Two Remarkable Twos for Inverses to Some Abelian Integrals
by Peter Lindqvist and Jaak Peetre
We refer to some interesting identities generalizing the familiar formula sin2 + cos2 = 1 as Ones. Such a formula was proved in 1879 by E. Lundberg for some "sines" and "cosines" arising from the inversion of an Abelian integral. The trigonometric One is a special case. These functions have applications to several branches of analysis.
Our paper is concerned with two related identities called Twos. Our first Two is a generalization of the formula (1 + sl2 )(1 + cl2 ) = 2 for the lemmiscate functions . The second Two generalizes a formula of A. Cayley from 1882. We also find a curious connection between Cayley's integral and the Weierstrass -function.
A Chaos Lemma
by Judy Kennedy, Sahin Kocak, and James a. York
The complicated behavior of a trajectory of a dynamical system can often be described in terms of its "itinerary", i.e., the order in which it passes through two (or more) sets, say SA and SB. One trajectory's sequence of A's and B's might be (A,B,B,A,A,A,É), while another's might be quite different. In a chaotic system many different sequences are possible. We begin by describing a simple version of the Smale Horseshoe example. There, the two sets are chosen so that every sequence corresponds to some trajectory. We give a new proof that is more elementary than previous proofs and extends easily to new examples of chaotic systems. The proof is based on families of sets that we call "expanders". It has the flavor of proofs for one-dimensional maps, but in those examples, usually only two expander sets are needed, while in our new proof infinitely many expander sets may be used.
Real Analyticity: From Calculus Class to the Grauert-Morrey Theorem
by Robert E. Greene
Because of the uniqueness of analytic continuation, real analytic functions have a kind of "rigidity": a real analytic function on the real line is determined by its values in a neighborhood of any given point, for example. Continuous and even infinitely differentiable functions are much more flexible--they can vanish identically on an open subset of the line without vanishing everywhere. The comparative rigidity of real analytic functions makes many of the standard constructions used in topology very difficult when they are attempted in the real analytic category. This article explains some details of this difficulty and how the difficulty can be overcome by using methods of complex analysis. We begin with a seemingly elementary problem that is easily stated in terms of familiar ideas from calculus. This problem is nonetheless representative of the general questions that arise and of the methods by which they can be treated. Building from this example, we give an outline of how complex analysis methods can be used to prove the Grauert-Morrey Theorem: Every real analytic manifold has a real analytic embedding in some Euclidean space.
Some Divergent Trigonometric Integrals
by Erik Talvila
Some years ago a rather famous mathematician made the following error. A convergent integral containing a parameter was differentiated under the integral sign with respect to the parameter without justification. This yielded a divergent integral that is listed even today in standard integral tables as converging. We give a simple proof that the resulting integral diverges and then trace its interesting history.
What Goes Up Must Come Down, Eventually
by Fred Brauer
Designing a Calculational Proof of Cantor's Theorem
by Edsger W. Dijkstra and Jayadev Misra
For Every e there Continuously Exists a d
by Giuseppe De Marco
The Lucas Circles of a Triangle
by Paul Yiu and Antreas P. Hatzipolakis
How to Integrate a Polynomial Over a Sphere
by Gerald B. Folland
THE EVOLUTION OF ...
Foundations of Mathematics in the Twentieth Century
by V. Wiktor Marek and Jan Mycielski
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PROBLEMS AND SOLUTIONS
Visual Mathematics, Illustrated by the TI-92 and TI-89.
By George C. Dorner, Jean Michel Ferrard, and Henri Lemberg
Reviewed by Yves Nievergelt
Mathematics Success and Failure Among African-American Youth.
By Danny Bernard Martin
Reviewed by David Scott