Showing papers in "American Mathematical Monthly in 1987"
TL;DR: In this paper, an Introduction to the Ising Model is presented, along with a discussion of its application in the context of algebraic geometry problems and its application to algebraic logic.
Abstract: (1987). An Introduction to the Ising Model. The American Mathematical Monthly: Vol. 94, No. 10, pp. 937-959.
325 citations
TL;DR: This chapter shows how tilings of the hyperbolic plane can help us visualize the Banach—Tarski paradox.
261 citations
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TL;DR: In this article, the Runge Example is used to define the runge example in the context of algebraic geometry. The American Mathematical Monthly: Vol. 94, No. 4, pp. 329-341.
Abstract: (1987). On the Runge Example. The American Mathematical Monthly: Vol. 94, No. 4, pp. 329-341.
149 citations
TL;DR: In this paper, the maximum order of an element of a finite symmetric group is analyzed in terms of the number of elements in the group and the order in which the elements are ordered.
Abstract: (1987). The Maximum Order of an Element of a Finite Symmetric Group. The American Mathematical Monthly: Vol. 94, No. 6, pp. 497-506.
65 citations
57 citations
TL;DR: In this article, a notice biographique de Wilhehn Jordan (1842-1899) (a ne pas confondre avec le mathematicien Camille Jordan 1838-1922) and introduction de sa methode comme sous le nom de methode de reduction de Gauss-Jordan
Abstract: Notice biographique de Wilhehn Jordan (1842-1899) (a ne pas confondre avec le mathematicien Camille Jordan 1838-1922) et introduction de sa methode comme sous le nom de methode de reduction de Gauss-Jordan
56 citations
TL;DR: In this article, the Geometric, Logarithmic and Arithmetic Mean Inequality (GMEI) were discussed. But they focused on geometric, logarithmically and arithmetic mean inequalities.
Abstract: (1987). The Geometric, Logarithmic, and Arithmetic Mean Inequality. The American Mathematical Monthly: Vol. 94, No. 6, pp. 527-528.
52 citations
TL;DR: In this article, 14 proofs of a result about tiling a rectangle are presented, and 14 proofs about the same result are discussed. The American Mathematical Monthly: Vol. 94, No. 7, pp. 601-617.
Abstract: (1987). Fourteen Proofs of a Result About Tiling a Rectangle. The American Mathematical Monthly: Vol. 94, No. 7, pp. 601-617.
41 citations
TL;DR: The Courant-Fischer Theorem is related to the Monotonicity Theorem and Cauchy's Interlace Theorem as discussed by the authors, as well as the Courant Fischer Theorem.
Abstract: (1987). The Monotonicity Theorem, Cauchy's Interlace Theorem, and the Courant-Fischer Theorem. The American Mathematical Monthly: Vol. 94, No. 4, pp. 352-354.
34 citations
TL;DR: The American Mathematical Monthly: Vol 94, No 2, pp 143-150, 1987 as discussed by the authors, was the first publication of Merlin's Magic Square, and was published in 1987.
Abstract: (1987) Merlin's Magic Square The American Mathematical Monthly: Vol 94, No 2, pp 143-150
33 citations
TL;DR: This is the revised and greatly expanded Second Edition of the hugely popular Numerical Recipes, with over 100 new routines (now well over 300 in all), plus upgraded versions of many of the original routines.
Abstract: From the Publisher:
This is the revised and greatly expanded Second Edition of the hugely popular Numerical Recipes: The Art of Scientific Computing. The product of a unique collaboration among four leading scientists in academic research and industry, Numerical Recipes is a complete text and reference book on scientific computing. In a self-contained manner it proceeds from mathematical and theoretical considerations to actual practical computer routines. With over 100 new routines (now well over 300 in all), plus upgraded versions of many of the original routines, this book is more than ever the most practical, comprehensive handbook of scientific computing available today. The book retains the informal, easy-to-read style that made the first edition so popular, with many new topics presented at the same accessible level. In addition, some sections of more advanced material have been introduced, set off in small type from the main body of the text. Numerical Recipes is an ideal textbook for scientists and engineers and an indispensable reference for anyone who works in scientific computing. Highlights of the new material include a new chapter on integral equations and inverse methods; multigrid methods for solving partial differential equations; improved random number routines; wavelet transforms; the statistical bootstrap method; a new chapter on \"less-numerical\" algorithms including compression coding and arbitrary precision arithmetic; band diagonal linear systems; linear algebra on sparse matrices; Cholesky and QR decomposition; calculation of numerical derivatives; Pade approximants, and rational Chebyshev approximation; new special functions; Monte Carlo integration in high-dimensional spaces; globally convergent methods for sets of nonlinear equations; an expanded chapter on fast Fourier methods; spectral analysis on unevenly sampled data; Savitzky-Golay smoothing filters; and two-dimensional Kolmogorov-Smirnoff tests. All this is in addition to material on such basic top
TL;DR: In this article, the authors presented a method for computing Binomial Coefficients and showed that it is possible to compute binomial coefficients with a fixed number of coefficients. The American Mathematical Monthly: Vol. 94, No. 4, pp. 360-365.
Abstract: (1987). Computing Binomial Coefficients. The American Mathematical Monthly: Vol. 94, No. 4, pp. 360-365.
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TL;DR: In this article, Ringing the cosets is discussed in the context of the ring-the-cosets problem and it is shown how to ring a set of cosets.
Abstract: (1987). Ringing the Cosets. The American Mathematical Monthly: Vol. 94, No. 8, pp. 721-746.
TL;DR: In this paper, the Diophantine Equation X2 + 7 = 2n is considered and the authors propose a solution to the problem of finding the number of nodes in 2n.
Abstract: (1987). The Diophantine Equation X2 + 7 = 2n. The American Mathematical Monthly: Vol. 94, No. 1, pp. 59-62.
TL;DR: The Hypergame Paradox as discussed by the authors offers the opportunity to recapture the sense of confusion and uncertainty that faced mathematicians in the early part of the twentieth century, including those common initial feelings that the eventual fix is a dishonest legalism which walks around a key question instead of answering it.
Abstract: A new paradox offers the opportunity to recapture the sense of confusion and uncertainty that faced mathematicians in the early part of the twentieth century, including those common initial feelings that the eventual fix is a dishonest legalism which walks around a key question instead of answering it. Also, it provides a new probe with which to explore the relationship between paradox and proof. I stumbled across the Hypergame Paradox when I was teaching a section on games and strategies for a "Mathematics for the Liberal Arts Major" course occasionally offered at Union College, and an idea for a bonus test question came to me. Let us define a game G to be totally finite if it satisfies the following conditions:
TL;DR: The Fascination of the Elementary as mentioned in this paper is a collection of essays about elementary education, focusing on the elementary subject of science and science education, with a focus on mathematics and technology.
Abstract: (1987). The Fascination of the Elementary. The American Mathematical Monthly: Vol. 94, No. 8, pp. 759-768.
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TL;DR: The Way of All Means as mentioned in this paper is a seminal work in the field of algebraic geometry, which is based on the way-of-all-means approach. The American Mathematical Monthly: Vol. 94, No. 6, pp. 519-522.
Abstract: (1987). The Way of All Means. The American Mathematical Monthly: Vol. 94, No. 6, pp. 519-522.
TL;DR: In this article, Extreme Points of Convex Sets in Infinite Dimensional Spaces (EPDS) are studied in the context of infinite dimensional spaces, where the points of convex sets are assumed to be convex.
Abstract: (1987). Extreme Points of Convex Sets in Infinite Dimensional Spaces. The American Mathematical Monthly: Vol. 94, No. 5, pp. 409-422.
TL;DR: In this paper, Sub-Gaussian techniques in proving strong laws of large numbers were used to prove strong numbers in the context of the construction of large number graphs, and the results showed that strong numbers can be computed by sub-Gaussians.
Abstract: (1987). Sub-Gaussian Techniques in Proving Strong Laws of Large Numbers. The American Mathematical Monthly: Vol. 94, No. 3, pp. 295-299.
TL;DR: In this paper, Brams et al. presented the Superior Beings: If They Exist, How Would We Know? By Steven J. Brams. The American Mathematical Monthly: Vol. 94, No. 1, pp. 92-95.
Abstract: (1987). Superior Beings: If They Exist, How Would We Know? By Steven J. Brams. The American Mathematical Monthly: Vol. 94, No. 1, pp. 92-95.
TL;DR: Turnbull and Aitken as mentioned in this paper gave a simple proof for the existence of the Jordan canonical form in linear algebra, which is known as the Turnbull-Aitken proof.
Abstract: To my way of thinking, it's a marvelously simple proof. It's over 50 years old, and when one uses the modem language of graph theory, it can be made very visual. The most difficult part of the proof is in achieving triangular form. After that the details are extremely easy to follow: elementary row operations followed by the corresponding elementary column operations. Thus for students who are more familiar with Gaussian elimination than with other aspects of linear algebra, it is a good way to introduce the Jordan canonical form. So where is this proof and why isn't it a standard proof? (I think it should be and I hope to convince you.) It's in a classical book [8] by H. W. Turnbull and A. C. Aitken entitled An Introduction to the Theory of Canonical Matrices. This book is well known to matrix theorists, so that one cannot claim that it's in an obscure book that has long been forgotten. The reason that it has been ignored may be due to the changes in mathematical rigor that have occurred since 1932 and the lack of the formalism provided by graph theory at that time. (Konig's classical book entitled Theorie der Endlichen und Unendlichen Graphen was published in 1936, although his classical paper [6] "Graphen und Matrizen" was published in 1931.) At first reading, Turnbull and Aitken's proof is somewhat obscure, and it is not clear that their proof is general. They refer to chains of nonzero elements in a matrix which now we would call paths in the digraph associated with the matrix. Whatever the reason, I hope-to revive their proof by publication of this article. In a recent article [2] in this journal, Fletcher and Sorensen describe a proof for the existence of the Jordan canonical form which is algorithmic in nature. This proof was adopted in [5]. The proof, like the one to be given here, proceeds in three steps: (I) reduce to upper triangular form; (II) further reduce to the case in which all eigenvalues are equal; (III) use induction to reduce an upper triangular matrix with equal eigenvalues to Jordan canonical form. As pointed out in [2], the only nonconstructive step is I. In [2] step II is accomplished by solving, using induction, a linear matrix equation of the form AX - XA = S, while step III is accomplished by induction and matrix factorizations. The Turnbull-Aitken approach accomplishes
TL;DR: In this paper, the soft file of the book is read and then download the book and get it. But this book is referred to read because it is an inspiring book to give you more chance to get experiences and also thoughts.
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TL;DR: In this paper, the product of the sum of the mi is defined as the smallest degree of any symmetric group in which it imbeds, and it behaves fairly nicely with respect to extensions.
Abstract: where the mi are prime powers. The (unordered) sequence of numbers mi, m2,..., ml completely determines G up to isomorphism, so any invariant of G is some symmetric function of the mi. The product of the mi is simply I GI, the order of G: in this note we consider another invariant, the sum of the mi. We show that it behaves fairly nicely with respect to extensions, and that it can be interpreted as the smallest degree of any symmetric group in which G imbeds. For any abelian group G with decomposition (1), let
TL;DR: In this article, a conjecture related to Chi-Bar-Squared Distributions is made, which is not supported by the results of the present paper, but is supported by several other papers.
Abstract: (1987). A Conjecture Related to Chi-Bar-Squared Distributions. The American Mathematical Monthly: Vol. 94, No. 1, pp. 46-48.
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TL;DR: In this paper, a modern fairy story is presented in the American Mathematical Monthly: Vol. 94, No. 3, pp. 291-295, with the title "A Modern Fairy Tale?"
Abstract: (1987). A Modern Fairy Tale? The American Mathematical Monthly: Vol. 94, No. 3, pp. 291-295.