Chaos, Cantor Sets, and Hyperbolicity for the Logistic Maps
by Roger L. Kraft
The family of logistic maps fr(x) = r x(1 - x) appears in almost every dynamical systems textbook. It is one of the simplest nonlinear systems that one can study, but it is amazingly rich in phenomena. It has a surprising number of connections to other topics in dynamical systems and applied mathematics, e.g., population dynamics, symbolic dynamics, complex analytic dynamics, the Mandelbrot set, period-doubling route to chaos, renormalization, universality, horseshoes, etc. Because of its simplicity, many introductory dynamical systems textbooks use it as a primary example, in particular as the primary example of a chaotic dynamical system. They prove that when r > 2 + 51/2 4.236, fr is chaotic on an invariant Cantor set. Most of these books then state without proof that Är is actually chaotic for all r > 4. The goal of this article is to give a simple proof of this fact.
An Elementary View of Euler's Summation Formula
by Tom M. Apostol
Euler's summation formula, which relates integrals and finite sums, is widely used in numerical analysis, analytic number theory, and the theory of asymptotic expansions. It contains Bernoulli numbers and periodic Bernoulli functions and is ordinarily discussed in courses in advanced calculus or real and complex analysis. Starting with a simple diagram, this paper shows how the general formula can be discovered using only elementary calculus, and it also shows how Bernoulli numbers and Bernoulli functions arise naturally along the way. The formula is used to calculate the first 8 digits in Euler's constant.
Marriage, Magic, and Solitaire
by David B. Leep and Gerry Myerson
Marriage: Hall's Marriage Theorem gives necessary and sufficient conditions for there to
be suitable partners for all the men and women in a given community.
Magic: A semi-magic square is a square matrix with non-negative integer entries, with all
row and column sums equal.
Solitaire: Deal out a deck of cards, face up, into a 4-by-13 array. Try to select
13 cards, one from each column, in such a way as to get one card of each denomination.
Marriage, Magic, and Solitaire: How do these connect?
The Isoperimetric Problem on Surfaces
by Hugh Howards, Michael Hutchings, and Frank Morgan
firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
The authors begin with their three favorite proofs that in the plane the circle minimizes perimeter for given area, including a slight twist on a magical proof of Gromov. In a paraboloid of revolution, the "obvious" solution turns out to be a recent result, though proved here using little more than calculus and the Gauss-Bonnet formula. There are also recent examples in hyperbolic surfaces.
What is a Closed-Form Number?
by Timothy Y. Chow
Can the root of the equation x = cos(x) be written in closed form? To answer this question we must first define "closed form". While much has been written about closed-form functions, scandalously little has been written about closed-form numbers, even though thousands of students have asked either explicitly or implicitly about them. We propose a definition of the term "closed-form number" that helps answer questions like these; we hope it will become standard. Possibly more interesting than the definition itself are its connections with Schanuel's conjecture in transcendental number theory and Tarski's theorem on the decidability of the first-order theory of the real numbers.
By the way, the x = cos(x) question remains an open problem. Perhaps an enterprising reader will solve it!
The Smallest Solution of (30n+1) <(30n) Is...
by Greg Martin
A Matrix Representation for Euler's Constant,
by Frank K. Kenter
More on a Mean Value Theorem Converse
by H. Fejzíc and D. Rinne
An Elegant Continued Fraction for p
by L. J. Lange
The Reciprocity Law for Dedekind Sums via the Constant Ehrhart Coefficient
by Matthias Beck
THE EVOLUTION OF...
Riemann's Dissertation and Its Effect
by Detlef Laugwitz
PROBLEMS AND SOLUTIONS
Life's Other Secret
By Ian Stewart
The Magical Maze
By Ian Stewart
Reviewed by Dan Schnabel
An Introductory Course in Commutative Algebra
By A. W. Chatters and C. R. Hajarnavis
Introduction to Algebra
By Peter J. Cameron
Reviewed by Cynthia Woodburn