is presented that provides a way to simplify and resolve a wide variety of zero-over-zero limits in terms of quotients of their derivatives.
Abstract: Zero divided by zero is arguably the single most important concept underlying calculus. For functions of more than one variable, methods of proof for indeterminate limits are not as familiar as for...
TL;DR: Mills proved that there exists a real constant A such that for all the values are prime numbers, and gives a first unconditional variant: is prime, where can be computed to millions of digits.
Abstract: Mills proved that there exists a real constant A > 1 such that for all n∈N the values ⌊A3n⌋ are prime numbers. No explicit value of A is known, but assuming the Riemann hypothesis one can choose A=...
TL;DR: The nineteenth-century version of the bounded convergence theorem is revisited, formulated by C. Arzelà in 1885 for Riemann integrable functions and by W. F. Osgood in 1897 for continuous functions.
Abstract: We revisit the nineteenth-century version of the bounded convergence theorem formulated by C. Arzela in 1885 for Riemann integrable functions and, independently, by W. F. Osgood in 1897 for continu...
TL;DR: The Kempner set K(S,b) as discussed by the authors is the set of all nonnegative integers whose base-b digital expansions contain only digits from S. This set was defined for integer b⩾2 and a set S⊂{0, S,b−1}.
Abstract: For an integer b⩾2 and a set S⊂{0,…,b−1} , we define the Kempner set K(S,b) to be the set of all nonnegative integers whose base-b digital expansions contain only digits from S. These well-studied ...
TL;DR: It is shown that the point set given by the values with is a Delone set in the complex plane, for any α, which complements Akiyama's recent observation that with forms aDelone set, if and only if α is badly approximated by rationals.
Abstract: Delone sets are locally finite point sets, such that (a) any two points are separated by a given minimum distance, and (b) there is a given radius so that every ball of that radius contains at leas...
TL;DR: Three proofs of Tychonoff’s theorem on the compactness of a product of compact topological spaces are given, each with a “categorical” flavor.
Abstract: We give three proofs of Tychonoff’s theorem on the compactness of a product of compact topological spaces. The first one proceeds “from scratch.” The second one relies on the characterization of co...
TL;DR: Some basic results of convex analysis and geometry are reviewed in the context of formulating a differential equation to track the distance between an observer flying outside a convex set K and K itself.
Abstract: We review some basic results of convex analysis and geometry in Rn in the context of formulating a differential equation to track the distance between an observer flying outside a convex set K and ...
TL;DR: The standard proof of the inequality of the title may appear a bit dry, and so I offer one that uses water.
Abstract: The standard proof of the inequality of the title may appear a bit dry, and so I offer one that uses water. Consider n cylindrical cans with different cross-sectional areas ak that are connected by...
TL;DR: It is shown that the Conway–Paterson–Moscow theorem can be proved and an application to a well-known number-theoretic problem is made.
Abstract: After disproving the celebrated Conway–Paterson–Moscow theorem [1], we prove that theorem and make an application to a well-known number-theoretic problem.
TL;DR: A combinatorial proof of two equivalent formulae on the weighted enumeration of spanning trees of an edge-weighted graph that results in a new proof of Foster’s first theorem.
Abstract: Let G be an electrical network graph with vertex set V and edge set E, and let Ω(i,j) be the effective resistance between vertices i and j of G. Foster’s first theorem states that ∑(i,j)∈EcijΩ(i,j)...
TL;DR: It is proved that the interior of every curve of bounded convex curvature contains an open unit disk.
Abstract: We say that a simple, closed curve γ in the plane has bounded convex curvature if for every point x on γ, there is an open unit disk Ux and ex>0 such that x∈∂Ux and Bex(x)∩Ux⊂Int γ. We prove that t...
TL;DR: A version of the well-known Riesz’s theorem on conjugate harmonic functions for Lumer's Hardy spaces on arbitrary domains Ω is obtained, with the best possible constant.
Abstract: In this note, we obtain a version of the well-known Riesz’s theorem on conjugate harmonic functions for Lumer’s Hardy spaces (Lh)2(Ω) on arbitrary domains Ω: If a real-valued harmonic function U∈(L...
TL;DR: It is shown that 17 is prime, and let n be the product of all primes except 17, which means that there are only finitely many primes.
Abstract: Suppose there are only finitely many primes. Observe that 17 is prime, and let n be the product of all primes except 17. Let S be the set of positive integers that are coprime to n, and let T be th...
TL;DR: It is shown that each positive rational number can be written as , where is Euler’s totient function and m and n are positive integers.
Abstract: In this note, we show that each positive rational number can be written as φ(m2)/φ(n2) , where φ is Euler’s totient function and m and n are positive integers.
TL;DR: A theorem implicit in the work of Schönberg is that these three conditions characterize a unique weight and for some constant functions.
Abstract: Let f:R→R be a continuous function for which we want to take local averages. Assuming we cannot look into the future, a commonly encountered problem is that the “average” g(t) at time t can only us...
TL;DR: This work examines the aggregate behavior of one-dimensional random walks in a model known as (one-dimensional) Internal Diffusion Limited Aggregation and shows the probability that k of the occupied sites are positive is given by an Eulerian probability distribution.
Abstract: We examine the aggregate behavior of one-dimensional random walks in a model known as (one-dimensional) Internal Diffusion Limited Aggregation. In this model, a sequence of n particles perform rand...
TL;DR: It is proved that the first k – 1 moments of their curvatures remain constant within a 1-parameter family of Steiner chains of length k, which follows from the Descartes circle theorem.
Abstract: A Steiner chain of length k consists of k circles tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that on...
TL;DR: Some maximum norm principles for scalar-valued analytic functions that do not appear to be widely known are discussed, and maximum and minimum principles for their singular values are deduced.
Abstract: To what extent is the maximum modulus principle for scalar-valued analytic functions valid for matrix-valued analytic functions? In response, we discuss some maximum norm principles for such functi...