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American Mathematical Monthly -August/September 2007

August/September 2007

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James Gregory's Calculus in the Geometriæ pars Universalis
By: Enrique A. González-Velasco
enrique_gonzalez@uml.edu
This paper contains a brief presentation of the main results obtained by James Gregory on the subject that would eventually become the calculus, in his book Geometriæ pars universalis of 1668. These include his method of tangents, a formula for arclength, the first version ever in print of the fundamental theorem of calculus, integration by substitution, and differentiation of rational powers.

Hilbert 90 for Biquadratic Extensions
By: Roman Dwilewicz, Ján Minác, Andrew Schultz, and John Swallow
romand@umr.edu, minac@uwo.ca, aschultz@stanford.edu, joswallow@davidson.edu
Hilbert's Theorem 90 is a classical result in the theory of cyclic extensions. The quadratic case of Hilbert 90, however, generalizes in noncyclic directions as well. Informed by a poem of Richard Wilbur, the article explores several generalizations, discerning connections among multiplicative groups of fields, values of binary quadratic forms, a bit of module theory over group rings, and even Galois cohomology.

Newton's Method Obeys Benford's Law
By: Arno Berger and Theodore P. Hill
arno.berger@canterbury.ac.nz, hill@math.gatech.edu
Floating-point numbers in computations based on Newton's method are not uniformly distributed, as might be expected, but instead follow a very specific logarithmic distribution known as Benford's law. This fact not only adds a surprising note to a notorious gem of mathematics folklore but also has important implications for the analysis of roundoff errors and, consequently, for estimates of average running times of algorithms. Geometric intuition helps explain why, with hindsight, the emergence of the century-old Benford's law from the three-century-old Newton's method should not have come as a complete surprise.

Tile-Makers and Semi-Tile-Makers
By: Jin Akiyama
fwjb5117@mb.infoweb.ne.jp
We mean by a development of a convex polyhedron a connected plane figure obtained by cutting the surface along any parts of its surface, not necessarily along its edges, and by an edge-development a development obtained by cutting along only its edges. A convex polyhedron (including dihedrons) P is said to be a tile-maker (respectively, a semi-tile-maker) if every development (respectively, edge-development) of P is a tiler, that is, a plane figure copies of which tile a plane. We determine all tile-makers and give a few observations and a conjecture on semi-tile-makers.

A Harmonic Measure Interpretation of the Arithmetic-Geometric Mean
By: Byron L. Walden and Lesley A. Ward
bwalden@math.scu.edu, ward@math.hmc.edu
The arithmetic-geometric mean M(a,b) of two positive real numbers a and b is defined by repeatedly replacing a and b with their arithmetic and geometric means and finding the common limit. The authors show a surprising connection between the arithmetic-geometric mean and harmonic measure, namely the harmonic measure of a portion of the boundary on a doubly-slit domain in the complex plane. By making this connection, the authors are able to recover three well-known elliptical integral formulas for M(a,b) due to Gauss. The techniques used in the article include Brownian motion; conformal mappings and conformal invariants; the Schwarz reflection principle; the Joukowski mapping; and Schwarz-Christoffel mappings.

Notes

An Interesting Property of the Evolute
By: Carlos A. Escudero and Agustí Reventós
carlos10@utp.edu.co, agusti@mat.uab.cat

Rotation in a Normed Plane
By: Jack Cook, Jonathan Lovett, and Frank Morgan
frank.morgan@williams.edu

On a Certain Lie Algebra Defined by a Finite Group
By: Arjeh M. Cohen and D. E. Taylor
a.m.cohen@tue.nl, d.taylor@maths.usyd.edu.au

Problems and Solutions

Reviews

Linearity, Symmetry, and Prediction in the Hydrogen Atom.
By: Stephanie Frank Singer
Reviewed by: Stephen A. Fulling
fulling@math.tamu.edu

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