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American Mathematical Monthly - January 1999


Large Torsional Oscillations in Suspension Bridges Revisited: Fixing an Old Approximation
by P. J. McKenna
mckenna@math.uconn.edu
When people discuss large torsional oscillations of suspension bridges like the one at the Tacoma Narrows, they invariably linearize the trigonometry out of the problem, unwittingly making a small-angle assumption. In this paper, I re-derive the equations for a torsionally oscillating plate, choosing the physical constants in accordance with the historical record.

I show how nonlinearity from the trigonometry embedded in the problem leads naturally to large amplitude motions sustained by small forces. These motions are an excellent match for the historical record. I also show how large vertical (nontorsional) motions can lead, via a sudden instability, into the purely torsional motions of the type recorded on famous historical film footage.

 

Inverse Conjugacies and Reversing Symmetry Groups
by Geoffrey R. Goodson
ggoodson@towson.edu
Some elementary group theory that arises in the theory of time-reversing dynamical systems is presented. Given a group G, and a, a member of G, then

 

C(a) = {x : ax = xa} and B(a) = {x : ax = x-1a}

are the centralizer and skew centralizer of a respectively. B(a) need not be a group and in fact is a group if and only if a is an involution, i.e., a2 = e, the identity of G. However, E(a), the union of C(a) and B(a) is always a group, called the reversing symmetry group of a. Properties of E(a) are given and several examples are studied.

 

More on Paperfolding
by Dmitry Fuchs and Serge Tabachnikov
fuchs@math.ucdavis.edu, serge@math.uark.edu
It is a common knowledge that folding a sheet of paper yields a straight line. We study paper folding along arbitrary smooth curves and describe the curves in space that can be obtained as the result of such folding.

 

Trigonometric Integrals and Hadamard Products
by L. R. Bragg
bragg@oakland.edu
The Hadamard product and a generalized version of it are employed to evaluate a variety of single and multiple trigonometric integrals and to substantiate binomial identities. They are also used to develop integral formulas for special functions including the Laguerre polynomials. Finally, they permit obtaining selector functions, which yield integral forms for the sums of coefficients of certain subterms of a given series or polynomial.

 

Cluster Primes
by Richard Blecksmith, Paul Erdös, and J. L. Selfridge
richard@math.niu.edu, selfridg@math.niu.edul
A cluster prime is a prime p such that every even number up to p - 3 is a difference of two primes not greater than p. The authors prove that the cluster primes are fairly sparsely distributed among all the primes. They describe an algorithm for efficiently finding the next cluster prime after a given one, and give some data on the cluster primes up to 1013.

 

NOTES

Fermat's Last Theorem for Gaussian Integer Exponents
by John A. Zuehlke
jaz@cpw.math.columbia.edu

On Rational Function Approximations to Square Roots
by M. J. Jamieson
mjj@dcs.glasgow.ac.uk

A Note on Jacobi Symbols and Continued Fractions
by A. J. van der Poorten and P. G. Walsh
alf@mpce.mq.edu.au, gwalsh@mathstat.uottawa.ca

THE EVOLUTION OF... The Birth of Literal Algebra
by I. G. Bashmakova and G. S. Smirnova

 

PROBLEMS AND SOLUTIONS

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Dummy View - NOT TO BE DELETED