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Showing papers in "American Mathematical Monthly in 1980"


Journal ArticleDOI
TL;DR: This book explains the development of the Matrix Model Framework and some types of Instability and discusses the Demographic Theory of Kinship and Microdemography.

768 citations


Journal ArticleDOI
TL;DR: The Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly as mentioned in this paper, which is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive.
Abstract: JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly.

399 citations


Journal ArticleDOI
TL;DR: The Unreasonable Effectiveness of Mathematics as mentioned in this paper is a classic work in the area of mathematics that deals with the problem of mathematics in general, and mathematics in particular, in general.
Abstract: (1980). The Unreasonable Effectiveness of Mathematics. The American Mathematical Monthly: Vol. 87, No. 2, pp. 81-90.

207 citations


Journal ArticleDOI
TL;DR: The Heart of Mathematics as discussed by the authors is a seminal work in the field of mathematics that addresses the problem of mathematical heart disease in the context of mathematics education and science, and includes the following:
Abstract: (1980). The Heart of Mathematics. The American Mathematical Monthly: Vol. 87, No. 7, pp. 519-524.

186 citations


Journal ArticleDOI
TL;DR: The Formula of Faa Di Bruno as discussed by the authors is a well-known formula in the mathematical community, and it has been used extensively in the past decade and a half in many applications.
Abstract: (1980). The Formula of Faa Di Bruno. The American Mathematical Monthly: Vol. 87, No. 10, pp. 805-809.

181 citations


Journal ArticleDOI
TL;DR: The Power and Generalized Logarithmic Means (PGLM) as discussed by the authors is a generalization of the generalized logarithm of the power-and-generalized linear mean.
Abstract: (1980). The Power and Generalized Logarithmic Means. The American Mathematical Monthly: Vol. 87, No. 7, pp. 545-548.

167 citations


Journal ArticleDOI
TL;DR: In this paper, Ramanujan's extensions of the Gamma and Beta functions are discussed. But they do not consider the Gamma function's extension of the Beta function. The American Mathematical Monthly: Vol. 87, No. 5, pp. 346-359.
Abstract: (1980). Ramanujan's Extensions of the Gamma and Beta Functions. The American Mathematical Monthly: Vol. 87, No. 5, pp. 346-359.

129 citations


Journal ArticleDOI
TL;DR: The Simple Analytic Proof of the Prime Number Theorem is a simple analytic proof of the prime number theorem, which is based on the simple analytic theorem of.
Abstract: (1980). Simple Analytic Proof of the Prime Number Theorem. The American Mathematical Monthly: Vol. 87, No. 9, pp. 693-696.

122 citations


Journal ArticleDOI
TL;DR: In this paper, teaching problem-solving skills is discussed in the context of Problem-Solving Skills for Mathematical Programming. The American Mathematical Monthly: Vol. 87, No. 10, pp. 794-805.
Abstract: (1980). Teaching Problem-Solving Skills. The American Mathematical Monthly: Vol. 87, No. 10, pp. 794-805.

110 citations





Journal ArticleDOI
TL;DR: In the early 19 century, J. Fourier was an impassioned advocate of the use of such sums, of course writing sines and cosines rather than complex exponentials as mentioned in this paper.
Abstract: In the early 19 century, J. Fourier was an impassioned advocate of the use of such sums, of course writing sines and cosines rather than complex exponentials. Euler, the Bernouillis, and others had used such sums in similar fashions and for similar ends, but Fourier made a claim extravagant for the time, namely that all functions could be expressed in such terms. Unfortunately, in those days there was no clear idea of what a function was, no vocabulary to specificy classes of functions, and no specification of what it would mean to represent a function by such a series. In hindsight, probably issues of pointwise and L convergence, unspecified to some degree, were confused with each other.


Journal ArticleDOI
TL;DR: It is shown that it is more reasonable to continue to regard all mathematical truths as a priori-no matter how they are arrived at-although it may indeed be necessary to introduce a new mathematical entity intermediate between a conjecture and a theorem.
Abstract: (1980). The Philosophical Implications of the Four-Color Problem. The American Mathematical Monthly: Vol. 87, No. 9, pp. 697-707.

Journal ArticleDOI
TL;DR: In this article, when is f(f(z)) =az2+bz+c?, and when is c = c + c?, where c is the cardinality of c.
Abstract: (1980). When is f(f(z))=az2+bz+c?. The American Mathematical Monthly: Vol. 87, No. 4, pp. 252-263.


Journal ArticleDOI
TL;DR: In this paper, modern multiplier rules are used to define Modern Multiplier Rules (MMR) and modern multipliers are used for the first time in the American Mathematical Monthly (AMM).
Abstract: (1980). Modern Multiplier Rules. The American Mathematical Monthly: Vol. 87, No. 6, pp. 433-452.

Journal ArticleDOI
TL;DR: In this paper, the Mathematical Sciences and World War II are discussed in the context of the American Mathematical Monthly: Vol. 87, No. 8, pp. 607-621.
Abstract: (1980). The Mathematical Sciences and World War II. The American Mathematical Monthly: Vol. 87, No. 8, pp. 607-621.

Journal ArticleDOI
TL;DR: The most important works in this respect were done primarily by Fourier, Cournot, Farkas and further by Gauss, Ostrogradsky and Hamel as mentioned in this paper.
Abstract: : The method of Lagrange for finding extrema of functions subject to equality constraints was published in 1788 in the famous book Mecanique Analytique. The works of Karush, John, Kuhn and Tucker concerning optimization subject to inequality constraints appeared more than 150 years after that. The purpose of this paper is to call attention to important papers, published as contributions to mechanics, containing fundamental ideas concerning optimization theory. The most important works in this respect were done primarily by Fourier, Cournot, Farkas and further by Gauss, Ostrogradsky and Hamel. (Author)

Journal ArticleDOI
TL;DR: A Less Strange Version of Milnor's Proof of Brouwer's Fixed-Point Theorem as mentioned in this paper is presented in the American Mathematical Monthly: Vol. 87, No. 7, pp. 525-527.
Abstract: (1980). A Less Strange Version of Milnor's Proof of Brouwer's Fixed-Point Theorem. The American Mathematical Monthly: Vol. 87, No. 7, pp. 525-527.

Journal ArticleDOI
TL;DR: An elementary proof of the Polar Decomposition Theorem is given in this article, where it is shown that the polar decomposition theorem can be proven in the presence of an elementary proof.
Abstract: (1980). An Elementary Proof of the Polar Decomposition Theorem. The American Mathematical Monthly: Vol. 87, No. 4, pp. 288-290.

Journal ArticleDOI
TL;DR: The postage stamp problem is the following: An envelope may carry no more than h stamps, and one has available k integer-valued stamp denominations; find the maximal integer n = n(h, k) such that all integer postage values from 1 to n can be made up.
Abstract: The postage stamp problem is the following: An envelope may carry no more than h stamps, and one has available k integer-valued stamp denominations. Given h and k, find the maximal integer n = n(h, k) such that all integer postage values from 1 to n can be made up. In addition, find all sets of k stamp denominations satisfying this condition. The problem statement is usually modified by augmenting the solution sets with a stamp of value zero, and requiring that a letter carry exactly h stamps. For example, if h = 2 and k = 3, then n(h, k) = 8. The unique solution set is {0, 1, 3, 4}. A construction of the integers 1, . . . , 8 is

Journal ArticleDOI
TL;DR: The Dirichlet problem for Harmonic Functions has been studied extensively in the literature, see as discussed by the authors for an overview. But it is not a deterministic problem, and it is hard to solve.
Abstract: (1980). The Dirichlet Problem for Harmonic Functions. The American Mathematical Monthly: Vol. 87, No. 8, pp. 621-628.


Journal ArticleDOI
TL;DR: In this paper, a copy of a cuneiform tablet measuring perhaps 3 inches by 5.5 inches was found in Akkad in the city of Nippur in theyear 1700, about 3700 years ago.
Abstract: Let me begin by clarifying the title “Sherlock Holmes in Babylon.” Lest some members of the Baker Street Irregulars be misled, my topic is the archaeology of mathematics, and my objective is to retrace a small portion of the research of two scholars: Otto Neugebauer, who is a recipient of the Distinguished Service Award, given to him by the Mathematical Association of America in 1979, and his colleague and long-time collaborator, Abraham Sachs. It is also a chance for me to repay both of them a personal debt. I went to Brown University in 1947, and as a new Assistant Professor I was welcomed as a regular visitor to the Seminar in the History of Mathematics and Astronomy. There, with a handful of others, I was privileged to watch experts engaged in the intellectual challenge of reconstructing pieces of a culture from random fragments of the past. (See [4], [5].) This experience left its mark upon me. While I do not regard myself as a historian in any sense, I have always remained a “friend of the history of mathematics”; and it is in this role that I come to you today. Let me begin with a sample of the raw materials. Figure 1 is a copy of a cuneiform tablet measuring perhaps 3 inches by 5. The markings can be made by pressing the end of a cut reed into wet clay. Dating such a tablet is seldom easy. The appearance of this tablet suggests that it may have been made in Akkad in the city of Nippur in theyear –1700, about 3700 years ago.

Journal ArticleDOI
TL;DR: Burnside's theorem on algebraic matrices was studied in this article, where it is shown that theorem on Algebras of Matrices is true for all matrices.
Abstract: (1980). Burnside's Theorem on Algebras of Matrices. The American Mathematical Monthly: Vol. 87, No. 10, pp. 810-810.

Journal ArticleDOI
TL;DR: The conditionally convergent series are those series that converge as written, but do not converge when each of their terms is replaced by the corresponding absolute value as discussed by the authors, i.e., the series that converges as written but does not converge if each of the terms of the series are replaced by an absolute value.
Abstract: Why are conditionally convergent series interesting? While mathematicians might undoubtably give many answers to such a question, Riemann's theorem on rearrangements of conditionally convergent series would probably rank near the top of most responses. Conditionally convergent series are those series that converge as written, but do not converge when each of their terms is replaced by the corresponding absolute value. The nineteenth-century mathematician Georg Friedrich Bernhard Riemann (1826-1866) proved that such series could be rearranged to converge to any prescribed sum. Almost every calculus text contains a chapter on infinite series that distinguishes between absolutely and conditionally convergent series. Students see the usefulness of studying absolutely convergent series since most convergence tests are for positive series, but to them conditionally convergent series seem to exist simply to provide good test questions for the instructor. This is unfortunate since the proof of Riemann's theorem is a model of clever simplicity that produces an exact algorithm. It is clear, however, that even with such a simple example as the alternating harmonic series one cannot hope for a closed form solution to the problem of rearranging it to sum to an arbitrary real number. Nevertheless, it is an old result that for any real number of the form ln r , where r is a rational number, there is a very simple rearrangement of the alternating harmonic series having this sum. Moreover, this rearrangement shares, at least in the long run, a nice feature of Riemann's rearrangement, namely that the partial sums at the breaks between blocks of different sign oscillate around the sum of the series.


Journal ArticleDOI
TL;DR: In this article, the probability that neighbors remain neighbors after random rearrangements is investigated. But the authors focus on the probability of neighbors staying neighbours after random re-arrangements.
Abstract: (1980). The Probability that Neighbors Remain Neighbors after Random Rearrangements. The American Mathematical Monthly: Vol. 87, No. 2, pp. 122-124.