Showing papers in "American Mathematical Monthly in 1981"
TL;DR: The object here is to prove that the algorithm for assigning students to universities gives each student the best university available in a stable system of assignments.
Abstract: Gale and Shapley have an algorithm for assigning students to universities which gives each student the best university available in a stable system of assignments. The object here is to prove that ...
632 citations
233 citations
TL;DR: In this paper, the authors present a rigid and flexible framework for building rigid structures with respect to rigid structures and flexible structures. The American Mathematical Monthly: Vol. 88, No. 1, pp. 6-21.
Abstract: (1981). Rigid and Flexible Frameworks. The American Mathematical Monthly: Vol. 88, No. 1, pp. 6-21.
152 citations
87 citations
TL;DR: • Bachelor of Arts with a major in Mathematics (BA) (http:// catalog.tamiu.edu/undergraduate-information/arts-sciences/mathematics-physics/ bachelor-arts-double-major-ba).
Abstract: • Bachelor of Arts with a Major in Mathematics (BA) (http:// catalog.tamiu.edu/undergraduate-information/arts-sciences/ mathematics-physics/bachelor-arts-major-mathematics-ba/) • Bachelor of Arts with a Major in Mathematics with Grades 7-12 Certification (BA) (http://catalog.tamiu.edu/undergraduateinformation/arts-sciences/mathematics-physics/bachelor-arts-majormathematics-grades-7-12-certification-ba/) • Bachelor of Arts with a Double Major (BA) (http://catalog.tamiu.edu/ undergraduate-information/arts-sciences/mathematics-physics/ bachelor-arts-double-major-ba/) • Bachelor of Science with a Major in Mathematics (BS) (http:// catalog.tamiu.edu/undergraduate-information/arts-sciences/ mathematics-physics/bachelor-science-major-mathematics-bs/)
86 citations
78 citations
TL;DR: In this article, the authors propose Iterating Analytic Self-Maps of Discs (ISMDS) for the problem of iterating analytic self-maps of discs.
Abstract: (1981). Iterating Analytic Self-Maps of Discs. The American Mathematical Monthly: Vol. 88, No. 6, pp. 396-407.
64 citations
TL;DR: The Uniformization Theorem as discussed by the authors is a well-known theorem in the field of mathematics, and it has been applied in many applications, e.g. in the following:
Abstract: (1981). The Uniformization Theorem. The American Mathematical Monthly: Vol. 88, No. 8, pp. 574-592.
62 citations
TL;DR: In this paper, the authors present Mathematical Models: A Sketch for the Philosophy of Mathematics, a sketch for the philosophy of mathematics with a focus on the mathematical models of the world.
Abstract: (1981). Mathematical Models: A Sketch for the Philosophy of Mathematics. The American Mathematical Monthly: Vol. 88, No. 7, pp. 462-472.
52 citations
TL;DR: In this article, Abel's theorem on the Lemniscate has been studied in the context of Abel's Theorem on the Lémoniscate (ABT) on the basis of the Abel Theorem.
Abstract: (1981). Abel's Theorem on the Lemniscate. The American Mathematical Monthly: Vol. 88, No. 6, pp. 387-395.
48 citations
TL;DR: In this paper, the Caustics of Plane Curves were studied in the context of plane curves and their relationship with the plane curve curvature. The American Mathematical Monthly: Vol. 88, No. 9, pp. 651-667.
Abstract: (1981). On Caustics of Plane Curves. The American Mathematical Monthly: Vol. 88, No. 9, pp. 651-667.
TL;DR: In this paper, the eigenvalues of the Laplacian and the Heat Equation were analyzed and the heat equation was shown to be a convex function of the Eigenvectors.
Abstract: (1981). Eigenvalues of the Laplacian and the Heat Equation. The American Mathematical Monthly: Vol. 88, No. 9, pp. 686-695.
TL;DR: The author examines the relationship between computer science, mathematics, and the Undergraduate Curricula in Both and the content of the courses taught in both schools.
Abstract: (1981). Computer Science, Mathematics, and the Undergraduate Curricula in Both. The American Mathematical Monthly: Vol. 88, No. 7, pp. 472-485.
TL;DR: The American Mathematical Monthly: Vol. 88, No. 6, pp. 413-419 as mentioned in this paper was the first issue of the first edition of the women and mathematics journal Fact and Fiction.
Abstract: (1981). Women and Mathematics: Fact and Fiction. The American Mathematical Monthly: Vol. 88, No. 6, pp. 413-419.
TL;DR: In this paper, a Permanent Inequality is discussed in the context of the American Mathematical Monthly: Vol. 88, No. 10, pp. 731-740, 1981.
Abstract: (1981). A Permanent Inequality. The American Mathematical Monthly: Vol. 88, No. 10, pp. 731-740.
TL;DR: The American Mathematical Monthly: Vol 88, No 5, No. 5, pp. 311-320 as mentioned in this paper is the earliest publication of the article. But it was published in 1981.
Abstract: (1981). Are There Coincidences in Mathematics? The American Mathematical Monthly: Vol. 88, No. 5, pp. 311-320.
TL;DR: The authors caracterise par la richesse des informations qu'il rassemble, la variete des publics auquel il s'adresse and la diversite des niveaux de ses articles.
Abstract: Ce dictionnaire de mathematiques est caracterise par la richesse des informations qu'il rassemble, la variete des publics auquel il s'adresse et la diversite des niveaux de ses articles. Son heterogeneite est assumee, outre un tres grand nombre de symboles et de formules, par un vocabulaire de pres de 8000 entrees et des renvois entre les articles.
TL;DR: In this paper, the authors give a complete description of all metric spaces that have the property that each continuous real-valued function on E is uniformly continuous and use this result to give a new necessary and sufficient criterion for compactness.
Abstract: Almost every textbook on analysis or topology contains a proof of the fact that on a compact metric space every continuous real-valued function is bounded and uniformly continuous. It would seem to be natural to ask which metric spaces E have the property that all continuous real-valued functions on E are bounded and which metric spaces E have the property that all continuous real-valued functions on E are uniformly continuous. In 1948 Hewitt answered the first question: A metric space E is compact if and only if every continuous function from E to R is bounded (cf. [1, p. 69]). In the present paper, we give a similar answer to the second question: In Theorem 1 we give a complete description of all metric spaces that have the property that each continuous real-valued function on E is uniformly continuous. In Theorem 2 we use this result to give a new necessary and sufficient criterion for compactness. In what follows, E will denote a metric space with metric d. For any x EE and any subset D of E we shall denote by d(x, D) the distance from x to D, that is, d(x, D)_ inf(d(x, y)Iy ED}. By d(x) we shall denote the distance from x to E \(x}. Remember that a point x EE is called an accumulation point of a subset S of E if d(x, S\(x})=O. Hence the set of accumulation points of E, which will be denoted by A, is just the set (xEEId(x)=O}. The set of isolated points is just the set (xEEId(x)>O}.
TL;DR: The American Mathematical Monthly: Vol. 88, No. 8, pp. 592-604 as discussed by the authors, was the first publication dedicated to black women in mathematics in the United States.
Abstract: (1981). Black Women in Mathematics in the United States. The American Mathematical Monthly: Vol. 88, No. 8, pp. 592-604.
TL;DR: In this paper, the authors proposed a game-theoretic approach to evaluate representation systems in terms of the extent to which they allocated "power" fairly, i.e., whether a change in an individual voter's choice from candidate A to candidate B or from candidate B to candidate A would alter the electoral outcome.
Abstract: 2. Background. In three articles that appeared in American law journals in the mid-1960's, a lawyer named John Banzhaf III proposed to evaluate representation systems in terms of the extent to which they allocated "power" fairly [1], [2], [3]. Banzhaf's analysis makes use of game-theoretic notions in which power is equated with the ability to affect outcomes. Consider a group of citizens choosing between two opposing candidates. To calculate the power of the individual voter, we generate the set of all possible voting coalitions among the district's electorate. If there are N voters in the district, then there will be 2N possible coalitions. Then we ask, for each of these possible coalitions, whether a change in an individual voter's choice from candidate A to candidate B (or from candidate B to candidate A) would alter the electoral outcome. If so, that voter's ballot is said to be decisive. The (absolute) Banzhaf index of a voter's power is defined as the number of the voter's decisive votes divided by 2X. The higher the percentage of voter coalitions in which a voter's vote is decisive, the higher that voter's power score. The Banzhaf index has considerable intuitive appeal; power is based on ability to affect outcome. For single-member district systems (smds) whose districts are of equal population, all voters have identical power. But what about the case of multiple-member district systems (mmds), with districts of more than one size? Here, since the voters who elect k representatives have k times as much impact as voters who can elect only one representative, we might think that to equalize voter power we should assign to each district a number of representatives proportional to the size of the district's population since, intuitively, we would expect a voter's ability to decisively affect outcomes should be inversely proportional to district population. Banzhaf [2] pointed out that this argument is mathematically incorrect. In a two-candidate/party contest where all voters have equal weight, in order for a voter to be decisive in a district of size N the rest of the voters (who are N- 1 in number) must split half for one candidate/party and half against. A straightforward combinatoric analysis reveals ([2], [7], [6]; Whitcomb v. Chavis (1970) 403 U.S. at 145 n. 23) that, if all combinations of vote outcomes are equally likely (i.e., if each voter is equally likely to vote for either candidate/party), then the number of each member's decisive votes, b, is given by:
TL;DR: In this article, the bounds for the zeros of polynomials are defined and analyzed in the context of polynomial numbers. But they do not consider the non-zero case.
Abstract: (1981). Bounds for the Zeros of Polynomials. The American Mathematical Monthly: Vol. 88, No. 3, pp. 205-206.
TL;DR: It is well known that the notion of topological degree, deg(F), of a continuous mapping F: M→N, where M and N are connected, oriented, compact n-manifolds (with triangulation), can be traced back t... as mentioned in this paper.
Abstract: It is well known that the notion of topological degree, deg(F), of a continuous mapping F: M→N, where M and N are connected, oriented, compact n-manifolds (with triangulation), can be traced back t...
TL;DR: Can We Make Mathematics Intelligible? The American Mathematical Monthly: Vol. 88, No. 10, pp. 727-731 as discussed by the authors was the first publication of this article.
Abstract: (1981). Can We Make Mathematics Intelligible? The American Mathematical Monthly: Vol. 88, No. 10, pp. 727-731.
TL;DR: In this paper, it is shown that the monotonie increasing and monotony decreasing with common limit π is the basis of Archimedes' method for approximating to π.
Abstract: Let p N and P N denote half the lengths of the perimeters of the inscribed and circumscribed regular N-gons of the unit circle. Thus \({p_3} = 3\sqrt 3 /2,{\kern 1pt} {p_3} = 3\sqrt 3 ,{p_4} = 2\sqrt 2\) , and P 4 = 4. It is geometrically obvious that the sequences {p N } and {p N } are respectively monotonie increasing and monotonie decreasing, with common limit π. This is the basis of Archimedes’ method for approximating to π. (See, for example, Heath [2].) Using elementary geometrical reasoning, Archimedes obtained the following recurrence relation, in which the two sequences remain entwined:
$$1/{P_{2N}} = \frac{1}{2}(1/{P_N} + 1{P_N})$$
(1a)
$${P_{2N}} = V({P_{2N}}{P_N})$$
(1b)
TL;DR: This exposition will be the study of biplanes and projective planes, focusing on certain interesting relationships between these two types of combinatorial designs, relationships which are uncovered by exploiting the techniques of coding theory.
Abstract: Design theory and algebraic coding theory have their origins in disparate fields of study: the statistical theory of the design of experiments and the theory of information transmission in electrical engineering. Yet each has enriched the other by providing tools capable of answering interesting and fundamental questions. A primary topic of this exposition will be the study of biplanes and projective planes, focusing on certain interesting relationships between these two types of combinatorial designs, relationships which are uncovered by exploiting the techniques of coding theory. We shall set the stage in Section 1 by presenting a fairly complete introduction to the theory of designs, thereby putting both planes and biplanes in proper perspective. Section 2 presents the terminology of coding theory and a few of its tools. Then we examine both biplanes and planes from the viewpoint of coding theory. Additional design-theoretic tools are developed in Section 3, and in Section 4 much of the preceding is brought to bear in elucidating an elegant coding-theoretic link between planes and biplanes.