Showing papers in "American Mathematical Monthly in 2002"
TL;DR: The authors argues that conceptual metaphor plays a central role in mathematical ideas within the cognitive unconscious, from arithmetic and algebra to sets and logic to infinity in all of its forms, and that abstract ideas for the most part arise via conceptual metaphor-metaphorical ideas projecting from the way we function in the everyday physical world.
Abstract: This book is about mathematical ideas, about what mathematics means-and why. Abstract ideas, for the most part, arise via conceptual metaphor-metaphorical ideas projecting from the way we function in the everyday physical world. Where Mathematics Comes From argues that conceptual metaphor plays a central role in mathematical ideas within the cognitive unconscious-from arithmetic and algebra to sets and logic to infinity in all of its forms.
1,843 citations
TL;DR: A restatement in terms of set partitions can be proved easily in a few lines, as the authors shall see in Section 2, though it still requires a bit of work to pass from that form to the form in (1.1).
Abstract: (2002). The Curious History of Faa di Bruno's Formula. The American Mathematical Monthly: Vol. 109, No. 3, pp. 217-234.
574 citations
TL;DR: In this article, the Descartes Circle Theorem has been extended to include the notion of beyond the circle theorem, and the authors present a proof of the theorem's correctness.
Abstract: (2002). Beyond the Descartes Circle Theorem. The American Mathematical Monthly: Vol. 109, No. 4, pp. 338-361.
137 citations
TL;DR: Kemeny's constant is revisited, generalized, derived upper and lower bounds on it, and given a novel interpretation in terms of the number of links a random surfer will follow to reach his final destination.
Abstract: We revisit Kemeny's constant in the context of Web navigation, also known as "surfing." We generalize the constant, derive upper and lower bounds on it, and give it a novel interpretation in terms of the number of links a random surfer will follow to reach his final destination.
131 citations
125 citations
TL;DR: In this article, the topology of domains in 3-space has been studied in the context of vector calculus, and the authors propose a topology for 3-Space vector calculus.
Abstract: (2002). Vector Calculus and the Topology of Domains in 3-Space. The American Mathematical Monthly: Vol. 109, No. 5, pp. 409-442.
113 citations
TL;DR: An Elementary Problem Equivalent to the Riemann Hypothesis as discussed by the authors is an elementary problem that is equivalent to the one we consider in this paper. The American Mathematical Monthly: Vol. 109, No. 6, pp. 534-543.
Abstract: (2002). An Elementary Problem Equivalent to the Riemann Hypothesis. The American Mathematical Monthly: Vol. 109, No. 6, pp. 534-543.
113 citations
TL;DR: The constant K has the same meaning as above, and one notices that the error bound for the midpoint rule is one half that of the trapezoidal rule.
Abstract: The constant K has the same meaning as above. 1CAS = Computer Algebra System. 2The interval [0, 2π] is for convenience only. Everything we say can easily be extended to an arbitrary interval [a, b]. 3One notices that the error bound for the midpoint rule is one half that of the trapezoidal rule; compare (2) with (3). For a pretty geometrical explanation of why one can expect the midpoint rule to be better by about a factor of two, the reader is referred to Stewart [13, p. 460].
110 citations
TL;DR: A very short proof of Kneser's conjecture is produced by combining the celebrated result of Lusterik, Schnirelman, and Borsuk on sphere covers with Gale's theorem concerning the even distribution of points on the sphere that does not rely on Gale's result.
Abstract: (2002). A New Short Proof of Kneser's Conjecture. The American Mathematical Monthly: Vol. 109, No. 10, pp. 918-920.
83 citations
TL;DR: The Hexagonal Economic Regions Solve the Location Problem is a posthumous publication based on a paper originally written by David I. Dickinson in 2002 and then edited by David C. Dickinson and published in 2002.
Abstract: (2002). Hexagonal Economic Regions Solve the Location Problem. The American Mathematical Monthly: Vol. 109, No. 2, pp. 165-172.
65 citations
TL;DR: The author examines the role of randomness in the construction of sequences in theorems, and concludes that sequences can be constructed in a number of ways that are random in nature.
Abstract: (2002). What Is a Random Sequence? The American Mathematical Monthly: Vol. 109, No. 1, pp. 46-63.
TL;DR: The sum of the distances of any interior point M from the sides of an equilateral triangle equals the altitude of the triangle, and applying Viviani’s Theorem solves the problem.
Abstract: (2002). The Fermat-Steiner Problem. The American Mathematical Monthly: Vol. 109, No. 5, pp. 443-451.
TL;DR: In this paper, Plimpton 322 is discussed, one of the world's most famous ancient mathematical artefacts, and the ways in which studying ancient mathematics is, or should be, different from researching modern mathematics are explored.
Abstract: 1. INTRODUCTION. In this paper I shall discuss Plimpton 322, one of the world's most famous ancient mathematical artefacts [Figure 1]. But I also want to explore the ways in which studying ancient mathematics is, or should be, different from researching modern mathematics. One of the most cited analyses of Plimpton 322, published some 20 years ago, was called \" Sherlock Holmes in Babylon \" [4]. This enticing title gave out the message that deciphering historical documents was rather like solving a fictional murder mystery: the amateur detective-historian need only pit his razor-sharp intellect against the clues provided by the self-contained story that is the piece of mathematics he is studying. Not only will he solve the puzzle, but he will outwit the well-meaning but incompetent professional history-police every time. In real life, the past isn't like an old-fashioned whodunnit: historical documents can only be understood in their historical context.
TL;DR: It is not difficult to see that the converse of the Buniakowski conjecture is true; namely, if a polynomial represents prime numbers infinitely often, then it is an irreducible polynomic, and this follows from Dirichlet's theorem on primes in arithmetic progressions.
Abstract: (2002). Prime Numbers and Irreducible Polynomials. The American Mathematical Monthly: Vol. 109, No. 5, pp. 452-458.
TL;DR: The topic seems ideal since it synthesizes results from number theory, algebra, and linear algebra, gives a natural and somewhat gentle introduction to deeper subjects not yet encountered by a beginning algebra student, has historical features, and could be a subject of continuing student research.
Abstract: Finite groups of matrices appear early as examples in a first course in abstract algebra, and most of the time these examples are given with integral entries. While these groups provide a setting in which to illustrate new concepts and to pose problems, they also have surprising and beautiful properties. For example, Minkowski proved the unexpected result that GL(n, Z), the group of n x n matrices having inverses whose entries are also integers, has only finitely many isomorphism classes of finite subgroups. As a consequence, there are only finitely many possible orders for elements of GL(n, Z); fortunately, the possible orders can be determined using linear algebra. In general, the number of possible orders increases as n increases, but even here we have the surprising result that no new possible orders are obtained when going from GL(2k, Z) to GL(2k + 1, 7). This paper is an exposition of these and other related results and questions. Although finite groups of integral matrices have a long history, most of the beautiful theorems concerning them do not appear in texts and are not known to many algebraists (for example, the first named author). While still an area of active research (see [7], [12], [23], and [24]), major parts are accessible to students; indeed, this paper has its origin in a term paper written by the second author for a beginning modem algebra course that used [9] as a text. One of our goals for this paper is to be a source of problems, readings, and projects for other students; the topic seems ideal since it synthesizes results from number theory, algebra, and linear algebra, gives a natural and somewhat gentle introduction to deeper subjects not yet encountered by a beginning algebra student, has historical features, and could be a subject of continuing student research. With this goal in mind, the authors have included exercises and have tried to organize the paper so that parts can be read independently, or given to students in outline form. Our second goal is to give the reader examples of current research in the area.
TL;DR: Condorcet's and Arrow's contributions are only the first two parts in a natural progression that is a trilogy-ending with the remarkable Gibbard-SatterthWaite Manipulability Theorem, culminating in the striking generalization recently proved by Duggan and Schwartz.
Abstract: (2002). The Manipulability of Voting Systems. The American Mathematical Monthly: Vol. 109, No. 4, pp. 321-337.
TL;DR: In this paper, the topology in a factory is described as a configuration space in the form of a topology, where the configuration space is composed of a set of configuration spaces.
Abstract: (2002). Finding Topology in a Factory: Configuration Spaces. The American Mathematical Monthly: Vol. 109, No. 2, pp. 140-150.
TL;DR: The dihedral angle of a regular n-dimensional simplex is the angle formed by a pair of intersecting faces that is known to equal cos-'(l/n) (see Coxeter [1]).
Abstract: The dihedral angle of a regular n-dimensional simplex is the angle formed by a pair of intersecting faces. In the case n = 2 we are describing the angle at the vertex of an equilateral triangle, while in the case n = 3 we are describing the angle formed by the faces of the regular tetrahedron. (This angle is cos-'(1/3), or approximately 70.5?.) In the general n-dimensional case, the dihedral angle is known to equal cos-'(l/n) (see Coxeter [1]). This fact has been proved elegantly by R. Krasnodebski in [3]. Krasnodebski bases his work on a construction used by Coxeter in proving Gosset's theorem (see Coxeter [2]), so Krasnodebski's proof is neither self-contained nor elementary. As far as we know, there is no simple, elementary proof in print. om the sual generati g function fo the Stirling numbers of the second kind,
TL;DR: (2002).
Abstract: (2002). Combinatorics of Higher Derivatives of Inverses. The American Mathematical Monthly: Vol. 109, No. 3, pp. 273-277.
TL;DR: This chapter discusses investment proportions in senior securities and equities under alternate holding periods, J. Levy and D. Gunthorpe (1999), which found that the optimal proportions were between 1.5% and 2.5%.
Abstract: (2002). The Friendship Theorem. The American Mathematical Monthly: Vol. 109, No. 2, pp. 192-194.
TL;DR: The path on which the investigations took us began with Mordell's book and proceeded to Diophantus, to the "Arithmetica," to the first appearance of those wonders known as elliptic curves, to a certain family of elliptic curve, and back to Mordell.
Abstract: (2002). Elliptic Curves from Mordell to Diophantus and Back. The American Mathematical Monthly: Vol. 109, No. 7, pp. 639-649.
TL;DR: A problem in finite groups whose solution relies heavily on techniques from elementary number theory is solved, and the main result is surprisingly a direct consequence of the fact that F 5 is composite.
Abstract: (2002). A Curious Connection Between Fermat Numbers and Finite Groups. The American Mathematical Monthly: Vol. 109, No. 6, pp. 517-524.
TL;DR: This article focuses on Laplace's basic result and his identity, and avoids popular techniques: "messy" contour integration and the theory of Fourier transforms, which invariably invokes the Fourier inversion formula.
Abstract: It is impossible to overplay this function's pervasive role in mathematics. From its humble origin as a complex-valued generalization of the shifted "factorial" function to its more sophisticated guise as the Mellin transform of e-x, the gamma function surfaces in the study of special functions everywhere. On the other hand it can be argued, following Weierstrass, that the reciprocal of the gamma function is a more natural player, one equally deserving of a central part. In this article we focus on Laplace's basic result and its numerous ramifications. Our first goal is to provide two simple proofs of his identity. We avoid popular techniques: "messy" contour integration (the kind that deals with branch points) and the theory of Fourier transforms, which invariably invokes the Fourier inversion formula. The only facts about the gamma function that we use follow from Euler's definition. These are its recurrence relation
TL;DR: 1. Friedman, Heesch tiles with surround numbers 3 and 4, Geombinatorics VIII, April 1999, Issue 4, 101-103; Griinbaum, personal correspondence, January 2001.
Abstract: (2002). On Stirling's Formula. The American Mathematical Monthly: Vol. 109, No. 4, pp. 388-390.
TL;DR: Inside the Levy Dragon as discussed by the authors is a book about inside the Levy dragon, and it is a classic example of non-convex combinatorial games with non-zero-sum games.
Abstract: (2002). Inside the Levy Dragon. The American Mathematical Monthly: Vol. 109, No. 8, pp. 689-703.