Showing papers in "American Mathematical Monthly in 1990"
503 citations
TL;DR: In this paper, the authors present a new topology from old via Ideals, which they call New Topologies from Old through Ideals (New Topology from Old via IBE).
Abstract: (1990). New Topologies from Old via Ideals. The American Mathematical Monthly: Vol. 97, No. 4, pp. 295-310.
483 citations
TL;DR: In this article, the Cayley graph of a group is defined as a graph whose edges are labeled by a finite set F such that there is exactly one incoming and one outgoing edge with each label at each vertex.
Abstract: John Conway discovered a technique using infinite, finitely presented groups that in a number of interesting cases resolves the question of whether a region in the plane can be tessellated by given tiles. The idea is that the tiles can be interpreted as describing relators in a group, in such a way that the plane region can be tiled, only if the group element which describes the boundary of the region is the trivial element 1. A convenient way to describe the construction is by means of the Cayley graph or graph of a group. If G is a group, then its graph F(G) with respect to generators g1, g2 . . ., gn is a directed graph whose vertices are the elements of the group. For each vertex v E F(G), there will be n outgoing edges, labeled by the generators, and n incoming edges: the edge labeled gi connects v to vgi. It is convenient to make a slight modification of this picture when a generator gi has order 2. In that case, instead of drawing an arrow from v to vgi and another arrow from vgi back to v, we draw a single undirected edge labeled gi. Thus, in a drawing of the graph of a group, if there are any undirected edges, it is understood that the corresponding generator has order 2. The graph of a group is automatically homogeneous: for every element g E G, the transformation v -4 gv is an automorphism of the graph. Every automorphism of the labeled graph has this form. This property characterizes graphs of groups: a graph whose edges are labeled by a finite set F such that there is exactly one incoming and one outgoing edge with each label at each vertex is the graph of a group if and only if it admits an automorphism taking any vertex to any other. Whenever R is a relator for the group, that is, a word in the generators which represents 1, then if you start from v EF rand trace out R, you get back to v again. If G has presentation
347 citations
218 citations
TL;DR: Osserman as mentioned in this paper has been at Stanford University since 1955 and will be spending half time as Deputy Director of MSRI in Berkeley for a period of three years, starting September 1, 1990.
Abstract: ROBERT OSSERMAN wrote a Ph.D. thesis on Riemann surfaces under the direction of Lars V. Ahlfors at Harvard University. He gradually moved from geometric function theory to minimal surfaces, differential geometry, isoperimetric inequalities, and some aspects of partial differential equations and ergodic theory. He has been at Stanford University since 1955 and will be spending half time as Deputy Director of MSRI in Berkeley for a period of three years, starting September 1, 1990.
181 citations
TL;DR: A one-sentence proof that every prime p ≡ 1 (mod 4) is a Sum of Two Squares is given in this paper, where every prime is a sum of two squares.
Abstract: (1990). A One-Sentence Proof That Every Prime p ≡ 1 (mod 4) Is a Sum of Two Squares. The American Mathematical Monthly: Vol. 97, No. 2, pp. 144-144.
84 citations
TL;DR: This article points out the close connection between MERLIN-type games and a class of cellular automata related to the ones described by Wolfram and gives a detailed analysis of the special case where the graph is a rectangular grid.
Abstract: 1. SummaryIn an article in this journal Don Pelletier discussed the mathematics involved in a little battery operated toy called Merlin (see [3], and also the “Addenda” in this Monthly, Dec. 1987, page 994). Several years ago Stephen Wolfram, in another article that appeared in the Monthly, analyzed a number of simple cellular automata and the fractal patterns generated by some of these automata (see [6]). In this article we point out the close connection between MERLIN-type games and a class of cellular automata related to the ones described by Wolfram. We introduce a game played on directed graphs and give a detailed analysis of the special case where the graph is a rectangular grid. Our analysis uses linear algebra as well as ideas from the theory of cellular automata.
64 citations
TL;DR: In this article, a methode elementaire d'approximation de Stirling du nombre n was presented, which utilise seulement des transformations d'integrales simples and l'integration terme par terme d'une serie.
Abstract: On presente une methode elementaire d'approximation de Stirling du nombre n! Le developpement en serie ne fait plus appel aux nombres de Berioulli ou a la formule de sommation d'Euler, mais utilise seulement des transformations d'integrales simples et l'integration terme par terme d'une serie
55 citations
54 citations
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TL;DR: Geometry is the visual study of shapes, sizes, patterns, and positions as mentioned in this paper and it occurred in all cultures, through at least one of these five strands of human activities: 1. building/structures (building/repairing a house, laying out a garden, making a kite,...) 2. machines/motion (using a pry-bar, riding a bike, sawing a board, swinging, etc.).
Abstract: Geometry is the visual study of shapes, sizes, patterns, and positions. It occurred in all cultures, through at least one of these five strands of human activities: 1. building/structures (building/repairing a house, laying out a garden, making a kite, ...) 2. machines/motion (using a pry-bar, riding a bike, sawing a board, swinging, ...) 3. navigating/star-gazing (How do I get from here to there?, using maps, ...) 4. art/patterns (designs, symmetries, representations, ...). 5. measurement (How big is it?, How far is it?, ...)
47 citations
TL;DR: Hoffman as discussed by the authors was the 1990 winner of the MAA Chauvenet Prize for Mathematical Programming. But he was not the winner of this prize in the 1990s.
Abstract: DAVID HOFFMAN is Professor of Mathematics and Co-Director of the Geometry, Analysis, Numerics and Graphics Center (GANG) at the University of Massachusetts, Amherst. He earned his Ph.D. in mathematics at Stanford, after receiving undergraduate degrees at the University of Rochester (in history and mathematics). He has pursued research and/or teaching at the Universities of Durham and Warwick (UK), Michigan and Stanford, as well as IMPA (Rio de Janeiro, Brazil) and the University of Paris VII. He is the 1990 recipient of the MAA Chauvenet Prize.
TL;DR: In this article, the stability properties of plane convex sets of inequalities are investigated and a discussion of the concepts and methods which are of importance in this area is presented. But the main focus of this paper is on stability properties.
Abstract: Geometric inequalities, like the isoperimetric inequality, have been a subject of intensive research for a long time. More recently, some properties of these inequalities, which may be called stability properties, have been investigated. Roughly speaking, these investigations concern the geometric implications if the inequalities are in a certain sense close to equalities. The present article is primarily an exposition of various stability results concerning inequalities for plane convex sets, including a discussion of the concepts and methods which are of importance in this area. Moreover, several new results and proofs are presented. 1. Introduction. Let RI denote n-dimensional Eucidean space. A bounded con- vex subset of R" will be called an n-dimensional convex body if it is closed and has interior points. We let qpn denote the class of all n-dimensional convex bodies. Two-dimensional convex bodies will be called convex domains. We consider primarily inequalities of the form
TL;DR: The study of how N celestial bodies move under gravitational forces is an old one as discussed by the authors, and it traces its origins to the earliest reaches of mankind, by acknowledging the work of ancient astrologers and shepherds.
Abstract: The study of how N celestial bodies move under gravitational forces is an old one. If one is willing to acknowledge the work of the ancient astrologers and shepherds ― two groups that carefully plotted the positions of the stars and planets ― then this subject area traces its origins to the earliest reaches of mankind. Indeed, had the title not been already preempted, one might suggest that the study of the N-body problem is «the world's oldest profession». If it isn't the oldest, then, most surely, it is «the second oldest»
TL;DR: In this paper, overdetermined systems of linear equations have been studied in the context of deterministic linear systems and linear systems of logic, and the American Mathematical Monthly: Vol 97, No. 6, pp. 511-513.
Abstract: (1990). Overdetermined Systems of Linear Equations. The American Mathematical Monthly: Vol. 97, No. 6, pp. 511-513.
TL;DR: In this paper, Lucas's theorem and some related results for Extended Pascal Triangles are discussed and discussed in detail, including the relation between extended Pascal triangulations and extended Pascal triangles.
Abstract: (1990). Lucas's Theorem and Some Related Results for Extended Pascal Triangles. The American Mathematical Monthly: Vol. 97, No. 3, pp. 198-204.
TL;DR: In this article, the square pyramid puzzle is solved in the form of a square pyramid, and the solution can be found in the following order: (1) The American Mathematical Monthly: Vol 97, No. 2, pp 120-124
Abstract: (1990). The Square Pyramid Puzzle. The American Mathematical Monthly: Vol. 97, No. 2, pp. 120-124.
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.
TL;DR: Don Zagier presents an extremely short, elegant, and elementary proof of the classic result that any prime p congruent to 1 (mod 4) is a sum of two squares and presents the algorithm and a proof of its correctness.
Abstract: (1990). Editor's Corner: The Euclidean Algorithm Strikes Again. The American Mathematical Monthly: Vol. 97, No. 2, pp. 125-129.
TL;DR: In this paper, the shape of the pedal triangle in a pedal sequence is shown to be ergodic, i.e., if all angles A, B, C are acute, then the triangle is not a pedal triangle.
Abstract: In an interesting article in this journal, Kingston and Synge discuss sequences of pedal triangles { T,, }, where T,,+1 is the pedal triangle of T7, i.e., its vertices are the feet of the altitudes of T,,. The authors observe that there are pedal sequences which are periodic in shape, and construct all of them. In this note we demonstrate that the shape of triangles in a pedal sequence is ergodic. Denote by A, B, C the angles of a given triangle; the angles A', B', C' of its pedal triangle are simply related to A, B, C: If all angles A, B, C are acute,
TL;DR: Schwarz as mentioned in this paper was born in Salzburg, Austria, and escaped to Palestine in 1938 after the Anschluss, and became a professor of statistics at the Hebrew University of Jerusalem.
Abstract: GIDEON E. SCHWARZ: Born 1933 in Salzburg, Austria. Escaped in 1938, after the Anschluss, to Palestine, today Israel. M.Sc. in Mathematics at the Hebrew University, Jerusalem in 1956. Ph.D. in Mathematical Statistics at Columbia University in 1961. Research fellowships: Miller Institute 1964-66, Institute for Advanced Studies on Mt. Scopus 1975-76. Visiting appointments: Stanford University, Tel Aviv University, University of California in Berkeley. Since 1961, Fellow of the Institute of Mathematical Statistics. Presently, Professor of Statistics at the Hebrew University.
TL;DR: The Solution of certain Integral Equations by means of Operators of Arbitrary Order (OIO) was proposed by as mentioned in this paper, who considered the problem of finding the solution of certain integral equations.
Abstract: (1990). The Solution of Certain Integral Equations by Means of Operators of Arbitrary Order. The American Mathematical Monthly: Vol. 97, No. 6, pp. 498-503.
TL;DR: In this article, the authors introduce the Magic Squares and Linear Algebra (MLA) algorithm for linear algebra and show that it can be used to solve linear algebra problems.
Abstract: (1990). Magic Squares and Linear Algebra. The American Mathematical Monthly: Vol. 97, No. 1, pp. 60-62.
TL;DR: Paul HALMOS has three degrees from the University of Illinois and is on the faculty of Santa Clara University since 1985, where his mathematical interests include ergodic theory, algebraic logic, and operators on Hilbert space.
Abstract: PAUL HALMOS has three degrees from the University of Illinois; soon after getting the last one he became, for a couple of years, assistant to John von Neumann. Since then he has taught at many universities (including Chicago, Michigan, and Indiana) and has visited many others (including Miami, Montevideo, Hawaii, Edinburgh, and Western Australia); he has been on the faculty of Santa Clara University since 1985. His mathematical interests include ergodic theory, algebraic logic, and operators on Hilbert space.
TL;DR: The Bitangent Sphere Problem (BSP) as mentioned in this paper is a generalization of the bitangent sphere problem, and it has been shown to be NP-hard in many applications.
Abstract: (1990). The Bitangent Sphere Problem. The American Mathematical Monthly: Vol. 97, No. 1, pp. 5-23.