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American Mathematical Monthly - October 1998

 

OCTOBER 1998

Mechanical Ideas in Geometry
by Tadashi F. Tokieda
tokieda@math.mcgill.ca
Traditionally, pure mathematics is expected to turn up unexpected uses in applied sciences of a later period. This short article samples some uses of an applied science (mechanics) in pure mathematics of an earlier period (euclidean geometry). Opening with a problem about polyhedra that is very hard to solve with mathematics but trivial with mechanics, it goes on to tell how PythagorasÕ theorem can be proved by the impossibility of perpetual motion, how a simple fact of statics implies the properties of the various classical centers of a triangle, and much more.

 

When Does f -1 = 1/ f?
by R. Cheng, A. Dasgupta, B. R. Ebanks, L. F. Kinch, L. M. Larson, and R. B. McFadden
rayc@ecihq.com
abhijit@ans.net
ebanks@marshall.edu
lfkinc01@ulkyvm.louisville.edu
llarson@louisville.edu
rbmcfa01@ulkyvm.louisville.edu
An unfortunate ambiguity in the standard notations for the reciprocal and the inverse of a function has led many students astray. This paper examines when the inverse and reciprocal of a function are equal. Characterizations of such functions are obtained on the real line, and on the complex plane, when the function is meromorphic.

 

A Golden Cantor Set
by Roger L. Kraft
roger@calumet.purdue.edu
You've read about the golden ratio. You've read about the Cantor set. Now read about them together in the same theorem. See how they unite to solve a geometric problem that arose from the study of dynamical systems. See how self-similarity provides the glue that bonds these two mathematical favorites together. See the technique of renormalization in action as it exploits self-similarity to arrive at proofs of all the needed theorems. See all this, and more, as you get a glimpse of some important ideas and techniques from dynamical systems and mathematical physics.

 

On Evaluating \int^{+\infty}_{-\infty} e^{ax(x-2b)}dx by Contour Integration Round a Parallelogram
by Darrell Desbrow
dd@maths.ed.ac.uk
A major difficulty with evaluation of an infinite integral by contour integration can be, for the student at least, that of determining an effective complex integrand and contour along which to integrate it. This is nicely illustrated by the probability and Fresnel integrals for which effective integrands and contours are not evident. The usual method for the probability integral uses a skew parallelogram as contour whereas a rectangle in standard position suffices for the Fresnel integrals. Neither integrand springs readily to mind. For the student it hardly satisfies, even if it suffices, to be bidden to integrate this integrand round that contour. Even if one is, the doubt lingers whether resort to a skew parallelogram is necessary for the probability integral when a rectangle serves so well for the Fresnel integrals. The purpose of this paper is to shed light on these matters but in relation to the more general integral of the title, which has the probability and Fresnel integrals as special cases. The primary aim is to demonstrate to the student, as regards the probability and Fresnel integrals in particular, that an evaluation by contour integration is possible, how an integrand and parallelogram contour suitable for the evaluation can be hit upon and to show why, given the tack taken, rectangular contours cannot work for the probability integral in particular.

It is shown, in two cases giving restrictions on the complex numbers a and b sufficient for convergence, that the integral evaluates to \pm i \fras{sqrt{\pi}}{\sqrt{a}} e^{-ab^{2}}, wherein \sqrt{a} is the principal square root of a and the sign is fixed by a definite rule.

 

On Generalizations of Conics and on a Generalization of the Fermat-Torricelli Problem
by C. Gross and T.-K. Strempel
strempel@mathematik.tu.darmstadt.de
gross@math.uni-augsburg.de
Conics are introduced usually in one of the two following ways: as a set of points in the plane that satisfy a polynomial equation in two variables of maximum degree 2, or as a set of points in the plane that have a constant weighted sum of distances to two so-called focal points.

Depending on the choice of the coefficients and weights ???±1 one gets ellipses, parabolas, and hyperbolas and it is possible to convert the curve representations into each other.

The authors generalize the second definition by allowing more than two focal points, weights other than ??? ±1, and point sets in higher dimensions in arbitrary vector norms. They show for positive weights that the interior of these generalized conics is convex, that the sets are ordered by inclusion, and that there always exists a smallest non-empty set with respect to this ordering, the set of so-called Fermat-Torricelli points. Furthermore, they show that generalized conics in differentiable norms are differentiable if and only if they don't contain the focal points.

For the Euclidean norm, generalized conics are classified analogously to the classical ones due to the weights and locations of the focal points. The authors also examine generalized conics with respect to several properties that hold for classical conics. For various mechanical and optical properties of classical conics, they show which of these properties apply to generalized conics.

The Fifty-Eighth William Lowell Putnam Mathematical Competition
by Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson

 

NOTES

Butterfly Hunting and the Symmetry of Mixed Partial Derivatives
by T. W. Körner
twk@dpmms.cam.ac.uk

A Note on the Non-existence of Regular Bases for Finite Groups
by Stephen M. Gagola, Jr.
gagola@mcs.kent.edu

The Irrationality of ex for Nonzero Rational x
by Joseph Amal Nathan
red@magnum.barc.ernet.in

A Note on Variable Replacement Rates in Urn Models
by Giles Warrack
warrack@aurora.ncat.edu

UNSOLVED PROBLEMS

Factoring Factorial n
by Richard K. Guy and John L. Selfridge

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