Showing papers in "American Mathematical Monthly in 2013"
TL;DR: In this mostly self-contained exposition, the reader is introduced to the interplay between the the study of direct-sum decompositions of modules and theStudy of factorizations in integral domains.
Abstract: Let R be a commutative ring with identity. It often happens that M1 ⊕ ⋯ ⊕ Ms ≅ N1 ⊕ ⋯ ⊕ Nt for indecomposable R-modules M1, …, Ms and N1, …, Nt with s ≠ t. This behavior can be captured by studying the commutative monoid {[M] ❘ M is an R-module} of isomorphism classes of R-modules with operation given by [M] + [N] = [M ⊕ N]. In this mostly self-contained exposition, we introduce the reader to the interplay between the the study of direct-sum decompositions of modules and the study of factorizations in integral domains.
72 citations
TL;DR: A proof of a 100-year-old conjecture: If the front wheel track of a unit bike is an oval with area at least π, then the respective monodromy is hyperbolic.
Abstract: We study a simple model of bicycle motion: A bicycle is a segment of fixed length that can move in the plane so that the velocity of the rear end is always aligned with the segment The sam
47 citations
TL;DR: A short proof is given of a comprehensive system of inequalities for including results obtained independently by Artin, Wendel, and Gautschi, including inequalities for binomial coefficients and the Bohr–Mollerup theorem.
Abstract: A short proof is given of a comprehensive system of inequalities for including results obtained independently by Artin, Wendel, and Gautschi Applications include inequalities for binomial coefficients and the Bohr–Mollerup theorem
43 citations
TL;DR: It is shown that evolutes and involutes of space curves are related to each other in different ways and the relationships between these properties and each other are explained by different mechanisms.
Abstract: Abstract We study evolutes and involutes of space curves.
31 citations
TL;DR: It is proved that no perfect division exists for more than 4 cuts and for an extension of this example to more than three players.
Abstract: A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake in which it is impossible to divide it among three players such that these three properties are satisfied, however many cuts are made. It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability but not both). We prove that no perfect division exists for an extension of the example for three or more players.
28 citations
Journal Article•
24 citations
TL;DR: Infinitesimals are natural products of the human imagination and their history goes back to Greek antiquity as mentioned in this paper. Their role in calculus and analysis has seen dramatic ups and downs; they have stimulat...
Abstract: Infinitesimals are natural products of the human imagination. Their history goes back to Greek antiquity. Their role in the calculus and analysis has seen dramatic ups and downs; they have stimulat...
24 citations
TL;DR: The history of the mathematics major and, more generally, collegiate mathematics, in the United States is described.
Abstract: This article describes the history of the mathematics major and, more generally, collegiate mathematics, in the United States. Interestingly, the Mathematical Association of America was organized about 100 years ago, around the same time that academic majors came into existence.
24 citations
TL;DR: It is pointed out that the Cut Axiom is equivalent to four “cornerstone theorems” of single-variable Real Analysis, namely, the Intermediate, Extreme, and Mean Value Theorems, as well as Darboux's Theorem.
Abstract: We first discuss the Cut Axiom, due to Dedekind, which is one of the many equivalent formulations of the completeness of the real numbers. We point out that the Cut Axiom is equivalent to four “cor...
22 citations
TL;DR: This paper was able to trace the origin of the secant method all the way back to the Rule of Double False Position described in the 18th-century b.c. Egyptian Rhind Papyrus, by showing that it predates Newton's method by more than 3,000 years.
Abstract: Many in the mathematical community believe that the secant method arose from Newton's method using a finite difference approximation to the derivative, most likely because that is the way that it i...
21 citations
TL;DR: Many of the theorems of real analysis are equivalent to Dedekind completeness, and hence can serve as completeness axioms for the reals, and the article offers a tour of some less-familiar ordered fields.
Abstract: Many of the theorems of real analysis, against the background of the ordered field axioms, are equivalent to Dedekind completeness, and hence can serve as completeness axioms for the reals. In the ...
TL;DR: In this paper, it has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers.
Abstract: It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the re...
TL;DR: It is argued that there are arithmetic statements that are true but not provable, but it is usually thought that they must necessarily be complicated.
Abstract: It has long been known that there are arithmetic statements that are true but not provable, but it is usually thought that they must necessarily be complicated. In this paper, I shall argue that th...
TL;DR: A number of classical proofs of the irreducibility of the nth cyclotomic polynomial ϕn(x), in both the original and a simplified form are presented.
Abstract: We present a number of classical proofs of the irreducibility of the nth cyclotomic polynomial ϕn(x). For n prime we present proofs due to Gauss (1801), in both the original and a simplified form, ...
TL;DR: It turns out that the classical calculus problem of describing the flow of brine in a system of tanks connected by pipes is closely related to the study of Laplacian matrices for directed graphs.
Abstract: We revisit the classical calculus problem of describing the flow of brine in a system of tanks connected by pipes. For various configurations involving an arbitrary number of tanks, we show that the corresponding linear system of differential equations can be solved analytically. Finally, we analyze the asymptotic behavior of solutions for a general closed system of tanks. It turns out that the problem is closely related to the study of Laplacian matrices for directed graphs.
TL;DR: In this paper, an elementary probabilistic proof of a binomial identity was given by computing the probability of a certain event in two different ways, yielding two different expressions for the binomial identities.
Abstract: We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for...
TL;DR: A short proof of the Coxeter–Conway classification theorem is presented and relations between Coxeter's frieze patterns and discrete Sturm–Liouville equations are studied.
Abstract: We study relations between Coxeter's frieze patterns and discrete Sturm–Liouville equations. We present a short proof of the Coxeter–Conway classification theorem.
TL;DR: It is shown that if A is an n × n complex matrix and λ is an eigenvalue of A with geometric multiplicity k, thenλ is in at least k of the n Geršgorin discs of A.
Abstract: If A is an n × n complex matrix and λ is an eigenvalue of A with geometric multiplicity k, then λ is in at least k of the n Gersgorin discs of A.
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TL;DR: In this paper, it was shown that two duelers with similar, lousy shooting skills (a.k.a. Galois duelers) will choose to take turns firing in accordance with the famous Thue-Morse sequence if they greedily demand their chances to fire as soon as the other's a priori probability of winning exceeds their own.
Abstract: We show that two duelers with similar, lousy shooting skills (a.k.a. Galois duelers) will choose to take turns firing in accordance with the famous Thue-Morse sequence if they greedily demand their chances to fire as soon as the other's a priori probability of winning exceeds their own. This contrasts with a result from the approximation theory of complex functions, which says what more patient duelers would do, if they really cared about being as fair as possible. We note a consequent interpretation of the Thue-Morse sequence in terms of certain expansions in fractional bases close to, but greater than, 1.
TL;DR: A generalization of both Pascal's Theorem and its converse, the Braikenridge–Maclaurin Theorem, that if two sets of k lines meet in k2 distinct points, then the remaining k(k − d) points lie on a unique curve S of degree k − d, which is d-constructible.
Abstract: We prove a generalization of both Pascal's Theorem and its converse, the Braikenridge–Maclaurin Theorem: If two sets of k lines meet in k2 distinct points, and if dk of those points lie on an irred...
TL;DR: Two simple mathematical results are discussed—the halftime price principle and the serious candidates principle—and corresponding data from baseball and the 2012 Republican Presidential nomination race are discussed.
Abstract: Prediction markets provide a rare setting where results of mathematical probability theory can be related to events of real-world interest and where one can compare theory to data. The paper discusses two simple mathematical results { the halftime price principle and the serious candidates principle { and corresponding data from baseball and the 2012 Republican Presidential Nomination race.
TL;DR: A classical result by Cauchy determines a disk containing all the zeros of a given polynomial, but this result is derived using only linear algebra techniques, and a twin disk is discovered that also contains all theZeros of thePolynomial.
Abstract: A classical result by Cauchy determines a disk containing all the zeros of a given polynomial. We derive this result using only linear algebra techniques, and, in the process, discover a twin disk ...
TL;DR: In this paper, the maximum number of leaf vertices contained in a spanning tree of a finite connected graph is defined as the number of vertices in the tree of the graph.
Abstract: Let G be a finite connected graph with minimum degree δ > 4. The leaf number L(G) of G is defined as the maximum number of leaf vertices contained in a spanning tree of G. We show that if δ ≥ L(G) ...
TL;DR: Fan's 1952 lemma on labelled triangulations of the n-sphere with n + 1 labels is equivalent to the Borsuk–Ulam theorem, and it is shown that this lemma directly implies Sperner's Lemma.
Abstract: We show that Fan's 1952 lemma on labelled triangulations of the n-sphere with nC 1 labels is equivalent to the Borsuk-Ulam theorem. Moreover, unlike other Borsuk-Ulam equivalents, we show that this lemma directly implies Sperner's Lemma, so this proof may be regarded as a combinatorial version of the fact that the Borsuk-Ulam theorem implies the Brouwer fixed-point theorem, or that the Lusternik-Schnirelmann-Borsuk theorem implies the KKM lemma. 1. INTRODUCTION. The Brouwer fixed-point theorem, the Knaster-Kuratowski- Mazurkiewicz (KKM) lemma, and Sperner's lemma are known to be equivalent. Equally powerful, they form a triumvirate of theorems whose interconnections have been exploited with great success in fixed point algorithms (15, 17) as well as in game theory (1). Similarly, the Borsuk-Ulam theorem, the Lusternik-Schnirelmann-Borsuk (LSB) theorem, and Tucker's lemma are another triumvirate of equivalent results. In each of these triples, the first is a topological result, the second is a set-covering result, and the third is a combinatorial result. Moreover, these triples are related to each other. Since the Borsuk-Ulam theorem implies the Brouwer fixed-point theorem, any theorem in the second triple must im- ply any theorem in the first. It is an interesting question to find direct proofs of each implication. For instance, a topological construction shows how a Brouwer fixed point follows from Borsuk-Ulam antipodes (13), and with set-coverings, the LSB theorem can be used to directly prove the KKM lemma (11). But in the combinatorial domain, we are unaware of a direct proof that Tucker's lemma implies Sperner's lemma. In this article, we show that another combinatorial lemma, Fan's NC 1 Lemma, may be a more natural combinatorial analogue to the Borsuk-Ulam theorem, and therefore more worthy to sit in the Borsuk-Ulam triumvirate than Tucker's lemma. In particular, in Section 3 we show that Fan's NC 1 Lemma is equivalent to the Borsuk- Ulam theorem, and in Section 4 we exhibit a direct proof that it implies Sperner's lemma (see Figure 1).
TL;DR: In this paper, a generalization of Routh's triangle theorem is presented, and a connection to Feynman's triangle is also given, which can be seen as a unification of Ceva and Menelaus.
Abstract: We prove a generalization of the well known Routh's triangle theorem. As a consequence, we get a unification of the theorems of Ceva and Menelaus. A connection to Feynman's triangle is also given.
TL;DR: The game of memory is played with a deck of n pairs of cards and it is determined that, when the game is played optimally, the expected number of moves is (3 − 2 ln 2)n + 7/8 − 2Ln 2 ≈ 1.61n.
Abstract: The game of memory is played with a deck of n pairs of cards. The cards in each pair are identical. The deck is shuffled and the cards laid face down. A move consists of flipping over first one car...
TL;DR: A large class of multiplicative submonoids of the natural numbers is presented, which includes congruence monoids as well as numerical monoids (by isomorphism) and the important factorization property of finite elasticity is characterized.
Abstract: A large class of multiplicative submonoids of the natural numbers is presented, which includes congruence monoids as well as numerical monoids (by isomorphism). For monoids in this class, the important factorization property of finite elasticity is characterized.
TL;DR: A parabolic analog of the arbelos is introduced, using theorems of Archimedes and Lambert to demonstrate seven properties of the parbelos, drawing analogies to similar properties ofThe arbelo, some of which may be new.
Abstract: The arbelos is a classical geometric shape bounded by three mutually tangent semicircles with collinear diameters. We introduce a parabolic analog, the parbelos. After a review of the parabola, we ...
TL;DR: It is shown that with at most one exception, all of these analogues of Euler's ᵞ constant are transcendental.
Abstract: We consider a class of analogues of Euler's ᵞ constant and use Baker's theory of linear forms in logarithms to study its arithmetic properties. In particular, we show that with at most one exceptio...
TL;DR: A factorization property about strong divisibility sequences in a unique factorization domain R is proved and it is proved that the dynamical analogues of cyclotomic polynomials over any unique factorized domain R are indeed polynmials in R[x].
Abstract: Almost all algebra texts define cyclotomic polynomials using primitive nth roots of unity. However, the elementary formula gcd(xm - 1, xn - 1) = xgcd(m,n) - 1 in ℤ[x] can be used to define the cycl...