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American Mathematical Monthly - October 1999


Some Fundamental Control Theory I: Controllability, Observability, and Duality
by William J. Terrell
This article introduces some fundamental mathematical ideas in control theory, approached via a natural question in elementary differential equations. We consider the question of when a linear n-dimensional state space system with input can be written as the simple n-th order linear equation, y(n)=v. The full answer to the question of equivalence introduces some central concepts of mathematical control theory. We point out the control-theoretic interpretations of the equivalence conditions, and we then derive some classical control-theoretic results that are accessible based on our discussion of the equivalence problem.


The Education of a Pure Mathematician
by Bruce Pourciau
Imagine yourself teaching the standard undergraduate course on the foundations of analysisÑyou know, the usual stuff: logic, sets, real numbers, least upper bounds, sequences, limits, and so on. Among the students in the class is Integrity Jane, a philosophy major and auditor, but not just any auditor: she's the Auditor from Hell. On the first day, she suggests three principles that you and the rest of the class agree should certainly be followed in the construction of any field of scientific inquiry:


  • Know what something means before you ask if it's true.

    Build in no clearly unwarranted assumptions.

    Move from the simple to the less simple.

These principles look innocuous--one student even calls them "spineless" --but during the early days of the course, Integrity's searching questions and complaints on behalf of her three principles undermine the traditional foundations of analysis, and, as a disturbing and disorienting consequence, your own understanding of mathematical existence and truth.

Such is the premise of The Education of a Pure Mathematician, a short play set in a university classroom, during the first few days of a course called Foundations of Analysis. The philosophical ideas are old, but the presentation is new: the particular relevance of these philosophical ideas to the way we teach the foundations of analysis, on the one hand, and the starring role played by Integrity's "Principles of Scientific Inquiry", on the other. I hope the play gets us thinking, or rethinking, about how a definite philosophical position is built into our lecture notes for the foundations of analysis course we teach, but also, more generally, about our sometimes calcified, even unconscious, classical attitude towards mathematical existence and truth that we absorbed in our own training as pure mathematicians. As Integrity Jane puts it early on, "at the very least, you would think that a scientist trained in a field of inquiry that violates some of these principles ought to be aware of this fact and be able to defend the violations."


Multivariable Calculus and the Plus Topology
by Daniel J. Velleman
The concepts of continuity and differentiability for a function of two variables are designed to tell us about the local behavior of the function near a point. Here "local" is defined by reference to the standard topology on the plane, in which a neighborhood of a point includes nearby points in all directions, not just the coordinate directions. On the other hand, the partial derivatives of a function at a point depend on the values of the function at nearby points in the coordinate directions, but not other directions, so the partial derivatives do not give us information about the local behavior of the function near the point in question--at least, not if "local" is defined by reference to the standard topology. But what if we use a different topology? Is there some topology on the plane that is appropriate for the study of continuity and differentiability? My purpose in this paper is to show that there is such a topology, and that the study of this topology can shed light on some of the subtleties of multivariable calculus.


The Forced Damped Pendulum: Chaos, Complication, and Control
by John H. Hubbard
We explore the dynamics of the forced pendulum, as described by the differential equation x'' + F x' + sin x = A cos t, for the parameter values F = 0.1, = 1, and A = 1.

It turns out that this pendulum has an attracting periodic oscillation of period 2p, but if you color the plane of initial conditions x(0), x«(0) according to how many times the pendulum goes over the top before settling down to this oscillation, the corresponding "basins of attraction" are extremely wild; in fact they form "lakes of Wada": every point in the boundary of one basin is in the boundary of all the infinitely many others.

We further show that given any preassigned sequence of gyrations, there exists an initial condition x(0), x'(0) that realize it; and discuss the resulting control theory of the pendulum.

This last part is proved by exhibiting a Smale horseshoe by computer calculation; the entire paper shows how a computer can yield amazing but rigorous results about differential equations using methods accessible to undergraduates. Such results can probably not be proved without technology, and the weaker perturbative results that can be proved use methods well beyond the undergraduate curriculum.



The Hyperbolic Pythagorean Theorem in the Poincaré Disc Model of Hyperbolic Geometry
by Abraham A. Ungar

Is the Composite Function Integrable?
by Lu Jitan

On the Generalized "Lanczos' Generalized Derivative"
by Jianhong Shen

A Stability Theorem
by Walter Rudin

Rationals and the Modular Group
by Roger C. Alperin

The Union of Vieta's and Wallis's Products for Pi
by Thomas J. Osler



The Four-Color Theorem
By Rudolph Fritsch and Gerda Fritsch

Reviewed by John A. Koch