Recovering a Function From a Dini Derivative
by John W. Hagood and Brian S. Thomson
john.hagood@nau.edu, thomson@cs.sfu.ca
A function can be recovered up to an additive constant from its derivative, but can the same be said of a single Dini derivative? Dini in 1878 knew that a continuous function was so determined, but he would have had no idea then how to recover the function, having little more than Riemann integration as a tool to work with. A half-century later, Lebesgue addressed the problem bringing in the heavy machinery of measure theory and arcane totalization methods. Our goal is to answer the question using an integral that arises from a Riemann sum process. The investigation is similar in some ways to the case using the derivative, but is more interesting, more varied, and even surprising at points. The path takes us into the integration process of Denjoy and Perron, now known more frequently as the Henstock-Kurzweil integral, and then into a modification of that integral. The presentation is seasoned with historical perspectives.
Some Graphical Solutions of the Kepler Problem
by Marc Frantz
mfrantz@indiana.edu
We typically crown the presentation of calculus by deriving the conic-section shapes of planetary orbits. To avoid spoiling the coronation, we don’t emphasize the fact that we can’t exactly answer the obvious question known as the “Kepler problem.” Namely, given a planet's position on the orbit at a given time, exactly where is it at a later time? Although the position cannot be expressed as an elementary function of the time, we can locate the planet at any time using graphs of elementary functions, thereby achieving a partial victory. The history of this endeavor is studded with the names of Kepler himself, John-Dominique Cassini (namesake of the “Cassini division” in the rings of Saturn) and his son Jacques, Johann Franz Encke (namesake of Comet Encke and the “Encke gap” in Saturn’s rings), Christopher Wren, John Wallis, and Isaac Newton. In this article we present a new approach that gives double solutions for both elliptic and hyperbolic orbits, includes both attractive and repulsive accelerations, and leads to interesting animations linking Keplerian motion with other physical processes. Throughout the article we encourage the reader to look at and enjoy these animations, which appear on the website
http://php.indiana.edu/~mathart/animated.
Notes
A Short Proof of the Simple Continued Fraction Expansion of e
A Proof of the Continued Fraction Expansion of e1/M
by Thomas J. Osler
osler@rowan.edu
A Continuous Differentiability of Solutions of ODEs with Respect to Initial Conditions
by Xiongping Dai
xpdai@nju.edu.cn, xiongpingdai@yahoo.com
Fastest Mixing Markov Chain on a Path
by Stephen Boyd, Persi Diaconis, Jun Sun, and Lin Xiao
boyd@stanford.edu, sunjun@stanford.edu, lxiao@caltech.edu
Evolution of…
The Poincaré Conjecture
by Pawel Strzelecki
Problems and Solutions
Reviews
The Pursuit of Perfect Packing
by Tomaso Aste and Denis Weaire.
Reviewed by Charles Radin
radin@math.utexas.edu
Kepler’s Conjecture
by George G. Szpiro
Reviewed by Charles Radin
radin@math.utexas.edu
Complexities: Women in Mathematics
Edited by Bettye Anne Case and Anne M. Leggett
Reviewed by Shandelle M. Henson
henson@andrews.edu