Showing papers in "American Mathematical Monthly in 1979"
331 citations
TL;DR: Bonnesen-style isoperimetric inequalities were studied in the American Mathematical Monthly (AMM) as discussed by the authors, where the Bonnesen style isomorphic inequalities were discussed.
Abstract: (1979). Bonnesen-Style Isoperimetric Inequalities. The American Mathematical Monthly: Vol. 86, No. 1, pp. 1-29.
275 citations
272 citations
219 citations
TL;DR: In this article, an Introduction to Ramanujan's "Lost" Notebook is given, along with a discussion of its relationship to the lost notebook and the lost notebook.
Abstract: (1979). An Introduction to Ramanujan's “Lost” Notebook. The American Mathematical Monthly: Vol. 86, No. 2, pp. 89-108.
175 citations
121 citations
TL;DR: In this paper, a proof of the Hairy Ball theorem is presented, and the proof is shown to hold for the case of the case where the balls do not change shape.
Abstract: (1979). A Proof of the Hairy Ball Theorem. The American Mathematical Monthly: Vol. 86, No. 7, pp. 571-574.
107 citations
TL;DR: In this paper, the Prisoners' Dilemma and Professional Sports Drafts were discussed in the context of the American Mathematical Monthly: Vol. 86, No. 2, pp. 80-88.
Abstract: (1979). Prisoners' Dilemma and Professional Sports Drafts. The American Mathematical Monthly: Vol. 86, No. 2, pp. 80-88.
62 citations
TL;DR: In this paper, the search and its optimisation is discussed in terms of Search and its Optimization, and the authors propose a search algorithm based on the American Mathematical Monthly: Vol 86, No. 7, pp. 527-540
Abstract: (1979). Search and its Optimization. The American Mathematical Monthly: Vol. 86, No. 7, pp. 527-540.
58 citations
TL;DR: The American Mathematical Monthly: Vol. 86, No. 7, No 7, pp. 540-551 as discussed by the authors, was the first publication of this paper. But it was published in 1979.
Abstract: (1979). Mathematics as an Objective Science. The American Mathematical Monthly: Vol. 86, No. 7, pp. 540-551.
TL;DR: The Banach-Tarski paradox has been studied extensively in the literature, e.g. as discussed by the authors. But it has not yet been studied in the mathematical domain, yet.
Abstract: (1979). The Banach-Tarski Paradox. The American Mathematical Monthly: Vol. 86, No. 3, pp. 151-161.
TL;DR: In this article, the authors present methods of nonstandard analysis at a level of formalism customary in other branches of mathematics and formulate a few simple and reasonably intuitive principles governing their behavior.
Abstract: Infinitely small and infinitely large quantities were systematically introduced into mathematics with the invention of calculus by Newton and Leibniz. The use of such quantities, however, was accompanied by logical contradictions, which mathematicians of the seventeenth and eighteenth centuries were unable to resolve. Although the method of infinitesimals generally yielded correct results, no one ever succeeded in formulating a precise, noncontradictory set of rules governing these objects; and infinitesimal quantities were gradually displaced (at least, in pure mathematics) by the familiar e-d calculus. A mathematically sound model of infinitely small and infinitely large objects became possible only after advances in mathematical logic in the twentieth century. Nonstandard Analysis, developed by A. Robinson in 1960, not only provided foundations for the calculus of infinitesimals in the classical spirit but also enabled mathematicians to use "nonstandard" objects in ways that could not be attempted on the basis of vague, intuitive understanding alone. Since then, interesting applications were found in various branches of mathematics, mathematical physics, and economics. Robinson's exposition in [10] and its subsequent simplifications unfortunately involve the cumbersome apparatus of mathematical logic. Our aim here is to present methods of Nonstandard Analysis at a level of formalism customary in other branches of mathematics. We view nonstandard objects as ideal, imaginary elements adjoined to the universe of the standard mathematics and formulate a few simple and reasonably intuitive principles governing their behavior. We then show, on examples selected to illustrate a variety of nonstandard constructions, how nonstandard mathematics can be developed from these principles. The basic framework for Nonstandard Analysis is presented in ?? 1-3; this system was introduced in [3], where its relative consistency with respect to the Zermelo-Fraenkel set theory is shown. We examine the real line and some concepts of general topology from our point of view in ??4-5; these results can be found in Robinson [10] and Luxemburg [7], [8]. Section 6 is devoted to nonstandard measure theory; our approach is basically that of Loeb [6] (except that we construct Loeb's extension in Theorem 3 of ?6 directly, rather than using Caratheodory's Theorem), with some ideas coming from Anderson [1]. The final ?7, part I, contains a more formal description of the logical foundations, and parts II and III discuss the relationship between our approach and the classical one based on higher-order nonstandard models, as well as some other axiomatizations of Nonstandard Analysis.
TL;DR: A Note on Partitions and Triangles with Integer Sides as discussed by the authors was the first publication of a column on columns and triangles with integer sides in the American Mathematical Monthly: Vol 86, No. 6, pp. 477-478.
Abstract: (1979). A Note on Partitions and Triangles with Integer Sides. The American Mathematical Monthly: Vol. 86, No. 6, pp. 477-478.
TL;DR: In this article, the authors discuss how not to deal with sex differences in mathematics: How Not to Deal with them, and how to avoid dealing with them in mathematics classes and conferences.
Abstract: (1979). Sex Differences in Mathematics: How Not to Deal with Them. The American Mathematical Monthly: Vol. 86, No. 3, pp. 161-168.
TL;DR: In this paper, Grassmann et al. discuss the creation of linear algebra and the history of linear algebras, including Grassmann's work on linear algebra, and their relationship with linear algebra.
Abstract: (1979). Hermann Grassmann and the Creation of Linear Algebra. The American Mathematical Monthly: Vol. 86, No. 10, pp. 809-817.
TL;DR: In this paper, the authors apply the Spline Notation Applied to a Volume Problem (SNTAP) to a volume problem and show that it works well for the volume problem.
Abstract: (1979). Spline Notation Applied to a Volume Problem. The American Mathematical Monthly: Vol. 86, No. 1, pp. 50-51.
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TL;DR: In this article, the Airy Transform was used for the first time in the context of the Manhattan algorithm, and the results showed that it can be computed in time O(n 2 ).
Abstract: (1979). The Airy Transform. The American Mathematical Monthly: Vol. 86, No. 4, pp. 271-277.
TL;DR: In this paper, the authors discuss the relevance of mathematics as a tool for economic understanding and the importance of mathematics in the field of population biology, including the four-color problem.
Abstract: Mathematics Today.- One.- Mathematics-Our Invisible Culture.- Two.- Number Theory.- Groups and Symmetry.- The Geometry of the Universe.- The Mathematics of Meteorology.- The Four Color Problem.- Three.- Combinatorial Scheduling Theory.- Statistical Analysis of Experimental Data.- What is a Computation?.- Mathematics as a Tool for Economic Understanding.- Mathematical Aspects of Population Biology.- Four.- The Relevance of Mathematics.- About the Authors.- Further Reading.
TL;DR: In this article, the Sums of Reciprocals of Integers Missing a Given Digit (SORD) is defined as the sum of all the Integers missing a given digit.
Abstract: (1979). Sums of Reciprocals of Integers Missing a Given Digit. The American Mathematical Monthly: Vol. 86, No. 5, pp. 372-374.
TL;DR: In this article, V-Flexing the Hexahexaflexagon is discussed, where the authors show that it is possible to bend the hexahexagonal shape of the hexagon.
Abstract: (1979). V-Flexing the Hexahexaflexagon. The American Mathematical Monthly: Vol. 86, No. 6, pp. 457-466.
TL;DR: The meaning of the word geometry changes with time and with the speaker as discussed by the authors, and the meaning of geometry can be summarized as follows: "Geometry consists of the logical conclusions drawn from a set of axioms".
Abstract: 1. Geometry. I believe I am expected to tell you all about geometry; what it is, its developments through the centuries, its current issues and problems, and, if possible, a peep into the future. The first question daes not have a clear-cut answer. The meaning of the word geometry changes with time and with the speaker. With Euclid, geometry consists of the logical conclusions drawn from a set of axioms. This is clearly not sufficient with the horizons of geometry ever widening. Thus in 1932 the great geometers 0. Veblen and J. H. C. Whitehead said, "A branch of mathematics is called geometry, because the name seems good on emotional and traditional grounds to a sufficiently large number of competent people" [1]. This opinion was enthusiastically seconded by the great French geometer Elie Cartan [2]. Being an analyst himself, the great American mathematician George Birkhoff mentioned a "disturbing secret fear that geometry may ultimately turn out to be no more than the glittering intuitional trappings of analysis" [3]. Recently my friend Andre Weil said: "The psychological aspects of true geometric intuition will perhaps never be cleared up. At one time it implied primarily the power of visualization in three-dimensional space. Now that higher-dimensional spaces have mostly driven out the more elementary problems, visualization can at best be partial or symbolic. Some degree of tactile imagination seems also to be involved" [4]. At this point it is perhaps better to let things stand and turn to some concrete topics.