# American Mathematical Monthly -December 2005

## December 2005

When a Mechanical Model Goes Nonlinear: Unexpected Responses to Low-Periodic Shaking
by Lisa D. Humphreys and P. Joseph McKenna
lhumphreys@ric.edu, mckenna@math.uconn.edu
This article studies the consequences of forcing a simple suspension bridge mechanical model with very-low-frequency periodic forcing. When the amplitude becomes big enough for the cables to slacken, a wide variety of interesting periodic responses are observed, many of which are quite counterintuitive. The most notable common theme is that as soon as the mass is forced into the nonlinear regime, the low-frequency forcing induces a response with a large high-frequency component.

Irreducible Quartic Polynomials with Factorizations modulo p
by Eric Driver, Philip A. Leonard, and Kenneth S. Williams
eric.driver@lmco.com, philip.leonard.asu.edu williams@math.carleton.ca
A simple arithmetic condition is given that is both necessary and sufficient for a biquadratic polynomial with integer coefficients to be irreducible over the integers but reducible modulo every prime. A similar condition is given for reducibility modulo every positive integer. A wealth of examples is provided for classroom use.

On the Zeroes of the Nth Partial Sum of the Exponential Series
by Stephen M. Zemyan
smz3@psu.edu
Many mathematicians think that the exponential function is the most important function in mathematics. The purpose of this paper is to learn more about it by studying the zeroes of its Nth partial sums. The first half of this paper reviews the known general properties of these zeroes (i.e., those found somewhere in the literature). In particular, we discuss their location and arrangement in the plane. These known results were established primarily by Szegö and others. The second half of this paper begins by showing that the minimum distance between any two adjacent zeroes of any Nth partial sum is bounded below independently of N. Then we study the moment sums of the zeroes (i.e., the sums of their integer powers). Surprisingly, these moment sums are integers for every N and each positive power p! Additionally, these integral moment sums seem to contain greater and greater powers of N as p increases. The techniques of this paper may be applied to the partial sums of other entire functions and to other sets of polynomials. The content of this paper is accessible to undergraduates. There are ten unsolved problems of analytic and number theoretic interest presented within the paper.

Notes

On Homeomorphism Groups and the Compact-Open Topology
by Jan J. Dijkstra
dijkstra@cs.vu.nl

Transpositions and Representability
by Zhibo Chen and Gary L. Mullen
zxc4@psu.edu, mullen@math.psu.edu

Generating Edge-Labeled Trees
by Oleg Pikhurko
pikhurko@andrew.cmu.edu

A Remark on the Erdös-Szekeres Theorem

On the Moduli of the Zeros of a Polynomial
by Seon-Hong Kim
shkim17@mail.chosun.ac.kr

Evolution ofÂ…

Nonstandard Analysis
by Leif Arkeryd

Problems and Solutions

Reviews

Functional Analysis
By Peter D. Lax
Reviewed by Gerald B. Folland
folland@math.washington.edu

Functional Analysis: An Introduction
By Yuli Eidelman, Vitaly Milman, and Antonis Tsolomitis
Reviewed by Gerald B. Folland
folland@math.washington.edu