Showing papers in "American Mathematical Monthly in 2007"
TL;DR: In this article, the authors introduce Automorphisms of Finite Abelian Groups (AFGs) for finite Abelian groups and show that they are automorphisms in the context of finite Abelians.
Abstract: (2007) Automorphisms of Finite Abelian Groups The American Mathematical Monthly: Vol 114, No 10, pp 917-923
147 citations
TL;DR: The Jordan curve theorem states that every simple closed pla-nar curve separates the plane into a bounded interior region and an unbounded exterior, and is hailed as a benchmark of mathematical rigor.
Abstract: (2007). The Jordan Curve Theorem, Formally and Informally. The American Mathematical Monthly: Vol. 114, No. 10, pp. 882-894.
119 citations
110 citations
TL;DR: This map-coloring game was invented about twenty-five years ago by Steven J. Brams with the hope of finding a game-theoretic proof of the Four Color Theorem, avoiding perhaps the use of computers, but has not been successful and is left with a new, intriguing map-Coloring problem: What is the fewest number of colors allowing a guaranteed win for Alice in the map- Coloring game in the plane?
Abstract: (2007). The Map-Coloring Game. The American Mathematical Monthly: Vol. 114, No. 9, pp. 793-803.
86 citations
TL;DR: In this paper, the Random Walker Ranking for NCAA Division I-A Football was used to evaluate the performance of the teams in the 2007 National Football Coaches Association (NFCA) Tournament.
Abstract: (2007). Random Walker Ranking for NCAA Division I-A Football. The American Mathematical Monthly: Vol. 114, No. 9, pp. 761-777.
82 citations
TL;DR: Can be found in closed form by means of certain transformations on generating func tions and the extraction of coefficients and is equivalent to the Lagrange inversion formula but is more compact and easy to remember.
Abstract: (2007). The Method of Coefficients. The American Mathematical Monthly: Vol. 114, No. 1, pp. 40-57.
72 citations
TL;DR: The Poncelet closure theo rem states that the polygonal line again closes up after n steps (see Figure 1), and these closed inscribed-circumscribed curves Poncelets polygons are called.
Abstract: (2007). The Poncelet Grid and Billiards in Ellipses. The American Mathematical Monthly: Vol. 114, No. 10, pp. 895-908.
67 citations
TL;DR: The Hausdorff Dimension, Its Properties, and Its Surprises: A Treatise on Uniformitarian Dimension Theory.
Abstract: (2007). Hausdorff Dimension, Its Properties, and Its Surprises. The American Mathematical Monthly: Vol. 114, No. 6, pp. 509-528.
63 citations
TL;DR: A q-analogue of Wolstenholme's harmonic series congruence is established and it is shown that the series can be described in terms of discrete numbers.
Abstract: (2007). A q-Analogue of Wolstenholme's Harmonic Series Congruence. The American Mathematical Monthly: Vol. 114, No. 6, pp. 529-531.
50 citations
TL;DR: The optimal strategy is described and the expected length of the game under optimal play for Random-Turn Hex and several other selection games is studied.
Abstract: The game of Hex has two players who take turns placing stones of their respective colors on the hexagons of a rhombus-shaped hexagonal grid. Black wins by completing a crossing between two opposite edges, while White wins by completing a crossing between the other pair of opposite edges. Although ordinary Hex is famously difficult to analyze, Random-Turn Hex–in which players toss a coin before each turn to decide who gets to place the next stone–has a simple optimal strategy. It belongs to a general class of random-turn games–called selection games–in which the expected payoff when both players play the random-turn game optimally is the same as when both players play randomly. We also describe the optimal strategy and study the expected length of the game under optimal play for Random-Turn Hex and several other selection games.
47 citations
TL;DR: The gamma function was subjected to intense study by almost every eminent mathematician of the nineteenth and early twentieth centuries and continues to interest the present genera tion from the integral representation numerous identities including the beta-gamma relation and the duplication formula of Legendre were deduced.
Abstract: (2007). The Gamma Function: An Eclectic Tour. The American Mathematical Monthly: Vol. 114, No. 4, pp. 297-315.
TL;DR: In this paper, the authors discuss tile-makers and semi-tile-makers in the context of the American Mathematical Monthly: Vol. 114, No. 7, pp. 602-609.
Abstract: (2007). Tile-Makers and Semi-Tile-Makers. The American Mathematical Monthly: Vol. 114, No. 7, pp. 602-609.
TL;DR: This work states that a very simple, hypothetical computer with only one decimal point precision, equipped with the IEEE Stand~d "Unbiase5!" Roundin& approximates the function f(x) = x 2 with a function f satisfying f(A) = .2, f(F) = 2, and 1(1.6) = A.
Abstract: 1. INTRODUCTION. In scientific calculations using digital computers and f1oating-point arithmetic, roundoff errors are inevitable, even with the most elementary of functions. For example, a very simple, hypothetical computer with only one decimal point precision, equipped with the IEEE Stand~d \"Unbiase5!\" Roundin& approximates the function f(x) = x 2 with a function f satisfying f(A) = .2, f(.5) = .2, and 1(.6) = A. While in this example the absolute value of the roundoff error never exceeds 0.05, not all values up to that bound may be equally likely in practical computations. As Knuth points out in his classic text The Art of Computer Programming [14,pp.253-255]:
TL;DR: Two heuristic arguments are discussed to suggest that the number of terms with a primitive divisor has a natural density, one using recent advances made about the distribution of roots of polynomial congruences.
Abstract: We consider primitive divisors of terms of integer sequences defined by quadratic polynomials. Apart from some small counterexamples, when a term has a primitive divisor, that primitive divisor is unique. It seems likely that the number of terms with a primitive divisor has a natural density. We discuss two heuristic arguments to suggest a value for that density, one using recent advances made about the distribution of roots of polynomial congruences.
TL;DR: An elementary proof of the Wallis product formula for pi is given, which does not require any integration or trigonometric functions.
Abstract: We give an elementary proof of the Wallis product formula for pi. The proof does not require any integration or trigonometric functions
TL;DR: Two simple proofs of Theorem 1 are given, one geometric and the other algebraic, as well as a small generalization of the theorem.
Abstract: (2007). A Property of Parallelograms Inscribed in Ellipses. The American Mathematical Monthly: Vol. 114, No. 10, pp. 909-914.
TL;DR: Sharkovsky's theorem states that, if / is a continuous map from a compact interval into itself that has a period-m point, then / also has aperiod-^ point whenever m < n in the Sharkovsky's ordering of the natural numbers.
Abstract: (2007) A Simple Proof of Sharkovsky's Theorem Revisited The American Mathematical Monthly: Vol 114, No 2, pp 152-155
TL;DR: This paper formulates somewhat vague questions more precisely, in terms of equations, and shows that they can be answered with surprisingly simple twodimensional geometric transformations, even when the cylinder is not circular.
Abstract: Now imagine the elliptical cross section replaced by any curve lying on the surface of a right circular cylinder. What happens to this curve when the cylinder is unwrapped? Consider also the inverse problem, which you can experiment with by yourself: Start with a plane curve (line, circle, parabola, sine curve, etc.) drawn with a felt pen on a rectangular sheet of transparent plastic, and roll the sheet into cylinders of different radii. What shapes does the curve take on these cylinders? How do they appear when viewed from different directions? A few trials reveal an enormous number of possibilities, even for the simple case of a circle. This paper formulates these somewhat vague questions more precisely, in terms of equations, and shows that they can be answered with surprisingly simple twodimensional geometric transformations, even when the cylinder is not circular. For a circular cylinder, a sinusoidal influence is always present, as exhibited in Figures
TL;DR: These two continuous transformations exemplify the group idea in this context: any sequence of two translations is also a translation, there is an identity translation (no translation), and for any translationthere is an inverse translation that undoes it (resulting in the identity translation).
Abstract: (2007). Solving Differential Equations by Symmetry Groups. The American Mathematical Monthly: Vol. 114, No. 9, pp. 778-792.
TL;DR: Is there a closed path along which one can roll the sphere (without slipping or twisting), starting with the first orientation, and return to the origin with the sphere in the second orientation?
Abstract: An old problem in the field of holonomy asks: Given a pair of orientations for a sphere resting on a plane, is there a closed path along which one can roll the sphere (without slipping or twisting), starting with the first orientation, and return to the origin with the sphere in the second orientation? (See Figure 1.) The answer is yes, and the goal of this article is to provide an elementary proof of this fact.
TL;DR: A short proof of the Transcendence of Thue-Morse Continued Fractions is presented, as well as a discussion of the implications of this proof for number theory.
Abstract: (2007). A Short Proof of the Transcendence of Thue-Morse Continued Fractions. The American Mathematical Monthly: Vol. 114, No. 6, pp. 536-540.
TL;DR: Newton's method can lead to chaotic domains of attraction, even for simple choices of F, which leads to the damped Newton's method, which consists of Hoping that z\, z2, ... converges to a zero of F.
Abstract: (2007). The Continuous Newton's Method, Inverse Functions, and Nash-Moser. The American Mathematical Monthly: Vol. 114, No. 5, pp. 432-437.
TL;DR: It is shown that at corner points, the Koch curve doesn’t even have one-sided tangent lines, which implies that no parametrization of the Koch Curve can have at any point a nonzero leftor right-hand derivative.
Abstract: (2007). A Characterization of Ellipses. The American Mathematical Monthly: Vol. 114, No. 1, pp. 66-70.
TL;DR: This article focuses on the property of diagonalizability for integer matrices and poses the question of the likelihood that an integer matrix is diagonalizable over the complex numbers, the real numbers, and the rational numbers, respectively.
Abstract: (2007). The Probability that a Matrix of Integers Is Diagonalizable. The American Mathematical Monthly: Vol. 114, No. 6, pp. 491-499.
TL;DR: This work begins with an experiment that requires minimal equipment: you just need your left and right fore fingers, and moves your right forefinger back and forth in a motion that could be graphed as shown in Figure 1.
Abstract: (2007). Nonlinear Oscillators at Our Fingertips. The American Mathematical Monthly: Vol. 114, No. 1, pp. 14-28.
TL;DR: In this paper, four different formulas for the roots of a general quartic are listed and derived there The derivation requires the solution of the general cubic (for which we give only hints at the derivation) Readers should be forewarned: this paper is a bit like a tourist trap, but you don't get to see it until you pass many of the souvenirs that are available along the way.
Abstract: By this, we really mean four different formulas each of which gives one root of the equation Each formula is expressible using only the operations of addition, subtrac tion, multiplication, division, and extraction of roots If you just want to know the formula or one way to derive it?and don't care about anything else?you can just skip to the last section of this paper The formulas for the roots of a general quartic are listed and derived there The derivation requires the solution of the general cubic (for which we give only hints at the derivation) Readers should be forewarned: this paper is a bit like a tourist trap There is a main attraction, but you don't get to see it until you pass many of the souvenirs that are available along the way The real goal of the paper is to expose readers to a number of mathematical tidbits related to the solution of the general quartic There is a mathematical notion of a "pencil" that is rather cool and has gained prominence in geometry and topology in recent years Pencils were also studied exten sively by algebraic geometers in the nineteenth century The geometric picture behind the solution to the quartic presented in the last section is a particular pencil associ ated with the quartic We begin with a description of a pencil from the view point of a topologist
TL;DR: Polyhedra F and G are equidecomposable if the polyhedron F can be suitably decomposed into a finite number of pieces that can be reassembled to give thepolyhedron G.
Abstract: (2007). A New Approach to Hilbert's Third Problem. The American Mathematical Monthly: Vol. 114, No. 8, pp. 665-676.
TL;DR: The automorphism group of finite abelian group is studied as a group of classes of congruent matrices with application to the group of isomorphisms of any abelsian group.
Abstract: (2007). On the Adjugate of a Matrix. The American Mathematical Monthly: Vol. 114, No. 10, pp. 923-924.