Showing papers in "American Mathematical Monthly in 1994"
TL;DR: In this paper, on Intervals, Transitivity = Chaos and Transitivity + Chaos = Chaos, the American Mathematical Monthly: Vol. 101, No. 4, pp. 353-355.
Abstract: (1994). On Intervals, Transitivity = Chaos. The American Mathematical Monthly: Vol. 101, No. 4, pp. 353-355.
161Â citations
TL;DR: In this paper, a stochastic approach to the Gamma function is presented, where the Gamma functions are modelled as a set of non-uniform functions, and the Gamma Function can be expressed as
Abstract: (1994). A Stochastic Approach to the Gamma Function. The American Mathematical Monthly: Vol. 101, No. 9, pp. 858-865.
141Â citations
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79Â citations
TL;DR: The Gersgorin circle theorem as discussed by the authors is a simple analytic theorem involving elementary inequalities derived from the basic algebraic eigenvalue/eigenvector equation, which is one of the rare instances of a theorem which is elegant and useful and which has a short, elementary proof.
Abstract: The Gersgorin circle theorem gives a region in the complex plane which contains all the eigenvalues of a square complex matrix. It is one of those rare instances of a theorem which is elegant and useful and which has a short, elementary proof. It is surprising that it hasn't made its way into many introductory texts on linear algebra. One reason for this neglect may be the fact that many (even most) mathematicians still regard linear algebra as being only about algebra. But modern linear algebra is more than algebra. It's linear systems, matrix theory and analysis, geometry, applications, numerics and, of course, algebra. The first course in linear algebra at most colleges and universities is to a great extent a service course for future scientists. Gersgorin's theorem is not an algebraic theorem. It is a simple analytic theorem involving elementary inequalities derived from the basic algebraic eigenvalue/ eigenvector equation. If A = [aij] is a complex matrix of order n and A is an eigenvalue of A, then there exists a nonzero vector x = (x1, x2,. . ., xn)T in Cn such that
78Â citations
TL;DR: In this article, the authors provide a proof of Le Cam's inequality using some basic facts from matrix analysis, including the sum of the squares of the pi as a quantity governing the quality of the Poisson approximation.
Abstract: where A = P1 + P2 + * .. + Pn Naturally, this inequality contains the classical Poisson limit law (Just set pi = A/n and note that the right side simplifies to 2A2/n), but it also achieves a great deal more. In particular, Le Cam's inequality identifies the sum of the squares of the pi as a quantity governing the quality of the Poisson approximation. Le Cam's inequality also seems to be one of those facts that repeatedly calls to be proved-and improved. Almost before the ink was dry on Le Cam's 1960 paper, an elementary proof was given by Hodges and Le Cam [18]. This proof was followed by numerous generalizations and refinements including contributions by Kerstan [19], Franken [15], Vervatt [30], Galambos [17], Freedman [16], Serfling [24], and Chen [11, 12]. In fact, for raw simplicity it is hard to find a better proof of Le Cam's inequality than that given in the survey of Serfling [25]. One purpose of this note is to provide a proof of Le Cam's inequality using some basic facts from matrix analysis. This proof is simple, but simplicity is not its raison d'etre. It also serves as a concrete introduction to the semi-group method for approximation of probability distributions. This method was used in Le Cam [20], and it has been used again most recently by Deheuvels and Pfeifer [13] to provide impressively precise results. The semi-group method is elegant and powerful, but it faces tough competition, especially from the coupling method and the Chen-Stein method. The literature of these methods is reviewed, and it is shown how they also lead to proofs of Le Cam's inequality.
78Â citations
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TL;DR: The n-Queens problem has been studied extensively in the literature, e.g. as discussed by the authors. But the n-queens problem is not a deterministic optimization problem.
Abstract: (1994). The n-Queens Problem. The American Mathematical Monthly: Vol. 101, No. 7, pp. 629-639.
77Â citations
TL;DR: In this article, the authors describe a method for Juggling drops and descents. But they do not discuss the evolution of the drop-and-descend method. The American Mathematical Monthly: Vol. 101, No. 6, pp. 507-519.
Abstract: (1994). Juggling Drops and Descents. The American Mathematical Monthly: Vol. 101, No. 6, pp. 507-519.
76Â citations
TL;DR: In this article, an application of the Fourier series to the most significant digit problem is presented, where the authors show that it can be used to solve the problem of the largest digit problem.
Abstract: (1994). An Application of Fourier Series to the Most Significant Digit Problem. The American Mathematical Monthly: Vol. 101, No. 9, pp. 879-886.
74Â citations
TL;DR: In this article, it was shown that there exists an object called the random triangulation of a circle, which is in a natural sense the n -* oo limit of the n-gon.
Abstract: (2m - 2)! cm - (m-1)!m! One of the interesting aspects of Polya's paper is that it exposed readers to his newly developed theory of "figurate series" We wish to consider the idea of letting n -*> oo and studying triangulations of the oo-gon, ie the circle This question doesn't make much sense as combinatorics, but we can shift viewpoint and consider random triangulations of the n-gon, in which each of the cn-1 possible triangulations is equally likely The purpose of this paper is to show that there exists an object "the random triangulation of a circle" which is in a natural sense the n -* oo limit of the random triangulation of the n-gon As with Polya [9], the exposition takes readers into some newly developed theory of the author Let's start by recalling a precise definition A triangulation of a finite set S is a collection of nonintersecting line segments with endpoints from S such that the convex hull of S is partitioned into triangular regions We shall be concerned only with the cases Sn consisting of the vertices of the regular n-gon inscribed in a fixed
71Â citations
TL;DR: The Paradox of Nontransitive Dice as discussed by the authors is a well-known topic in non-convex games and games, and it has been studied extensively in the literature.
Abstract: (1994). The Paradox of Nontransitive Dice. The American Mathematical Monthly: Vol. 101, No. 5, pp. 429-436.
61Â citations
TL;DR: In this paper, it was shown that for the case of complete sequences, p < 2 except that p = 3 for the example in the right column of Table 1, and a similar result holds for sequences that are not complete.
Abstract: QUESTIONS. (i) Give formulae or meaningful bounds for the preperiod io. (As for the period p, it turns out that for the case of complete sequences, p < 2 except that p = 3 for the example in the right column of Table 1. A similar result holds for sequences that are not complete.) (ii) Is there an infinite sequence S0 for which every successor is defined, such that Si+1 differs from Si in infinitely many elements for some i? (iii) Is there an infinite sequence S0 such that every successor is defined and 9 is not ultimately periodic?
TL;DR: In this paper, the authors present three problems in search of a measure and three solutions to the problems of finding a measure in the problem of measuring the distance between two sets of points.
Abstract: (1994) Three Problems in Search of a Measure The American Mathematical Monthly: Vol 101, No 7, pp 609-628
TL;DR: In this paper, the authors considered the problem of estimating the shape of a deflated mylar balloon when it is fully inflated with either air or helium and found that the balloon shape is not spherical, which is surprising for it is well known that the sphere gives the maximal volume for a given surface area.
Abstract: Mylar balloons of different shapes and sizes have become popular as gifts or in bouquets. The most common balloons are comprised of two circular sheets of mylar, fused together at the circumference. A small opening on the circumference allows the balloon to be inflated with either air or helium. These balloons are not spherical, which is at first surprising, for it is well known that the sphere gives the maximal volume for a given surface area. Thus, these balloons suggest the following mathematical problem: Given a circular mylar balloon of deflated radius r, what will be the shape of the balloon when it is fully inflated? In particular, we could ask
TL;DR: In this article, it was shown that two fundamental constants of the geometry of the plane are equal, i.e., the probability that if four points are chosen independently uniformly at random in R, then their convex hull is a quadrilateral, and the rectilinear crossing number of the complete graph on n vertices.
Abstract: We prove that two fundamental constants of the geometry of the plane are equal. First, if R is an open set in the plane with finite Lesbesque measure, let q(R) denote the probability that if four points are chosen independently uniformly at random in R, then their convex hull is a quadrilateral. Let qt be the infimum of q(R) over all such sets R. Second, let v(Kn) denote the rectilinear crossing number of the complete graph on n vertices, i.e., the minimum number of intersections in any drawing of Kn in
TL;DR: In this article, the authors present a theorem concerning the arc length on the Riemann sphere of the image of the unit circle under a rational function, which they call the "Buffon needle problem".
Abstract: In this paper we present a theorem concerning the arc length on the Riemann sphere of the image of the unit circle under a rational function. But our larger purpose is to tell a story. We thought at first that the story began in 1962 with the Kreiss matrix theorem, the application that originally motivated us. However, our arc length question turns out to be more interesting than that. The story goes back to the famous "Buffon needle problem" of 1777.
TL;DR: In this paper, every number is expressible as the sum of how many polygonal numbers, and every number can be expressed as a sum of the number of polygons.
Abstract: (1994). Every Number is Expressible as the Sum of How Many Polygonal Numbers? The American Mathematical Monthly: Vol. 101, No. 2, pp. 169-172.
TL;DR: In this article, the introduction to Fermat's Last Theorem is presented, and a discussion of its application in the context of algebraic geometry is presented. The American Mathematical Monthly: Vol. 101, No. 1, pp. 3-14.
Abstract: (1994). Introduction to Fermat's Last Theorem. The American Mathematical Monthly: Vol. 101, No. 1, pp. 3-14.
TL;DR: Literacy in the Language of Mathematics: A Review of the American Mathematical Monthly: Vol. 101, No. 8, pp. 735-743, as mentioned in this paper.
Abstract: (1994). Literacy in the Language of Mathematics. The American Mathematical Monthly: Vol. 101, No. 8, pp. 735-743.
TL;DR: In this paper, the Rational Periodic Points of the Quadratic Function Qc(x) = x2 + c. The American Mathematical Monthly: Vol. 101, No. 4, pp. 318-331.
Abstract: (1994). Rational Periodic Points of the Quadratic Function Qc(x) = x2 + c. The American Mathematical Monthly: Vol. 101, No. 4, pp. 318-331.
TL;DR: In this article, Hypatia and Her Mathematics is discussed. But the authors focus on the problem of finding a solution to the Hypatian problem in terms of the number of elements.
Abstract: (1994). Hypatia and Her Mathematics. The American Mathematical Monthly: Vol. 101, No. 3, pp. 234-243.
TL;DR: In this paper, the authors consider the problem of finding a homeomorphism of the projective plane to itself that simultaneously straightens all the members of a finite family of simple closed curves.
Abstract: 1. INTRODUCTION. Let r be a finite family of simple curves in the plane. When is there a homeomorphism of the plane to itself that takes all the curves in r to straight lines? In the Euclidean plane, E2, we are faced with the fact that two non-intersecting curves in our family must map to two parallel lines. This introduces extraneous technical complications that only distract from the essence of the problem. As with many other geometric questions, it is much simpler to avoid the special cases caused by parallel lines by moving to the projective plane. The real projective plane p2 iS the Euclidean plane E2 with an extra "line at infinity" adjoined, each point of which represents a parallel direction in E2. p2 has the virtue of simplicity: every pair of points determines a unique line which is topologically a circle (i.e., a simple closed curve), and every two lines meet at a unique point. Thus our question becomes: When is there a homeomorphism of p2 to itself that simultaneously straightens all the members of a finite family r of simple closed curves?
TL;DR: The American Mathematical Monthly: Vol. 101, No. 9, pp. 819-832 as mentioned in this paper, is a collection of articles about Georg Cantor and Transcendental Numbers.
Abstract: (1994). Georg Cantor and Transcendental Numbers. The American Mathematical Monthly: Vol. 101, No. 9, pp. 819-832.
TL;DR: In this paper, the difference between Cantor sets is discussed and a comparison of the two classes of sets is made. But the authors focus on the difference in the two sets of sets.
Abstract: (1994). What's the Difference Between Cantor Sets? The American Mathematical Monthly: Vol. 101, No. 7, pp. 640-650.
TL;DR: In this paper, a characterisation of Solvable Quintics x5 + ax + b is presented. The American Mathematical Monthly: Vol. 101, No. 10, pp. 986-992.
Abstract: (1994). Characterization of Solvable Quintics x5 + ax + b. The American Mathematical Monthly: Vol. 101, No. 10, pp. 986-992.
TL;DR: In this article, the authors explore the rich and elegant interplay that exists between the two main currents of mathematics, the continuous and the discrete, and present over 400 problems from probability, analysis, and number theory.
Abstract: The purpose of this book is to explore the rich and elegant interplay that exists between the two main currents of mathematics, the continuous and the discrete. Such fundamental notions in discrete mathematics as induction, recursion, combinatorics, number theory, discrete probability, and the algorithmic point of view as a unifying principle are continually explored as they interact with traditional calculus. The book is addressed primarily to well-trained calculus students and those who teach them, but it can also serve as a supplement in a traditional calculus course for anyone who wants to see more. The problems, taken for the most part from probability, analysis, and number theory, are an integral part of the text. There are over 400 problems presented in this book.
TL;DR: The role of paradoxes in the evolution of mathematics has been discussed in this article, where the authors discuss the role of contradiction in the development of mathematics and its evolution in general.
Abstract: (1994). The Role of Paradoxes in the Evolution of Mathematics. The American Mathematical Monthly: Vol. 101, No. 10, pp. 963-974.
TL;DR: An elementary proof of the square summability of the Discrete Hilbert Transform is given in this paper, where it is shown that the number of squares in the transform is square.
Abstract: (1994). An Elementary Proof of the Square Summability of the Discrete Hilbert Transform. The American Mathematical Monthly: Vol. 101, No. 5, pp. 456-458.
TL;DR: For every (doubly infinite) line l, where s denotes length measure, if f must be identically zero, then f must have vanishing integrals on all lines as discussed by the authors.
Abstract: for every (doubly infinite) line l, where s denotes length measure, then f must be identically zero. This has been known for many years (see [5] for references), but only comparatively recently was it shown that this result fails in the absence of the global integrability condition. In fact, Zalcman [S; pp. 243-244], using a theorem of Arakelian [1; p. 1189] concerning tangential holomorphic approximation on unbounded sets, constructed a non-constant entire function t satisfying (1) on every line 1. The purpose of this note is to give a construction, similar to Zalcman's, but using only elementary complex analysis. As a bonus all the derivatives of the function we produce also have vanishing integrals on all lines. The main idea is to construct a non-constant entire function g whose derivatives satisfy
TL;DR: The concept of piecewise circular curves (PC) curves as discussed by the authors was introduced in the early 1970s as a way of approximating the tangent lines at the points of a smooth curve.
Abstract: In this article we would like to promote a class of plane curves that have a number of special and attractive properties, the piecewise circular curves, or PC curves. (We feel constrained to point out that the term has nothing to do with Personal Computers, Privy Councils, or Political Correctness.) They are nearly as easy to define as polygons: a PC curve is given by a finite sequence of circular arcs or line segments, with the endpoint of one arc coinciding with the beginning point of the next. These curves are more versatile than polygons in that they can have a well-defined tangent line at every point: a PC curve is said to be smooth if the directed tangent line at the end of one arc coincides with the directed tangent line at the beginning of the next. (In particular, in a smooth PC curve, no arc degenerates to a single point.) In the literature of descriptive geometry and more recently in computer graphics, PC curves have been used to approximate smooth curves so that the approximation is not only pointwise close, as in the case of an inscribed polygon, but also has the property that the tangent lines at the points of the smooth curve are approximated by the tangent lines of the PC curve. Given a pair of nearby points on a smooth curve together with their tangent directions, there will not in general be a single circular arc through the points with those directions at its endpoints, but there will be a family of biarcs meeting these boundary conditions, PC curves composed of two tangent circular arcs. (See [M-N] for a discussion of this construction.) EXAMPLES OF PC CURVES. PC curves arise naturally as the solutions of a number of variational problems related to isoperimetric problems. A classical problem is to find the curve of shortest length enclosing a fixed area, and the solution is a circle. If the curve is required to surround a fixed pair of points, then the curve of shortest length enclosing a given area will be either a circle or a lens formed by two arcs of circles of the same radius meeting at the two points. More generally Besicovitch has shown that a curve of fixed length surrounding a given convex polygon and enclosing the maximum area must be a PC curve with all radii of arcs equal [Be]. One such curve is the Reuleaux "triangle", a three-arc PC curve enclosing an equilateral triangle, with each radius equal to the length of a side of the triangle. Such three-arc PC curves, and many far more elaborate examples can be found in the tracery of gothic windows [A]. If we require that a curve of fixed length L surround a given pair of discs of the same radius, then, for a certain range of values of L, the curve that encloses the greatest area is a smooth convex PC curve consisting of two arcs on the boundary circles of the discs and two arcs of equal radius tangent to both discs. Such four-arc convex PC curves have long been used in engineering drawing for approximating ellipses, and we call such a curve a PC ellipse [FIGURE 1]. One special PC ellipse is