# American Mathematical Monthly -August-September 2005

## August-September 2005

The Automorphism Groups of Domains
by Kang-Tae Kim and Steven G. Krantz
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We study the group of conformal or biholomorphic self-mappings of a domain in the plane or in higher-dimensional complex space. We learn how algebraic and topological properties of the group reflect geometric properties of the domain, and vice versa. Of particular interest are domains that have noncompact automorphism group. In one complex dimension there is a nearly complete characterization of these domains, while in several complex variables there are many open problems. A number of important and enlightening examples of domains with remarkable automorphism group behavior are provided. At the end of the paper, some open problems and research directions are proposed.

Finding Factors of Factor Rings over the Gaussian Integers
by Greg Dresden and Wayne Dymàcek
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We are all familiar with modular arithmetic in the integers: this is simply arithmetic in the factor ring Zn. These rings can be factored quite easily into a product of rings, each of the form Zpe . What if we extend this to the Gaussian integers and look at the factor ring Z[i]a+bi? It turns out that these, too, can be factored into smaller (but not always simpler) rings and this can help us gain insight into the structure of these factor rings. A connection is made to the theory of ramified primes, and a strange ring of size eight makes a surprise appearance.

Fibonacci, Chebyshev, and Orthogonal Polynomials
by Dov Aharonov, Alan Beardon, and Kathy Driver
[email protected], [email protected], [email protected]
The Fibonacci numbers satisfy a second-order linear recurrence relation, and a variety of identities, and they have the property that the mth term divides the nth term precisely when m divides n. These properties are also shared by some Chebyshev polynomials, some solutions of second-order linear recurrence relations with constant coefficients, and some solutions of some linear recurrence relations with variable coefficients. In this expository article, we attempt to explain why this is so.

Tolstoy’s Integration Metaphor from War and Peace
by Stephen T. Ahearn
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In War and Peace Leo Tolstoy employs some striking mathematical metaphors to illustrate his theory of history and to explain the naiveté and arrogance of placing the responsibility of history’s direction on the shoulders of the leaders of armies and nations. These metaphors are unlike any other mathematical references I have seen in literature. They are not numerology nor has Tolstoy simply appropriated mathematical terms. These metaphors are rich and deep, requiring knowledge of some mathematics to fully comprehend their meaning. And they do what good metaphors should do: they enhance and clarify your understanding of Tolstoy’s theory. In this essay I explore these mathematical metaphors which Tolstoy uses to describe his theory of history. I focus on the mathematical ideas Tolstoy draws on to illustrate his theory, specifically integral calculus and the use of the discrete to stand for the continuous. At the end of the essay I discuss the origin of Tolstoy’s mathematical metaphors and briefly describe my use of Tolstoy’s metaphors in calculus class.

Notes

Eigenvalues, Almost Periodic Functions, and the Derivative of an Integral
by Mark Finkelstein and Robert Whitley
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An Elementary Proof That Every Singular Matrix Is a Product of Idempotent Matrices
by J. Araújo and J. D. Mitchell
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A Group Theoretic Approach to a Famous Partition Formula
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An Elementary Proof of Lyapunov’s Theorem
by David A. Ross
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Problems and Solutions

Reviews

Analytic Theory of Polynomials
By Qazi Ibadur Rahman and Gerhard Schmeisser
Reviewed by Kenneth B. Stolarsky
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Complex Polynomials
By Terry Sheil-Small
Reviewed by Kenneth B. Stolarsky
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