Showing papers in "American Mathematical Monthly in 1995"

An envy-free cake division protocol

Abstract: (1995). An Envy-Free Cake Division Protocol. The American Mathematical Monthly: Vol. 102, No. 1, pp. 9-18.

How to Write a Proof

Abstract: (1995). How to Write a Proof. The American Mathematical Monthly: Vol. 102, No. 7, pp. 600-608.

On the Geometry of Halley's Method

Abstract: According to Traub [Tra64], Halley's iteration function (I.F.) "must share with the secant I.F. the distinction of being the most frequently rediscovered I.F. in the literature." Halley's method is a close relative of Newton's method, an iterative technique depicted as a sequence of tangent lines with zeros converging to a root of a function. The usual derivation of Halley's method, however, lacks any obvious geometric interpretation. We present a derivation of Halley's method having such an interpretation, and give a brief history of Halley's work and the method that bears his name.

An Abstract Algebra Story.

Abstract: We agree. And we think there's a fairly wide consensus on this among experienced abstract algebra instructors, and an even wider one among experienced students. Statement: There's little the conscientious math professor can do about it. The stuff is simply too hard for most students. Students are not well-prepared and they are unwilling to make the effort to learn this very difficult material. We disagree. But we suspect that many experienced abstract algebra instructors hold such beliefs. This is especially true for some excellent instructors: Their lectures are truly masterpieces, surely you can't improve much on that; so if the students still fail, that's too bad, but it can't really be helped. We claim that, far from being an immutable fact of nature resulting from inadequacies of the student, this failure is, at least in part, an artifact of a too narrowly conceived view of instruction. In fact, replacing the lecture method with constructive, interactive methods involving computer activities and cooperative learn- ing, can change radically the amount of meaningful learning achieved by average students. In this paper we would like to paint a picture of such an alternative approach, which we and others have been developing and using in our classes over the last several years. We are painfully aware of the limitations inherent in any attempt to give such a description by means of the written text only. It would have been much better if you could actually visit our classes and observe the dynamics of the students' interactions with both the computer and their peers. By way of compro- mise, we will try to simulate such a visit by organizing our paper around several classroom "scenarios" and some commentary on the events depicted in each scenario. As a matter of principle, we have tried to make the scenarios as realistic as space limitation permits.

Force and Geometry in Newton's "Principia"

Abstract: This text introduces one to the reading of Newton's "Principia" in its own terms. The path of access that De Gandt proposes leads through the study of geometrization of force, resulting in a meditation on the sources and meaning of Newton's magnum opus. In Chapter I, De Gandt presents a translation of and detailed commentary on an earlier and simpler version of what in 1687 became Book I of the "Principia". Chapter II places these dynamics in the context of earlier efforts - the first seeds of celestial dynamics in Kepler, Galileo's theory of accelerated motion, and huygen's quantification of centrifrugal force - and evaluates Newton's debt to these thinkers. Chapter III is a study of the mathematical tools used by Newton and their intellectual antecedents in the works of Galileo, Torricelli, Barrow and other 17th-century mathematicians. The conclusion discusses the new status of force and cause in the science that emerges from Newton's "Principia".

Adding Distinct Congruence Classes Modulo a Prime

Abstract: The Cauchy-Davenport theorem states that if A and B are nonempty sets of congruence classes modulo a prime p, and if |A| = k and |B| = l, then the sumset A + B contains at least min(p, k + l − 1) congruence classes. It follows that the sumset 2A contains at least min(p, 2k − 1) congruence classes. Erdős and Heilbronn conjectured 30 years ago that there are at least min(p, 2k − 3) congruence classes that can be written as the sum of two distinct elements of A. Erdős has frequently mentioned this problem in his lectures and papers (for example, Erdős-Graham [4, p. 95]). The conjecture was recently proven by Dias da Silva and Hamidoune [3], using linear algebra and the representation theory of the symmetric group. The purpose of this paper is to give a simple proof of the Erdős-Heilbronn conjecture that uses only the most elementary properties of polynomials. The method, in fact, yields generalizations of both the Erdős-Heilbronn conjecture and the Cauchy-Davenport theorem.

Drums That Sound the Same

Abstract: (1995). Drums That Sound the Same. The American Mathematical Monthly: Vol. 102, No. 2, pp. 124-138.

Curves of Constant Precession

Abstract: 1. INTRODUCTION. Given initial position and direction, the flight-path of a ship in Euclidean space is completely determined by how much it turns and how much it twists at each odometer reading. This is an intuitive interpretation of the Fundamental Theorem for Space Curves, which states that curvature K and torsion , as functions of arclength s, determine a space curve uniquely up to rigid motion. This statement of the Fundamental Theorem ([14], §1-8) should be tempered with the reservations expressed by Nomizu [12] and Wong & Lai [15]. Given a parametric space curve, there are well-known formulae for the arclength, curvature, and torsion (as functions of the parameter). Given two functions of one parameter (potentially curvature and torsion parametrized by arc-length) one might like to find a parametrized space curve for which the two functions are the curvature and torsion. This activity, called "solving natural equations" ([14], §1-10), is generally achieved by solving Riccati equations like dw/ds = -iz/2 iKW + i7W /2. Although the solution generally exists, it usually cannot be obtained explicitly. Euler [6] found explicit integral formulae for plane curves (where z - O) through direct geometric analysis. Hoppe [9] developed a general method for solving the natural equations for space curves by solving Riccati equations through a complicated sequence of integral transformations. He digressed to obtain formulae for the tangent, normal, and binormal indicatrices for general helices and essentially for curves of constant precession. Enneper [5] obtained explicit closed-form solutions for helices on revolved conic sections through direct geometric analysis. A curve of constant precession is defined by the property that as the curve is traversed with unit speed, its centrode revolves about a fixed axis with constant angle and constant speed. In this paper we obtain an arclength-parametrized closed-form solution of the natural equations for curves of constant precession through direct geometric analysis. As part of this analysis, we obtain a new theorem for curves of constant precession analogous with Lancret's Theorem for general helices. We provide the first rendering of a curve of constant precession. We also note for the first time that curves of constant precession lie on circular hyperboloids of one sheet and have closure conditions that are simply related to their arclength, curvature, and torsion. These are 3-type curves, except one family of closed 2-type curves (when Z = 4,u; see [2], [3], and [1]). Given a closed C3 curve in space, it is rather obvious that the curvature and torsion functions will be periodic functions of the arclength, with period equal the total arclength. This is a necessary condition but, as the circular helices (K and z both constant) show, not a sufficient condition that integral curves be closed. Efimov [4] and Fenchel [7] independently formulated The Closed Curve Problem. Find (explicit) necessary and sufficient conditions that determine when, given two periodic functions K(S) and z(s) with the same period L, the integral curve is closed.

Introduction to hyperbolic geometry

Abstract: This text for advanced undergraduates emphasizes the logical connections of the subject. The derivations of formulas from the axioms do not make use of models of the hyperbolic plane until the axioms are shown to be categorical; the differential geometry of surfaces is developed far enough to establish its connections to the hyperbolic plane; and the axioms and proofs use the properties of the real number system to avoid the tedium of a completely synthetic approach. The development includes properties of the isometry group of the hyperbolic plane, tilings, and applications to special relativity. Elementary techniques from complex analysis, matrix theory, and group theory are used, and some mathematical sophistication on the part of students is thus required, but a formal course in these topics is not a prerequisite.

Derivative polynomials for tangent and secant

Abstract: (1995). Derivative Polynomials For Tangent and Secant. The American Mathematical Monthly: Vol. 102, No. 1, pp. 23-30.

The Angle Between Complementary Subspaces

Abstract: Usually the discussion stops right there, and extensions to angles between subspaces of higher dimensions are, more or less tacitly, shoved under the rug. Perhaps this is because most instructors feel that such extensions are difficult to understand, or that further effort in this direction is not worthwhile. Indeed, this makes sense for angles between general subspaces because one would have to introduce concepts like gap or distance between subspaces [7, 12], principal (or canonical) angles [1, 2, 15, 12], the CS decomposition [11, 4, 10, 6, 12], and so on. These topics are better off in a more advanced course. However, angles between complementary subspaces are easier to deal with. The purpose of our article is to draw attention to some simple, though not very well known, expressions for the angle between complementary subspaces which are easily derived from the fundamental theorem of linear algebra [14] and elementary facts about matrix norms and projectors. Angles between complementary subspaces are not just academic. They arise, for instance, in the context of controller robustness [9, 16]. Roughly speaking, the spaces associated with the controller and the plant (a system described by a set of differential equations) are complementary subspaces. The robustness of the controller is defined by the smallest perturbation that renders the system unstable, which means that the associated subspaces are no longer complementary. The system remains stable as long as perturbations are smaller than the distance between the complementary subspaces. One measure of distance is the sine of the angle between the spaces.

Turán’s graph theorem

Abstract: One of the fundamental results in graph theory is the theorem of Turan from 1941, which initiated extremal graph theory. Turan’s theorem was rediscovered many times with various different proofs. We will discuss five of them and let the reader decide which one belongs in The Book.

Injective Polynomial Maps Are Automorphisms

Abstract: (1995). Injective Polynomial Maps Are Automorphisms. The American Mathematical Monthly: Vol. 102, No. 6, pp. 540-543.

Polygonal Rooms Not Illuminable from Every Point

Abstract: This problem has been attributed to Ernst Straus in the early 1950's, and has remained open for over forty years. It was first published by Victor Klee in 1969 [1]. It has since reappeared on various lists of unsolved problems, notably Klee again in 1979 [2] and in two recent books on unsolved problems, one by Klee and Wagon in 1991 [3] and one by Croft, Falconer and Guy, also in 1991 [4]. In this article, we will settle the above problem in the negative. We will as well give elementary techniques for constructing rooms, both in the plane and in three-space, which are not illuminable from every point. In particular, we will show that if the two people are located in a two-dimensional planar room as shown in Figure 2, then they cannot see each other.

A note on entire solutions of the eiconal equation

Abstract: A1< 2| [u2(sin2s-COS2S)2uusinscoss-u2sin2s+1]ds. (S) The product 2uu is the derivative of u2; integrating by parts changes (5) into 1 v A1< 21 (1-u2sin2s)ds, clearly < Fr/2. Equality holds only if u O, which makes y(s) constant sin s. Since equality in (3) holds only if y = x = 21 _ y2, y(5) _ sin s, x(s) + cos s + constant. This is a semicircle. Q.e.d. Courant Institute of Mathematical Sciences New York Uniuersity 251 Mercer Street New York, NY 10012

The Words of Mathematics: An Etymological Dictionary of Mathematical Terms used in English

Abstract: The Words of Mathematics explains the origins of over 1500 mathematical terms used in English. While other dictionaries of mathematics define technical terms, this book concentrates on where those terms came from and what their literal meanings are. The words included here range from simple to advanced. This dictionary is easy to use. Although some of the entries are highly technical, the book explains them in plain English. The introduction gives an overview of how the ancient language known as Indo-European developed into Latin, Greek, French, and English, the languages from which most of our mathematical vocabulary has been derived. Another section discusses the many ways in which mathematicians have borrowed and created their specialized vocabulary over the centuries. A glossary explains historical and linguistic terms used throughout the book.

Coin-Weighing Problems

Abstract: (1995). Coin-Weighing Problems. The American Mathematical Monthly: Vol. 102, No. 2, pp. 164-167.

Down with determinants

Abstract: (1995). Down With Determinants! The American Mathematical Monthly: Vol. 102, No. 2, pp. 139-154.

Calculating normal probabilities

Abstract: (1995). Calculating Normal Probabilities. The American Mathematical Monthly: Vol. 102, No. 1, pp. 46-49.

Niels Hendrik Abel and Equations of the Fifth Degree

Abstract: (1995). Niels Hendrik Abel and Equations of the Fifth Degree. The American Mathematical Monthly: Vol. 102, No. 6, pp. 495-505.

Football Pools A Game for Mathematicians

Abstract: have some nice, relevant properties. At the same time another group of people combinatorialists are busy trying to produce many different types of arrays of numbers with special features. We have the feeling that the two groups are not well acquwainted with each other's work, although recently some mathematical journals have published papers reporting results obtained by the playing community, and we hope that this paper will contribute to increasing the mathematicians' interest in these problems. The problems are natural and mathematically easy to formulate, but are highly nontrivial and can be attacked using powerful combinatorial machinery.

Alan turing and the central limit theorem

Abstract: Because the English mathematician Alan Mathison Turing (1912–1954) is remembered today primarily for his work in mathematical logic (Turing machines and the “Entscheidungsproblem”), machine computation, and artificial intelligence (the “Turing test”), his name is not usually thought of in connection with either probability or statistics. One of the basic tools in both of these subjects is the use of the normal or Gaussian distribution as an approximation, one basic result being the Lindeberg-Feller central limit theorem taught in first-year graduate courses in mathematical probability. No-one associates Turing with the central limit theorem, but in 1934 Turing, while still an undergraduate, rediscovered a version of Lindeberg's 1922 theorem and much of the Feller-Levy converse to it (then unpublished). This paper discusses Turing's connection with the central limit theorem and its surprising aftermath: his use of statistical methods during World War II to break key German military codes. 1 Introduction Turing went up to Cambridge as an undergraduate in the Fall Term of 1931, having gained a scholarship to King's College. (Ironically, King's was his second choice; he had failed to gain a scholarship to Trinity.) Two years later, during the course of his studies, Turing attended a series of lectures on the Methodology of Science, given in the autumn of 1933 by the distinguished astrophysicist Sir Arthur Stanley Eddington. One topic Eddington discussed was the tendency of experimental measurements subject to errors of observation to often have an approximately normal or Gaussian distribution.

The rise, fall, and possible transfiguration of triangle geometry: a mini-history

Abstract: (1995). The Rise, Fall, and Possible Transfiguration of Triangle Geometry: A Mini-history. The American Mathematical Monthly: Vol. 102, No. 3, pp. 204-214.

A spigot algorithm for the digits of pi

Abstract: (1995). A Spigot Algorithm for the Digits of π. The American Mathematical Monthly: Vol. 102, No. 3, pp. 195-203.

Areas of polygons inscribed in a circle

Abstract: Heron of Alexandria showed that the areaK of a triangle with sidesa,b, andc is given by $$K = \sqrt {s(s - a)(s - b)(s - c)} ,$$ wheres is the semiperimeter (a+b+c)/2. Brahmagupta gave a generalization to quadrilaterals inscribed in a circle. In this paper we derive formulas giving the areas of a pentagon or hexagon inscribed in a circle in terms of their side lengths. While the pentagon and hexagon formulas are complicated, we show that each can be written in a surprisingly compact form related to the formula for the discriminant of a cubic polynomial in one variable.

Pick's Formula via the Weierstrass ℘-Function

Abstract: (1995). Pick's Formula via the Weierstrass ℘-Function. The American Mathematical Monthly: Vol. 102, No. 5, pp. 431-437.